Download Dimensionless Physical Quantities in Science and

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Relative density wikipedia, lookup

Thomas Young (scientist) wikipedia, lookup

Navier–Stokes equations wikipedia, lookup

Time in physics wikipedia, lookup

Heat wikipedia, lookup

Accretion disk wikipedia, lookup

Convection wikipedia, lookup

Superfluid helium-4 wikipedia, lookup

Derivation of the Navier–Stokes equations wikipedia, lookup

Thermal conductivity wikipedia, lookup

Bernoulli's principle wikipedia, lookup

History of fluid mechanics wikipedia, lookup

Turbulence wikipedia, lookup

Thermal conduction wikipedia, lookup

Transcript
Dimensionless Physical Quantities in Science
and Engineering
Dimensionless Physical
Quantities in Science
and Engineering
Josef Kuneš
Department of Physics
University of West Bohemia
Plzeň
Czech Republic
AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD
PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO
Elsevier
32 Jamestown Road, London NW1 7BY
225 Wyman Street, Waltham, MA 02451, USA
First edition 2012
Copyright r 2012 Elsevier Inc. All rights reserved
No part of this publication may be reproduced or transmitted in any form or by any means,
electronic or mechanical, including photocopying, recording, or any information storage and
retrieval system, without permission in writing from the publisher. Details on how to seek
permission, further information about the Publisher’s permissions policies and our
arrangement with organizations such as the Copyright Clearance Center and the Copyright
Licensing Agency, can be found at our website: www.elsevier.com/permissions
This book and the individual contributions contained in it are protected under copyright by
the Publisher (other than as may be noted herein).
Notices
Knowledge and best practice in this field are constantly changing. As new research and
experience broaden our understanding, changes in research methods, professional practices,
or medical treatment may become necessary.
Practitioners and researchers must always rely on their own experience and knowledge in
evaluating and using any information, methods, compounds, or experiments described
herein. In using such information or methods they should be mindful of their own safety and
the safety of others, including parties for whom they have a professional responsibility.
To the fullest extent of the law, neither the Publisher nor the authors, contributors, or
editors, assume any liability for any injury and/or damage to persons or property as a matter
of products liability, negligence or otherwise, or from any use or operation of any methods,
products, instructions, or ideas contained in the material herein.
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Cataloging-in-Publication Data
A catalogue record for this book is available from the Library of Congress
ISBN: 978-0-12-416013-2
For information on all Elsevier publications
visit our website at elsevierdirect.com
This book has been manufactured using Print On Demand technology. Each copy is
produced to order and is limited to black ink. The online version of this book will show
color figures where appropriate.
I dedicate this book to the memory of my teachers
Prof. Vladimı´r Marcelli
and
Prof. Josef Hošek.
They took me and my workplace,
half a century ago,
on the path of modelling and experiment.
Preface
All things are numbers.
Pythagoras of Samos (570 495 BC)
The century of informatics, characterized by the remarkable growth of information,
is closely connected with the increasing importance of the model experiment. The
specific important has computer modelling, in practically all scientific spheres.
Modelling is a powerful tool not only in terms of the scale assignment of a model
and its original, as it was for a long time in the case of physical models, but also in
terms of the compression of the information flow on a model and by the processing
of the modelling results. In this sense, the importance of dimensionless quantities
and especially of similarity criteria has, in fact, increased. The dominant opinion,
still held by many even today, that these quantities are connected primarily with
physical modelling, is mistaken. On the contrary, their importance has grown
significantly in today’s dominant mathematical and computer modelling. The
dimensionless formulation of a mathematical or computer model does not lose its
physical importance; on the contrary, it is magnified. Simplification and generalization of modelling results to the dimensionless forms are also other significant attributes. One example is the generation of experimental mathematical models used
whenever it is not possible to use an exact mathematical model. It follows that
dimensionless quantities and similarity theory have an even wider importance in the
contemporary development of modelling than they did in the past.
The use of dimensionless quantities in science matches only the growth of
globalization in the world. Its use in science is, above all, in interlinking dimensional physical and other quantities in dimensionless groups and sinks so growth of
information flow in modelling.
Dimensionless Physical Quantities in Science and Engineering presents in nine
chapters approximately 1200 dimensionless quantities from several types of fields
in which modelling plays an important role. It is probably the most extensive collection of these quantities involving both classic and newly developing fields. In
addition to traditional fields like fluid mechanics and heat transfer, in which using
dimensionless quantities has long been common, this book includes many others,
in particular newly developing fields such as solid phase mechanics, electromagnetism, physical macro- and nanotechnology, technology and mechanical engineering,
geophysics and ecology. Each dimensionless quantity is presented with both its
physical characteristics and its significance in the relevant field.
This book is not only a simple summary of these quantities, but also features
a clarification of their physical principles, areas of use and other specific
properties. The book also facilitates the retrieval of dimensionless quantities for
x
Preface
practical use. Furthermore, it provides citations to important sources, also facilitating the use of dimensionless quantities in their appropriate fields.
The wide range of different spheres in which dimensionless quantities have
extraordinary importance presented considerable difficulty. First of all, for their
specific dissimilarities in such physically different spheres. Moreover, there are
many problems with terminology, and the unsystematic designation of single nontraditional dimensionless quantities by various authors. The book presents the
established or most often used designations, and recommends preferred
designations.
The book is an attempt to systematically present dimensionless physical quantities from different scientific and engineering fields. It concerns especially the fields
and processes related to physical substance. Therefore, this book does not include
some fields, such as economics, in which modelling has considerable importance,
but in which the use of dimensionless quantities is out of physical line. The use of
dimensionless quantities in medicine, biology, physiology and other spheres is
increasing along with the fast development of modelling in those fields. However,
this book is mostly concerned with typical dimensionless quantities or their modifications. A similar situation exists in other fields.
In this book, the presented bibliography is divided into four parts, including a
part discussing books and proceedings (A) and section on articles in journals (B).
Many information sources are presented on Internet sites (C), and there is also a
list (D) of some simple calculators for determination of dimensionless quantities,
and information related to simple software.
A mutual connection of relevant dimensionless quantities by means of crossreferences is an important advantage of this book.
Josef Kuneš
Foreword
The most fascinating feature of the book Dimensionless Physical Quantities in
Science and Engineering is the presentation of about 1200 dimensionless quantities
relevant for modelling in several research areas, both the traditional and newly
developing ones, which hopefully will contribute to the further development of
model experiments. For this reason, we are indeed fortunate to have Professor
Kuneš to guide the thoughts and activities of many scientists and students.
The reason for presenting this relatively great number of dimensionless quantities in this book is their extraordinary significance for modelling. They are relevant
not only to scale models but also to almost all types of models, including computer
models. The prevalent opinion that these quantities are primarily connected to
physical modelling is erroneous. On the contrary, we cannot ignore the fact that
their importance continues to grow, especially in today’s predominant mathematical and computer modelling. With the dimensionless expressions used in a mathematical or computer model, the physical meaning is not lost, but is, in fact,
intensified. Moreover, the process investigated becomes more transparent, and the
number of variable quantities and the intensity of the flow of information are significantly reduced. This is also true of the number of output quantities according to
which the modelling results are generalized. The dimensionless quantities have fundamental significance for experimental mathematical models (phenomenological
mathematical models) generally, when more exact asymptotic mathematical models
cannot be used.
The English edition of this book will thus certainly make a significant contribution to possible future achievements in the relevant fields of research.
Professor J. Vlček
Head of the Department of Physics
University of West Bohemia
Plzeň, Czech Republic
1 Introduction
The main goal of physics is to describe a maximum of phenomena with a minimum
of variables.
CERN Courier [A40]
The dimensionless quantity expresses either a simple ratio of two dimensionally
equal quantities (simple) or that of dimensionally equal products of quantities in the
numerator and in the denominator (composed). The dimensionless quantities can
be divided into several groups. The most important group consists of the physical
similarity criteria obtained by some of the similarity theory methods. They are also
called generalized variable quantities. The dimensionless physical constants belong
to another group. In addition, the approximate ratio quantities can also be included
among the dimensionless quantities. They usually come from experimental results
and from the experimenter’s intuition, i.e. without using any of the similarity theory
methods. Usually, the extent of the validity of these quantities is limited only to
a certain area. Other dimensionless ratio quantities can be created as well, which do
not have any full-value importance from the modelling point of view.
Each of the similarity criteria can be expressed in the form of a mutual relation
between, for example, two forces, momentums or energies acting in a process.
Therefore, by observing the size of the criterion, an idea can be obtained from the
character of the investigated process. This fact is well known, for example, with
the Reynolds number Re, which expresses the dynamic-to-viscous force ratio and
characterizes viscous fluid flow. According to the value of the Re number, the flow
can be distributed into three fundamental characteristic types: laminar, transient
and turbulent. This is similar to the use of the Weber number for single-phase and
two-phase fluids, where this number expresses the ratio of the surface tension force
to the inertial one. More details are in [A23] and [A24], where examples using the
Weber number are given for condensation, boiling and motion of gas bubbles in a
fluid or interaction of a drop with a warm wall.
Sometimes, however, it seems as though the transition from dimensional physical
quantities to dimensionless ones would obscure the view of the investigated process. In
fact, the contrary is true because the reduced number of variable quantities expressed
in the dimensionless way enables one to understand the mutual physical contexts in
the investigated process more deeply. A good example is the Fourier number, which
is very often used to express dimensionless time. In fact, it is used in all unsteady
processes occurring in various fields. For example, in thermomechanics
in the
case of heat conduction it expresses the coupling of time with the characteristic
Dimensionless Physical Quantities in Science and Engineering. DOI: 10.1016/B978-0-12-416013-2.00001-4
© 2012 Elsevier Inc. All rights reserved.
2
Dimensionless Physical Quantities in Science and Engineering
geometrical dimension and the thermal diffusivity. It is described by a single
dimensionless variable which is expressive of the influence of all three dimensional
quantities on the temperature field.
Because of a misunderstanding of the physical significance of the similarity
criteria and the functions thereof as generalized variables, dimensional quantities
are often applied even if an experiment or other research should result in an expression of the information obtained in a condensed and generalized way, so that it can
be utilized for all physically similar systems. With dimensional quantities, on the
contrary, the immediate perception of the process is advantageous in measurement
and identification of an investigated system. The dimensionless similarity criteria
impose a limitation on repeating solutions for tasks of similar character.
Physical similarity criteria are divided into composed criteria and simple (parametrical) criteria. Among other things, the importance of similarity criteria is that they
keep a deep physical significance despite the fact that they are outwardly dimensionless. That is to say, they express the ratios of diverse physical quantities such as forces,
energies and momentums, which especially enable one to understand the acting mechanism of individual quantities in analyzing complex physical processes. The physical
similarity criteria can be obtained either by the dimensional analysis method or by the
similarity analysis method, either of a phenomenological physical model or by exact
mathematical model analysis. These methods have been described in the author’s
book Similarity and Modeling in Science and Engineering (CISP, Cambridge, 2012)
in more detail.
This book is primarily focused on physical similarity criteria in various fields
which are characteristics of the development of contemporary science and engineering. Eight chapters summarize about 30 independent fields or spheres in which
modelling plays an important role. The most widespread application of similarity
criteria has already occurred in the era of pre-computer modelling in fluid mechanics and in heat transfer. The present number of criteria in these fields corresponds
to this, and so do the systematic descriptions and surveys presented in the literature,
such as [B11] and [B12]. Of course, the origin of new fields and the increasing
importance of existing ones, together with the entrance of computer modelling, has
resulted in the emergence of many other similarity criteria and modifications of
original ones. However, the literature lacks both adequate descriptions and analyses
of criteria in individual fields and magnetism are partial exceptions. A survey of
the similarity criteria of several fields is given in [A32] and [A33].
The dimensionless quantities from tens of fields are summarized in eight chapters.
Among them, new original and modified dimensionless quantities are presented,
which have been introduced and used in the workplaces of the author.
This especially concerns chapters 5, 6, 7 and 8. Included in each of the chapters
are brief profiles of many important scientists and engineers who worked in the fields
surveyed and have similarity criteria named after them. This should contribute to
the recognition that the dimensionless physical quantities have a human intellectual
dimension in addition to their physical significance.
2 Physics and Physical Chemistry
The basic laws of physics and chemistry are like each other.
Dmitri Ivanovich Mendeleev (18341907)
2.1
Physics, Mathematics and Geometry
In physics, dimensionless physical quantities and constants have been widely used,
in thermodynamics, optics, radiation and other spheres of physics, especially in applications in various natural scientific and technical branches, and have become
an important tool in their development. In mathematics, dimensionless quantities
have their theoretical base in the theory of groups and also in linear algebra and
matrix calculus. At the same time, the fundamental theorem to determine the similarity criteria is the dimensional homogeneity of equations of mathematical physics
as defined by Fourier. The similarity criteria are practically important in numerical
mathematics and computer modelling, e.g. not only in generalized dimensionless
expressions of numerical solution stability of mathematical physics equations
but also in other spheres of mathematics. Among the best known physical dimensionless quantities are the following numbers: Abbe´, Fresnel and Snellen numbers in
optics; Bejan, Boyle, Carnot, Gay-Lussac, Pitzer and Van der Walls numbers in thermodynamics; and the Planck number for radiation.
In mathematics, for example, diverse dimensionless numbers express the
stability conditions of the numerical solution, such as the Courant, Damkőhler,
Neumann and Pe´clet numbers and other mathematical and geometrical dimensionless numbers.
2.1.1
Abbé Number V
V5
nD 21
nF 2 nC
nD, nF, nC () refractive indices of the material at the wavelength of the
Fraunhofer D-, F- and C spectral lines (589.2, 486.1 and 656.3 nm, respectively).
It is used to classify the glasses in the dispersion measurement in the visible radiation band. Low-fracture glasses have high values of V, e.g. for lead crystal glass it is
V , 50, whereas for crown glass it is V . 50. For heavy flint glasses, the common
extent of V is about 20. Very light crown glasses have values of V up to 60.
Info: [C2].
Dimensionless Physical Quantities in Science and Engineering. DOI: 10.1016/B978-0-12-416013-2.00002-6
© 2012 Elsevier Inc. All rights reserved.
4
Dimensionless Physical Quantities in Science and Engineering
Ernst Abbé (23.1.184014.1.1905), German physicist and
astronomer.
He was engaged in physics, mathematics and meteorology but in optics and astronomy above all. He deduced the
mathematical theory of a light microscope. He designed and
fabricated high-quality lenses for scientific purposes. He
manufactured special instruments. He was the co-founder
of the Carl Zeiss works in Jena.
2.1.2
Energy Accommodation Coefficient rE, σ, α
rE 5
Ein 2 Ere
Ein 2 EW
Ein, Ere (J) incident and reflected energy flux; EW (J) reflected energy flux
obtained if the molecules are in thermal equilibrium.
It characterizes the mutual energetic effects of gas molecules with a solid body
surface with heat passing in diluted gases. It expresses the energy of that part
of the total number of gas molecules which come in contact with the surface
and the energy which, after rebound or reemission, is reduced because of being
accommodated to the surface temperature. Besides, it is the measure of the
thermal energy transfer perfection. With the complete transfer, rE 5 1 is valid;
and with a complete (mirrored) reflection of the energy, rE 5 0. Its size depends
on the physical properties of the surroundings and usually does not differ very
much from the number one.
Info: [A33].
2.1.3
Avogadro Number NA
NA 5 6:0220 3 1023 mol21
The Avogadro number expresses the number of particles (e.g. atoms, molecules
or ions) in a chemically homogeneous body with the substance quantity of one
mole (mol). The mole is the substance quantity of the set which contains exactly
as many elementary individual units (e.g. atoms, molecules or other particles) as
the atoms in 0.012 kg of the nuclide of the carbon isotope 126 C: It is widely applied
in physics, chemistry and other branches.
Info: [C5].
Physics and Physical Chemistry
5
Lorenzo Romano Amedeo Carlo Avogadro (9.8.1776
9.7.1856), Italian mathematician, physicist and chemist.
He was engaged in statistics, physics and chemistry, and
in meteorology as well. In the year 1811, he published the
hypothesis known later as the Avogadro law, which expresses
the fact that equal gas volumes contain equal numbers of
molecules under equal temperature and pressure. However,
Avogadro could not prove the hypothesis precisely by experiment and did not live to see its acceptance. To honour his
work in the molar mass theory, the number of molecules in
one mole of particles was named after him.
2.1.4
Bejan Thermodynamical Number Be
Be 5
S1
S1 1 S2
S1 (J K21) entropy generation contribution by heat transfer; S2 (J K21) entropy
generation contribution by fluid friction.
It expresses the ratio of heat transfer unreturnability to the total unreturnability
caused by heat transfer and fluid friction.
Info: [C6]
Adrian Bejan (born 24.9.1948), American engineer of
Romanian origin.
His research activity involves a wide area of thermal
engineering and thermodynamics. He was engaged in
the entropy minimization problem; the energy conversion
analysis (exergy); natural convection; convection in porous
materials; heat and mass transfer; and problems of turbulence, melting, solidification, condensation, contamination,
solar energy conversion, cryogenic engineering, applied
superconductivity and tribology. He created the constructive
form and structure theory in nature.
2.1.5
Boltzmann Distribution, Boltzmann Factor NB, Pn
NB 5
γBh
Ni
5 e2 kT
N
Ni (m23) number of states having energy Ei; N (m23) total number of particles;
γ () magnetogyroscopic ratio; B (T) magnetic induction; h (J s) Planck
constant; k (J K21) Boltzmann constant; T (K) sample temperature.
6
Dimensionless Physical Quantities in Science and Engineering
In physics, it represents the prediction of the function of particle distribution
of which each one has the energy Ei. Alternatively, it expresses the volume magnetizing vector as well.
Info: [C8].
Ludwig Boltzmann (p. 205).
2.1.6
Boyle Number Bo
Bo 5
TB
Tcrit
TB (K) Boyle temperature; Tcrit (K) critical temperature.
This number expresses the ratio of the thermodynamic temperature of Boyle’s
point, corresponding to the zero isobar, to the critical temperature. BoAh2:3; 3i:
Info: [C11].
Robert Boyle (25.1.162730.12.1691), Irish chemist and
mathematician.
He is called the father of chemistry. He applied experimental and quantitative methods. He was the first to deliver
the modern definition of chemical elements, and he used
it to measure the acidity of colour indicators. He discovered
the indirect proportionality between gas pressure and volume
under constant temperature. This is Boyle’s law.
2.1.7
Bulk Concentration Nbc
Nbc 5
cb
R
cb (kg m23) concentration of bulk particles; R (kg m23) liquid density.
It characterizes the relative concentration of solid particles in a solution. Filtration.
2.1.8
Carnot Number Ca
Ca 5
T2 2 T1
T2
T1, T2 (K) absolute temperatures.
Physics and Physical Chemistry
7
It characterizes the theoretical efficiency of the Carnot circulation occurring
between two thermal states, limited by the thermodynamic temperatures T1 and T2.
Thermodynamics.
Info: [B20],[C16].
Nicolas Léonard Sadi Carnot (1.6.179624.8.1832), French
physicist.
He was engaged in thermodynamics above all. In the year
1824, he elaborated the thermal machines theory in his work
Re´flections sur la Puissance Motrice de la Feu (Considerations
on the Driving Power of Fire). He designed the Carnot reversal
thermal circulation and found that thermal machine efficiency
depends only on the inlet and outlet temperatures.
2.1.9
Coefficient of Variation C
C5
σ
μ
σ () standard deviation2; μ () mean value.
In probability and statistical theories, it expresses the dispersion size of the
probability distribution. It is often used to evaluate the normal distribution with
a positive mean value and with a standard deviation which is less than the mean
deviation expressively. In compliance with the distribution character of the standard
deviation, the variation coefficient size can be greater or less than one. Mathematics,
statistics.
Info: [C21].
2.1.10 Compressibility Factor Z
Z5
pv
rT
ð1Þ;
Zcrit 5 Wa21
ð2Þ
p (Pa) pressure; v (m3 kg21) specific volume; r (J kg21 K21) specific gas
constant; T (K) temperature; Wa () Van der Waals number (1.) (p. 32).
This factor characterizes the mutual molecular coupling in a substance for a
certain thermodynamic state. At the thermodynamic critical point, the relation (2)
is valid. In ideal gases with Z 5 1, the deviations from this value express the size of
the intermolecular coupling.
Info: [C83].
8
Dimensionless Physical Quantities in Science and Engineering
2.1.11 Courant Numerical Number Cou, CFL, ν
Cou 5
wΔτ
, C;
Δx
where C # 1
ð1Þ;
wx Δτ
wy Δτ
1
, C ð2Þ;
Δx
Δy
ffi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
w 2 w 2
x
y
1
Cou 5 Δτ
# 1 ð3Þ;
Δx
Δy
Cou 5
CFL 5
Δτ
,C
Δx4
ð4Þ
w (m s21 velocity; Δτ (s) time step; Δx, Δy (m) mesh increment in the
x and y axis; C () constant (time stepping parameter) depends on the particular
equation to be solved and not on Δτ and Δx; wx, wy (m s21) flow (fluid) velocity
component in the x and y axis.
It expresses the stability criterion of a numeric process solution of one-dimensional fluid flow (1), two-dimensional (2) fluid flow and two-dimensional flow
through a porous material (3). For certain fourth-order partial differential equations,
the form (4) can be used. The CFL condition can be a very limiting constraint on
the time step Δτ. Numerical mathematics.
Info: [B114].
2.1.12 Courant Wave Number Cou
Cou 5
w2 Δτ 2
1
# ;
2
Δ2
if Δ 5 Δx 5 Δy
w (m s21) velocity of wave propagation; Δτ (s) time step; Δ, Δx, Δy (m) mesh
increment.
It expresses the numerical solution stability condition for wave propagation.
Numerical mathematics.
Info: [A11]
2.1.13 CourantFriedrichsLewy Numerical Number CFL, NCFL, Cou
Δτ w
wBL ð1Þ;
k Δx
Δτ wx
wy
wz
CFL 5
1
1
wBL
k Δx Δy Δz
CFL 5
ð2Þ
Physics and Physical Chemistry
9
Δτ (s) time step; k (m s21) filtration coefficient; w, wx, wy, wz (m s21) flow
velocity and its components; Δx, Δy, Δz (m) space step; wBL (m s21) saturation
velocity.
This number represents the numerical criterion to determine optimal time
step in a non-stationary, two-phase, one-dimensional (1) filtering flow or in a
three-dimensional (2) one. The stability condition is CFL # 1.
Info: [B114].
2.1.14 CrankNicolson Parameter μC
μC 5 0:5
ð1Þ; μC 5 0
ð2Þ;
μC 5 1 ð3Þ
unj () dimensionless-dependent variable; ΔFo () Fourier difference number
(p.13); j () geometric nodal point; n () time level in the numeric solution.
It occurs in the CrankNicolson differential diagram for a one-dimensional
solution of the parabolic Fourier’s equation:
h
i
n
n11
n11
n11
n
n
n
2
u
5
μ
ΔFo
ðu
2
2u
1
u
Þ
1ðu
2
2u
1
u
Þ
un11
C
j
j
j11
j
j21
j11
j
j21 :
In case relation (1) is valid, the CrankNicolson solution is stable unconditionally.
With relation (2) holding, the explicit solution (FTCS) is obtained. In the case of (3),
the implicit solution is gained.
Info: [C24].
John Crank (6.2.19163.10.2006), English mathematician.
Together with the mathematician Phyllis Nicolson, he
worked in the sphere of the numerical solution of secondorder partial differential equations especially those of
unsteady heat conduction and the solution thereof by the
finite differences method. They were engaged in the problem
of numerical solution instability and of the related choice
of optimal geometric and time steps. The CrankNicolson
method is stable numerically, but a simple system of linear
equations must be solved at every time level.
Phyllis Nicolson (21.9.19176.10.1968), English mathematician.
Together with the mathematician John Crank, she worked
on the solution of second-order partial differential equations
numerically. As a mathematician, she was engaged in the
magnetron theory and its interpretation.
10
Dimensionless Physical Quantities in Science and Engineering
2.1.15 Damkőhler Numerical Number Danum
Danum 5
kΔx
w
k (s21) disintegration unit rate; Δx (m) mesh increment; w (m s21) flow velocity.
This number expresses the stability criterion in a digital process of the onedimensional modelling of a chemical conversion in fluid flow. It characterizes
the hydrodynamic influence in chemical reactions.
Info: [C98].
Gerhard Friedrich Damkőhler (19081944), German physical chemist.
2.1.16 Dimensionless Heat Capacity NC
NC 5
C
C
5
nR Nk
C (J K21) heat capacity of a body; n (mol) amount of matter in the body;
R (J mol21 K21) molar gas constant; N () number of molecules in the body;
k (J K21) Boltzmann constant.
It expresses the body heat capacity in a dimensionless shape. Thermomechanics.
Info: [C123].
2.1.17 Eddington Number Ed
Ed 5 136 3 2256 1:575 3 1079
It expresses the exact number of protons in the Universe, where 136 represents
the inverse value of the fine structure constant α (p. 13) as it could be stated by
measurement in the given time.
Info: [C43].
Arthur Stanley Eddington (28.12.188222.11.1944),
American physicist and astronomer.
Probably, he is the most prominent astrophysicist of the
twentieth century. He examined the stellar system interior. The
basic contribution consisted in the verification of gravitational
curvature influence on the bending of rays around the Sun.
He explained the pulsation of stars theoretically. He calculated
the Sun’s nucleus temperature in millions of Kelvin. He was
one of the first who defined Einstein’s relativity theory with
more precision.
Physics and Physical Chemistry
11
2.1.18 EddingtonDirac Number EdD
EdD 1040
This hypothetical number follows from the fact that: (i) the ratio of the mutual
force of the electron and the proton equals 2.27 3 1039, (ii) the ratio of the elementary length (radius of the electron) to the radius of the Universe is 3 3 1040 and
(iii) the ratio of the elementary time (electron radius to light velocity) to the
Universe’s age is 6 3 1040. It represents only the mysterious dimensionless number
in the extent of 1040 approximately, which was presented by Dirac with respect to
digital relations between the microscopic and macroscopic scales and the effects
of various forces. Considering Dirac’s hypothesis of extensive numbers, obviously
this numerical relation between very different phenomena was given a deeper
cosmologic significance. At present, this idea is not set greater physical store
mostly.
Info: [C129].
Arthur Stanley Eddington (see above).
2.1.19 Energy Efficiency ηt
ηt 5
W
;
E
ηt Ah0; 1i
W (J) mechanical work or energy released by the process; E (J) quantity of work
or energy used as input to run process.
It is an important technical indicator for economic utilization of processes
and facilities. According to thermodynamic law, the efficiency ηt 5 1 cannot be
reached.
Info: [C46].
2.1.20 Entropy Generation Number NS
NS 5
L2 T0 EG
;
λðTw 2 T0 Þ2
where EG 5
η
λ ðrx TÞ2 1 ðry TÞ2 1 1 ðry uÞ2
2
T0
T0
L (m) characteristic length (wall thickness); T0 (K) input fluid temperature;
EG (W m23 K21) entropy change (volume density of heat flux) by temperature change 1 K; λ (W m21 K21) wall thermal conductivity; Tw (K) wall
12
Dimensionless Physical Quantities in Science and Engineering
temperature; η (Pa s) dynamic viscosity of the fluid; rxT, ryT (K m21) temperature gradient in the direction of x and y axis; ryu (s21) velocity gradient in the
direction of y axis.
It characterizes the fluid entropy change in laminar streaming of viscous
incompressible fluid through an inclined canal with isothermic walls. It was
determined from the analysis of the second law of thermodynamics. Fluid
mechanics.
Info: [B76].
2.1.21 Feigenbaum Delta δ
δ 5 lim
Tn 2 Tn21
2 Tn
n-N Tn11
Tn (s) value of the nth bifurcations period.
It characterizes the convergence velocity of the cascade redoubling period. It
is determined experimentally and has the value δ . 1. It is used in the dissipation
theory of non-linear systems in the phase transfer measurement for example,
in electronic circuits, lasers, chemical reactions and in fluid mechanics if
approaching the turbulent state.
For example, Feigenbaum showed that all non-linear dynamic systems, showing
periodic doubling, tend towards chaos and usually have a value of δ 5 4.669.
Chaos theory.
2.1.22 Fermi’s Paradox K
K 5 e τ 1043 3 10
T
6
T (year) age of the Universe (T 5 1010); τ (year) specific time of exponential
development of our civilization (τ 5 102 years).
This gigantic dimensionless number, exceeding the framework of theoretical
physics see Eddington number Ed (p. 10) characterizes the growth of technological civilization during the Universe’s existence. This number is so large that the total
number of elementary particles in the Universe is very small as compared to it.
According to Fermi, ‘If there were any civilisations in the Universe, their spaceship
would have been in our Solar system long ago’. The absence probability of ‘space
miracles’ is 10243 3 106 in our Universe, or virtually zero. Unfortunately, nobody
has discovered them. Fermi’s paradox consists in the idea that our miracleless world
is fantastic and does exist.
Info: [C54].
Physics and Physical Chemistry
13
2.1.23 Fine Structure Constant, Sommerfeld Fine-Structure Constant α
α5
e2
4 π ε0 h̄ c
e (C) elementary charge (1.60219 3 10219 C); ε0 (F m21) permittivity of vacuum;
h̄ (J s) Planck constant h̄ 5 h (2π)21 (1.0545887 3 10234 J s); c (m s21) speed
of light.
It is the basic dimensionless physical constant characterizing the electromagnetic
interaction intensity. It was introduced into physics by A. Sommerfeld in the year
1916. It is used in analyzing Feynman’s quantum electrodynamic diagrams. Its
exact value, determined on a physical basis is α21 5 137 but α21 5 137.0399976
if determined by experimental procedure.
Info: [A29].
Arthur Stanley Eddington (p. 10).
2.1.24 Fourier Difference Number ΔFo, Fomesh
ΔFo 5 Fomesh 5
ΔFo #
1
4
ð3Þ;
aΔτ
ðΔxÞ2
ΔFo 5
ð1Þ;
1
6
ΔFo , 0:5
ð2Þ;
ð4Þ
a (m2 s21) thermal diffusivity; Δτ (s) time step; Δx (m) finite increase of
distance coordinate x.
In equation (1), it expresses the relation between the amount of time and the
geometric steps in the numerical solution of the parabolic Fourier equation. For
the explicit 1-D task, condition (2) is valid for solution stability. For the 2-D task,
condition (3) is valid; for the 3-D task, condition (4) is valid. Numerical mathematics.
The final difference method.
Info: [A10].
Jean Baptiste Joseph Fourier (p. 175).
2.1.25 Fresnel Number F
F5
L2
λs
L (m) characteristic size (radius) of the aperture; λ (m) wavelength; s (m) distance of the screen from the aperture.
14
Dimensionless Physical Quantities in Science and Engineering
It characterizes Frauenhofer diffraction equation (F{1) and that of Fresnel
(F $ 1). Optics.
Info: [C59].
Augustin Jean Fresnel (10.5.178814.7.1827), French
physicist and mathematician.
Due to his works in optics, he became one of the founders of light wave theory. He showed the reason for optical
diffraction, consisting in transversal light undulation. He
created the mathematical theory of refraction and polarization in anisotropic materials. From this theory, conical
refraction was predicted and discovered soon afterwards.
Joseph von Fraunhofer (6.3.17877.6.1826), German
physicist.
In the year 1814, he investigated the solar spectrum and
discovered dark spectral lines called the Fraunhofer lines.
He is well known for his work on light diffraction in systems
with small Fresnel numbers. This is called Fraunhofer’s
diffraction to honour him.
2.1.26 f-Stop Number Nf
Nf 5
f
D
f (m) focal length of the lens or mirror; D (m) aperture diameter;
In optics, the Nf 5 2f ; 3f; . . . is used currently. In film exposition, it is P~N12 :
f
Optics, film, photography.
2.1.27 Gay-Lussac Number Gc
Gc 5
1
βΔT
β (K21) coefficient of bulk expansion; ΔT (K) temperature difference.
In the dimensionless form, it characterizes the relative thermal volume expansibility of substances.
Info: [A29].
Physics and Physical Chemistry
15
Joseph Louis Gay-Lussac (17781850), French chemist
and physicist.
For ideal gases, he expressed the law of gas pressure
dependence on temperature, so-called the Gay-Lussac law.
Later, this led to the introduction of the thermodynamic
temperature scale and to the formulation of the ideal gas
state equation. When only a 26-year-old chemist, he executed
many courageous high-altitude atmospheric measurements
by means of a balloon and with instruments he designed.
He studied terrestrial magnetism as well.
2.1.28 Geometric Coordinates X, Y, Z
X5
x
;
L
Y5
y
;
L
Z5
z
L
x, y, z (m) dimensional coordinates; L (m) characteristic length; S (m2) surface
area; V (m3) volume.
They express the ratio of the coordinate of a point in space to the characteristic
length of the system. In the dimensionless form, it characterizes the position of the
place M (X, Y, Z) in space. For a plate, the semi-thickness (symmetrical case) or
the thickness (unsymmetrical case) is usually chosen as the characteristic length.
In the case of a cylinder or a ball, the radius is chosen.
Sometimes for a general shape, it is convenient to choose the volumesurface
ratio of a body (module) L 5 V S21. In the case of a plate, an unlimited cylinder
and a ball, the ratio 3:1.5:1 is obtained, which is also the ratio of mutually corresponding process times according to the analytic heat transfer theory. However, the
relative length, determined in this manner, does not agree with the geometric coefficient influence. Therefore, the generalized relative length L 5 KtV S21 is used
sometimes where Kt is the relative shape coefficient. For a ball, an unlimited plate
and a cylinder, Kt 5 1 holds, with Kt . 1 for other bodies.
Info: [A23].
2.1.29 Gravitational Coupling Constant αg
2
Gm2e
me
5
αg 5
1; 752 3 10245
h̄c
mP
G (N m2 kg22) Newtonian constant of gravitation; me (kg) electron mass; h (J s) Planck constant; c (m s21) speed of light in vacuum; mP (kg) Planck mass.
It represents a basic physical constant and characterizes the gravitational force
between typical elementary particles. It is related to gravitation as the fine structure
constant α (p. 13) is to electromagnetism and quantum electrodynamics.
Info: [C69],[C68].
16
Dimensionless Physical Quantities in Science and Engineering
2.1.30 Guldberg Number Gu
Gu 5
Tn
Tcrit
Tn (K) thermodynamic temperature of saturation by pressure p 5 105 Pa; Tcrit (K) critical thermodynamic temperature.
This number expresses the relation between the thermodynamic saturation
temperature at an atmospheric isobar and the critical thermodynamic temperature.
GuAh0.37; 0.8i.
Info: [A23].
Cato Maximilian Guldberg (p. 42).
2.1.31 Hadamard Number Hd
Hd 5
3ηb 1 3ηf
3ηb 1 2ηf
ηb, ηf (Pa s) dynamic viscosity of fluid in bubble (b) and viscosity of ambient
fluid (f).
In the dimensionless form, it expresses the resulting dynamic viscosity of a
mixture of a fluid and bubbles contained therein.
Info: [A29].
Jacques Salomon Hadamard (8.12.186517.10.1963),
French mathematician.
He introduced the correct task concept in the partial differential equation theory. Hadamard’s matrices and the Hadamard
transformation are named for him, representing an example
of the generalized class of Fourier transformations. He edited
many publications, e.g. publications on geodesy and matrix
theory. After the WWII, his long and eventful life led to peace
activities and support for mathematicians all over the world.
2.1.32 Hamilton Numeric Number H
H5
N
N
X
1 2kRij X
e
1
R2i ;
R
i.j ij
i
where
ri
;
Ri 5
rref
rffiffiffiffiffiffiffiffiffiffiffi
2q2
3
;
rref 5
mεω2
k5
rref
lse
Physics and Physical Chemistry
17
N () number of the particles; k () inverse dimensionless screening length;
Rij () interparticle relative position of ith and jth particle; Ri () relative
position of the ith particle; ri (m) position of the ith particle; rref (m) reference
position; q (C) particle charge; m (kg) mass of the particle; ε (F m2 1) dielectric constant of the medium; ω (s21) angular frequency of the particle;
lse (m) screening length.
It characterizes the interaction of particles due to the Coulomb potential, depending on physical parameters and plasma background, and can have various forms.
Description by the Hamiltonian (H) starts from the isotropic potential between
particles and from the purely repelling and approximated potential. With k 5 0,
the interaction between particles is purely the Coulomb potential. Mathematical
physics.
2.1.33 Kolmogorov Microscale λKol
rffiffiffiffiffi
3
4 ν
λKol 5
ε
ð1Þ;
τ5
ν
ε
ð2Þ
ν (m2 s21) kinematic viscosity; ε (m2 s23) speed of energy dissipation relative
to the mass unit; τ (s) time.
It expresses the nominal length with which the viscous dissipation occurs in
three-dimensional turbulent flow. Together with the timescale (2), it characterizes
the turbulence beginning in a certain place in the flow. Physics. Fluid mechanics.
Info: [C75].
Andrey Nikolaevich Kolmogorov (25.4.190320.10.1987), Russian mathematician (p. 104).
2.1.34 Lautrec Number for Fluid NL
L
NL 5 ;
δf
rffiffiffiffiffiffiffi
2af
where δf 5
ω
L (m) separated half-thickness of the plate; δf (m) sound penetration in fluid
material; af (m2 s21) thermal diffusivity of fluid; ω (s21) angular frequency of
the acoustic oscillations.
It characterizes the thermoacoustic process originating in the thermal and acoustic energy transformation in a fluid. It expresses the influences in the acoustics, in
which the heat transfer and fluid entropy changes play an important role.
Thermoacoustics.
18
Dimensionless Physical Quantities in Science and Engineering
2.1.35 Lautrec Number for Solid NL
rffiffiffiffiffiffiffi
2af
where δs 5
ω
L
NL 5 ;
δf
L (m) plate half-thickness; δf (m) sound penetration in solid material; as
(m2 s21) thermal diffusivity of material; ω (s21) angular frequency of the
acoustic oscillations.
This number expresses the thermoacoustic process of acoustic wave penetration
into a material. Alternatively, it describes the thermal and acoustic energy transformation in a material. Thermoacoustics.
2.1.36 Lobachevsky Number Lo
Lo 5
1
OA
ð
ðOA Þ
dbmed
dli
dbloc
ð1Þ;
Lo 5
1
SA
ð
ðSA Þ
dbmed
dSi
dbloc
ð2Þ
OA (m), SA (m2) circumference and area of cross section; bmed, bloc (m) mean and
local distance between isolines; dli (m) elementary length; dSi (m2) elementary
area.
In heat and mass transfer and in aero-hydrodynamics, it characterizes the isolines
shape. It represents the ratio, centred on the line of force, of the elementary distance
between the isolines to the local distance between two points of the same lines of
force. With the isolines and isoareas not changing their shape, the Lo number keeps
an equal value. In heat and mass transfer, it expresses the influence of a capillary
porous body on the heat and mass transfer.
Info: [A23],[A24].
Nikolay Ivanovich Lobachevsky (1.12.179224.2.1856),
Russian mathematician.
He founded non-Euclidean geometry, a geometry in which
Euclid’s fifth postulate is not true. He published a whole
range of books, starting with the Foundation of Geometry
(18351838) to Pangeometry (1855). Geometry with a constant negative curvature is called Lobachevskian geometry.
2.1.37 Logarithmical Decrement Λ, ϑ
Λ 5 ln
sðtÞ
sðt 1 TÞ
s(t) (m) displacement of oscillation; t (s) time; T (s) time period of oscillation.
Physics and Physical Chemistry
19
It is the logarithm of the ratio of the vibrating movement deviation s(t) to a
deviation in the time t 1 T.
Info: [C28].
2.1.38 Lorentz Number Lo, L
Lo ðΔUÞ2 γ
ΔT λ
ΔU (V) voltage difference; γ (S m21) specific electrical conductivity; ΔT (K) temperature difference; λ (W m21 K21) specific thermal conductivity.
It expresses the dependence between thermal and electrical conductivities
for metals. It starts with the WiedemannFranz law, according to which
both conductivities are proportional to the absolute temperature and depend
on the movement of free electrons. Physics. Thermomechanics. Electrical
engineering.
Info: [C79].
Hendrik Antoon Lorentz (p. 323).
2.1.39 Ludolph’s Number π
π5
U
D
U (m) circle circumference; D (m) circle diameter.
It expresses the circumference-to-diameter ratio of a circle. Probably, it is
the oldest parametric criterion, known about the year 2000 BC, and approximated
by the Egyptians afterwards. Mathematics. Geometry.
Info: [C82].
Ludolph van Ceulen (28.1.154031.12.1610), Dutch mathematician and engineer.
In the year 1600, he was appointed as the first professor
of mathematics at Leyden University. He calculated the number π to 35 decimal places by applying the same methods as
Archimedes about 2000 years earlier. With this irrational
transcendental number, he confirmed that the classic circle
quadrature task is unsolvable. He wrote several works, of
which On the Circle (Van den Circkel) is the most popular.
The number is called Ludolph’s number after him.
20
Dimensionless Physical Quantities in Science and Engineering
2.1.40 Mass-to-Charge Ratio MZ
MZ 5
m
z
m () atom (molecule) mass in atomic mass units; z () atom (molecule)
electric charge.
This is an important quantity used in mass spectrometry, for example, in
depicting the ion signal dependence on this dimensionless parameter. In the case of
multi-degree mass spectrometry, this method can be applied to chemical structure
determination.
Info: [C86].
Joseph John Thomson (18.12.185630.8.1940), English
physicist. Nobel Prize in Physics, 1906.
He discovered the electron in cathode radiation and
presented the atom as having its own internal structure.
He clarified the properties of ions. He determined the massto-charge ratio of the electron. He observed the first data
confirming the existence of isotopes. He received the Nobel
Prize for his research in electricity conduction in gas.
2.1.41 Mechanical Efficiency ηmech
ηmech 5
P2
uF
5
P1
P1
P2 (W) work output; P1 (W) mechanical advantage (work input); u (m s21) speed; F (N) force.
It expresses the effective input power ratio.
Info: [C88].
2.1.42 Mendeleev Number Me
Me 5
patm
pcrit
patm (Pa) atmospheric pressure; pcrit (Pa) critical pressure.
It expresses the atmospheric to critical pressure ratio on a static surface.
MeAh0.430 3 1022; 0.443 3 1022i.
Info: [A23].
Physics and Physical Chemistry
21
Dmitriy Ivanovich Mendeleev (7.2.18342.2.1907), Russian
chemist.
He assembled and published (1869) 63 elements,
unknown before, in the periodic table based on the atomic
number. Thus, he became the discoverer of the periodic law
of the elements, which was confirmed later by subsequent
discovery of other elements and by studying X-ray spectra
and quantum mechanics. He wrote about 400 published
works, concerning physics and technology, besides chemistry.
He wrote the textbook Foundations of Chemistry.
2.1.43 Moment of Inertia Ratio γ
γ5
I
m r2
I (kg m2) moment of inertia of the body; m (kg) mass of the body; r (m) radius.
The ratio γ represents the normalized dimensionless inertia moment.
Info: [C93].
2.1.44 MoninObukhov Length L
L52
Rcp w3
gαK
R (kg m23) air density; cp (J kg21 K21) specific heat capacity; w (m s21) air
velocity; g (m s22) gravitational acceleration; α (K21) coefficient of linear
thermal expansion; K (m kg K22) Kármán constant.
The length L corresponds to the measurement at which the Richardson
number Ri (p. 83) is Ri 5 1 for the flow near a heated wall with free and forced
convections.
Info: [C95],[C94].
2.1.45 Neumann Numerical Number Neu
Neu 5 D
1
1
Δτ # χ
1
ðΔxÞ2
ðΔyÞ2
D (m2 s21) diffusion coefficient; Δx, Δy (m) grid spacing in both coordinate
axes; Δτ (s) time step; χ () time stepping parameter.
22
Dimensionless Physical Quantities in Science and Engineering
This number expresses the stability criterion for a numerical solution, for
example, of two-dimensional diffusive flow (Neu # 1). Together with the Courant
numerical number Cou (p. 8), it characterizes the oscillating convection flow
through porous material. Numerical mathematics. Two-phase flow.
Info: [B44].
John von Neumann (28.12.19038.2.1957), American
mathematician of Hungarian origin.
He was one of the greatest mathematicians of the twentieth
century. He substantially influenced the development of the
structural and functional principles of computers. His scientific
sphere of interest was extraordinarily wide. He participated in
elaborating theoretical foundations for atomic energy utilization. He founded the theory of sets, quantum theory and theory
of games, too, which represent important areas of mathematics
and economics.
2.1.46 Normalized Frequency, V number V
V5
2πr
λ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n21 2 n22
r (m) core radius of optical fibre; λ (m) wavelength in vacuum; n1 () maximum refractive index of the core; n2 () refractive index of the homogeneous
cladding.
It is important in telecommunication, to quantify the optical fibres especially,
to determine the actual to relative or nominal frequencies ratio. It is applied in
optoelectronics and telecommunications. For single-mode operations it is V , 2 and
for multi-mode ones V . 5. Optoelectronics, physics and telecommunications.
Info: [C97].
2.1.47 Optical Thickness τ, h
τ5k L
ð1Þ;
x
Ix;λ 5 I0;λ exp 2
τ
ð2Þ
k (m21) monochromatic absorption coefficient; L (m) characteristic length,
thickness; Ix,λ, I0,λ () luminous intensity at the depth x and on the material
surface at the wavelength λ; x (m) depth below the surface; λ (m) wavelength.
It expresses the material depth at which the radiation intensity (of various wavelengths, e.g. of visible light) of a given frequency is reduced by a coefficient 1e.
In the optical thickness depth, about 23 of the radiation is absorbed. Physics. Optics.
Atmospheric radiation.
Info: [C99].
Physics and Physical Chemistry
23
2.1.48 Péclet Difference Number Penum
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2 wx
wy
Dxx
Dyy
Dxy 21
Penum 5
1
1
2
Δx
Δy
ðΔxÞ2
ðΔyÞ2 Δx Δy
wx, wy (m s21) flow velocity components in pores; Δx, Δy (m) mesh increments
in x and y axes; Dxx, Dyy (m2 s21) components of the dispersion coefficient;
Dxy (m2 s21) transverse dispersion coefficient.
This number expresses the stability criterion of a numerical solution, for example,
of an underground water flow through dispersive surroundings. Physically, it is about
dynamic and dispersive forces acting. Numerical mathematics. Two-phase flow.
Info: [C98].
Jean Claude Eugène Péclet (p. 180).
2.1.49 Péclet Grid Number Pemesh
w Δx
, 2 ð1Þ;
D
wx;max Δx wz;max Δz
;
Pemesh 5 max
D
D
Pemesh 5
ð2Þ
w (m s21) velocity; wx,max, wz,max (m s21) maximum value of the velocity
components; Δx, Δz (m) mesh increments in the x and z axes; D (m2 s21) diffusion coefficient.
Equation (1) is the stability criterion of a numerical process solution for onedimensional flow or transfer. Equation (2) is the limiting condition to choose a difference step for solution a two-dimensional diffusive process with a moving member. Numerical fluid mechanics. Mathematics.
Info: [A47],[B77],[B44].
Jean Claude Eugène Péclet (p. 180).
2.1.50 Pitzer Number Pi
Pi P0:7 5
p
T
5 0:7 at Tred 5
5 0:7
pref
Tref
p, pref (Pa) pressure, reference pressure; T, Tref (K) temperature, reference
temperature; P0.7 () reduced pressure; Tred () reduced temperature.
It serves to determine the reduced steam pressure at reduced temperature.
Thermodynamics.
24
Dimensionless Physical Quantities in Science and Engineering
Kenneth Sanborn Pitzer (6.1.191426.12.1997), American
theoretical chemist.
Pitzer was the founder of modern theoretical chemistry.
He used quantum and statistical mechanics to explain the
thermodynamic and conformational properties of molecules,
and he pioneered quantum scattering theory for describing
chemical reactions at the most fundamental level. He also
made contributions to relativistic effects in chemical bonding
and the theory of fluids and electrolyte solutions.
2.1.51 Planck Number, Radiation Parameter Pl
Pl 5
λβ
3
0 Tref
4n2 σ
λ (W m21 K21) intrinsic ambient thermal conductivity; β (m21) parameters
vector of the absorption coefficient; n () refractive index; σ0 (W m22 K24) StefanBoltzmann constant; Tref (K) reference temperature.
This number expresses radiation effects on a convective boundary layer with a
uniformly heated surface. An increasing Planck number leads to the reduction of
the thickness of the layer. On the contrary, an increasing temperature increases the
thickness of the layer. For gases, the Pl number depends on the Prandtl number Pr
(p. 197) and the temperature and is in the range of 100150. For water steam with
the temperature 100500 C and Pr 5 1, it is PlAh30; 200i. It is analogous to the
radiation number (2.) N (p. 211).
Info: [B3].
Max Karl Ernst Ludwig Planck (23.4.18584.10.1947),
German physicist. Nobel Prize in Physics, 1918.
He was engaged in research related to the radiation of the
perfect black body and discovered the Planck radiation law,
which determines the dependence of the volume radiation
density of this body on the radiation frequency and body temperature. This law is based on the hypothesis of discontinuous
radiation of electromagnetic energy in doses. For his discovery
of the electric quanta, he won the Nobel Prize. He was one of
the first who accepted and evolved Einstein’s relativity theory.
2.1.52 Porosity Φ, ϕ, p, n, e
Φ5
Vf
Vt
Vf (m3) volume of void space filled up by fluid; Vt (m3) total or bulk volume
of material including the solid and void components.
Physics and Physical Chemistry
25
It characterizes the ability of a porous material to receive a fluid. The porosity
ΦAh0; 1i or ΦAh0; 100%i. Fluid mechanics. Geophysics. Sedimentation. Drying.
Info: [C105].
2.1.53 Reduced Boiling Temperature Nrbt
Nrbt 5
Pi
D
RePr
4
L
D (m) circular pipe diameter; L (m) characteristic length; Pi () Pitzer number
(p. 23); Re () Reynolds number (p. 81); Pr () Prandtl number (p. 197).
It expresses the reduced normal boiling temperature. It is used for steam transfer
through a pipeline. Thermodynamics.15
2.1.54 Reduced Pressure pred
pred 5
p
pcrit
p (Pa) liquid pressure; pcrit (Pa) critical pressure.
It serves to predict the properties (p, v, T) of gases from critical values.
Thermodynamics.
Info: [C110].
2.1.55 Reduced Volume vred
vred 5
v
vcrit
v (m3 kg21) gas or vapour specific volume; vcrit (m3 kg21) gas or vapour specific volume at critical state.
It serves to compare and predict the properties (p, v, T) of gases from critical
values. Thermodynamics.
2.1.56 Relative Atomic (Molecular) Mass Ar, Mr
Ar 5
mA
u
ð1Þ;
Mr 5
mM
u
ð2Þ
mA (kg) atomic mass; mM (kg) molecular mass; u (kg) atomic mass unit
(1.66057 3 10227 kg); mM (kg mol21) molar mass.
26
Dimensionless Physical Quantities in Science and Engineering
The relative atomic mass of an atom is a dimensionless number indicating
how many times the atomic mass is greater than the atomic mass unit. Similarly,
the relative molecular mass of a molecule is a dimensionless number indicating
how many times the molecular mass is greater than the atomic mass unit. The
relative molecular mass can be calculated as a sum of all relative atomic masses of
all atoms of a molecule. The atomic mass unit is defined as 1/12 of the mass of the
carbon isotope nuclide 126 C;
m 126 C
:
u5
12
The molar mass gives the 1-mole quantity substance mass. The pure substance
molar mass can be calculated as a quotient of its mass and its mass quantity or as a
product of the atom (molecular) mass and the number of atoms (molecules) in one
mole of the substance,
Mm 5 mM NA 5 Mr uNA 5 0:000999995Mr
which is valid for the molar mass in the kg mol21. For the molar mass in g mol21
units, the relation Mm 5 Mr is valid. Therefore, the relative molecular (atomic)
mass corresponds numerically to the molar mass expressed in the g mol21 units.
For example, for the relative molecular mass of carbon dioxide CO2, one obtains
(12 12 3 15.9994) 5 43.9988. Hence, the CO2 molecule has the molecular mass of
43.9988u and the molar mass of 43.9988 g mol21.
The relative molecular (atomic) masses and the molar masses can be used especially in stochiometric calculations.
Info: [C91].
2.1.57 Relative Humidity ϕ
ϕ5
φ
φv
φ (kg kg21) specific absolute humidity; φv (kg kg21) saturation-state-specific
absolute humidity.
It expresses the relation of the gas humidity to the saturation-state humidity, specifically under the same temperature and pressure. If saturated, the gas has the dew
point temperature.
Info: [A24].
2.1.58 Riedel Number Rie
T dp
Rie 5
p dT crit
T (K) temperature; p (Pa) pressure.
Physics and Physical Chemistry
27
It expresses the temperaturepressure relation of fluids under critical
conditions.
2.1.59 Shape Equivalence of System A
A5
S
Sekv
S (m2) heat transfer surface of warmed-up or cooled-down body; Sekv (m2) surface area of equivalent body by the same volume; O, Oekv (m) cross section
perimeters of the body; Vekv (m3) sphere equivalent volume.
It characterizes the relation between the geometric shape of a body and its
thermal field. It is used to replace geometrically complicated bodies with an unlimited finite thickness plate, a cylinder or a ball. For an infinitely long cylinder,
21
5 O O21
A 5 S Sekv
ekv , and for a ball
S
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A5 p
3
2
36πVekv
21
For example, for a boundary condition, it is αekv 5 α S Sekv
.
Info: [A23].
2.1.60 Size Factor x
x5
2πa
5 ka
λ
a (m) particle size; λ (m) wavelength; k (m21) wave number.
It expresses the relation between the size of a moving particle, the wavelength
and the wave number.
Info: [C121].
2.1.61 Snellen Number Sn
Sn 5
a
b
ð1Þ;
Sn 5
a
a
5 291 3 1026
b
h
ð2Þ
a (m) distance at which an object can be recognized; b (m) distance at which the
same object can be seen at a subtended angle of 1/60 to the eye; h (m) object size.
It expresses the perfection of sight. For perfect sight it is Sn 5 1 and for imperfect
sight it is Sn , 1. Optics. Ophthalmology.
Info: [A43].
28
Dimensionless Physical Quantities in Science and Engineering
Herman Snellen (19.2.183418.1.1908), Dutch ocularist
and physicist.
Above all, he was engaged in the physiology of the
eye. He created a very complex body of work in this special
sphere. Together with F.C. Donders he did great numbers
of experiments and participated in the development of ophthalmological instruments. He focused on corrective eye
glasses calculations. The visual Snellen chart, consisting of
variously sized letters, is still used to determine the quality
of sight all over the world.
2.1.62 Sound Emissivity NI
NI 5
Ia
Ii
Ia, Ii (W m22) intensity of absorbed (a) and incident (i) sound.
It expresses the ratio of the absorbed sound intensity to that of the incident
sound.
Info: [A24].
2.1.63 Sound Pressure Level L
L 5 20 log
p
I
5 10 log
p0
I0
p (Pa) acoustic pressure; p0 (Pa) reference pressure (p0 5 2 3 1025 Pa);
I (W m22) acoustic intensity; I0 (W m22) threshold acoustic intensity.
It expresses the logarithm of the acoustic-to-reference pressure ratio. Acoustics,
noise measurement.
2.1.64 Specific Heat Ratio k, γ
k5
cp
cv
cp (J kg21 K21) specific heat capacity at constant pressure; cv (J kg21 K21) specific heat capacity at constant volume.
It expresses the ratio of the heat capacity at constant pressure to the heat capacity at constant volume. Alternatively, it expresses the ratio of the enthalpy to the
internal energy. It characterizes thermodynamic relations in compressible fluid
flow. It is called also the adiabatic criterion, Poisson’s constant or the adiabatic
exponent. For air, it is γ 5 1.4. Physics. Mechanics. Aerodynamics.
Physics and Physical Chemistry
29
Info: [C108].
Siméon Denis Poisson (p. 143).
2.1.65 Strehl Ratio S
S 5 expð22ð2πσÞ2 Þ
σ () root-mean-square deviation of the wavefront measured in wavelengths.
The Strehl ratio is expressed by the relation of the focus intensities peak in
deviated and ideal points of diffraction. Optics, light interference and diffraction.
Info: [C125].
2.1.66 Surface Elasticity Number NSE
NSE 5 2
CS L @σ
DS η @CS
CS (K s21) surface concentration of surfactant in undisturbed state; DS (m2 s21) surface diffusivity; L (m) film thickness; η (Pa s) dynamic viscosity;
σ (N m21) surface tension.
This number expresses the surface elasticity of solutions in diffusive mass
transfer.
Info: [A29].
2.1.67 Surface Tension Number NST
NST 5
η2
hσR
η (Pa s) dynamic viscosity; h (m) ratio of surface area to perimeter; σ (N m21) surface tension; R (kg m23) density.
It characterizes the surface tension in mass transfer.
Info: [A31].
2.1.68 Surface Viscosity Number NSV
NSV 5
ηS
ηh
ηS (N s m21) surface viscosity; η (Pa s) dynamic viscosity; h (m) film depth
(thickness).
30
Dimensionless Physical Quantities in Science and Engineering
This number is used if there is a convective transfer zone in a fluid layer with
a surface-active substance.
Info: [A29].
2.1.69 Temkin Number Es
pffiffiffi
S
LS
p
Es 5 3 ffiffiffiffi 5
LV
V
ð1Þ;
O
Es 5 pffiffiffi
A
ð2Þ
S (m2), V (m3) surface and volume of thermal system; LS, LV (m) characteristic
dimensions; O (m), A (m2) perimeter and cross-sectional area.
It characterizes the body shape influence on its thermal field. However, in
comparing diverse bodies, the difference of physical parameters is not considered.
For limited bodies, expression (1) is valid and so is expression (2) for in-one-direction
non-limited bodies which have two finite shapes.
The lengths LS and LV depend on the prolongation and shape of the body. To
consider also the body size in the Es number, the auxiliary parameter Es0 5 L L21
V
is used.
In general, the Es number is not a similarity criterion because the set of bodies
corresponds to its one value. However, from the equality of the criteria, no conclusions can be made about their physical similarity even if their shapes are subject to
the isomorphism condition. The mutually unambiguous relation between the body
shape and the Es criterion can be determined if the body shape and size are the
functions of two physical dimensions only. For example, Es 5 2.35 is valid for a
cylinder of a radius r and height
pffiffiffi 2r. For a ball of radius r, Es 5 2.21 and Es 5 2, 51
for a cone of the height of r 3 and a base of a radius r.
With the same values of the Biot number Bi (p. 173) and Fourier number Fo
(p. 175), the increasing of the number Es leads to faster cooling of the body, to
decreasing its surface temperature and to increasing the temperature non-uniformity.
The influence of the Es number on the temperature field of bodies, other than convex
ones, appears most expressively. It is also called the shape criterion or the geometric
cooling down criterion.
Info: [A23].
A.G. Tĕmkin.
2.1.70 Timescale Number Nτ, Go
Nτ 5
wτ
L
ð1Þ;
Go 5
τD
τR
ð2Þ
w (m s21) speed; τ (s) time scale; L (m) length scale; τ D (s) dynamic time;
τ R (s) radiation time.
Physics and Physical Chemistry
31
In expression (1), it is the dimensionless expression of time of a moving body
or fluid. The controlling timescale number for the atmosphere or the interior
of non-rotating planets (2) is called the Golitsyn number Go (p. 397). Dynamic processes. Astronomy.
Info: [C134].
2.1.71 Tortuosity τ
τ5
l
L
l (m) mean length of the free path; L (m) characteristic length.
It characterizes permeable substances and expresses the ratio of a true
mean length of free path, for example, of the flow, to the characteristic length
(thickness) of the substance. Besides the application in fluid mechanics in flow
through crystallic or porous materials, it is used in hydrogeology, in physical
technologies, in mathematics in functional minimizing by cubic splines and in
solving curvature problems. For example, in physical technologies, it expresses the
ratio of the molecule path, which must pass through the deposited layer tortuously,
to the thickness of the layer. Mathematics. Physics. Fluid mechanics. Geophysics.
Physical technology.
Info: [C135].
2.1.72 Tzou Number, Stability and Convergency Criterion Tz
Tz
Tz
Δx
5 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ;
2aΔtð2τ T 1 ΔtÞ
2τ q 1 Δt
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sffiffiffiffiffi
Δx
Δt
a
5
; v5
11
vΔt
2τ q
τq
ð2Þ
Δx (m) mesh increment; a (m2 s21) thermal diffusivity; Δt (s) time step;
τ T (s) thermalization time; τ q (s) relaxation time; v (m s21) thermal wave
speed.
For the finite differences method it determines the stability and convergence zone
of a solution if Tz $ 1. In the case of unbalanced heat propagation with the defined
relaxation time τ q and the thermalization time τ T, the criterion is in form (1). In the
case of wave-diffusive heat propagation (τ T 5 0), the criterion can be expressed in
form (2). In the case of balanced diffusive heat propagation (τ T 5 τ q), the criterion
is reduced on the CrankNicolson parameter (p. 9).
Info: [A46],[B116].
32
Dimensionless Physical Quantities in Science and Engineering
2.1.73 Van der Waals Number (1.) Wa
Wa 5
rRcrit Tcrit
rTcrit
5
pcrit
pcrit vcrit
r (J kg21 K21) specific gas constant; Rcrit (kg m23) substance density at critical
state; Tcrit (K) critical thermodynamic temperature; pcrit (Pa) critical pressure;
vcrit (m3 kg21) critical specific volume.
It is the fundamental criterion of thermodynamics. It expresses the state law of
fluids for a critical point and characterizes one substance or a whole group thereof in
accordance with their thermodynamical similarity. For various substances, its value
is various in general. However, in van der Waal’s equation its value is Wa 5 8/3, but
it is WaAh3.3; 4.5i alternatively. It is called the critical coefficient as well.
Info: [A23].
Johannes Diderik van der Waals (23.11.18379.3.1923),
Dutch physicist. Nobel Prize in Physics, 1910.
He was engaged in research in gaseous and liquid states
of a mass. He started from the idea that the ideal gas state
equation can be deduced from the kinetic gas theory, provided the volume of molecules and the force between them
is neglected. With it, he succeeded in creating a new theory.
Subsequently, in the year 1881, he formulated a new state
equation named the van der Waals equation, the validity of
which he extended to all substances.
2.1.74 Van der Waals Number (2.) Wa
Wa 5
A
kT
A (J) Hamaker constant; k (J K21) Boltzmann constant; T (K) absolute
temperature.
It characterizes the ratio of the van der Waals interaction energy to the heat
energy of particles. Thermodynamics of two-phase fluids. Filtration. Deposition.
Info: [A42].
Johannes Diderik van der Waals (see above).
2.1.75 Void Fraction ε
ε5
Va
Vm
Va (m3) volume of void space; Vm (m3) total or bulk volume of material
including the solid and void components.
Physics and Physical Chemistry
33
It expresses the ratio of the void space volume in the material to the total material
volume. Physics. Fluid mechanics. Flow through a porous material. Drying.
2.1.76 Wave Number K, k
K5
L
λ
ð1Þ;
K5
2πL ωL
5
λ
a
ð2Þ;
K 5 Kref n ð3Þ
L (m) characteristic length; λ (m) wavelength; ω (s21) angular frequency;
a (m s21) sound velocity; Kref () wave number in the reference medium;
n () medium’s index of refraction.19
In a dimensionless shape it expresses the wave length. The expressions (1) and
(2) are different forms only. For example, the expression (3) is used with electromagnetic waves. In acoustics, it is called the substance wave number (medium
wave number).
Info: [A2],[B7],[C139].
2.1.77 Zhukovsky Shape Number Ju
Ju 5
A 2 2
Es Es0
16
ð1Þ;
1
Julam
64
ð3Þ
Jurel 5
Julam 5 32LoΨ
ð2Þ;
Es, Es0 () Temkin number and parameter (p. 30); A () shape equivalence
of system (p. 27); Lo () Lobachevsky number (p. 18).
It expresses the influence of clear area on the hydraulic resistance for fluid flow in
non-circular canals and pipelines. By using the Lobachevsky number Lo (p. 18) and
the non-uniformity criterion of the rate field Ψ 5 wmaxw21, the Zhukhovski laminar
number (2) is obtained. The expression (3) is the relative shape criterion. The correspondence between the expressions (1) and (3) is given by various modifications of
HagenPoiseuille’s law for the laminar flow in a circular, cross-sectional tube.
Info: [A23],[A33].
Nikolay Jegorovich Zhukovsky (17.1.184717.3.1921),
Russian physicist and mechanicist.
He founded the Russian school of hydromechanics and
aerodynamics and is called the father of Russian aeronautics.
He was engaged in improving the wing profile (Zhukhovski’s
profile) and published two works (1906) in which he introduced
a mathematical expression for wing lift (KuttaZhukhovski’s
theorem). He devoted himself also to high-velocity aerodynamics and to hydraulic shocks. In mathematics, he is known thanks
to his conformal mapping in a complex plane, which is known
as Zhukhovski’s transform.
34
Dimensionless Physical Quantities in Science and Engineering
2.2
Physical Chemistry
In physical chemistry, the dimensionless quantities mostly express the physical
and chemical conversions in homogeneous or heterogeneous processes or systems.
The focus is the non-isothermic or isothermic processes such as burning and
combustion, dynamics of flame propagation, baking, and heat and mass transfers
with chemical or phase reactions. Further, there are mixing, drying and corrosion
processes and physical and chemical processes in chemical reactors, mass diffusion
processes in catalyzers, granulation processes, dissociation of electrolytes in solutions, separation of solid particles in solutions and others.
The Arrhenius, Damkőhler, Frank-Kamenetskii, GuldbergWaage, Knudsen
diffusion and Prater numbers are among the best known dimensionless numbers.
2.2.1
Arrhenius Number (1.) Energetic Ah1
Ah1 5
E
RT
ð1Þ;
Ah1 5 U21 5
E
RTN
ð2Þ
E (J mol21) molar activation energy; R (J mol21 K21) molar gas constant;
T (K) absolute temperature; TN (K) ambient temperature.
This number expresses the activation-to-potential energy ratio of a fluid. It
characterizes the energetic conditions in a chemicalthermal reaction. For example,
it relates to the ability of a fuel to self-ignite. It is used, for example, to describe the
heat transfer in combustion chambers. Usually, the inverse value of equation (2) is
called the combustion parameter U (p. 35).
Info: [A29],[A33],[B17].
Svante August Arrhenius (19.2.18592.10.1927), Swedish
physical chemist. Nobel Prize in Chemistry, 1903.
He was engaged in chemical reaction description and
discovered the Arrhenius velocity law, which describes the
speed of chemical reactions. In addition, he elaborated on
the theory of electrolytes and made clear the electrical
conductivity of ion solutions. He received the Nobel Prize
for his electrolytic dissociation theory.
2.2.2
Arrhenius Number (2.) Velocity Ah2
Ah2 Bd Pem 5
wL
Da
w (m s21) speed; L (m) characteristic length; Da (m2 s21) axial diffusivity;
Bd () Bodenstein number (p. 244); Pem () Péclet mass number (p. 258).
Physics and Physical Chemistry
35
It characterizes the mass diffusion process in chemical reactors. It is the measure
of the convective mass transfer related to the molecular transfer in flowing surroundings. It expresses the degree to which an investigated system approaches an
idealized model of the reacting flow structure. With Ah2-N, it relates to a reactor
or system with total separation. With Ah2-0, it relates to a completely mixing system. In intermediate models, N . Ah2 . 0 is valid. It concerns diffusive, combined
or cellular models. Often, it is called the Bodenstein number Bd (p. 244). In
essence, it is the modified Pe´clet mass number Pem (p. 258).
Info: [A33].
Svante August Arrhenius (see above).
2.2.3
Arrhenius Number (3.) Time Ah3
Ah3 Fo21
ch 5
uL2
L2
5
RDa
Da τ ch
u (kg m23 s21) chemical reaction rate; L (m) characteristic length; R (kg m23) density; Da (m2 s21) axial diffusivity; τ ch (s) chemical reaction time;
Foch () Fourier chemical number (p. 39).
In the dimensionless shape, its reciprocal value characterizes the chemical reaction
time. Alternatively, it is also the Fourier chemical number Foch (p. 39). Sometimes it
is called the diffusion contact number Ko (p. 38), namely in a mononuclear reaction
in a stationary process with a simple conversion.
Info: [A33].
Svante August Arrhenius (see above).
2.2.4
γ
Burnout Parameter γ
2
RTN
cV R
E Q Ci
R (J mol21 K21) molar gas constant; TN (K) ambient temperature; E (J mol21) molar activation energy; cV (J kg21 K21) specific heat capacity; R (kg m23) density;
Q (J) heat reaction effect; Ci (m23) reagent numerical molecule concentration.
It characterizes the burnout in fuel combustion.
Info: [A23].
2.2.5
Combustion Parameter U
U Ah21
1 5
RTN
E
R (J mol21 K21) molar gas constant; TN (K) ambient temperature; E (J mol21) molar activation energy; Ah1 () Arrhenius number (1.) energetic (p. 34).
36
Dimensionless Physical Quantities in Science and Engineering
This parameter characterizes the temperature originating in a thermal system as
a chemical reaction result. See the Arrhenius number (1.) energetic Ah1 (p. 34).
Info: [A23].
2.2.6
Chemical Reaction Rate Γ
Θ
Γ 5 δγη exp
1 1 UΘ
δ () Frank-Kamenetskii number (p. 35); γ () burnout parameter (p. 40); η ()
critical reagent concentration; Θ () reagent temperature (p. 47); U () combustion parameter (p. 35).
It characterizes the reaction kinetics in a physical and chemical conversion
process.
Info: [A23].
2.2.7
Damkőhler Number Da
Da 5
τD
τR
τ D (s) characteristic diffusion timescale; τ R (s) characteristic reaction timescale.
It expresses the relation of diffusion and relaxation times in chemical processes.
It is used for turbulent combustion. With Da{1, the turbulence is much faster than
the chemical reaction.
Info: [A23],[A33].
Gerhard Friedrich Damkőhler (19081944), German physical chemist.
2.2.8
Damkőhler Number (1.) Hydrodynamic Da1
Da1 5
uL
wRi
ð1Þ;
Da1 5
τc
τr
ð2Þ
u (kg m23 s21) chemical reaction rate; L (m) characteristic length; w (m s21) flow velocity; Ri (kg m23) mass concentration; τ c, τ r (s) chemical conversion
and reaction time.
This number expresses the ratio of chemical conversion rate to that of the fluid
flow. Alternatively, it is the ratio of the number of moles entering the reaction to
that of moles induced by the fluid flow. It characterizes the hydrodynamic influence
in chemical reactions. Alternatively, together with other Damkőhler numbers, it
characterizes the reaction time in general with transport phenomena.
Info: [A2],[A26],[A29],[B11].
Gerhard Friedrich Damkőhler (see above).
Physics and Physical Chemistry
2.2.9
37
Damkőhler Number (2.) Diffusion Da2
Da2 5
u L2
D Ri
u (kg m23 s21) chemical reaction rate; L (m) characteristic length; D (m2 s21) diffusivity; Ri (kg m23) mass concentration.
It characterizes the diffusion relations in chemical reactions. It expresses the
ratio of the chemical conversion rate in a fluid flow to the molecular mass diffusion
rate; alternatively, it is the ratio of the number of reacting moles in a fluid flow to
the number of moles brought by the molecular diffusion.
Info: [A2],[A23],[A26],[A29],[B11].
Gerhard Friedrich Damkőhler (see above).
2.2.10 Damkőhler Number (3.) (1. Heat) Da3
Da3 5
huL
cp RwT
h (J kg21) specific enthalpy; u (kg m23 s21) chemical reaction rate; L (m) characteristic length; cp (J kg21 K21) specific heat capacity; R (kg m23) density;
w (m s21) flow velocity; T (K) temperature.
This number expresses the ratio of the heat liberated in a chemical reaction in
the fluid flow to that transferred by convection. It characterizes the heat transfer by
convection in chemical reactions in fluids.
Info: [A2],[A23],[A26],[A29],[B11].
Gerhard Friedrich Damkőhler (see above).
2.2.11 Damkőhler Number (4.) (2. Heat) Da4
Da4 Th2 5
huL2
λT
ð1Þ;
Da4 PoV Os Th2 huL2
λTref
ð2Þ
h (J kg21) specific enthalpy; u (kg m23 s21) chemical reaction rate; L (m) characteristic length; λ (W m21 K21) thermal conductivity; T (K) temperature;
Tref (K) reference temperature; Th () Thiele modulus (1.) (p. 267); PoV () Pomerantsev heat number (p. 181); Os () Ostrogradsky number (p. 179).
It expresses the ratio of the heat liberated by a chemical reaction to that transferred by conduction. It characterizes the heat transfer process in chemical reactions.
Substantially, the matter is the Pomerantsev heat number PoV (p. 181), also
38
Dimensionless Physical Quantities in Science and Engineering
called the Ostrogradsky number Os (p. 179) in Russian literature. Its second root
corresponds to the Thiele modulus (1.) Th (p. 267).
Info: [A2],[A23],[A26],[A29],[B11].
Gerhard Friedrich Damkőhler (see above).
2.2.12 Damkőhler Number (5.) Hydrodynamic Da5
Da5 Re 5
wL
v
ð1Þ;
Da2
5 Sc ð2Þ;
Da1 Da5
Da4
5 Pr
Da3 Da5
ð3Þ
w (m s21) flow velocity; L (m) characteristic length; ν (m2 s21) kinematic
viscosity; Re () Reynolds number (p. 81); Sc () Schmidt number (p. 263);
Pr () Prandtl number (p. 197); Da1 () Damkőhler number (1.) hydrodynamic (p. 36); Da2 () Damkőhler number (2.) diffusion (p. 37); Da3 () Damkőhler number (3.) (1. heat) (p. 37); Da4 () Damkőhler number (4.) (2.
heat) (p. 37).
For flowing fluid, it expresses the inertia-to-friction ratio. It characterizes the
hydrodynamic influence of the inertia and friction forces of the streaming fluid in
a chemical reaction. The relation of the Schmidt number (p. 263) and the Prandtl
number (p. 197) to the Damkőhler numbers (p. 36, 38) is obvious from the expressions (2) and (3).
Info: [A4],[A23],[A26],[A29],[B11].
Gerhard Friedrich Damkőhler (see above).
2.2.13 Diffusion Contact Number Ko
See Arrhenius number (3.) time Ah3 (p. 35).
2.2.14 Djakonov Number (1.) Contact Dj, Ko
Dj 5 k1 ϕ1 ðck Þτ 5
τ
τr
k1 ϕ1 (ck) (s21) unit speed at a given concentration; k1 () virtual speed constant
of direct reduction process; ck () concentration; τ (s) contact time; τ r (s) decay time.
Together with the thermodynamic equilibrium of chemical reduction Nreac
(p. 48), it expresses the general similarity condition for the physical and chemical
conversion process. It is the measure of the contact-to-decomposition time ratio
of exiting products of the chemical conversion. The contact time means the time
during which the corresponding mixture composition is in the reaction zone.
Info: [A13].
Physics and Physical Chemistry
39
German Konstantinovich Djakonov (11.6.19071953),
Russian physical chemical engineer.
His work represents a new dimension in complex process
modelling. He was engaged especially in the theory and
methods of modelling physical and chemical processes and
conversions. He solved the problem of thermodynamic equilibrium. From the analysis of all gained similarity conditions
and criteria, he determined that they cannot be fulfilled at
the same time. Therefore, he was engaged in approximate
modelling and created an original method of solution.
2.2.15 Djakonov Number (2.) Thermodynamic Dj, Po
Dj 5
k2 ϕ2 ðck Þ
k1 ϕ1 ðck Þ
k2 ϕ2 (ck) (s21) unit rate of indirect change; k1 ϕ1(ck) (s21) unit rate of direct
change; k1, k2 () virtual rate constant of direct and indirect reaction; ck () critical concentration.
It expresses the thermodynamic equilibrium in the physical and chemical conversion process. It represents the return-to-direct conversion rates ratio. Together
with the Djakonov number (1.) contact Dj (p. 38), it represents the condition for
the general physical and chemical conversion.
Info: [A13].
German Konstantinovich Djakonov (see above).
2.2.16 Djakonov Number (3.) Thermodynamic Di
Di 5
q
n
q (m3 s21 m23) specific volume flux; n (s21) rotational frequency.
It expresses the ratio of the substance, mixed in a unity reaction volume during
a time unit, to the amount of the substance conveyed into a reacting volume. It
characterizes the physical and chemical conversion process in the mixing of fluids.
Alternatively, it can also be expressed as the ratio of the substance conversion rate
to the rate of its feeding into the chemical fluid mixing facility.
Info: [A13].
German Konstantinovich Djakonov (see above).
2.2.17 Fourier Chemical Number Foch
See Arrhenius number (3.) time Ah3 (p. 35).
Jean Baptiste Joseph Fourier (p. 175).
40
Dimensionless Physical Quantities in Science and Engineering
2.2.18 Fraction Granulation Number Pfrac
Pfrac 5
V
VV
V (m3) particle volume; VV (m3) vessel volume.
It represents the parameter expressing the ratio of the volume of particles to that
of the vessel in the granulation process. Physical chemistry. Granulation. Mixing.
Info: [C26].
2.2.19 Frank-Kamenetskii Number δ
Q k Ci L2 E
E
qv L2 E
E
5
exp 2
exp 2
λ TN RTN
λTN RTN
R TN
RTN
PoV
1
exp 2
5 PoV Ah1 expð2Ah1 Þ
5
U
U
δ5
Qk (J s21) volume reaction heat flow rate; Ci (m23) initial numerical concentration
of reagent molecules; L (m) characteristic length; λ (W m21 K21) thermal conductivity of reacting mixture; TN (K) ambient temperature; E (J mol21) activation
energy per unit molar mass; R (J mol21 K21) molar gas constant; qV (W m23) volume heat flux at the beginning of chemical reaction; PoV () Pomerantsev
heat number (p. 181); U () combustion parameter (p. 35); Ah1 () Arrhenius
number (1.) energetic (p. 34).
In the dimensionless shape, this number characterizes the reaction heat and the
heat transfer in reacting systems, for example, in burning.
Info: [A29],[A35].
David Albert Frank-Kamenetskii (born 1910), Russian engineer.
2.2.20 Frank-Kamenetskii Temperature Φ
Φ5
EðT 2 T0 Þ
2
RTN
E (J mol21) molar activation energy of reaction; T (K) temperature after
reaction; T0 (K) initial temperature of the reacting chemicals; R (J mol21 K21) molar gas constant; TN (K) reference temperature.
It characterizes the temperature or temperature stains (hot spots, higher temperature zones) originating in an exothermic reactive filling of a reservoir in which a
thermal explosion can occur if the heat generated in the reservoir is greater than
that let out of it. Physical chemistry. Ecology.
Info: [B106].
David Albert Frank-Kamenetskii (see above).
Physics and Physical Chemistry
41
2.2.21 Froude Granulation Number Frgran
Frgran 5
n2 d
g
n (s21) engine revolutions; d (m) engine diameter; g (m s22) acceleration
of gravity.
It expresses the centrifugal-to-gravitational energy ratio in mixing and granulating
processes. The criterion was deduced from empirical experience. Physical chemistry.
Granulation.
Info: [C67].
William Froude (p. 63).
2.2.22 Granulation Fluid Number Ngran
Ngran 5
mτ
VR
m (kg s21) quantity of granulating fluid added at time unit; τ (s) time; V (m3) volume of particles; R (kg m23) density of particles.
This number expresses the specific quantity of granulating fluid in granulation.
Physical chemistry. Granulation.
Info: [C67].
2.2.23 GuldbergWaage Number Gw
n2
m1 m2
Gw 5 yn1
1 y2 . . . 2ðx1 x2 . . .Þ
1
k
y1, y2, ... () mole fractions of reaction products; x1, x2,... () mole fractions
of reactants; n1, n2, ... () stoichiometric coefficients of individual reaction
products; m1, m2, ... () stoichiometric coefficients of individual reactants;
k () dimensionless partial equilibrium constant.
It characterizes the similarity of chemical reactions in progress, for example,
in blast furnaces. It is expressed by the relation of volumes of reacting gases and
products of the reaction.
Info: [A23],[A29].
Cato Maximilian Guldberg (11.8.183614.1.1902), Norwegian
mathematician.
Together with his cousin Peter Waage, in the year 1864, he
formulated the law of mass behaviour in chemical reactions.
This law expresses in detail the influence of concentration,
mass and temperature on the speed of chemical reactions.
42
Dimensionless Physical Quantities in Science and Engineering
Peter Waage (29.6.183313.1.1900), Norwegian chemist.
In the year 1864, together with his cousin Cato
Maximilian Guldberg, he formulated the law of mass
behaviour in chemical reactions. He was also engaged in
mineralogy and crystallography.
2.2.24 Hatta Number Ha, Ht
γ
;
Ha 5
tanh γ
Ha 5
k1 D
β2
1
2
ðkn Rn21
i DÞ
where γ 5
β
ð1Þ;
ð2Þ
kn (m3 (n21) kg2 (n21) s21) reaction rate constant for nth order chemical reaction;
Ri (kg m23) mass concentration; D (m2 s21) diffusivity; β (m s21) mass
transfer coefficient.
Generally, in form (1), it characterizes the gas absorption with a chemical
reaction being under way. In form (2), it is about the first order of a chemical reaction, for example, in cleaning (washing) the gas in fluid. Flow in geologic layers,
silts and deposits.
Info: [A39],[A50].
Shirōji Hatta (born 1895), Japanese chemist.
2.2.25 Helmholtz Pulsation Number, Helmholtz Resonator Group K
3 1
L 2 21
K5
M
V
L (m) characteristic length, diameter; V (m3) volume; M () Mach number
(p. 73).
It characterizes the processes in pulsating combustion.
Info: [A43].
Hermann
Ludwig
Ferdinand
von
Helmholtz
(31.8.18218.9.1894), German physiologist, physicist and
philosopher.
In the year 1847, he explained the energy conservation law
mathematically and was first to extend Joule’s results into a
general principle. In the year 1850, he measured the impulse
propagation velocity in a nerve tissue. He was engaged in the
physiology of eyesight and hearing. Furthermore, he devoted
himself to electromagnetism, acoustics and thermodynamics.
Physics and Physical Chemistry
43
2.2.26 Henry Number He
He 5
R1
R2
R1, R2 (kg m23) mass concentration.
In a chemical reaction, it expresses the ratio of the mass concentration and the
phase interface.
Info: [A33].
William Henry (12.12.17752.9.1836), English chemist.
He was focused on research on the chemistry of gas, especially on gas absorption in water under diverse temperatures
and pressures. He formulated Henry’s law. At only 23 years
of age, he published his best works describing the experiments he executed. In other works, gas analysis, humidity
in burning and other chemical problems are treated. The
book Elements of Experimental Chemistry (1799) was his
most important work.
2.2.27 Hess Number Hs
Hs 5
kn L2 Rn21
ini
am
kn (m3 (n21) kg2 (n21) s21) reaction rate constant for nth order of chemical
reaction; L (m) characteristic length; Rini (kg m23) initial concentration of
solid material; am (m2 s21) mass diffusivity of reaction products.
It characterizes the heat and mass transfer in processes with chemical and phase
reactions.
Info: [A35].
Germain Henri Hess (18021850), Russian chemist of Swiss origin.
2.2.28 Karlowitz Number (1.) Ka
Ka 5
dw L
;
dy w
where L 5
λ
cp R g w
w (m s21) gas velocity; y (m) coordinate; L (m) characteristic length;
λ (W m21 K21) thermal conductivity; cp (J kg21 K21) specific heat capacity;
Rg (kg m23) gas density.
In the burning process, it characterizes the flame extending or increasing its
surface in flowing gas, the rate of which is strongly higher than the burning rate.
Due to the surface extending, the front of the flame front is curved. The criterion
44
Dimensionless Physical Quantities in Science and Engineering
expresses the critical degree of the flame divergence with which the equilibrium
between heat generation and thermal loss, in the reactive zone, can be destroyed
and the flame extinguished. With the burning wave extended on the path L being
longer than the critical path, the flame is extinguished.
Info: [A23],[A33].
2.2.29 Karlowitz Number (2.) Ka
Ka 5
τ char
w2
5 Kol
;
τ Kol
w2lam
where wKol 5
p
ffiffiffiffiffi
4
vε
τ char (s) characteristic flame timescale; τ Kol (s) smallest Kolmogorov turbulence
time; wKol (m s21) Kolmogorov velocity; wlam (m s21) laminar flame velocity;
ν (m2 s21) kinematic viscosity; ε (m2 s23) turbulence dissipation velocity.
It characterizes a thin reactive zone in the combustion area. It is used if the least
turbulent whirling in the flow is less than the flame width.
Info: [A103].
2.2.30 Kármán Ratio N
N5
λ κ M um
at a2
22
5
5 Le21
;
p M
2
ap w2
R cp R w
where
p
ap 5
;
Mum
λ
;
at 5
cp R
sffiffiffiffiffi
kp
a5
R
λ (W m21 K21) gas thermal conductivity; Rg (kg m23) gas density;
cp (J kg21 K21) specific heat capacity; M (kg mol21) molar density;
um (mol m23 s21) molar rate of chemical reaction; w (m s21) tributary velocity
of fuel mixture; at, ap (m2 s21) temperature and pressure diffusivity; a (m s21) sound velocity; p (Pa) pressure; k () specific heat ratio (p. 28); Lep 5 ap at21
() Lewis pressure number; M () Mach number (p. 73).
This ratio characterizes the steady-state burning of a gas mixture flow. It indicates
the relation between the flame propagation rate and the quantities that characterize
burning.
Info: [A23].
Theodore von Kármán (p. 67).
Physics and Physical Chemistry
45
2.2.31 Knudsen Number (2.) Diffusion Kn
K n Sm21 5
μ DAB
ζ DKA
μ () porosity; DAB (m2 s21) coefficient of binary volume diffusion of AB
system; ζ () diffusion tortuosity; DKA (m2 s21) Knudsen diffusion coefficient;
Sm () Smoluchowski number (p. 86).
It expresses the ratio of the volume diffusion to that of Knudsen diffusion in
a granulated layer. It characterizes the diffusion in continuous granulated layers
of a catalyzer. It is especially applied in chemical technology. Its inverse value is
called the Smoluchowski number Sm (p. 86).
Info: [A29],[A35].
Martin Hans Christian Knudsen (p. 420).
2.2.32 Mass Fraction M
mA
M5 P
mi
mA (kg) mass of substance A; mi (kg) mass of individual substances.
It characterizes the mass fraction of partial components, for example, in mixing.
Physical chemistry.
Info: [C85].
2.2.33 Mole Fraction xi
xi 5
ni
Ni
5
n
N
ni () number of moles of ith substance; n () total number of moles in a
mixture; Ni () number of molecules of ith substance; N () total number of
molecules in a mixture; NA () Avogadro number (p. 4).
The mole fraction is one of the ways to express the concentration of various
chemical components in a mixture. It is about approximation to an ideal mixture,
but corrections (active coefficients) must be introduced in practice.
n5
X
j
nj ;
N i 5 ni N A ;
X
i
xi 5 1
46
Dimensionless Physical Quantities in Science and Engineering
Physical chemistry. Thermodynamics.
Info: [C90].
2.2.34 Prater Number Pra, β
Pra 5
2ΔH De CA0
T λe
ΔH (kJ kmol21) heat of reaction; De (m2 s21) effective diffusivity; CA0
(kmol m23) concentration of species A; T (K) temperature; λ (W m21 K21) effective thermal conductivity.
This number characterizes the non-isothermic chemical reaction with diffusion
and thermal transfer resistance in pores or the isothermic internal reaction with
the diffusion resistance in pores. It expresses the joining of physical and chemical
processes such as diffusive and convective heat and mass transfers. Above all, it is
important in designing, starting and controlling chemical reactors.
Info: [A49].
2.2.35 Reaction Enthalpy Number N
N5
ΔhA ΔcA
cp ΔT
ΔhA (J kg21) specific reaction enthalpy change of A product; ΔcA () change
of mass fraction of A product; cp (J kg21 K21) specific heat capacity; ΔT (K) temperature difference.
It expresses the ratio of the reaction energy change to the thermal energy
change. It characterizes the inter-phase transfer with chemical reactions.
Info: [A35].
2.2.36 Reaction Enthalpy Number (2.) Nre
Nre 5
ðΔHÞA ΔMA
CΔT
(ΔH)A (J kg21) reaction enthalpy; ΔMA (kg) mass of fraction A; C (J K21) heat capacity; ΔT (K) temperature difference.
It expresses the ratio of reactive energy change to that of thermal energy.
A phase transfer with a chemical reaction.
Info: [A29].
Physics and Physical Chemistry
47
2.2.37 Reagent Concentration η, χ
η5
C
Cini
ð1Þ;
χ5
RQ C
E R cV
ð2Þ
C, Cini (m23) numerical initial molecule concentration of reagent; R (J mol21 K21) molar gas constant; E (J mol21) molar activation energy; Q (J) thermal effect
of reaction; Rg (kg m23) density; cV (J kg21 K21) specific heat capacity.
In burning, it characterizes the reagent concentration or the chemical enthalpy.
Info: [A48].
2.2.38 Reagent Temperature Θ
Θ5
E
ðT 2 TN Þ
2
RTN
E (J mol21) molar activation energy; R (J mol21 K21) molar gas constant;
TN (K) ambient temperature; T (K) reagent temperature.
It characterizes the reagent temperature in fuel burning.
Info: [A23].
2.2.39 Rejection Coefficient R
R5
cr 2 cp
;
cr
where RAh0; 1i
cr (kg m23) separating and suspension with a solid concentration; cp (kg m23) at the same time measure the solid concentration of the permeate.
It characterizes the solid substances separation process in suspensions.
Essentially, it expresses the diaphragm (of filters) efficiency. Physical chemistry.
Separation and filtration of materials.
Info: [C81],[C112].
2.2.40 Semenov Number N
N5
β
wch
β (m s21) mass transfer coefficient; wch (m s21) chemical reaction progress rate.
It characterizes the reaction kinetics in a physical and chemical conversion
process.
Info: [A23].
48
Dimensionless Physical Quantities in Science and Engineering
Nikolay Nikolajevich Semenov (3.4.189625.9.1986),
Russian physicist and chemist. Nobel Prize in Chemistry, 1956.
He was engaged in the mechanisms of chemical transformation involving the complete analysis of chain theory application
for diverse chemical reactions and combustion processes. He
proposed the degenerate branching theory, which explained the
phenomenon related to inducing periods in oxidation processes.
He wrote great numbers of works in the sphere of molecular
physics and investigated shock wave propagation and electron
phenomena.
2.2.41 Thermal Capacity of Combustion Chamber N
N5
σ T 4k
ku
σ (W m22 K24) specific thermal absorption; T (K) flue gas temperature;
k (m21) coefficient of environment reduction in combustor; k (J kg21) specific
combustion heat; u (kg m23 s21) chemical reaction rate.
It expresses the ratio of the heat absorbed in a chamber filled with the mediums to
all heat released by fuel combustion. It characterizes the complicated heat transfer in
combustion chambers with physical and chemical processes of fuel burning.
Info: [A23].
2.2.42 Thermodynamic Equilibrium of Chemical Reduction Nreac
k2 ϕ2 ðRÞ
Nreac 5
k1 ϕ1 ðRÞ
ð1Þ;
P
α2i21
k2 R2i
P
Nreac 5
α1i21
k1 R1i
ð2Þ
k1, k2 () constants of chemical reaction; ϕ1(R), ϕ2(R) () mass concentration
function of chemical reaction products; R, R1i, R2i (kg m23) densities of individual
components; α1i, α2i () stoichiometric number of component change.
In the form (1), it expresses the ratio of outlet products of physical and chemical
reduction to inlet ones. It characterizes the internal process thereof. It represents
a local criterion of the thermodynamic equilibrium or the deviation degree from
the state of the thermodynamic equilibrium alternatively. For simple homogeneous
processes, (2) is valid. For quasi-static processes, Nreac 5 1 holds and so does
Nreac , 1 for dynamic ones. For continuously changing states of the internal equilibrium, it is Nreac-1. For the process rate approaching the change rate, Nreac-0
holds, whereas Nreac 5 0 is valid for limiting irreversible processes.
Info: [A33].
Physics and Physical Chemistry
49
2.2.43 Thermochemical Bond Ntch
Ntch 5
lRm
cp RT
l (J kg21) latent heat; Rm (kg m23) mass concentration; cp (J kg21 K21) specific
heat capacity; R (kg m23) density; T (K) temperature.
It expresses the ratio of thermochemical heat reduction to that caused by a molar
movement of reacting surroundings. It characterizes the bond between the thermal
and concentrating fields. Heat transfer in physical and chemical processes.
Info: [A33].
2.2.44 Thiele Modulus (2.) Φ, Th
rffiffiffiffiffi
k1 pffiffiffiffiffiffiffiffi
5 Da2
Φ5L
D
L (m) effective particle diameter of porous catalyst (equals 6/S)26; k1 (s21) reaction rate constant; D (m2 s21) diffusivity; Da2 () Damkőhler number (2.)
diffusion (p. 37).
It expresses the isothermic reaction with the diffusion resistance in pores.
Diffusion in porous catalyzers.
Info: [A33],[A49].
Ernest William Thiele (p. 267).
2.2.45 ThringNewby Number Tn
Tn 5
Qmt 1 QmP r
QmP
L
Qmt (kg s21) fluid mass flux in nozzle; QmP (kg s21) ambient mass flux; r (m) equivalent nozzle radius; L (m) characteristic length, wall half-thickness.
It characterizes the fuel combustion process in combustion spaces.
Info: [A23],[A35].
Meredith Wooldridge Thring (17.12.191515.9.2006), English engineer.
Maurice Purcell Newby (born 1917), English physicist.
2.2.46 Time of Chemical Reaction Kτ
Kτ 5
τ ch
τL
τ ch (s) chemical reaction time necessary to reach the equilibrium state in flowing
fluid; τ L (s) time necessary to fluid flow through the characteristic length L.
50
Dimensionless Physical Quantities in Science and Engineering
It characterizes the time to reach the equilibrium state in chemical reactions
in flowing fluids. For a flow with homogeneous chemical reactions, with Kτ 5 0,
the chemical equilibrium state occurs in the flow. It is the case in which the
chemical reaction rates are so high that the equilibrium state occurs in every
place of the surroundings with given mass densities. With Kτ -N, the chemical
reactions are so small that in every place of the surroundings, the mixture
composition is determined by mass flows of various components. It is the case
of frozen flow.
In flow with heterogeneous reactions with Kτ 5 0, the chemically reacting
mixture, adhering to the surface, is in a chemical equilibrium state with the
temperature and pressure on the surface. This situation is called the ideally catalytic surface. The non-catalytic surface case occurs for Kτ -N, in which the
catalytic reaction rate is so small that the mixture surface composition is determined by diffusion and convection processes not coupled with the reactions on
the wall.
Info: [A23].
2.2.47 Van ’t Hoff Factor i, Ni
i 5 1 1 aðq 21Þ
a () degree of ionization; q () number of ions formed for each dissociated
molecule.
It expresses the number of moles in a water solution which are created by a
mole of a solid admixture. The matter is the electrolyte dissociation into ions in the
solution. For example, NaCl-Na1 1 Cl2 (i 5 2), MgBr2-Mg21 1 2Br2 (i 5 3).
For glucose it is i 5 1 because one mole of it dissolved in water leads to the result
of one diluted mole.
Info: [C137].
Jacobus Henricus van ’t Hoff, Jr. (30.8.18521.3.1911),
Dutch physical chemist. Nobel Prize in Chemistry, 1911.
He was engaged in research on chemical kinetics, chemical equilibrium, osmotic pressure and crystallography. He
contributed to elaborating the branch of physical chemistry.
He was engaged also in organic chemistry. In the year 1874,
he described the optical activity phenomenon. He studied
three-dimensional chemical structures and other problems
of stereochemistry. In addition, he worked out Svante
Arrhenius’ theory of electrolytic dissociation.
Physics and Physical Chemistry
51
2.2.48 Volume Fraction ϕ
VA
ϕ5 P
Vi
VA (kg) volume of the A component; Vi (kg) volumes of individual components in a mixture.
It characterizes the volume fraction of partial components, for example, in mixing. Physical chemistry.
Info: [C138].
2.2.49 Wagner Number Wa
Wa 5
γ @U
d @JA
γ (S m21) electric conductivity of solid phase; U (V) upper potential for reduction of O2; JA (A m22) surface density of electric current; d (m) thickness of
catalytic layer.
In catalytic layers it characterizes the ohmic drop, specifically by comparing
the charge transfer with the ohmic resistances. For Wa , 1, the ohmic resistance
predominates. It is used to design diaphragmelectrolyte systems, to solve cathodic
catalytic layers and gas diffusers and to fuel elements if need be. It can also be
applied to determine the current density distribution in electrochemical cells.
Info: [B42].
2.2.50 Weisz Modulus Wz, Ψ0
Wz 5 ηk Φ2 5 ηk Da2 ;
where ηk 5
uef
uðTS ; CS Þ
uef (s21) effective chemical reaction rate; u (s21) chemical reaction rate; TS (K) outside surface temperature of catalyzer; ηk () degree of required porosity; Φ () Thiele modulus (2.) (p. 49); Da2 () Damkőhler number (2.) diffusion (p. 37);
CS () gas constant for catalyzer surface.
It is applied to characterize the isothermal reaction with the diffusive resistance
of pores. Catalyzers. Diffusion in porous catalyzers.
Info: [A49].
3 Fluid Mechanics
The quantitative character of objects, one of the most general laws of the
existence.
Leonhard Euler (17071783)
3.1
One-Phase Fluid Mechanics
In this field, the range of similarity criteria used is great. It involves the flow of
potential, compressible and incompressible fluids; viscous and non-viscous flows;
steady and unsteady flows; and laminar, turbulent and other flows. Some criteria
concern the related phenomena arising in flow, such as the boundary layer, friction,
acting forces, stroke waves, the sound barrier, surface tension and swirls. Among
the fundamental and most widespread similarity criteria are the Froude, Knudsen,
Mach, Prandtl, Richardson and Strouhal numbers.
3.1.1
Acceleration Number Ac
Ac 5
E3
5 Re2 Fr 2 Ho23
R g 2 η2
E (Pa) modulus of elasticity; R (kg m23) density; g (m s22) gravitational
acceleration; η (Pa s) dynamic viscosity; Re () Reynolds number (p. 81);
Fr () Froude number (1.) (p. 62); Ho () Hooke number (p. 138).
This number characterizes the acceleration flow. It depends on physical properties and gravitational acceleration only.
Info: [B20].
3.1.2
Archimedes Hydrodynamic Number Ar
Ar 5
g L3 Rs 2 R
R 2R
5 Ga s
v2
R
R
ð1Þ;
Ar 5
g L3 εB
v2 1 2 ε B
ð2Þ
g (m s22) gravitational acceleration; L d (m) particle diameter; ν (m2 s21) fluid kinematic viscosity; Rs, R (kg m23) particle and fluid density; εB () volume fraction; Ga () Galileo number (p. 123).
Dimensionless Physical Quantities in Science and Engineering. DOI: 10.1016/B978-0-12-416013-2.00003-8
© 2012 Elsevier Inc. All rights reserved.
54
Dimensionless Physical Quantities in Science and Engineering
It expresses the ratio of product of a floating body weight and the inertia force
to the square of the viscosity force. It characterizes the free flow of fluid caused by
the density difference of partial components. For example, it is about the floating of
solid particles, drops or bubbles (2) or possibly the whole composition thereof caused by uplift forces in the fluid volume. It is an unusual modification of the
Galileo number Ga (p. 123). It occurs in the heat transfer between the particles,
for example, in fluidization and in material bundles. The characteristic feature is
the heat transfer by convection and conduction in a wall-particle system.
Info: [A14],[A29],[B20].
Archimedes of Syracuse (287 BC212 BC), Greek mathematician.
From our present point of view, he was a physicist and
engineer. He formulated the physical principle of upward
force and the weight of drawn fluid. This theorem became
known as the Archimedes law. He utilized Euclidean geometry to calculate the surfaces and volumes of bodies. He determined the approximate value of π.
3.1.3
Bairstow Number NBa
See Mach number M (p. 73).
Info: [A29].
Leonard Bairstow (18801963), English aerospace engineer.
3.1.4
Bejan Convective Number Be
Be 5
ΔpL2
ηa
Δp (Pa) pressure drop; L (m) characteristic length; η (Pa s) dynamic viscosity; a (m2 s21) thermal diffusivity.
In fluid mechanics, it represents the dimensionless expression of the pressure
drop along an L-long canal. In convection, its role resembles the Rayleigh number
(2.) (heat instability) Ra2 (p. 187) in natural convection.
Info: [C6].
Adrian Bejan (p. 9).
3.1.5
Beranek Number Be
Be 5
w3s R2l
ηl gðRs 2 Rl Þ
Fluid Mechanics
55
ws (m s21) fall velocity of solid particles; Rl, Rs (kg m23) density of liquid and
solid particles; ηl (Pa s) dynamic viscosity of liquid; g (m s22) gravitational
acceleration.
It expresses the ratio of the inertia force square to the product of the viscosity
and the gravitational force. It characterizes the bypassing of free falling particles in
immobile unlimited fluid.
Info: [A29].
Josef Beránek, Czech chemist.
3.1.6
Blasius Shape of Boundary Layer δ=x
δ
4; 9
5 pffiffiffiffiffiffiffi ;
x
Rex
where Rex 5
wc
v
δ (m) boundary layer thickness; x (m) distance from entering edge; w (m s21) flow velocity; ν (m2 s21) kinematic viscosity; Rex () local Reynolds number
relative to the x position.
It characterizes the viscous fluid flow in a laminar boundary layer above a floating plate. The expression is used to monitor the boundary layer development and to
simplify the solution.
Info: [A21].
3.1.7
Boussinesq Number Bs
Bs 5
w2
2gL
w (m s21) flow velocity; g (m s22) gravitational acceleration; L (m) characteristic length, hydraulic radius.
It expresses the inertia-to-weight forces ratio. Weight influence on surface
waves in open canals.
Info: [A29], [B17], [B20].
Valentin Joseph Boussinesq (13.3.184219.2.1929), French
physicist and mathematician.
He was engaged in many areas of mathematical physics,
but especially in hydrodynamics, the undulation and flowing
of fluids, resistance of bodies in flowing fluid, the cooling
down effect of fluids and the problems of turbulence and
fluid elasticity.
56
Dimensionless Physical Quantities in Science and Engineering
3.1.8
Boussinesq Approximation Parameter Bs
Bs 5
g0
R 2 R2
5 1
g
Rref
g0 (m s22) approximated gravitational acceleration; g (m s22) gravitational
acceleration; R1, R2 (kg m23) density of warm and cold liquid; Rref (kg m23) reference density.
Compared to the often used parametric expression R1/R2, the flow tasks become
more precise by making use of this parameter. Fluid mechanics.
Info: [C109].
Valentin Joseph Boussinesq (see above).
3.1.9
Buoyancy Number NB
NB 5
L2 FβΔT
ηVw
L (m) characteristic length; F (N) buoyancy force; β (K21) volume thermal
expansion coefficient; ΔT (K) temperature difference; η (Pa s) dynamic viscosity; V (m3) volume; w (m s21) velocity.
It is the buoyancy-to-viscous forces ratio. It describes the influence of viscous
fluid natural convection on buoyancy.
3.1.10 Buoyancy Parameter NB
NB 5
ΔT gL
ΔT 21
5 Gr1 Re 22 5
Fr
T w2
T
ΔT (K) temperature difference; T (K) environment temperature; g (m s22) gravitational acceleration; L (m) characteristic length; w (m s21) flow velocity;
Gr1 () Grashof heat number (p. 185); Re () Reynolds number (p. 81);
Fr () Froude number (1.) (p. 62).
It characterizes the buoyancy in free and forced convections. It is analogous to
the Archimedes thermodynamic number Ar2 (p. 184).
Info: [A35].
3.1.11 Capillary Number (2.) Ca
Ca 5
η2 E ηw2
5
5 We21 Ho21 Re22
Rσ2
σ
Fluid Mechanics
57
η (Pa s) dynamic viscosity; E (Pa) bulk modulus of fluid; R (kg m23) liquid
density; σ (N m21) surface tension; w (m s21) flow velocity; We1 () Weber number (1.) (p. 91); Ho () Hooke number (p. 138); Re () Reynolds
number (p. 81).
It represents the influence of the surface tension acting on the flowing fluid and
its capillarity. It depends on physical properties of the fluid.
Info: [A29],[A35].
3.1.12 Centrifuge Number Ncf
Ncf 5
Rr 2 hω2
σ
R (kg m23) density; r (m) radius; h (m) fluid depth; ω (s21) angular frequency; σ (Pa) surface tension.
It equals the centrifugal-to-capillary forces ratio. The flow in curved canals. The
centrifugal forces of stroke and surface waves. Hydromechanics.
Info: [B20].
3.1.13 Coefficient of Velocity ϕ
ϕ5
w
;
wt
whereas
ϕ,1
w (m s21) real velocity; wt (m s21) theoretic velocity.
This coefficient expresses the ratio of the actual fluid flow velocity, with friction
and without contraction, to the theoretical velocity without friction. It depends on
the bypassed wall roughness. Usually ϕAh0, 95; 1i. Together with the contraction
coefficient α (p. 57), it determines the outflow coefficient μ (p. 77). Aerohydrodynamics.
Info: [A39].
3.1.14 Contraction Coefficient α, ε
α5
A
A0
A, A0 (m2) cross section of nozzle spray and outlet area.
It is the ratio of the out-flowing beam diameter to the orifice area.
Hydromechanics.
Info: [A39].
58
Dimensionless Physical Quantities in Science and Engineering
3.1.15 Correlation Coefficient R, .
wx1 wx2
R 5 pffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffi
wx1 2 wx2 2
wx1, wx2, wx1 ; wx2 (m s21) pulsation velocities measured at the same time in positions 1 and 2 and their mean values.
It expresses the flow turbulence measure. Aerodynamics.
Info: [C22].
3.1.16 Crispation Number Ncr
Ncr 5
ηa
σL
η (Pa s) dynamic viscosity; a (m2 s21) thermal diffusivity; σ (Pa m) undisturbed surface tension of the layer; L (m) characteristic length, layer thickness.
This number characterizes the influence of the penetration of viscous fluid convective flow in the layer thickness while considering the thermal diffusion and the
surface stress. One-phase fluid mechanics.
Info: [C25].
3.1.17 Crocco Number Cr
212
w
2
Cr 5
5 11
wmax
ðk 21ÞM 2
w (m s21) local flow velocity; wmax (m s21) velocity at adiabatic expansion;
κ () specific heat ratio (p. 28); M () Mach number (p. 73).
This number expresses the ratio of local flow rate to the highest possible fluid
rate in outlet into a vacuum (adiabatic expansion). It characterizes the compressible
fluid flow. It is also called the modified Mach number M.
Info: [A29], [B20].
3.1.18 Darcy Number (1.) Dc
Dc 5
2ghd
w2 L
g (m s22) gravitational acceleration; h (m) pressure loss; d () pipe diameter; w (m s21) flow velocity; L (m) characteristic pipe length of constant cross
section.
Fluid Mechanics
59
It expresses the pressure loss of fluid flowing through a pipeline. The Darcy
friction number fD (p. 119) or the Fanning friction number fF (p. 163) are modifications of this. Fluid mechanics. Hydraulics. Tribology.
Info: [A26], [B20].
Henry Philibert Gaspard Darcy (p. 98).
3.1.19 Darcy Number (2.) Dc
Dc 5
2ghrh
w2
ð1Þ;
Dc 5
8ghrh
w2
ð2Þ
g (m s22) gravitational acceleration; h (m) mean hydraulic depth; rh () hydraulic gradient; w (m s21) flow velocity.
It characterizes the fluid flows in open canals and over falls. It is analogous to
the Darcy friction number fD (p. 119). Hydraulics and hydromechanics.
Info: [A21],[B20].
Henry Philibert Gaspard Darcy (p. 98).
3.1.20 Dean Number Dn
wd
Dn 5
v
rffiffiffi
rffiffiffi
r
r
5 Re
R
R
w (m s21) flow velocity; r 5 d/2 (m) canal radius; ν (m2 s21) kinematics
viscosity; R (m) radius of canal curvature; Re () Reynolds number (p. 81).
It expresses the mutual relation between centrifugal, friction and inertia forces
and characterizes the flow in curved canals.
Info: [A14],[B17],[B20].
William Reginald Dean (18961973), British mathematician and physicist.
3.1.21 Degree of Turbulence E
E5
w
wT
w (m s21) mean square velocity of pulsation; wT (m s21) mean velocity of turbulence flow.
This degree expresses the ratio between the mean-quadratic pulsation velocity
and the mean turbulent flow velocity. Aero-hydrodynamics.
60
Dimensionless Physical Quantities in Science and Engineering
3.1.22 Density Number N.
NR 5
ΔR
R0
ΔR (kg m23) density difference of matter; R0 (kg m23) initial density.
It expresses the ratio of the density difference, before and after loading, to the
initial density of solid, liquid or gaseous surroundings. For example, it is used in
fluid mechanics with the Boussinesq approximation when the original gravitational
acceleration g is transformed to g 5 g(ΔR/R) in controlled buoyancy fluid flow.
Mechanics. Fluid mechanics. Buoyancy flow.
Info: [C10].
3.1.23 Derjaguin Number De
rffiffiffiffiffiffi
Rg
De 5 L
2σ
L (m) characteristic length, film thickness; R (kg m23) density; g (m s22) gravitational acceleration; σ (N m21) surface tension.
It expresses the ratio of the film thickness to the capillary length. It characterizes
the process of coatings and films generation of fluids on a body surface. Fluid
mechanics. Physical technology.
Info: [A35].
3.1.24 Drag Coefficient CD
CD 5
FD
1 2
Rw A
2
ð1Þ;
CD 5
ðRS 2 RF ÞLS g
RF w2
ð2Þ
FD (N) drag force on body; R, RS, RF (kg m23) density, density of solid particles and fluid; w (m s21) flow velocity; A (m2) cross-sectional area of body
perpendicular to flow velocity; LS (m) characteristic small dimension of particles; g (m s22) gravitational acceleration; m_ (kg s21) mass flux.
In expression (1), this coefficient expresses the ratio of the drag force FD acting
perpendicularly to the dynamic force at a stagnation point on a transversal area A. It
characterizes the aerodynamic or hydrodynamic resistance in fluid flow. The expression (2) is valid in case of flow with small particles and steady distribution thereof.
From the relation (1), an equation expressing the drag force follows, which is
necessary to overpower the aerodynamic resistance and to set an object into
movement.
Fluid Mechanics
61
1
FD 5 CD Rw2 A
2
ð3Þ
The expression (3) for the drag force is valid provided the flow is turbulent,
where CD is approximately constant. For laminar flow, it is
_
FD mw
ð4Þ
and the force is, therefore, proportional to the velocity. With a turbulent flow, the
power is
1
P 5 FD w 5 Rw3 A
2
ð5Þ
The drag coefficient CD is important especially in the aircraft and car industries.
One tries to reach the least value of CD. For example, for automobiles it is CDAh0.35;
0.45i and for sports cars CDAh0.25; 0.30i. For standard cars, it is CD0.5, for a racing bicycle CD0.9, and for a motorcycle CD1.8. The coefficient CD is the physical
similarity criterion which enables in the case of its equality between the model and
the object (CDM 5 CD), the provision of the behaviour similarity condition, for example, with various geometric dimensions of the model and the object.
Info: [A21],[A35].
3.1.25 Euler Number (1.) Eu
Eu 5
Δp
F
5
2
Rw
Rw2 L2
ð1Þ;
Eu 5
1
κM 2
ð2Þ
Δp (Pa) pressure drop due to friction; R (kg m23) density; w (m s21) flow
velocity; F (N) frictional force; L (m) characteristic length; κ () specific
heat ratio (p. 28); M () Mach number (p. 73).
This number expresses the pressure gradient due to friction related to dynamic
pressure; alternatively, the ratio of the pressure surface force to the inertia force. It
characterizes the hydrodynamic pressure, pressure loss, hydraulic resistance and
fluid friction, and their influence on the proceeding process. With a compressible
environment flow, the expression (2) is used. For the pressure flow, it is Eu 5 Ne.
Info: [A29],[A35],[A43],[B20].
Leonhard Euler (15.4.170718.9.1783), Swiss mathematician and physicist.
He shifted the boundaries in modern analytical geometry
and trigonometry. He was characterized by his extraordinary
memory and, despite his initially partial and later total blindness, he was most prolific mathematician of all time with his
more than 800 works. Great numbers of his works concern
optics, mechanics, electricity and magnetism.
62
Dimensionless Physical Quantities in Science and Engineering
3.1.26 Euler Curved Number Eucurv
Eucurv 5
8Δpr 3 R
Dn2 η2 R
Δp (Pa) pressure drop in the curved pipe; r (m) pipe diameter; R (kg m23) liquid density; η (Pa s) dynamic viscosity; R (m) curvature radius; Dn () Dean number (p. 59).
It expresses the ratio of the pressure gradient due to friction in a curved tube to
the dynamic pressure. It characterizes the pressure loss, hydraulic resistance and
friction in curved canals. Fluid mechanics. Flow in curved canals.
Leonhard Euler (see above).
3.1.27 Force Coefficient CF
CF 5
τw
F
5
5 2Ne
1 2
1 2
Rw
Rw A
2
2
τ w (Pa) stress component of circumfluenced body; R (kg m23) fluid density; w
(m s21) flow velocity; F (N) force; A (m2) drag area; Ne () Newton
number (p. 75).
It is important mainly in aerodynamics and expresses the resistance-to-inertia
forces ratio. As a vector, the force F has a drag component FD and that of uplift FL.
Info: [A20].
3.1.28 Fourier Hydrodynamic Number (Fourier Flow Number) Foh
Foh Zh 5
vτ
ητ
5 2
2
L
RL
ν (m2 s21) kinematic viscosity; τ (s) time; L (m) characteristic time; η (Pa s) dynamic viscosity; R (kg m23) density; Zh () Zhukovsky number (p. 93).
In the dimensionless shape, it characterizes the non-stationary flow time of a
viscous incompressible fluid. It is called the Zhukovsky number Zh (p. 93).
Info: [A29].
Jean Baptiste Joseph Fourier (p. 175).
3.1.29 Froude Number (1.) Fr
Fr 5 Ri 21 5
w2
5 Ga21 Re2
gL
ð1Þ;
Fr 5
w2 R
gLΔR
ð2Þ
Fluid Mechanics
63
w (m s21) flow velocity; g (m s22) gravitational acceleration; L (m) characteristic length; R (kg m23) liquid density; ΔR (kg m23) density difference
between liquid and other phase; Ri () Richardson number (p. 83); Ga () Galileo number (p. 123); Re () Reynolds number (p. 81).
This number expresses the ratio of the inertia force to the gravitational or uplift
force; alternatively, the ratio of the kinetic energy to the potential energy. For
example, it characterizes the surface phenomena in free-level fluid flow. It is
applied in hydraulics, wherever the weight influence must be considered. It can be
neglected in gases. With the bubble floating process in a fluid, the bubble radius
(L r) is inserted for the characteristic length. The inverse value of the Froude
number (1.) is called the Richardson number Ri (p. 83) or the Reech number Ree
(p. 81) sometimes. The Boussinesq number (1.) Bs (p. 55) is its analogue, as well.
Info: [A7],[B20].
William Froude (28.11.18104.5.1879), English engineer
and naval architect.
He was engaged in hydrodynamics and hydraulics and
especially in surface phenomena in fluid flowing on free surfaces and in open canals, influenced by gravity. He also studied processes of the rising of bubbles in a fluid and other
phenomena connected to fluid flow and floating bodies.
3.1.30 Froude Number (2.) External Fr
pffiffiffiffiffiffiffi
w
Fr 5 pffiffiffiffiffiffi 5 Fr ð1Þ;
gL
w
w
Fr 5 pffiffiffiffiffi 5
c
gh
ð2Þ
w (m s21) flow velocity; c (m s21) velocity of the water level wave propagation at the position, where is the sea depth h; Fr () Froude number (1.) Fr
(p. 63); for other quantities see Froude number (1.) Fr (p. 62).
It expresses the relation of the inertia force to the gravitation force; alternatively,
the ratio of the flow rate in an open canal to the propagation rate of slight gravitation waves. It characterizes the free-level fluid flow in open canals. It is also used
to determine the character of the flow. With Fr , 1, the flow is subcritical, with
Fr 5 1, it is critical and with Fr . 1, it is supercritical or accelerated.
Info: [A43],[B20].
William Froude (see above).
3.1.31 Froude Internal Number (1.) Frint
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Rw2
Frint 5
gL2 ðdR=dzÞ
64
Dimensionless Physical Quantities in Science and Engineering
R (kg m23) unperturbed density; w (m s21) flow velocity; g (m s21) gravitational acceleration; L (m) characteristic length; dR/dz (kg m24) vertical density
gradient.
This number expresses the influence of the kinetic and potential energy and of
the vertical density gradient in sedimentation.
Info: [C73].
William Froude (see above).
3.1.32 Froude Internal Number (2.) Frint
Frint 5
w
w
5
cg
fB h
w (m s21) flow velocity; cg (m s21) velocity of gravitation wave propagation;
fB (s21) buoyancy frequency; h (m) depth.
It expresses the ratio of the current fluid acceleration to the wave-induced
acceleration.
Info: [C73].
William Froude (see above).
3.1.33 Froude Rotation Number Frrot
Frrot 5
Ln2
g
L (m) characteristic length, L d diameter of pipe, bubble, drop, shaft or
wheel; n (s21) rotation speed; g (m s22) gravitational acceleration.
It characterizes the forced fluid flow due to mixing, vibrating or other
movement.
Info: [A29].
William Froude (see above).
3.1.34 Gőrtler (Goertler) Number, Parameter Goe
rffiffiffi
δ
ð1Þ;
Goe 5 Reδ
R
whereas
Reδ 5
wδ
v
ð2Þ
δ (m) thickness of momentum boundary layer; R (m) radius of wall edge; w
(m s21) flow velocity; ν (m2 s21) kinematic viscosity; Reδ () Reynolds
boundary layer number (p. 82).
In the form (1), it characterizes the instability of laminar boundary layers on
curved concave walls. It expresses the origin of Kármán swirls in the bypassing of
Fluid Mechanics
65
solid bodies. It is a modification of the Reynolds boundary layer number Reδ
(p. 82) given by the expression (2). Therefore, it is also called the flow stabilization
criterion in a boundary layer.
Info: [A29].
Henry Gőrtler (born 1909), German mathematician.
3.1.35 Goucher Number Go
Go 5
r 2 Rg
2σ
r (m) radius of body, wire, etc.; R (kg m23) density; g (m s22) gravitation
acceleration; σ (N m21) surface tension.
It is the gravitation-to-surface-tension forces ratio. The surface phenomena in
fluids. The tension. Making of films and coatings. See the Deryagin (p. 60), Bond
(p. 95) and Eőtvős (p. 99) numbers.
Info: [A29],[B20].
Frederick Shand Goucher (30.9.188821.8.1973), Canadian physicist.
3.1.36 Hagen Number Hg
Hg 5 2 R
dp L3
dx v2
rp 5 2 dp/dx (Pa m21) pressure gradient, for natural convection holds
rp 5 gRβΔT; L (m) characteristic length; ν (m2 s21) kinematic viscosity;
g (m s22) gravitational acceleration; R (kg m23) liquid density; β (K21) coefficient of volume expansion at constant pressure; ΔT (K) temperature difference.
It is used for forced flow. The Hg number approaches the Grashof heat number
Gr (p. 185) for a natural flow (Hg-Gr).
Info: [C70].
3.1.37 HagenPoiseuille Number Ha
Ha 5 2
dp L2
ds ηw
dp/ds (Pa m21) pressure gradient; L (m) characteristic length; η (Pa s) dynamic viscosity; w (m s21) flow velocity.
This number characterizes laminar viscous fluid flow. It is also called the
Poiseuille number.
Info: [A35].
Jean Louis Marie Poiseuille (p. 78).
66
Dimensionless Physical Quantities in Science and Engineering
3.1.38 Hodgson Number Hs
Hs 5
Vf Δp
Q V pS
V (m3) system volume; f (s21) flow gas frequency; Δp (Pa) pressure fall
due to friction loss, etc.; pS (Pa) mean static pressure; QV (m3 s21) mean volume flux.
It expresses the non-uniformity of the delivered amount in pulsating flow.
It states the ratio of a set time constant to the pulsation period time. Pulsating flow.
Info: [A29],[A33],[B20].
John Lawrence Hodgson (18811936), English engineer.
3.1.39 Hydraulic Resistance Group NRH
NRH 5
Δp
5 Eu Fr
Rl gL
Δp (Pa) pressure drop across liquid on distillation tray; Rl (kg m23) fluid density; g (m s22) gravitational acceleration; L (m) depth of liquid layer; Eu () Euler number (1.) (p. 61); Fr () Froude number (1.) (p. 62).
This number expresses the pressure drop, for example, in distillation columns.
Info: [A29].
3.1.40 Jeffrey Number Je
Je 5
RgL2
5 Re Fr 21 5 Stk121
ηw
R (kg m23) density; g (m s22) gravitational acceleration; L (m) characteristic
length; η (Pa s) dynamic viscosity; w (m s21) velocity; Re () Reynolds
number (p. 81); Fr () Froude number (1.) (p. 62); Stk1 () Stokes number
(p. 130).
It expresses the slow fluid flow.
Info: [A26].
3.1.41 Kármán Number (1.) Ka
dp 22 U Rpd3
1=2
Ka 5 Rd3 2
η 5 2 5 2fF Re2
η L
dx
Fluid Mechanics
67
R (kg m23) density; d (m) pipe diameter; dp/dx (Pa m21) pressure gradient;
η (Pa s) dynamic viscosity; L (m) characteristic pipe length; fF () Fanning
friction number (p. 163); Re () Reynolds number (p. 81).
It expresses the flow in a pipeline with friction.
Info: [A35],[B17].
Theodore von Kármán (11.5.18816.5.1963), American
engineer of Hungarian origin.
His scientific work is characterized by the use of applied
mathematics in engineering. His work is very extensive and
includes fluid mechanics, turbulence theory, supersonic
flights, aircraft structures and wind erosion of soil. He was
engaged in the theory of missile movement in the atmosphere
and was a co-founder of magneto-hydrodynamics.
3.1.42 Kármán Number (3.) Ka
Ka 5
h
v
rffiffiffiffiffi
τs
R
h (m) mean height of surface roughness; ν (m2 s21) kinematic viscosity;
τ s (Pa) shear stress near the wall surface; R (kg m23) density.
It expresses the ratio of the shear stress and the viscous force in a fluid. It characterizes the friction in floating fluids.
Info: [A29].
Theodore von Kármán (see above).
3.1.43 Kármán Turbulent Number Katur, Tu
Katur
sffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
w2x 1 w2y 1 w2z
2e
5
Tu 5
2
3w2r
3wr
e (m2 s22) specific kinetic energy of turbulent flow; wr (m s21) characteristic
flow velocity, for example, input flow; wx, wy, wz (m s21) pulsation velocity components in any points of turbulent flow; Tu () turbulence number (p. 90).
It expresses the ratio of a mean-quadratic component of a local flow rate pulsation vector to the mean value of the relative flow rate. It characterizes the flow turbulence and its influence on the heat transfer within forced convection. It is also
called the turbulence number Tu (p. 90).
Info: [A33].
Theodore von Kármán (see above).
68
Dimensionless Physical Quantities in Science and Engineering
3.1.44 KármánTsien Correction Factor Cp
Cp 5
Cp0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 1
1 2 MN
!
2
MN
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C2p0
2
1 1 1 2 MN
Cp0 () pressure coefficient for incompressible fluid, see pressure coefficient Cp
(p. 80); MN () Mach number for high subsonic flow, see Mach number M
(p. 73).
This rule expresses the compressibility correction for the pressure distribution on
a bypassed body surface in high subsonic flow. Fluid mechanics. Aerodynamics.
Info: [A21].
Theodore von Kármán (see above).
3.1.45 Keulegan Number Ke
w
ffiffiffiffiffiffiffiffiffiffiffiffi
Ke 5 pffiffiffiffiffi p
3 gv 5 3 Fr Re
w (m s21) flow velocity; g (m s22) gravitational acceleration; ν (m2 s21) kinematic viscosity; Fr () Froude number (1.) (p. 62); Re () Reynolds
number (p. 81).
This number expresses the dimensionless fluid flow rate, hydrodynamic stability
and structure and dynamics of the near-surface layer of the ocean. Hydraulics.
Oceanography.
Info: [A41].
3.1.46 KeuleganCarpenter Number (1.) Kec, KC
Kec 5
2πA
L
A (m) amplitude of sinusoidal flow; L (m) characteristic length, height of the
circumfluenced body.
It characterizes the ratio of drag force to inertia for a bluff object in oscillatory
flow. It is analogous to the Strouhal number Sh (p. 87). A low value of KecAh0.4; 2i
shows the inertial effect to be more important than the viscosity effect.
Hydrodynamics and aerodynamics. Oceanography. Flow oscillations in turbines and
pipelines.
Info: [A8],[A21].
Fluid Mechanics
69
3.1.47 Kirchhoff Number Kh
Kh 5
RpL2
5 Eu Re2
η2
R (kg m23) density; p (Pa) pressure; L (m) characteristic length; η (Pa s) dynamic viscosity; Eu () Euler number (1.) (p. 61); Re () Reynolds
number (p. 81).
It expresses the pressure-to-molecular-friction forces ratio. It characterizes the
fluid flow process in flow-through canals.
Info: [A23],[A33].
Gustav Robert Kirchhoff (12.3.182417.10.1887), German
physicist.
He was engaged in mechanics and electricity. In the year
1859, he constructed a microscope by means of which he discovered two new elements. In addition to applied spectral
analysis, he was engaged in electric circuits and in thermal
radiation. He formulated two Kirchhoff laws enabling the
calculation of the currents and voltages in electric circuits.
His work on black body radiation contributed to the development of quantum theory.
3.1.48 Kirpitchev Hydrodynamic Number Kih
sffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffi
3 RFR
5 3 Re2 CF
Kih 5
2
η
R (kg m23) density; FR (kg m s22) hydrodynamic resistance; η (Pa s) dynamic
viscosity; Re () Reynolds number (p. 81); CF () force coefficient (p. 62).
This number expresses the flow around obstacles and the resistance acting on a
body submerged in a viscous fluid flow.
Info: [A26],[A29].
Michail Viktorovich Kirpichev (18791955), Russian engineer and physicist
(p. 177).
3.1.49 Knudsen Number (1.) Kn
Kn 5 Sm 21 5
Kn 5 M Re21
pffiffiffi
κη
lm
5 1:28
aRL
L
ð2Þ;
ð1Þ;
1
Kn 5 M Re2 2
ð3Þ;
70
Dimensionless Physical Quantities in Science and Engineering
pffiffiffi M
Kn 5 1:28 κ
Re
ð4Þ
lm (m) mean free path; L (m) characteristic length; η (Pa s) dynamic viscosity;
a (m2 s21) thermal diffusivity; R (kg m23) density; Sm () Smoluchowski
number (p. 86); κ () specific heat ratio (p. 28); M () Mach number (p. 73);
Re () Reynolds number (p. 81).
It is also called the flow continuity criterion. It expresses the ratio of the average
free path of molecules in gas to the characteristic length dimension of a body.
It characterizes the aerodynamic gas dilution degree or, alternatively, its deflection
from the continuum state. It determines the conditions under which the phenomena
connected to gas dilution occur. With Kn . 10 the molecules flow freely, with
0.1 , Kn , 10 transient flow occurs, and with 0.01 , Kn , 0.1 slipping on a wall
sets in. With Kn , 0.01, a continual flow is achieved. The flow of artificial satellites in an orbit without collisions above the earth is an example for Knc1. With
Re # 1 the expression (2) is applied, as is expression (3) for Rec1. The inverse
value of the Knudsen number is called the Smoluchowski number Sm (p. 86). In
kinetic theory, this number is used in the form (4).
Info: [A21],[A43],[B11],[B20].
Martin Hans Christian Knudsen (p. 420).
3.1.50 Lagrange Number (3.) Lg3
Lg3 5
ΔpL
5 Eu Re
wη
Δp (Pa) pressure difference; L (m) characteristic length; w (m s21) mean
velocity; η (Pa s) dynamic viscosity; Eu () Euler number (1.) (p. 61);
Re () Reynolds number (p. 81).
This number expresses the ratio of the pressure force, under the action of the
hydraulic resistance, to the viscosity force. With fluid flowing in direct smooth and
even rough flow canals, it characterizes the relation between the pressure and
velocity fields. It is used in magneto-hydrodynamics especially.
Info: [A1],[A4],[A43],[B110].
Joseph-Louis Lagrange (25.1.173610.4.1813), French
mathematician and physicist of Italian origin.
He was the greatest mathematician of the nineteenth century. He was engaged in the theory of numbers, algebra, the
theory of analytic functions, differential equations, variation
calculus, celestial mechanics and mathematical cartography.
The Lagrange equations represent the foundation of theoretical
mechanics. His Me´canique Analytique (Analytic Mechanics,
1788) was his principal work.
Fluid Mechanics
71
3.1.51 Laplace Number Lp
Lp 5
ΔpL
5 Eu We1
σ
Lp 5
rffiffiffiffiffiffiffiffiffiffi
1
2
σ
5 Fr We1 2
2
RgL
ð1Þ;
Lp 5
σRL
5 Ca Re
η2
ð2Þ;
ð3Þ
Δp (Pa) pressure difference; L (m) characteristic length; σ (N m21) surface
tension; R (kg m23) density; η (Pa s) dynamic viscosity; g (m s22) gravitational acceleration; Eu () Euler number (1.) (p. 61); Ca () capillary number (2.) (p. 96); Re () Reynolds number (p. 81); Fr () Froude number (2.)
external (p. 63); We1 () Weber number (1.) (p. 91).
In expression (1), it expresses the ratio of the pressure force to the surface
stress. In expression (2), it expresses the ratio of the surface stress to the momentum transfer in a fluid or to the gravitation force in a flowing mixture. In case (3),
it describes the gasfluid mixture flow in a vertical pipeline or in distillation
plants.
Info: [A29],[A35],[C76].
Pierre-Simon, Marquis de Laplace (23.3.17495.3.1827),
French physicist, mathematician, astronomer and statesman.
He was engaged in mathematical analysis, probability
theory and celestial mechanics, in which he summarized
the results of his predecessors in the five-volume work
Me´canique Ce´leste (Celestial Mechanics). He formulated a
hypothesis on the origin of the solar system from a rotating
nebula. He was engaged in the integral transform known
today as the Laplace transform. He contributed to the calorimetric theory and the determination of the thermal capacities of many materials. He discovered the gravitational
potential.
3.1.52 Laval Number La
La 5
w
acrit
ð1Þ;
La 5 w
2κ
rT
κ11
2 1
2
ð2Þ
w (m s21) flow velocity; acrit (m s21) critical sound velocity; r (J kg21 K21) specific gas constant; T (K) temperature; κ () specific heat ratio (p. 28).
It expresses the ratio of the flow rate to the critical sound velocity. It characterizes incompressible fluid flow. It is also called the critical Mach number Mcrit
and sometimes is denoted as M . It is analogous to the Crocco number Cr (p. 58).
Info: [A26],[A29],[A35].
72
Dimensionless Physical Quantities in Science and Engineering
Carl Gustaf Patrik de Laval (9.5.18452.2.1913), Swedish
engineer of French origin.
In 1887, he built small steam turbines to show how these
plants could be constructed on a large scale. Then, in the
year 1890, he developed the Laval nozzle enabling the
increase of the steam velocity entering a turbine. At present,
the principle of this nozzle is still utilized, for example, in
missile construction. He contributed significantly to the
development of centrifugal separators and other devices.
3.1.53 Lift Coefficient CL
CL 5
FL
1 2
Rw A
2
FL (N) lift force representing vertical aerodynamic drag; R (kg m23) fluid density; w (m s21) flow velocity; A (m2) cross-sectional area of body.
Its main importance is in aerodynamics. It expresses the ratio of the vertical
component of the resistance force to the inertia force. It depends strongly on the
inclination angle. For the lift force perpendicular to the drag force, the following
equation for object lifting is valid:
1
FL 5 CL Rw2 A
2
Info: [A7],[A20],[A21].
3.1.54 Lift-to-Drag Ratio CLD
CLD 5
FL
CL
5
FD
CD
FL (N) buoyancy force; FD (N) drag force; CL () lift coefficient (p. 72);
CD () drag coefficient (p. 60).
It expresses the efficiency measure of an aerodynamic profile. The greater this
number is, the better the wing is.
Info: [A8],[A21].
3.1.55 Loss Factor ζ, ζ c , K
ζ5
ez
ek
ð1Þ;
ζc 5 2
n
X
ezi
w2
i51
ð2Þ;
ζ5
Δp
1
2
2Rw
ð3Þ
ez (J kg21) specific loss energy; ek (J kg21) specific kinetic energy; w (m s21) flow velocity; Δp (Pa) head loss; R (kg m23) fluid density.
Fluid Mechanics
73
This factor expresses the ratio of the specific loss energy to the specific kinetic
energy (1). In expression (2), it is about the relation of the total energy loss to the mean
kinetic energy. In equation (3), it is the loss coefficient for pipelines. Fluid mechanics.
Info: [A21].
3.1.56 Lyashenko Number Ly
Ly 5 Re3 Ar 21
Re () Reynolds number (p. 81); Ar () Archimedes hydrodynamic number
(p. 53).
This number expresses the influence of the inertia force, gravitation force and
viscosity. Fluidization.
Info: [A4],[A35].
3.1.57 Mach Number M, Ma
w
w
ð1Þ; M 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2Þ;
a
ðκ 21Þcp T
pffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffi
where a 5 E=R for any liquid, a 5 kRT for ideal gas; w (m s21) local flow
velocity; a (m s21) local sound velocity; κ () specific heat ratio (p. 28); cp
(J kg21 K21) specific heat capacity; T (K) gas temperature; E (Pa) modulus
of elasticity; R (kg m23) density.
In form (1), it expresses the ratio of the local flow rate to the sound velocity:
elastic deformation propagation or the mean velocity of the thermal molecule
movement in gases. It characterizes the fluid flow in a given place or, alternatively,
the enthalpy conversion into the kinetic energy in a flowing ideal gas. For M , 1 it
U
1 about sonic ones, for M . 1 about superis about subsonic velocities, for M 5
sonic ones, for 0.8 , M , 1.3 about transonic ones and for Mc1 about hypersonic
flow. With a hypersonic flow (M . 5) in the flow core, the fluid starts to behave as
a plasma due to high temperature. In the subsonic zone, it is a measure of the ideal
gas compressibility (2). The compressibility influence appears over M . 0:3:
~
In France, it is also called the Sarrau number Sa (p. 85) or the Bairstow number
NBa (p. 54) in England, though this is not yet used in general practice. Sometimes,
it is called the Maievski number Ma (p. 74).
Info: [A2],[B20].
M5
Ernst Mach (18.2.183819.2.1916), Austrian physicist of
Moravian origin.
He influenced considerably the development of science in the
twentieth century. The idea that recognition of the laws of nature
is mediated by human senses forms the basis of his philosophy.
The Mach principle expresses inertia effects as a property of the
whole universe. Using this concept, Einstein began to formulate
the general relativity theory. He was engaged, too, in experimental physics and the construction of devices.
74
Dimensionless Physical Quantities in Science and Engineering
3.1.58 Mach Angle μ
μ5
1
;
sinðM 21 Þ
for
M-1
it holds
μ-90
M () Mach number (p. 73).
It expresses the Mach cone angle in supersonic fluid flow. High velocity
aerodynamics.
Info: [A21].
Ernst Mach (see above).
3.1.59 Mach Characteristic Number Machar, M0
Machar 5
w
a0
w (m s21) flow velocity; a0 (m s21) sound velocity at Mchar 5 1.
The Mchar is not the usual sonic Mach number M (p. 73), but the number related
to the sound velocity reached with the sonic Mach number. This Mach number form
is used to simplify the basic equations only; see the Prandtl relation M2 (p. 79).
Info: [A21].
Ernst Mach (see above).
3.1.60 Mach Number Across a Shock M
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðγ 21ÞM 2 1 2
M5
2γM 2 2 ðγ 2 1Þ
γ () specific heat ratio (p. 28); M () Mach number of inclination wave,
see Mach number M (p. 73).
For a normal shock wave, the Mach number across a shock expresses the
changes across the shock, which are a function of the drain Mach number only. For
the Mach number of inclination wave
M 5 1,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi the Mach number across a shock is
M 5 2. For M -N, it is M 5 ðγ 21Þ=2γ : Fluid mechanics. Aerodynamics of
high velocities.
Info: [A21].
Ernst Mach (see above).
3.1.61 Maievski Number Ma
See Mach number M, Ma (p. 73).
Nikolaj Vladimirovič Majevskij (18231892), Russian physicist and
mechanist.
Fluid Mechanics
75
3.1.62 Mass Ratio Nm
Nm 5
m
πRL3
m (kg) mass; R (kg m23) fluid density; L (m) characteristic length.
It characterizes the ratio of a submerged body weight to that of the surrounding
fluid.
Info: [A29],[B20].
3.1.63 Moment Coefficient CM
CM 5
M
1Rw2 Ac
2
M (N m) total moment acting on wing; R (kg m23) fluid density; w (m s21) flow velocity; A (m2) drag area c (m) mean chord of aerofoil section.
This moment is important in the aerodynamic stability of aircraft and other
objects. For the moment necessary for tilting, it is
1
M 5 CM Rw2 Ac
2
Info: [A20].
3.1.64 Momentum Number NM
NM 5
MN δR
ηΔw
MN (m2 s22) specific energy of momentum flux; δ (m) layer thickness; R (kg m23)
density; η (Pa s) dynamic viscosity; Δw (m s21) velocity difference.
It expresses the ratio of the specific energy of a momentum flux to the momentum change of viscous fluid.
Info: [B20].
3.1.65 Newton Number Ne
Ne 5
F
5 0; 5CF
Rw2 L2
F (N) hydrodynamic force; R (kg m23) fluid density; w (m s21) flow velocity; L (m) characteristic length; CF () force coefficient (p. 62).
76
Dimensionless Physical Quantities in Science and Engineering
This number expresses the loading-to-inertia forces ratio. It characterizes the
friction in flowing fluids. In the case of viscous fluid flow, it expresses the turbulization process in flowing fluid. In a pressure flow, it is Ne 5 Eu; in a gravitation
flow, it is Ne 5 Fr 2 1, and when considering the internal friction the Ne 5 Re 2 1 is
valid. It also expresses dynamic force processes in vibration, mixing, material
deformation and other processes. The Fanning friction number fF (p. 163) and
the force coefficient CF (p. 62) are also analogues to the Newton number.
Hydrodynamics.
Info: [A4],[A29],[A35],[B17].
Isaac Newton (4.1.164331.3.1727), English physicist,
mathematician, astronomer and philosopher.
He laid the foundations of differential and integral calculus. His works on optics and gravitation made him one of
the greatest scientists of the world. In his book, Principia
(Principles), he formulated three movement laws: of inertia,
of action and reaction, and of acceleration proportional to
force. He was ingenious not only in mathematics but also in
experimentation, which enabled him to formulate new and
simpler laws of mechanics and hydromechanics.
3.1.66 Number of Velocity Heights Nvh
F
Nvh 5
RL2
w2
2
21
F (N) force; R (kg m23) density; L (m) characteristic length; w (m s21) fluid velocity.
It expresses the ratio of the utilized pressure height to the velocity height.
Friction in pipelines and other water systems.
Info: [A35].
3.1.67 Nusselt Film Thickness Parameter Nuft
rffiffiffiffiffi
1
3 g
Nuft 5 L 2 Ga3
v
L (m) characteristic length (layer thickness); g (m s2) gravitational acceleration; ν (m2 s21) kinematic viscosity; Ga () Galilei number (p. 123).
This parameter represents a special case of the Galilei number Ga (p. 123) for
thin film creation.
Info: [B20].
Ernst Kraft Wilhelm Nusselt (p. 196).
Fluid Mechanics
77
3.1.68 Oberbeck Number Ob
Ob 5
L3 w2 R
Q
L (m) characteristic length; w (m s21) flow velocity; R (kg m23) fluid density; Q (J) heat.
It expresses the ratio of flowing fluid kinetic energy to heat transfer.
Info: [A6].
3.1.69 Outflow Coefficient μ
μ 5 αϕ;
whereas μ , 1
α () contraction coefficient (p. 57); ϕ () coefficient of velocity (p. 57).
Using this, the velocities and contractions are considered in fluid outflow or passage. Aero-hydrodynamics.
3.1.70 Pipe Friction Coefficient Λ, Cf, cf
Λ5
ez D
ek L
ez (J kg21) friction loss energy; ek (J kg21) specific kinetic energy; D (m) pipe diameter; L (m) pipe length.
It expresses the friction loss in fluid flow in pipelines. Hydraulics.
Hydrodynamics.
3.1.71 Pipeline Parameter Np, .
Np 5
wp wini
2ghs
wp (m s21) pressure wave velocity; wini (m s21) initial fluid velocity; g (m s22) gravitational acceleration; hs (m) static head.
It expresses the ratio of the highest pressure in a water stroke to the static pressure. It characterizes the pressure originating in a pipeline during the water stroke.
Hydraulics.
Info: [B11],[B20].
78
Dimensionless Physical Quantities in Science and Engineering
3.1.72 Pohlhausen Number Ph
Ph 5
δ2 dp
vRwN dL
δ (m) boundary layer thickness; ν (m2 s21) kinematic viscosity; R (kg m23) density; wN (m s21) rise velocity; p (Pa) pressure; L (m) characteristic length.
In tasks on a boundary layer, it expresses the influence of the outer flow properties on the velocity distribution in a certain boundary layer cross section and on the
velocity profile change in the flow direction. It is the measure of the pressureto-viscous forces ratio in flowing fluid.
Info: [A19].
E. Pohlhausen.
3.1.73 Poiseuille Number (1.) Ps
Ps 5
D2 Δp
ηwL
D (m) pipe diameter; Δp (Pa m 2 1) loss of pressure along the pipe; η (Pa s) dynamic viscosity; w (m s21) flow velocity; L (m) characteristic length.
It expresses the pressure-to-friction forces ratio in fluid laminar flow in a pipeline.
Info: [B20].
Jean Louis Marie Poiseuille (22.4.179726.12.1869),
French doctor and physiologist.
He was educated in physics and mathematics. He was
engaged in blood flowing in narrow tubes. In 1840 and 1846,
he formulated and published the Poiseuille law, which he
had deduced by experiment in 1838. This describes the
stationary flowing of incompressible viscous fluid through
cylindrical tubes. He applied the results to blood flowing
through capillaries, arteries, pulmonary alveolas and the like.
3.1.74 Prandtl Dimensionless Distance X
X5
pffiffiffiffiffiffiffiffi
x Rτ w
η
x (m) distance from wall surface.; R (kg m23) fluid density; τ w (Pa) surface
tension on the wall; η (Pa s) dynamic viscosity.
It is used to study turbulent flow.
Info: [A29].
Ludwig Prandtl (p. 197).
Fluid Mechanics
79
3.1.75 PrandtlGlauert rule Cp, CL, CD
Cp0
Cp 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
1 2 MN
ð1Þ;
CL0
CL 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
1 2 MN
ð2Þ;
2ϑ
Cp 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 21
MN
ð3Þ;
4α
CL 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 21
MN
ð4Þ;
4α2
CD 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 21
MN
ð5Þ;
where
MN 5
wN
a
ð6Þ
Cp0 () uncorrected pressure coefficient for incompressible fluid, see pressure
coefficient Cp (p. 80); CL0 () uncorrected buoyancy coefficient for incompressible fluid, see lift coefficient CL (p. 72); α () incidence angle of thin aerofoil
section; ϑ () fluid incidence angle; wN (m s21) velocity of free-stream conditions; a (m s21) local sound velocity; MN () Mach number of free-stream
conditions, see Mach number M (p. 73).
The dimensionless quantities Cp, CL and CD are the pressure, lift and drag coefficients, respectively, corrected for a compressible fluid. See the pressure coefficient Cp (p. 80), the lift coefficient CL (p. 72) and the drag coefficient CD (p. 60).
The expressions (1) and (2) express the corrected relations for the fluid compressibility in high subsonic flow which is characterized by the extent (0.3 , MN , 0.8),
where MN is the Mach number of free-stream conditions, defined by the relation
(6). The expressions (3)(5) are valid for supersonic flow.
Info: [A21].
Ludwig Prandtl (see above).
3.1.76 Prandtl Relation M2
M1;crit M2;crit 5 1
M1;crit 5
w1
acrit
ð1Þ;
ð2Þ;
M2;crit 5
w2
acrit
ð3Þ
w1, w2 (m s21) flow velocity before and after shock; acrit (m s21) critical sound
velocity; M1,crit () Mach number before shock considering the critical value
instead of local sound velocity, see Mach number M (p. 73); M2,crit () Mach
number after shock considering the critical value instead of local sound velocity,
see Mach number M (p. 73).
It expresses the case of a fluid hit by a permanent shock wave. It simplifies both
Mach numbers (2) and (3) determination significantly because the sound velocity is
80
Dimensionless Physical Quantities in Science and Engineering
equal on both sides of the shock, compared to the local velocity, and the velocities
can be measured very easily. In comparison to it, the pressure, temperature, density,
enthalpy and entropy increase. The expressions (1)(3) are valid for a perfect gas,
and for supersonic and subsonic conditions. Fluid mechanics. High velocity aerodynamics. Shocks.
Info: [A21].
Ludwig Prandtl (see above).
3.1.77 Prandtl Velocity Ratio Pr, u1
rffiffiffi
R
Pr 5 w
τ
w (m s21) flow velocity; R (kg m23) density; τ (Pa) shear stress.
This is the ratio of the inertia force to the shear force arising on a wall by the
action of a flowing fluid. It characterizes the degree of fluid flow turbulence.
Hydromechanics.
Info: [A29],[B20].
Ludwig Prandtl (see above).
3.1.78 Pressure Coefficient Cp
Cp 5
Δp
1
2
2RN wN
5
p 2 pN
1
2
2RN wN
Δp (Pa) pressure difference in fluid; p (Pa) static pressure; pN (Pa), RN
(kg m23), wN (m s21) pressure, density and velocity of free flow.
It expresses the ratio of the overpressure in a fluid to the dynamic pressure in a
free flow. It is a special case of the Newton number Ne (p. 75).
Info: [A7],[A21],[A29].
3.1.79 Pressure Number (1.) Np
p
Np 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
gηðRl 2 Rg Þ
p (Pa) total pressure in system; g (m s22) gravitational acceleration; η (Pa s) dynamic viscosity; Rl, Rg (kg m23) liquid and gas density.
It expresses the ratio of the total system pressure to the pressure gradient on a
fluid surface.
Info: [A29].
Fluid Mechanics
81
3.1.80 Pressure Recovery Coefficient Cp,rec
Cp;rec 5
pout 2 pin
1 2
R
2 in
pout (Pa) output pressure; pin (Pa) inlet flow pressure; R (kg m23) liquid
density; win (m s21) inlet flow velocity.
It expresses the ratio of the difference of static inlet and outlet pressures to the
inlet dynamic pressure. It characterizes the pressure rise in flow which is typical
for diffusers. Fluid mechanics. Aeromechanics.
Info: [A21].
3.1.81 Reech Number Ree
Ree 5
gL
5 Fr 21
w2
g (m s22) gravitational acceleration; L (m) characteristic length; w (m s21) velocity; Fr () Froude number (1.) (p. 62).
This number expresses the weight-to-inertia forces ratio. It can be neglected for
gases. It represents the movement of solid bodies in fluid, waves and surface phenomena. Gravitation influence on the movement of surface ships.
Info: [A29],[A35].
Ferdinand Reech (18051880), French naval instructor.
3.1.82 Relative Amplitude of Fluid Displacement NA
NA 5
2wmax
D Remax
5
ωL
2L Va
wmax (m s21) maximal velocity (amplitude) at fluid flow oscillation; ω (s21) angular frequency; L (m) characteristic pipe length; D (m) pipe diameter; Remax
() Reynolds maximum number (p. 83); Va () Valensi number (p. 90).
It characterizes the pulsation flow in a pipeline. With NA . 1, all initial fluid
passes outwards from a pipeline during one cycle. With NA 5 1, all initial fluid is
moving in a pipeline. With NA , 1, a part of the initial fluid volume contained in a
pipeline does not leave the pipeline during one cycle.
Info: [A1],[A19].
3.1.83 Reynolds Number Re
Re 5
wLR wL
5
η
v
ð1Þ;
Rerot 5
ωL2
RnL2
5
v
η
ð2Þ
82
Dimensionless Physical Quantities in Science and Engineering
w (m s21) flow velocity; L (m) characteristic length; R (kg m23) density; η
(Pa s) dynamic viscosity; ν (m2 s21) kinematic viscosity; ω (Hz) angular
frequency; n (s21) rotational frequency.
This number expresses the ratio of the fluid inertia force to that of molecular
friction (viscosity). It characterizes the hydrodynamic conditions for viscous fluid
flow. It determines the character of the flow (laminar, turbulent and transient flows).
For a laminar flow Re , 2000 is valid, for a transient flow 2000 , Re , 4000, and
for a turbulent flow it is Re . 4000. With low values of the Re number, the viscous
friction muffles the originating dynamic influence of the flow relatively quickly and
intensively, due to which the streamlines and elementary fluid volumes cannot be
deformed substantially and the flow remains laminar. With large Re numbers, the
dynamic flow effect cannot be equalized by viscous friction and the flow stability is
lost, which is manifested by swirls and turbulence in the fluid. The expression (2) is
valid for flow in rotating canals and is often called the Reynolds rotary number.
Info: [A21],[A23],[A29],[A35],[B20].
Osborne Reynolds (23.8.184221.2.1912), English engineer
and physicist.
He opened the era of convective heat transfer (1838). He
showed that the flow of fluid through a tube depends on the
relation of the inertial force to the viscosity (Re number);
so he expressed the basic criterion for forced viscous fluid
convection and determined its critical value. In addition,
Reynolds’s studies relate to vapour condensation and fluid
boiling, heat transfer between solid materials and fluid,
lubrication theory (1886) and resistance laws in streaming
(1883).
3.1.84 Reynolds Boundary Layer Number Reδ
Reδ 5
wδ
v
w (m s21) velocity; δ (m) layer thickness; ν (m2 s21) kinematic viscosity.
It characterizes the viscous fluid flow in the boundary layer near a bypassed
body surface.
Osborne Reynolds (see above).
3.1.85 Reynolds Curved Number Re
wf L
;
Re 5
2v
rffiffiffiffi
τ
where wf 5
Rl
Fluid Mechanics
83
wf (m s21) friction velocity; L (m) characteristic length, curvature radius of layer;
ν (m2 s21) kinematics viscosity; τ (Pa) shear stress; Rl (kg m23) fluid density.
It expresses the friction influence on the heat transfer in a curved fluid layer.
Osborne Reynolds (see above).
3.1.86 Reynolds Entry Number ReE
ReE 5
L
Re
D
L (m) input length; D (m) diameter; Re () Reynolds number (p. 81).
It is applied to solve the flow in inlet flow-through parts.
Info: [B34].
Osborne Reynolds (see above).
3.1.87 Reynolds Maximum Number Remax
Remax 5
wmax L 4Va
5
v
Sh
wmax (m s21) maximal velocity amplitude at oscillation flow; L (m) characteristic length, pipe diameter; ν (m2 s21) kinematic viscosity; Va () Valensi
number (p. 90); Sh () Strouhal number (p. 87).
It characterizes a pulsating flow. Fluid mechanics.
Info: [A1].
Osborne Reynolds (see above).
3.1.88 Reynolds Turbulence Number Returb
Returb 5
e2
vε
e (J kg21) turbulent specific kinetic energy; ν (m2 s21) kinematic viscosity;
ε (s21) dissipative velocity of turbulence.
For a turbulent flow, it expresses the kinetic-to-friction energy ratio. Fluid
mechanics.
Info: [A1].
Osborne Reynolds (see above).
3.1.89 Richardson Number Ri
Ri 5
gh
1
5
2
w
Fr
ð1Þ;
Ri 5
gred h
w2
ð2Þ
84
Dimensionless Physical Quantities in Science and Engineering
g (m s22) gravitational acceleration; h (m) characteristic length, vertical;
w (m s21) characteristic flow velocity; gred (m s22) reduced gravitational
acceleration; Fr () Froude number (1.) (p. 62).
This number expresses the potential-to-kinetic energies ratio. More often, its
inverse value is used: the Froude number (1.) Fr (p. 62). With small flow density
changes (for example in the ocean or in the atmosphere), the Richardson number is
used in expression (2).
Info: [A21].
Lewis Fry Richardson (p. 403).
3.1.90 Roshko Number Ro
Ro 5 Sh Re 5
fL2
v
f (s21) whirl frequency; L (m) characteristic length, for example, hydraulic
radius; ν (m2 s21) kinematic viscosity; Sh () Strouhal number (p. 87);
Re () Reynolds number (p. 81).
It describes the mechanism of a swirl oscillating flow. Compared to the Strouhal
number Sh (p. 87), it describes the character and viscosity of the fluid flow. Its
validity has been verified by bypassing a cylinder in the range ReAh40; 10000i.
Fluid mechanics. Turbulence.
Info: [A37],[C118].
Anatol Roshko (born 1911), American physical engineer of
Canadian origin.
He is an authority in aircraft and space engineering, especially in the spheres of turbulence, swirl flows, fluid mechanics and gas dynamics. In turbulence research, he initiated
new trends primarily with his work on coherent structures. As
a co-author, he wrote the book Elements of Gasdynamics
(1956), which is well known all over the world. He worked
for 40 years as a professor at the California Institute of
Technology and gave remarkable lectures on fluid mechanics.
3.1.91 Russell Number Ru
w
Ru 5 ;
fh
where
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g dR
f5 2
R dz
w (m s21) wind velocity; f (s21) natural vertical frequency of the fluid part;
h (m) height of obstruction; g (m s22) gravitational acceleration; R (kg m23) density; z (m) vertical coordinate.
Fluid Mechanics
85
This number expresses the inertia-to-uplift forces ratio. It concerns buoyancy,
gravitation effects, two-phase flow, and shock and surface waves in hydromechanics and waves in stratified flow.
Info: [B20].
3.1.92 Sarrau Number Sa
See Mach number M (p. 73).
Jacques Rose Ferdinand Emile Sarrau, French engineer.
3.1.93 Shear Stress Number NS
NS 5
Rw
IF
R (kg m23) fluid density; w (m s21) flow velocity; IF (kg m22 s21) force
impulse.
It characterizes the shear stress in fluid flow.
Info: [A49].
3.1.94 Schiller Number (1.) Sch
sffiffiffiffiffiffiffiffi rffiffiffiffiffiffi
R2
3
3 Re
5
Sch 5 wL
2ηF
CD
w (m s21) flow velocity; L (m) characteristic length; R (kg m23) gas or
liquid density; η (Pa s) dynamic viscosity; F (N) resistance force; Re () Reynolds number (p. 81); CD () drag coefficient (p. 60).
This number characterizes the flow around obstacles and expresses the ratio of
velocity to the force acting on a submerged body. It also characterizes the inertia
force of certain sized particles in relation to the resistance coefficient. It also
expresses the ratio of forces acting on a diving body.
Info: [A25],[A29],[A35].
L. Schiller.
3.1.95 Slosh Time NS
rffiffiffiffiffiffiffi
σ
NS 5 τ 3 3
gr
τ (s) time; σ (N m21) surface tension; g (m s22) gravitational acceleration;
r (m) pipe radius.
86
Dimensionless Physical Quantities in Science and Engineering
It expresses the dimensionless time of fluid strokes in pipelines and in surface
oscillation. Hydraulics.
Info: [B 20].
3.1.96 Smoluchowski Number Sm
It expresses the low pressure gas flow, see Knudsen number (1.) Kn (p. 69).
Info: [A29],[A35].
Marian Smoluchowski (28.5.18725.9.1917), Polish physicist.
He was engaged in the phenomena connected with the
thermal movement of molecules and atoms. Independently of
Einstein, he formulated the expression explaining the Brown
movement. He worked on the statistical expression of the
second law of thermodynamics. The whole range of his work
was an important contribution to science. It was not only
about physics, but also about physical chemistry, geophysics
and geology. He was the first to calculate the zeta potential
of electrokinetic movement.
3.1.97 Speed Number Pw, W
Pw 5
w
wmax
w, wmax (m s21) local and maximal velocity.
It expresses the local-to-maximum velocities ratio. Aero-hydrodynamics.
Info: [A24].
3.1.98 Stokes Number (6.) Stk
R R3
Stk 5 1:042mF gR 1 2
R F η2
mF (kg) float mass; g (m s22) gravitational acceleration; R (kg m23) fluid
density; RF (kg m23) float density; R () pipe and float radii ratio; η (Pa s) dynamic viscosity.
It is used to calibrate the rotameters used in measuring fluid passage. Measuring
techniques.
Info: [A35].
George Gabriel Stokes (p. 131).
3.1.99 Stratification Parameter S
S5
ω
2ωC
Fluid Mechanics
87
ω (s21) angular frequency; ωC (s21) Coriolis frequency.
It characterizes the fluid flow with rotation and action of the Coriolis force. For
S{1, stratified flow occurs (laminar flow, etc.). For Sc1, the rotation influence
prevails. Fluid mechanics. Geophysics.
Info: [C124].
3.1.100 Strouhal Number Sh
wτ
L
ð1Þ;
ð3Þ;
Sh 5
Sh Th 5
Sh 5
fL
w
Sh 5
w
fL
w
nL
ð2Þ;
ð4Þ
w (m s21) velocity; τ (s) time; L (m) characteristic length; n (s21) rotational frequency; f (Hz) frequency; Th () Thomson number (p. 90).
This number describes the oscillating flow mechanism and expresses the vibrationto-flow velocities ratio; alternatively, it relates the ratio of local force, caused by a
non-stationary process, to the inertia force. It represents the universal dynamic similarity criterion of non-stationary processes in systems which are similar in terms of
geometry and kinematics. It determines the non-stationary to convective momentum
transfer in a system. It is the measure for movement non-stationarity. With Sh $ 1, the
local accelerations are commensurate with the convective ones and the movement is
non-stationary. On the contrary, in stationary movement, Sh 5_ 0 is valid.
It is known as the concept of reduced frequency. Sometimes, it is called
the Thomson number Th (p. 90), or the acceleration (accelerating) ratio in expression (2).
Info: [A29],[A43],[A95],[B20].
Vincenc Strouhal (10.4.185023.1.1922), Czech experimental physicist.
He was engaged in hydrodynamic phenomena, acoustics
and electric and magnetic properties of steel. In the sphere of
gas and fluid dynamics, his experimental work led to the
determination of the dimensionless parameter which is called
the Strouhal number. He wrote several publications on experimental physics: Mechanika (Mechanics, 1901), Akustika
(Acoustics, 1902), Thermika (Thermics, 1908) and Optika
(Optics, 1919).
3.1.101 Strouhal Turbulence Number Shturb
Shturb 5
jðwrÞwj
τ
τ
w cor ~ cor
@w
Lturb τ turb
@τ 88
Dimensionless Physical Quantities in Science and Engineering
w (m s21) flow velocity; τ (s) running time; τ cor (s) correlation time; τ turb
(s) fluctuation time; Lturb (m) characteristic length of turbulence.
In the dimensionless form, it expresses the ratio of the correlation time to the
fluctuation time of turbulent flow. Fluid mechanics. Astrophysics.
Info: [B58].
Vincenc Strouhal (see above).
3.1.102 Swirl Number Sw
Sw 5
w
u
w (m s21) tangential velocity component; u (m s21) axial velocity component.
It expresses the swirl intensity in flow. It is given by the tangential-to-axial
momentum ratio. The tangential velocity is the mean tangential velocity component
in the outlet plane of a passage canal. The axial velocity is the longitudinal movement velocity in the canal plane.
Info: [C127].
3.1.103 Szebehely Number Sz, Σ
Sz 5
[email protected][email protected]τj
jðwUrÞwj
w (m s21) flow velocity; τ (s) time.
The Szebehely number characterizes the unsteadiness measure of a fluid flow. It
expresses the ratio of the fluid flow local acceleration to the convective acceleration. Conforming to this criterion, the flow can be divided into the following
categories:
steady flow with acceleration (Sz 5 0),
unsteady flow without acceleration (Sz 5 1),
steady flow without acceleration (Sz not determined),
unsteady flow with acceleration (0 , Sz # N, Sz 6¼ 1). Sz 5 N corresponds to unsteady
flow with zero convective acceleration.
The Szebehely number follows from two definitions for the velocity field:
a steady flow is that for which @[email protected]τ 5 0 is valid and
a flow without acceleration is that for which Dw=Dt 5 ð@[email protected]Þ 1ðwUrÞw 5 0:
Therefore, an arbitrary velocity field can be assigned to one of the four categories together with the corresponding Szebehely number value.
Info: [B5].
Fluid Mechanics
89
Victor G. Szebehely (192113.9.1997), American space
engineer of Hungarian origin.
He was among the key persons in the US Apollo space
program, which resulted in the landing of a man on the
moon. He was engaged in applied mathematics, especially in
the dynamics of seaborne and space ships. In 1956, he used
the dimensionless number to express time-dependent nonstationary flowing, which is named the Szebehely number.
Similarly, the Szebehely equation, which determines the
gravitational potential of the earth, planets, satellites and
galaxies, is named after him.
3.1.104 Taylor Number Ta
Ta 5
2wL2
v
2
1
ð1Þ;
f 2 L4
5 Re2rot
v2
ð2Þ;
3
ωr 2 ðr2 2 r1 Þ2
Ta 5 1
v
where
Ta 5
ð3Þ;
1 2
Ta 5 Re2 d3 v21
ð4Þ;
Tacrit 5 41:3
ω (Hz) angular frequency; L (m) characteristic length (thickness of flow gap
or liquid); ν (m2 s21) kinematic viscosity; f (s21) frequency; r2, r1 (m) outer
and inner gap radii; d (m) gap width between the cylinders (d 5 r2 2 r1); Rerot
() Reynolds rotation number (p. 383).
This number expresses the centrifugal and viscous forces; alternatively, the second power of the Coriolis force and the viscous force. It characterizes the rotation
influence on free convection. It is used in expressions (1) or (2). In expression (3),
it expresses a flow with viscous fluid swirl instability originating in a ring canal
between two concentric circular cylinders of which the inner one rotates and the
outer one is stable. In this case, with Ta , 41.3 the flow is laminar, for
41.3 # Ta , 400 the flow is laminar with steady origination of Taylor swirls, and
for Ta . 400 the flow is turbulent.
Info: [A21],[A35],[B11],[B20].
Geoffrey Ingram Taylor (7.3.188627.6.1975), English
mathematician, physicist, aerodynamicist and meteorologist.
Having started with the theoretical study of shock waves, he
continued with the work of J.J. Thompson. Taylor was engaged
in his proposals and verified quantum theory experimentally.
His other spheres of interest included dynamic meteorology including his work on atmospheric turbulence and
oceanography.
90
Dimensionless Physical Quantities in Science and Engineering
3.1.105 Thien Parameter NT
NT 5 Maϑ
ϑ () flow angle; Ma () local Mach number (p. 73).
It characterizes the hypersonic flow for cases of sin ϑ-0 (NT{1). In these cases,
it enables linearization of complicated non-linear equations of the supersonic flow.
Info: [A21].
3.1.106 Thoma Number (1.) Th
See Cavitation number Ncav (p. 366).
Dieter Thoma (18811942), German hydraulic engineer.
3.1.107 Thomson Number Th
Th Sh 5
wτ
L
w (m s21) flow velocity; τ (s) time; L (m) characteristic length; Sh () Strouhal number (p. 87).
It describes oscillation flow. It is identical with the Strouhal number Sh (p. 87).
Info: [A29],[A35].
James Thomson (18821992), Irish engineer.
3.1.108 Transiency Group Nt
@p
@
1
@L
Nt 5
@p @ðFoÞ
@L
ð1Þ;
Nt 5
1 @ðReÞ
Re @ðFoÞ
ð2Þ
@[email protected] (Pa m21) pressure gradient in the flow direction; Fo () Fourier
number (p. 175); Re () Reynolds number (p. 81).
It expresses transient changes of flowing fluid behaviour.
Info: [A35].
3.1.109 Turbulence Number Tu
See Kármán turbulent number Ka, Tu (p. 67).
3.1.110 Valensi Number Va
Va 5
r2 ω
v
Fluid Mechanics
91
r (m) inner pipe radius; ω (s21) angular oscillation frequency; ν (m2 s21) kinematic viscosity.
This number expresses the ratio of the inner tube radius to the viscous diffuse
length in fluid flowing through a tube with a periodic boundary condition. Fluid
dynamics.
Info: [A1],[A29],[A35].
Jacques Valensi (born 1903), French engineer.
3.1.111 Vedernikov Number Ve
Ve 5
ζξw
wA 2 w
ζ () exponent of hydraulic radius; ξ () form factor of cross section;
w (m s21) mean flow velocity; wA (m s21) absolute velocity of wave
disturbance.
It describes the flow instability in an open canal. Hydraulics.
Info: [A33],[A35].
3.1.112 Wave Period NT
NT 5
Twα
h
T (s) period of water wave; w (m s21) mean velocity of steady flow; α () canal slope; h (m) steady water depth in canal.
It is applied to determine the opening time of irrigation canal gates. Hydraulics.
Info: [A47].
3.1.113 Weber Number (1.) and (2.) We
Rw2 L
We1 5
σ
ð1Þ;
1
We2 5 ðWe1 Þ2
1
RL 2
5w
σ
ð2Þ
R (kg m23) fluid density; w (m s21) movement velocity; L (m) characteristic
length; σ (N m21) surface tension.
These numbers express the ratio of the inertia force to the surface stress force.
The Weber number is connected with surface stress waves, as the Froude number
(2.) external Fr (p. 63) is with gravitational waves. The We number characterizes
the originating process of bubbles or drops during fluid boiling or steam condensation. Sometimes, it is used in expression of the second root (2) and is called the
Weber number (2.).
Info: [A7],[A21],[A43],[B20].
92
Dimensionless Physical Quantities in Science and Engineering
Ernst Heinrich Weber (24.6.179526.1.1878), German
psychophysicist, anatomist, physiologist.
He laid the foundation for psychophysics, which involves
psychological reactions to physical stimuli. Together with his
younger brother, Wilhelm Eduard Weber, he was engaged in
research on fluid flow. The well-known Weber criterion for
viscous liquid flow is named after them. Together, they
wrote the book Wave Theory and Fluidity.
Wilhelm Eduard Weber (24.10.180423.6.1891), German
physicist.
Already at only 20 years of age, he was engaged in fluid
flow research, with his elder brother Ernst Heinrich Weber.
He also devoted himself to the study of magnetism and magnetic tension measurement. He formulated a logical system
of units for electricity and tried to unify electricity and magnetism into a unique basic law of forces. He described electric current as a flow of electrons. The magnetic flux unit
was named after him.
3.1.114 Weber Rotation Number Werot
Werot 5
n2 L3 R
σ
n (s21) rotation frequency; L (m) characteristic length, diameter; R (kg m23) fluid density; σ (N m21) surface tension.
It is the inertia-to-capillary forces ratio. It characterizes the process of forced
fluid mixing. Hydromechanics. Blade machines.
Info: [A35].
Ernst Heinrich Weber (see above).
Wilhelm Eduard Weber (see above).
3.1.115 Womersley Number Wo
rffiffiffiffi
ω pffiffiffiffiffiffiffiffiffiffiffiffi
5 Re Sh
Wo 5 L
v
ð1Þ;
rffiffiffiffiffiffiffiffiffi
2πfn
Wo 5 L
ð2Þ
v
L (m) characteristic length, half-thickness; ω (s21) angular frequency (ω 5 2πf);
ν (m2 s21) kinematic viscosity; f (s21) pulsation frequency; n () degree of
harmonic frequency; Re () Reynolds number (p. 81); Sh () Strouhal number
(p. 87).
Fluid Mechanics
93
It characterizes the non-stationary oscillating flow which manifests itself as the
response to pressure gradient oscillations. With composed polyharmonic pulsating
flow, the expression (2) is valid. The Wo number shows whether the flow is quasistationary or not. It is also important in biomechanics in the flow of biologically
significant fluids such as air, water and blood. With Wo , 1, a quasi-stationary
flow can be presumed, with an oscillating pressure gradient and a parabolic velocity profile with the greatest amplitude. With Wo . 1 the flow is phase shifted in
time as compared with pressure gradient oscillations, and the fluid oscillation
amplitude can either grow or drop. Fluid mechanics. Aeroelasticity. Biomechanics.
Info: [B69].
3.1.116 Zhukovsky Number Zh
See Fourier hydrodynamic number Foh (p. 62).
Nikolay Yegorovich Zhukovsky (p. 33).
3.2
Multiphase Fluid Mechanics
Usually, this concerns the similarity criteria for two-phase flow in which mutual
chemical interference does not occur. It is about various combinations of two
phases, such as, for example, watersteam, waterair and solid particles (dust) in
the air or in fluid. Flow with more than two phases occurs, for example, with explosions. The criteria concern such things as the solid particle movement in fluid, the
dynamics of increasing transfer and collapse of bubbles in fluids, flow accompanied
with boiling or condensation and granulation flow. The Bagnold, Camp, Darcy,
gravitational, capillary, Kolmogorov, Leibenzonov, Leverette, Morton, Ohnesorge
and Stokes numbers are among the wide range of similarity criteria applied.
3.2.1
Accelerating Frequency Nf
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ΔRω2 L3
Nf 5
σ
ΔR (kg m23) density difference; ω (s21) angular velocity; L (m) characteristic length; σ (N m21) surface tension.
This frequency characterizes the fluid flow acceleration in microgravitation
conditions. Essentially, it is an analogue to the Weber rotation number Werot
(p. 92), which compares inertia and capillary forces. Two-phase flow.
3.2.2
Bagnold Number (1.) Bg
Bg 5
3CD Rg w2
4dgRs
94
Dimensionless Physical Quantities in Science and Engineering
CD () drag coefficient (p. 60); Rg, Rs (kg m23) density of fluid and solid
particles; w (m s21) flow velocity; d (m) particle diameter; g (m s22) gravitational acceleration.
It is the ratio of the front resistance of bodies (particles) to their weight. It characterizes the carrying along of the solid particles by a fluid. Hydromechanics.
Ecology. Soil erosion.
Info: [A26],[B20].
Ralph Alger Bagnold (18961990), English geophysicist,
soldier and traveller.
His observations of sand movement while travelling in the
African desert led to his interest in the physical substance
and movement of sand dunes. His efforts resulted in the
monograph The Physics of Blown Sand and Desert Dunes
(1941). Later he also devoted himself to solid material water
stream transport. He was noted not only for his study of sand
dunes on earth, but also those on Mars.
3.2.3
Bagnold Number (2.) Bg
0
VS RS L τ_
Bg 5
1 2 VS η
2
ð1Þ;
B
Bg 5 @
1
VS3
1
3
Vmax
11
2
2
C Rs L τ_
A
1
η
3
ð2Þ
2 VS
VS (m3) volume of granular particles; RS (kg m23) particle density; L (m) characteristic length, grain diameter; τ_ (s21) unit speed of shear deformation;
η (Pa s) fluid dynamic viscosity in pores; Vmax (m3) maximal possible volume
grain concentration.
It expresses the relation between the number of collisions and the viscous tension
in steady and uniform shear flow. With Bg , 40, the macroviscous mode occurs, in
which the tensions are proportional to the shear tension velocity. With Bg . 200,
the collision mode prevails and the tension value is proportional to the second
power of the shear tension velocity. The expression (2) is the improvement of (1).
Info: [A7],[B13].
Ralph Alger Bagnold (see above).
3.2.4
Blake Number Bl
Bl 5
wRL
ηð1 2 eÞ
w (m s21) flow velocity; R (kg m23) density; L 5 V S 2 1 (m) characteristic
length of particle; V (m3) volume of particle; S (m2) surface of particle;
η (Pa s) dynamic viscosity; e () porosity (p. 24).
Fluid Mechanics
95
This number expresses the inertia-to-viscous forces ratio in momentum and mass
transfers through loose material layers. It characterizes the flow in loose material
transport. Essentially, it is a modification of the Reynolds number Re (p. 81).
Info: [A35],[B20].
J.R. Blake.
3.2.5
Bond Number Bo
Bo 5 Eo 5
ΔRgL2
5 We1 Fr 21
σ
ΔR 5 Rl 2 Rg (kg m23) difference between liquid and gas or vapour density; g
(m s22) gravitational acceleration; L 5 dB (m) diameter of water drop or bubble; σ (N m21) surface tension; Eo () Eőtvős number (p. 99); We1 () Weber number (1.) (p. 91); Fr () Froude number (1.) (p. 62).
It expresses the ratio of the gravity force to the surface stress force. It characterizes the movement of free bubbles or little drops in a stationary fluid, further the
vapour strokes arising due to energy release in fast mixing of heated and cold evaporating fluids. It applies to the processes in nuclear reactors and in combustion
engines, for example. Generally, it characterizes the fluid spraying process in a
two-phase environment flow. It is often called the Eőtvős number Eo (p. 99).
See also Goucher number Go (p. 65). Fluids atomization. Capillary flow. Stroke
and surface waves. Bodies immersed in fluid. Spraying of fluids.
Info: [A35],[B20].
Wilfrid Noel Bond (18971937), English physicist.
3.2.6
Boussinesq Number (2.) Bob
Bob 5
ηs
ηrb
ηs (Pa m s) surface shear viscosity; η (Pa s) liquid dynamic viscosity; rb (m) bubble diameter.
It expresses the characteristic radius of bubbles in flowing foamed fluid. Twophase flow.
Info: [B109].
Valentin Joseph Boussinesq (p. 55).
3.2.7
BrownellKatz Number BK
BK 5 Ca Bo 5
ηwRgL2
σ2
96
Dimensionless Physical Quantities in Science and Engineering
η (Pa s) dynamic viscosity; w (m s21) flow velocity; R (kg m23) density;
g (m s22) gravitational acceleration; L (m) characteristic length, diameter;
σ (N m21) surface tension; Ca () capillary number (1.) (p. 97); Bo () Bond number (p. 95).
This number characterizes the flow through the porous material. It expresses the
combination of the capillary number (1.) Ca (p. 97) and the Bond number Bo (p. 95);
alternatively, it is the ratio of viscous and gravitational forces product to the second
power of the surface stress force. Two-phase flow. Filtration. Draining. Drying.
Info: [C38].
3.2.8
Camp Number Ca
sffiffiffiffiffiffiffiffi
PV
Ca 5
ηQ2
P (W) energy dissipation in fluid volume V ; V (m3) fluid volume; η (Pa s) dynamic viscosity; Q (m3 s21) volume flow.
It expresses the ratio of the elapsed time to the average shear off value in a
fluid. It is the criterion for free clustering of suspended particles.
Info: [A29].
Thomas Ringgold Camp (born 1895), American engineer.
3.2.9
Capillary Buoyancy Number CaB
CaB 5
gη4
5 We31 Fr 21 Re 24
Rσ3
g (m s22) gravitational acceleration; η (Pa s) dynamic viscosity; R (kg m23) density; σ (N m21) surface tension; We1 () Weber number (1.) (p. 91);
Fr () Froude number (1.) (p. 62); Re () Reynolds number (p. 81).
It characterizes the influences of the surface tension, viscosity and acceleration
in two-phase flow when fluid balls move in another fluid. It depends on physical
properties only. See the Morton number (p. 107). In the case of solid particles in
flowing fluid, the modified number (property number) K is used, which provides
better correlation with the experiment. Capillary flow.
Info: [A29],[B20].
3.2.10 Capillary Number (1.) Ca
Ca 5
pffiffiffi
σ k
ηwL
Fluid Mechanics
97
σ (N m21) surface tension; k (m2) permeability; η (Pa s) dynamic viscosity;
w (m s21) velocity; L (m) characteristic length dimension.
It is the capillary-to-filtration forces ratio. Capillary flow and porous materials.
Info: [B20].
3.2.11 Capillary Number (1.) Ca
Ca 5
where
ηw
5 We1 Re 21
σ
pc 5
ð1Þ;
Ca 5
ηw
2cos ϑ
pc r
ð2Þ;
2σ cos ϑ
r
η (Pa s) dynamic viscosity; w (m s21) characteristic velocity; σ (N m21) surface or half-surface tension between two fluid phases; pc (Pa) capillary pressure; r (m) diameter of pore throat; ϑ () wetting angle, for hard
wettable material is valid cos ϑ 5 1; We1 () Weber number (1.) (p. 91);
Re () Reynolds number (p. 81).
This number expresses the ratio of the viscosity force to the surface stress which
acts across the interface between a fluid surface and a gas or between two mixable
fluids. It characterizes the fluid spraying process in the flow of two-phase surroundings, in capillary tubes and porous material layers. With the pore structure and wettability considered, expression (2) is valid. Capillary flow and porous materials.
Info: [A29],[B20].
3.2.12 Capillary Number (2.) Ca
Ca 5
L2 Rg
σ
L d (m) diameter (e.g. of pipe, particle, bubble, impeller or shaft); R (kg m23) density; g (m s22) gravitational acceleration; σ (N m21) undisturbed surface
tension.
This number equals the ratio of the gravitational force to the surface tension
force. Two-phase medium flow.
Info: [B6].
3.2.13 Capillary Multiphase Number Ca
Ca 5
wD η
;
pC R
where
pC 5 2σ cos ϑR 21
wD (m s21) characteristic velocity (Darcy velocity); η (Pa s) dynamic viscosity; pC (Pa) capillary pressure; R (m) mean pore radius; σ (N m21) surface
tension; ϑ () contact angle.
98
Dimensionless Physical Quantities in Science and Engineering
It characterizes the capillary flow in porous material. For a fully wetted material,
it is cos ϑ 5 1. With constant velocity, the capillary number is almost constant.
With the velocity changing in time, the number Ca becomes the dynamic parameter. An example is petroleum driven out with gas.
Info: [A29].
3.2.14 Darcy Granulation Number (1.) Dc
Dc 5
η
VS RS τ_ k
η (Pa s) fluid dynamic viscosity; VS (m3) volume of grain particles;
RS (kg m23) density of solid particles; τ_ (s21) unit speed of shear strain;
k (m21) hydraulic permeability.
It expresses the ratio of tension caused by interaction to that due to particle inertia.
Info: [A29],[B13].
Henry Philibert Gaspard Darcy (10.6.18033.1.1858),
French engineer.
He executed research on flow and friction loss in a pipeline. This research resulted in the DarcyWeisbach equation.
He improved the pitot tube structure and was the first to
express the boundary layer in flowing fluid. In the last years
of his life, he carried out many experiments leading to the
Darcy law for flow in sand. Since that time, this law has been
generalized and extended to unsaturated and multiphase flow.
3.2.15 Darcy Granulation Number (2.) Dc
Dc 5
wL
D0
w (m s21) flow velocity; L (m) characteristic length; D0 (m2 s21) permeability of granulated material.
It expresses the inertia-to-permeability forces ratio. It characterizes the flow in
porous material. It is an analogy of the Reynolds number Re (p. 81) and the Pe´clet
heat number Pe (p. 180).
Info: [A29].
Henry Philibert Gaspard Darcy (see above).
3.2.16 Driftage Elements Number ND
ND 5
wv w
gL
Fluid Mechanics
99
wv (m s21) free fall velocity of particle; w (m s21) flow velocity; g (m s22) gravitational acceleration; L (m) characteristic length.
This number describes the drifting and depositing of fine particles in a fluid
flow. Creation of sediments in canals and pipelines. Pneumatic transport.15
3.2.17 Eőtvős Number Eo
See the Bond number Bo (p. 95).
Info: [A29],[B7].
Lóránd Baron von Eőtvős (27.7.18488.4.1919),
Hungarian physicist.
Initially, he was engaged in capillary phenomena. Later
he devoted himself to gravitation. He used torsion balance to
prove that the ratio of the gravitational and inertia mass is
constant with the accuracy of 5 3 1029, which is fundamentally important for general relativity theory because it confirms that both masses are equivalent.
3.2.18 Expansion Bubbling Number Ex
Ex 5
gd Rl 2 Rv
5 Fr 21 PR
w2 Rl
g (m s22) gravitational acceleration; d (m) diameter of gas or vapour bubble;
w (m s21) bubble velocity; Rl, Rv (kg m23) liquid and gas density; Fr () Froude number (1.) (p. 62); PR () relative change of density, equivalent of
Boussinesq approximation number Bs (p. 56).
It expresses the lift-to-inertia forces ratio. It characterizes the process of gas or
vapour bubble propagation in a fluid.
Info: [A29].
3.2.19 Filtration Energy Gradient J
J 52
@H
w2
f ðReÞ
5
gL
@x
H (m) energetic height; x (m) axis x; w (m s21) filtration velocity; g (m s22) gravitational acceleration; L (m) characteristic length of grains; Re () Reynolds
filtration number, see Reynolds number (p. 81).
It expresses the energy gradient in the x-axis direction. Two-phase flow.
Hydraulics. Filtration.
Info: [A48].
100
Dimensionless Physical Quantities in Science and Engineering
3.2.20 Fluidization Number Nfluid
ζVs ðRs 2 Rf Þ
Nfluid 5
ηVf
rffiffiffi
g
L
ζ (m2) permeability of porous material; Vs, Vf (m3) volume of solid particles
and fluid; Rs, Rf (kg m23) density of solid particles and fluid; η (Pa s) dynamic
viscosity; g (m s22) gravitational acceleration; L (m) characteristic length, particle diameter.
It expresses the fluidization measure in relation to the velocity. For Nfluid{1, it
is used for the majority of flows. Two-phase fluid mechanics.
Info: [B54].
3.2.21 Frequency Number Nf
Nf 5
ωL
5 2πSh
w
ω (s21) angular frequency; L (m) characteristic length, thickness; w (m s21) velocity of fluid infiltration; Sh () Strouhal number (p. 87).
It characterizes the flow in condensed or suspension layers.
Info: [A29].
3.2.22 Friction Number Nf
Nf 5
Vs nðRs 2 Rf ÞgL tan ϕ
1 2 Vs
τ_ η
Vs (m3) volume of solid particles; n () grain number on the grain surface; Rs,
Rf (kg m23) density of solid part and fluid; g (m s22) gravitational acceleration; L (m) characteristic length (diameter); ϕ () angle of inner friction; τ_
(s21) unit velocity of shear deformation; η (Pa s) dynamic viscosity.
It expresses the ratio of friction tensions, originating with steady contact of
grains, to viscous shear tensions. It resembles the Bingham number Bm (p. 118),
but the characteristic tensions originate for solid and liquid parts differently.
Dominating viscous forces arise with Nf . 100.
Info: [B13].
3.2.23 Froude Multiphase Number Frmp
wg
Frmp 5 pffiffiffiffiffiffi
gL
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Rg
Rl 2 Rg
Fluid Mechanics
101
wg (m s21) velocity of moist gas flow; g (m s22) gravitational acceleration;
L (m) characteristic length; Rl, Rg (kg m23) density of fluid and gas.
It expresses the inertia-to-gravitation forces ratio in two-phase fluid flow.
Multiphase fluid mechanics. Wet gas flow.
Info: [D3].
3.2.24 Gibbs Number Gb
Gb 5
E
kT
E (J) local energy of perturbation; k (J K21) Boltzmann constant; T (K) absolute temperature.
This number expresses the kinetic theory of molecules. For example, it characterizes cavitation processes and bubble dynamics in fluids. It is about the final
period in the homogeneous nucleon theory, in which the deposition energy E acts
and the bubbles originate. Two-phase flow. Boiling. Cavitation.
Info: [A7].
Josiah Willard Gibbs (11.2.183928.4.1903), American
mathematician, theoretical physicist and chemist.
He became an authority in theoretical thermodynamics. He
was a Professor in the Department of Mathematical Physics at
Yale University. He deduced the Gibbs phase rule. Free
enthalpy is denoted as the Gibbs thermodynamic potential.
Gibbs distribution, which is a basic law of statistical physics,
is connected with his name as well. His most important book
is Elementary Principles in Statistical Mechanics (1902).
3.2.25 Granulometric Number Gm
dmax
Gm 5
dekv
"
0:6
;
where
dekv 5
n
X
i51
# 21
ðgi di21 Þ
dmax (m) maximal particle diameter; dekv (m) equivalent particle diameter of layer
that polydisperses; gi (), di (m) weighted part and diameter of i-kind particles.
It expresses the fluid flow in dispersed layers.
Info: [A33].
3.2.26 Gravitational Number (1.) Ng
Ng 5
k gΔR k gðRp 2 Rf Þ
5
ηw
ηw
102
Dimensionless Physical Quantities in Science and Engineering
k (m2) permeability of environment; g (m s22) gravitational acceleration;
ΔR (kg m23) density difference between particles and fluid; Rp, Rf (kg m23) density of particles and fluid; η (Pa s) dynamic viscosity; w (m s21) flow
velocity.
It expresses the gravitational-to-filtration forces ratio. Two-phase flow. Filtration.
Info: [B20],[B115].
3.2.27 Gravitational Number (2.) Ng
Ng 5
4πrp4 ðRp 2 Rf Þg
3kT
rp (m) particle radius; Rp, Rf (kg m23) density of particle and fluid; g (m s22) gravitational acceleration; k (J K21) Boltzmann constant; T (K) fluid absolute
temperature.
It expresses the ratio of the gravitational potential of a particle to its thermal
energy. Physically chemical filtration in saturated porous materials.
Info: [B115].
3.2.28 Gravitational Number (3.) Ng
Ng 5
ηw
Ca
5
2
gΔRL
Bo
η (Pa s) fluid dynamic viscosity; w (m s21) flow velocity; g (m s22) gravitational acceleration; ΔR (kg m23) density difference between two non-miscible
liquids; L (m) characteristic length, thickness, diameter; Ca () capillary
number (1.) (p. 97); Bo () Bond number (p. 95).
It expresses gravitational force as the viscous-to-capillary forces ratio. It acts in
the penetration of two non-miscible fluids through a porous material. Two-phase
porous flow. Filtration.
3.2.29 Gravity Number NG
NG 5
k gΔR
ηw
ð1Þ;
NG 5
2 r 2 gðRp 2 Rf Þ
9
ηw
ð2Þ
k (m2) permeability; g (m s22) gravitational acceleration; ΔR (kg m23) density difference of phases; η (Pa s) dynamic viscosity; w (m s21) flow velocity;
r (m) particle radius; g (m s22) gravitational acceleration; Rp, Rf (kg m23) density of particle and fluid.
Fluid Mechanics
103
This number denotes the gravity-to-filtration forces ratio or, alternatively, the
ratio of the Stokes rate of particle sedimentation to the fluid rate. It is expressed in
expression (2) as well. Flow through porous material. Two-phase flow. Filtration.
Info: [B20].
3.2.30 HabermanMorton Number Hab, Hm
mp
gη4l
We31
5
Hab 5
1
2
Rl σ3
Rl Vp
Fr 2 Re4
g (m s22) gravitational acceleration; ηl (Pa s) fluid dynamic viscosity;
Rl (kg m23) fluid density; σ (N m21) surface tension; mp (kg) particle mass;
Vp (m3) particle volume; We1 () Weber number (1.) (p. 91); Fr () Froude number (1.) (p. 62); Re () Reynolds number (p. 81).
This number expresses the deformation of moving particles, as drops, bubbles or
fine deforming (granular) materials. In the case of bubbles mp{Rl, mp =ðRl Vp Þ-0
holds. Then the bubble dimension influence on the number Hab can be neglected.
For filtered water Hmi 5 0:25 3 10210 ; for mineral oil Hmi 5 1:45 3 1022 and for
syrup Hmi 5 0:92 3 106 :
Info: [A2],[A7],[A8].
3.2.31 Inertia Parameter Ninert
Ninert 5
d2 RP w
18ηL
d (m) diameter (such as of pipe, particle, bubble or drop); RP (kg m23) particle
density; w (m s21) fluid velocity; η (Pa s) dynamic viscosity; L (m) characteristic length (such as of channel width, distance from wall or height of liquid
layer).
It expresses the inertia to friction forces in particle flow in a fluid.
Info: [C72].
3.2.32 KeuleganCarpenter Number (2.) KC
KC 5
wA τ p
2L
wA (m s21) speed of body oscillating; τ p (s) time period; L (m) characteristic
length, diameter.
It characterizes non-steady oscillation phenomena joined with the particle movement in fluid. For KC , 5, the inertial phenomena prevail. Two-phase flow.
Info: [A4],[A7].
104
Dimensionless Physical Quantities in Science and Engineering
3.2.33 Knudsen Number (3.) Diffusion Kn
Kn 5
3 hDAB
4 ζKOA wA
h (m) height of roughness; DAB (m2 s21) binary bulk diffusion coefficient for
system AB; ζ () diffusion tortuosity; KOA (m) Knudsen coefficient of permeability; wA (m s21) equilibrium mean molecular speed of species A.
In contrast to the Knudsen number (2.) diffusion Kn (p. 45), it involves the influence of surface roughness and diffusive tortuousness. It expresses the gas diffusion
in bundles.
Info: [A29].
Martin Hans Christian Knudsen (p. 420).
3.2.34 Kolmogorov Number Kol
Rp 2 R gwg @w 21
Kol 5
;
R w2dyn @z
where
rffiffiffi
τ
wdyn 5
R
Rp, R (kg m23) density of particles and liquid; g (m s22) gravitational acceleration; wg (m s21) free fall velocity of a particles in liquid; wdyn (m s21) dynamic
@w 21
velocity;
(s ) vertical gradient of velocity flow; τ (m21 kg s22) shear [email protected]
tion tension.
It expresses the relative loss of the turbulent energy expended to maintain the
equilibrium of particles in a fluid flow. It is a criterion of the dynamic balancing
activity, for example, the balanced particles’ influence on the flowing fluid dynamics. It is analogous to the Kolmogorov parameter Kol (p. 399).
Info: [B6],[C75].
Andrey Nikolaevich Kolmogorov (25.4.190320.10.1987),
Russian mathematician.
He contributed substantially to the development of probability theory, topology and other mathematical branches.
However, he also worked in the sphere of classic mechanics,
especially on turbulence theory. He formulated the algorithmic complex theory. Beginning in the 1940s, he was engaged
in dynamic systems theory in relation to interplanetary flight
and other problems in space science. Besides mathematics,
he had an extraordinary interest in poetry, especially the
Russian poet Pushkin.
Fluid Mechanics
105
3.2.35 Kozeny Number Kz, k
Kz 5
Δp μ3 S
ηL ð1 2 ϕ2 Þw
Δp (Pa) piezometric pressure drop in the thickness L of porous material; η (Pa s) dynamic viscosity; L (m) characteristic length, thickness; ϕ () porosity (p. 24);
S (m2) area; w (m s21) mean velocity component in direction L.
It expresses the ratio of the pressure and volume deformation forces to the fluid
momentum. It characterizes the mass transfer through a porous material.
Info: [A29].
Josef Alexander Kozeny (born 1889), Austrian engineer of Czech origin.
3.2.36 Kubo Number of Percolation Kub
Kub 5
w
Lω
w (m s21) characteristic velocity; L (m) characteristic length, necking; ω (s21) characteristic perturbation frequency.
It expresses the accidentality level in the percolation process of a moderately
compressible fluid under turbulent transfer conditions. Two-phase fluid mechanics.
Turbulent diffusion.
Info: [B4],[B53].
3.2.37 Leibenson Number (2.) Lb
Lb 5
RR3H Δp
η2 L
R (kg m23) fluid density; RH (m) hydraulic radius; Δp (Pa) pressure drop;
η (Pa s) dynamic viscosity; L (m) characteristic length.
It characterizes the fluid filtration in a porous environment. It expresses the
porous environment layer resistance, which depends on its structure and the physical properties of the filtering environment. It is a measure of the ratio of pressure
forces under the action of hydraulic resistance to the viscosity forces.
Info: [A33].
Leonid Samuilovich Leibenzon (18791951), Russian physicist and
mechanist.
106
Dimensionless Physical Quantities in Science and Engineering
3.2.38 Leverett Number Lt
sffiffiffi
k pc
Lt 5
pσ
k (m2) permeability; p () porosity (p. 24); pc (Pa) capillary pressure;
σ (Pa) surface tension.
It expresses the ratio of the characteristic dimension of an inter-area curvature to
the characteristic dimension of pores. Two-phase flow in porous materials.
Info: [A29],[B20].
Miles Corrington Leverett (born 1910), American chemical engineer.
3.2.39 LockhartMartinelli Parameter χ
ml
χ5
mg
rffiffiffiffiffi
Rg
Rl
ml, mg (kg s21) liquid and gas mass flux; Rl, Rg liquid and gas density.
This parameter is used for internal two-phase flow. It characterizes the influence
of the fluid fraction portion on the flow. It occurs mainly in pressure spraying and
heat transfer in boiling and condensation.
Info: [A12],[C9].
3.2.40 Lyashenko Number Lj
Lj 5 Re3 Ar 21
Re () Reynolds number (p. 81); Ar () Archimedes hydrodynamic number
(p. 53).
This number characterizes the hydraulic effect of the fluid flowing through a
granular material (fluidization).
Info: [A29].
3.2.41 Martinelli Parameter X
21
dp
dp
X 5 dz fr liq dz fr g
2
dp
(N m23) pressure gradient; subscripts: fr friction; liq liquid; g gas or
dz
vapour.
Fluid Mechanics
107
It expresses the ratio of the pressure caused by the friction of drops and the gas
flow. It is used for two-phase models to express the pressure of diversely shaped
drops.
Info: [A7].
3.2.42 Mass Number Nm
Nm 5
VS RS
Vf Rf
VS, Vf (m3) volume of solid and liquid components; RS, Rf (kg m23) density of
solid and liquid components.
It expresses the volume to density fractions ratio of solid and liquid particles or,
alternatively, the inertia thereof in mixing. It quantifies the effect of the volume tolerance. For Nm . 1, the momentum transfer of the granules in the fluid prevails.
Two-phase fluid mechanics.
Info: [A4],[B53],[B54].
3.2.43 Mobility Number of Grains Nmov
Nmov 5
Rw2
fV L
R (kg m23) fluid density; w (m s21) characteristic flow velocity; fV (N m23) specific grain force relative to volume unit; L (m) characteristic length of grains.
In a two-phase flow, it characterizes the ratio of the fluid flow dynamic force to
the dynamic force of flowing grains. Two-phase flow. Hydrodynamics. Filtration.
Info: [A48].
3.2.44 Morton Number Mo
Mo 5
gη4
We31
5
Rσ3
Fr 2 Re4
Mo 5
gη4 ðRf 2 Rs Þ
R2f σ3
ð1Þ;
Mo 5
gη4 ðR1 2 Rg Þ
R2l σ3
ð2Þ;
ð3Þ
g (m s2) gravitational acceleration; η (Pa s) dynamic viscosity; Rl, Rg, Rf, Rs
(kg m23) density of liquid, gas, fluid and solid particles; σ (N m21) surface
tension; W e1 () Weber number (1.) (p. 91); Fr () Froude number (1.)
(p. 62); Re () Reynolds number (p. 81).
108
Dimensionless Physical Quantities in Science and Engineering
This number expresses the ratio of the gravitational acceleration to the molecular
acceleration of a fluid. It depends on fluid properties only. In expression (1), it is
called the Morton fluid number, which expresses the relations arising in the process
of immersing solid bodies into viscous fluid. It corresponds to the capillary buoyancy number CaB (p. 96). Expression (2) describes the free bubble movement in
a steady state fluid. Expression (3) describes the solid particle movement in the
fluid. It is also called the property groups NP (p. 110) or the HabermanMorton
number Hab (p. 103). For water at 20 C, its value is approximately 3 3 10211.
Info: [A26],[B20].
3.2.45 Non-dimensional Thermal Diffusivity Na
Na 5
ag f
c2g
ag (m2 s21) gas thermal diffusivity; f (s21) bubble frequency; cg (m s21) gas
sound velocity; κ () specific heat ratio (p. 28).
It expresses the internal thermal conductivity, including the thermal dumping
influence of bubbles, in two-phase surroundings. The effective polytropic exponent
k depends on internal thermal conductivity, and 1 , k , κ holds. For high frequencies, the mean free path in gas can be comparable with the bubble dimension and
the k can be outside the range mentioned as well.
3.2.46 Ohnesorge Number Oh, Z
1
η
Oh Z 5 pffiffiffiffiffiffiffiffiffi 5 We21 Re 21
RLσ
ð1Þ;
1
Oh 5 Su2
ð2Þ
η (Pa s) dynamic viscosity; R (kg m23) density; L (m) characteristic length;
σ (Pa) surface tension; We1 () Weber number (1.) (p. 91); Re () Reynolds number (p. 81); Su () Suratman number (p. 115).
It is the ratio of the viscous force to the surface stress force. It characterizes the
processes of fluid spraying (atomization) in a two-phase flow. Sometimes, the
Suratman number Su (p. 115) is used instead of it. Stroke and surface waves.
Spraying. Splashing.
Info: [A26],[B20],[B56].
Wolfgang von Ohnesorge (8.9.190126.5.1976), German hydrodynamics
specialist.
3.2.47 Pavlovsky Number Pa
Pa 5
d2 Δp
wηδ
Fluid Mechanics
109
d (m) particle diameter; Δp (Pa) pressure drop; w (m s21) characteristic
flow velocity; η (Pa s) dynamic viscosity; δ (m) layer thickness.
It characterizes the fluid flow through a granular material layer. It expresses the
pressure force, with consideration of the hydraulic resistances, to the viscosity force.
Info: [A33].
3.2.48 Poiseuille number (2.) Ps
Ps 5
wv
ðRs 2 Rf ÞgD2
w (m s21) flow velocity; ν (m2 s21) kinematic viscosity; Rs, Rf (kg m23) density of solid particles and liquid; g (m s22) gravitational acceleration; D (m) diameter of solid particle.
It expresses the fraction and gravitation forces in a fluid flow with mass particles. Two-phase flow.
Info: [A29].
Jean Louis Marie Poiseuille (p. 78).
3.2.49 Porous Flow Number NP
wηL
NP 5 pffiffiffi
kσ cos ϑ
w (m s21) flow velocity; η (Pa s) dynamic viscosity; L (m) characteristic
length; k (m2) permeability; σ (N m21) surface tension; ϑ () angle of capillary depression.
It is the viscous-to-capillary pressures ratio. It is the inverse value of the capillary number (1.) Ca (p. 56). Filtration with a porous material.
Info: [B20].
3.2.50 Porous Pressure Number NP
D
NP 5 2
h
sffiffiffi
L
g
D (m2 s21) porous pressure diffusivity; h (m) height; L (m) characteristic
length; g (m s22) gravitational acceleration.
This number expresses the ratio of the gravity flow time to the time of the pore
pressure diffusion which is perpendicular to the flow direction. The small value
Npor{1 shows that the porous pressure and the diffusion act at different times.
Two-phase fluid mechanics.
110
Dimensionless Physical Quantities in Science and Engineering
3.2.51 Property Groups NP
NP 5
gη4 ðRf 2 Rs Þ
R2f σ3
g (m s22) gravitational acceleration; η (Pa s) dynamic viscosity; Rf, Rs (kg m23) density of fluid, solid particles or bubbles; σ (N m21) surface tension.
It is a widely used and specifying modification of the capillary buoyancy
number CaB (p. 96).
Info: [A29].
3.2.52 Rabin Number Rab
We1
Rab 5 pffiffiffiffiffiffi
Re
ð1Þ;
sffiffiffiffiffiffiffiffiffi
3
4 We
1
ð2Þ
Rab 5
Lp
d (m) drop diameter; R (kg m23) drop density; σ (N m21) surface tension;
η (Pa s) dynamic viscosity; We1 () Weber number (1.) (p. 91); Re () Reynolds number (p. 81); Lp () Laplace number (p. 71).
In expression (1), it serves to determine various drop decomposition. It can be rewritten into expression (2), which is essentially analogous to the Weber number (1.) We1
(p. 91), with certain corrections for the drop dimension expressed by the Laplace
number Lp (p. 71). Two-phase flow. Condensation. Getting frostbitten. Spraying.
Info: [A15].
3.2.53 Radial Frequency Parameter (1.) Pfr
Pfr 5
ωr D
w2
ωr (s21) angular frequency; D (m2 s21) mass diffusion or dispersion;
w (m s21) fluid velocity.
It expresses the cyclic transfer of molecular mass diffusion with dynamic fluid
acting on a rotating system. It serves to monitor the resonating frequency and is
related to control. With its value increasing, the radial characteristics strengthen
repeatedly. Fluidized coating.
Info: [A35].
3.2.54 Radial Frequency Parameter (2.) Pfr
Pfr 5
ωr L2
a
Fluid Mechanics
111
ωr (s21) angular frequency; a (m2 s21) thermal diffusivity; w (m s21) fluid
velocity.
It expresses the heat diffusion cyclic transfer with dynamic fluid acting on rotating systems. It is also called the wave thermal parameter. It serves to monitor the
resonating frequency. An increasing value leads to repeated strengthening.
Fluidized coating.
Info: [A35].
3.2.55 Radial Frequency Parameter (3.) Pfr
Pfr 5
ω2r DL
5 ðPfr Þ2 Pem21
w3
ωr (s21) angular frequency; D (m2 s21) mass diffusivity or dispersion; L (m) characteristic length; w (m s21) fluid velocity; Pfr () radial frequency parameter (1.) (p. 110); Pem () Péclet mass number (p. 258).
It characterizes the cyclic transfer of molecular mass diffusion with dynamic
fluid acting on a rotating system. Fluidized coating.
Info: [A35].
3.2.56 Radial Frequency Parameter (4.) Pfr
Pfr 5 L
rffiffiffiffiffiffi
ωr
2D
L (m) characteristic length; D (m2 s21) diffusivity or dispersion; ωr (s21) angular frequency.
It characterizes the cyclic transfer of molecular mass diffusion. Fluidized
coating.
Info: [A35].
3.2.57 Rayleigh Pore Flow Number Rapor
Rapor 5
ϕgβL3 ΔT
av
ϕ () porosity (p. 24); g (m s22) gravitational acceleration; β (K21) volume
thermal expansion coefficient; L (m) characteristic length, thickness; ΔT (K) temperature difference; a (m2 s21) thermal diffusivity; ν (m2 s21) kinematic
viscosity.
112
Dimensionless Physical Quantities in Science and Engineering
This number expresses the viscous fluid infiltration through a porous layer
caused by the thermal gradient, gravitation force and diffusion heat propagation. It
is analogous to the Rayleigh modified number Ramod (p. 382).
Info: [A35].
Lord Rayleigh (p. 187).
3.2.58 Relative Filtration Velocity Pw
Pw 5
c
w
c (m s21) velocity propagation of surface wave; w (m s21) filtration velocity.
It expresses the influence of the water wave propagation velocity on the filtration velocity. Two-phase flow. Filtration. Geophysics.
Info: [A48].
3.2.59 Reynolds Quasi Number Req
Req 5
pffiffiffiffiffiffi
Rh gL
Vf η
R (kg m23) density; h (m) layer thickness; L (m) characteristic length;
g (m s22) gravitational acceleration; Vf (m3) fluid volume; η (Pa s) dynamic
viscosity.
In infiltration, it represents the dynamic coefficient analogous to the Reynolds
number Re (p. 81) in Newtonian flow. Usually, it reaches the value Req . 106.
Two-phase fluid mechanics.
Info: [B12],[B13],[B505].
Osborne Reynolds (p. 82).
3.2.60 Savage Number Sav
Sav 5
_ sL
γR
ðRs 2 Rf Þgh
ð1Þ;
Sav 5
_ sL
γR
nðRs 2 Rf Þgh tan ϕ
ð2Þ
γ_ (s21) unit speed of shear deformation; Rs, Rf (kg m23) density of particles
and fluid; L (m) characteristic length, grain diameter; g (m s22) gravitational
acceleration; h (m) layer thickness; n () number of surface grains; ϕ () angle of inner friction.
This number expresses the ratio of tensions caused by grain collision to gravitation tensions arising due to the contact friction of grains. With Sav . 0.1 in the
usual layer depth, the tensions originating due to collision have great influence.
Info: [B13].
Fluid Mechanics
113
3.2.61 Schiller Number (2.) Sch
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffi
3R
Reδ
3
Sch 5 w
5
W
4gvðRs 2 RÞ
where
W5
wδ
;
w
Reδ 5
wδ
v
w (m s21) fluid flow velocity; R, Rs (kg m23) density of fluid and solid particles; g (m s22) gravitational acceleration; ν (m2 s21) kinematic viscosity; wδ
(m s21) boundary layer velocity; Reδ () Reynolds boundary layer number,
see Reynolds number Re (p. 81); W () relative velocity.
It expresses the equilibrium rate of ball-shaped particles floating in flowing
fluid. Fluidization.
Info: [A25],[A35].
L. Schiller.
3.2.62 Spray Number NA
NA 5
gðR 2 Rf ÞL2
σ
g (m s22) gravitational acceleration; R, Rf (kg m23) density of bubble and
ambient fluid; L (m) characteristic length; σ (N m21) surface tension.
It expresses the ratio of the weight force to the surface stress force. It corresponds to the Bond number Bo (p. 95). Atomization of fluids.
Info: [B7].
3.2.63 Stokes Foam Drainage Number St
St 5
ηws
5 fðε; ReÞ
Rgrb2
η (Pa s) dynamic fluid viscosity; ws (m s21) liquid surface velocity; R (kg m23) density; g (m s22) gravitational acceleration; rb (m) bubble diameter; ε () volume liquid fraction in foam; Re () Reynolds number (p. 81).
It expresses the fluid foaming in steady state flowing through sewers or other
canals, and the fact that the flow is influenced by the relative volume fluid fraction
and the Reynolds number Re (p. 81). Two-phase fluid mechanics.
Info: [B109].
George Gabriel Stokes (p. 131).
114
Dimensionless Physical Quantities in Science and Engineering
3.2.64 Stokes Particle Number St
τ rx
R d 2
St 5
5
τk
18 η
where
Rd2
;
τ rx 5
18v
rffiffiffi
v
τk 5
;
ε
3 1
η 5 v4 ε4
τ rx (s) relaxation time of particle; τ k (s) Kolmogorov time; R () density
ratio of particle and fluid; d (m) particle diameter; η (m) Kolmogorov length;
ν (m2 s21) kinematic viscosity; ε (m2 s21) dissipation rate of turbulent energy.
This number characterizes the particles moving by inertia in a turbulent flow
and clustering in a low swirl zone of the fluid due to the inertia equilibrium among
denser particles and the lighter surrounding fluid. The clustering peaks occur with
St1. Particle clustering in turbulent aerosols. Two-phase flow.
George Gabriel Stokes (see above).
3.2.65 Stokes Time Number Stk
Stk 5
τp
Rd 2 wCc
5
τf
18ηdc
τ p (s) time response; τ f (s) characteristic flow time; R (kg m23) density; d (m) particle diameter; w (m s21) flow velocity; Cc () CunninghamStokes correction
factor; η (Pa s) dynamic viscosity; dc (m) cylinder diameter.
It expresses the ability of aerosol particles to collide. It represents the ratio of
the response time to the characteristic slow time. Sometime, it is called the inertia
parameter.
George Gabriel Stokes (see above).
3.2.66 Strain Rate of Grain N, B
rffiffiffiffiffiffiffiffiffiffi
Rs D3
N 5 ε_
k
ε_ (s21) speed of relative elongation; Rs (kg m23) particle density; D (m) particle diameter; k (kg s22) linear stiffness of particle.
It characterizes the deformation rate of grains in the two-phase slide flow of
granulated fluid where the deformation can be understood as plastic or viscous.
Usually, real granulated flow is between these two extremes. Two-phase flow.
Info: [B4].
Fluid Mechanics
115
3.2.67 Suratman Number Su
Su 5 Oh 22 5
Rl Lσ
5 Re2 We121
η2l
Rl (kg m23) fluid density; ηl (Pa s) liquid dynamic viscosity; L (m) characteristic
length; σ (N m21) liquid surface tension; Oh () Ohnesorge number (p. 108);
Re () Reynolds number (p. 81); We1 () Weber number (1.) (p. 91).
It expresses the ratio of the product of the inertia force and the surface stress to
the square of the viscous force. Alternatively, it expresses the relation between the
surface stress and the friction of plunged particles. It characterizes the fluid spraying process. Sometimes, the Ohnesorge number Oh (p. 108) is used instead of it.
Hydromechanics. Dynamics of particles.
Info: [A29],[B20].
3.2.68 Thermal Diffusivity of Gas Ag
Ag 5
ag ω
c2g
ag (m2 s21) gas thermal diffusivity; ω (s21) bubble frequency; cg (m s21) gas sound velocity.
This diffusivity expresses the influence of the frequency of the bubbles of
diverse gases in a fluid on the change of thermal diffusivity and the polytropic gas
exponent k. At low frequencies, the time is sufficient for thermal isothermal diffusion (k 5 1). With higher frequencies, the time for the thermal diffusion is insufficient and the behaviour causes an isentropic tendency. With a constant high
frequency, the mean free path in the gas is comparable to the bubble size and
the polytropic exponent can reach a value outside the range kAh1; κi, where κ is
the specific heat ratio (p. 28). Thermodynamics. Multiphase fluids mechanics.
Cavitation.
Info: [A2].
3.2.69 Two-Phase Flow Number NP2
NP2 5
ηdw
σL
η (Pa s) dynamic viscosity; d (m) bubble or particles diameter; w (m s21) flow velocity; σ (N m21) surface tension; L (m) characteristic length.
It is the ratio of the viscous (friction) force to that of surface stress. Capillary
flow. Dispersion materials.
Info: [B20].
116
Dimensionless Physical Quantities in Science and Engineering
3.2.70 Two-Phase Porous Flow Number NPP
wη
NPP 5 pffiffiffiffiffiffiffiffiffi
k1 k2 gΔR
w (m s21) flow velocity; η (Pa s) dynamic viscosity; k1, k2 (m2) longitudinal
and transversal permeability; g (m s22) gravitational acceleration; ΔR (kg m23) density difference of both phases.
It is the viscous-to-gravitational pressures ratio. Capillary flow in porous
materials.
Info: [B20].
3.2.71 Valensi Two-Phase Number Va
Va 5
ωL2
v
ω (s21) angular oscillation frequency of object in fluid with zero dynamic viscosity; L (m) characteristic length; ν (m2 s21) kinematic viscosity.
Physically, it characterizes the ratio of the viscous diffusion time L2/(4ν) to the
oscillating frequency ω 2 1. It expresses the oscillation of objects, drops and bubbles
in a fluid especially.
Info: [A35].
Jacques Valensi (born 1903), French engineer.
3.2.72 Viscous Inverse Number Nvis
pffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffi
3
gD3t
4 Eo
5
Nvis 5
v
Mo
g (m s22) gravitational acceleration; Dt (m) characteristic pipe diameter;
ν (m2 s21) kinematic viscosity; Eo () Eőtvős number (p. 99); Mo () Morton number (p. 101).
This number characterizes the influence of the viscosity and other properties of
a two-phase fluid with the generation of individual Taylor bubbles and formation
of wake in a vertical pipeline or canal. With high gas velocities (Nvis . 1500), the
liquid part can be pushed to the pipeline wall and the gas flows through the core
continuously. With 500 , Nvis , 1500, the end of bubbles is nearly in plane and
the wake part is closed immovably. It tends to lose symmetry with respect to the
tube axis and provides periodic waves, the frequency of which grows with increasing Nvis. With Nvis , 500, the wake part is divided into two closed ring swirls
which act mutually on each other.
Info: [B112].
Fluid Mechanics
117
3.2.73 Weber Spray Number Wespray
Wespray 5
Rq2V L
σ
R (kg m23) fluid density; qV (m s21) volume flux density; L (m) characteristic
length, diameter; σ (N m21) surface tension.
It characterizes the shower sprinkler spraying of a fluid. Among other things, it
applies to cooling in electrical engineering, diverse industrial, apartment or office
rooms, gardening and agriculture.
Info: [B7].
Ernst Heinrich Weber (p. 92).
Wilhelm Eduard Weber.
3.3
Rheology
In rheology, the similarity criteria express the deformation and flow of nonNewtonian fluids and plastic materials with various resistances thereof. It concerns
the mechanical properties of these substances, such as relations between stress and
deformation and deformation rate. The applications of rheology involve a wide
range of fields, from engineering to geophysics to physiology. These include, for
example, diverse polymer and biopolymer systems, terrestrial lava movement and
blood flow. The Bingham number, characterizing the flow of Bingham plastics, is a
fundamental similarity criterion. As for other criteria, these include the Deborah,
elastic, Elvis, Galilei, Hedstrőm and Weissenberg numbers.
3.3.1
API Gravity Degree API
API 5
Rref
2131:5
R
Rref (kg m23) reference density (Rref 5 1.415 3 105 kg m23); R (kg m23) liquid
density at temperature 15.55 C.
The API gravity degree characterizes the density of liquid petroleum products.
Info: [A43].
3.3.2
Bingham Compression Number Bn
Bn 5
σmin H 2
F
ð1Þ;
Bn 5
σmin H n
K
ð2Þ
σmin (Pa) minimal yield strength; H (m) initial height of material sample;
F (N) constant loading force; n () constant of time independent deformation
speed; K (N) dynamic loading force.
118
Dimensionless Physical Quantities in Science and Engineering
It expresses the state of a semi-solid Bingham material if compressed. In
expression (1), it is about a constant load; in expression (2), it is about the condition with a constant deformation rate. Rheology. Bingham fluid.
Info: [B2],[C7].
Eugene Cook Bingham (18781945), American physicist.
3.3.3
Bingham Growth Exponent M
rffiffiffiffiffiffiffiffiffiffi
F
n
M5m
ð1Þ;
H2K
M5
mu
H
ð2Þ
m (s) exponent of stress raising; n () power index of non-linear fluid behaviour; F (N) loading force; H (m) initial sample height; K (m21 kg s21) consistent index corresponding to dynamic viscosity; u (m s21) compression speed.
This exponent characterizes the rheological behaviour dynamics of semi-solid
pulps. The expression (1) holds for a constant load; the expression (2) is valid for a
constant deformation rate. The index n has the value of n , 1 for a thinner fluid
layer and n . 1 for a thicker one. For the Bingham model, n 5 1 is valid.
Info: [B2],[C7].
Eugene Cook Bingham (see above).
3.3.4
Bingham Number Bm
Bm P 5
σk L
ηw
σk (Pa) yield strength; L (m) channel width; η (Pa s) dynamic viscosity of
plastic; w (m s21) plastic flow rate; P () plasticity number (p. 129).
This number expresses the ratio of the yield point stress to the viscous tension.
It characterizes the creep of Bingham plastics and the rheological phenomena in
viscous material flow, and material creep and forming. It is connected with the
Hedstrőm number (1.) Hd1 (p. 124). Sometimes, it is called the plasticity number
P (p. 129). Its inverse value is the plastic deformation K (p. 128).
Info: [A13],[B20],[C7].
Eugene Cook Bingham (see above).
3.3.5
Brinkman Rheological Number Br
Br 5
w2 η
λΔT
ð1Þ;
Br Gn 5
ηw2
λΔTproc
ð2Þ
Fluid Mechanics
119
w (m s21) flux velocity; η (Pa s) dynamic viscosity; λ (W m21 K21) thermal
conductivity; ΔT (K) reference temperature difference; ΔTproc (K) temperature difference between fluid and boundary; Gn () heat build-up number
(p. 124); Pr () Prandtl number (p. 197).
It characterizes the mutual relation between the dissipation and heat conduction
in non-isothermal rheological systems, such as molten polymers. With the relative
temperature gradient represented by the difference in a thermal process
(ΔT 5 ΔTproc), it is called the Heat build-up number Gn (p. 124) or the Eckert
rheological number Ecrh (p. 121), Ecrh 5 GnPr 2 1. Large values of the numbers
Gn or Ecrh mean that, in the flowing polymer, the heat development is determined
by the viscous dissipation, but very little by the conduction.
Info: [A23].
Henri Coenraad Brinkman, German physicist.
3.3.6
Darcy Friction Number fD
fD 5 2
fD 5
dp DH
dx R w2
8τ w
R w2N
ð3Þ;
ð1Þ;
fD 5 4 fF 5
where
DH 5
8 Δp DH
R w2 L
ð2Þ;
4A
U
dp
(Pa m21) pressure gradient; DH (m) hydraulic diameter; R (kg m23) dx
liquid density; w (m s21) flow velocity; Δp (Pa) pressure difference; L (m) characteristic length; A (m2) cross-section area; U (m) wetted perimeter of
cross-section area; τ w (Pa) shear stress on the wall; wN (m s21) free flow
velocity; fF () Fanning friction number (p. 163).
In expression (1) or (2), it describes the pressure loss caused by the friction in
the flow in a constant cross-section pipeline. In expression (3), it is analogous to
the DarcyWeisbach friction coefficient. See Fanning friction number fF (p. 163).
Info: [A2],[A35].
Henry Philibert Gaspard Darcy (p. 98).
3.3.7
Deborah Generalized Number De
De 5 τ
pffiffiffiffiffiffiffiffiffiffiffiffiffi
Ie 2 I v
τ (s) natural time of viscoelastic material; Ie (s22) invariant of rate of strain
tensor; Iv (s22) invariant of vorticity tensor.
In the dimensionless form, it characterizes the viscoelastic process time. See
also the Weissenberg generalized number Ws (p. 132).
Info: [A29],[A35].
120
Dimensionless Physical Quantities in Science and Engineering
3.3.8
Deborah Mixing Number Demix
Demix 5
τ mix
τ pol
τ mix (s) mixing time; τ pol (s) polymerization time.
It expresses the ratio of the characteristic mixing time to that of a polymerizing
process. It is analogous to the Deborah number (1.) De (p. 120). Rheology. Mixing
of melted polymers.
Info: [A29].
3.3.9
Deborah Number (1.) De
De 5
τ rx
Δτ
ð1Þ;
De 5
τm
Δτ
De{1
ðliquid-likeÞ ð3Þ;
De-0
ðviscoelasticÞ ð4Þ;
Dec1
ðsolid-likeÞ ð5Þ;
ð2Þ
τ rx (s) relaxation time; Δτ (s) natural time process interval (experimental or
observation time); τ m (s) molecular time.
This number expresses the ratio of the relaxation time to the process observation
(experimental) time interval. It characterizes the elastic behaviour depending on
the relaxation time length of the polymeric fluid. A long relaxation time is connected with further stress preservation, whereas a short relaxation time means purely
viscous behaviour of the fluid. A Newtonian fluid is the limit case (De 5 0). With a
great value of the number De, the material behaves as elastically solid; with extremely
small De numbers, it behaves as a purely viscous fluid. In expression (2), for example,
it holds for the flow of nematic liquid crystals. Rheology of viscoelastic fluids.
Info: [A29].
3.3.10 Deborah Pressurization Number Dep
Dep 5
τ rx v
h
ð1Þ;
Dep;crit 5 τ rx ωcrit
ð2Þ
τ rx (s) relaxation time; v (m s21) moving plate velocity; h (m) gap width
between two plates; ωcrit (s21) critical angular velocity.
In expression (1), it characterizes the dynamics of viscoelastic fluid pressurization in a gap between two parallel plates. The critical value is given by expression
(2). Rheology.
Info: [A35],[B98].
Fluid Mechanics
121
3.3.11 Eckert Rheological Number Ecrh
See Heat build-up number Gn (p. 124).
Ernst Rudolf Georg Eckert (p. 192).
3.3.12 Elasticity Number (1.) El
El 5
τ rx v
5 De Re 21 Forx
L2
τ rx (s) relaxation time; ν (m2 s21) kinematic viscosity; L (m) characteristic
length; De () Deborah number (1.) (p. 120); Re () Reynolds number
(p. 81); Forx () Fourier viscoelastic relaxation number (p. 121).
This number expresses the elastic-to-inertia forces ratio. Alternatively, it is the
ratio of the relaxation (polymerization) time to the diffusion time. It characterizes
the non-isotropic flow of polymer fluids. Elc1. Rheology.
Info: [A26],[B20].
3.3.13 Elasticity Number (2.) Nelast, El
Nelast 5
cp R
5 Gc Ho Du 21
βK
cp (J kg21 K21) specific heat capacity; R (kg m23) density; β (K21) volume
thermal expansion coefficient; K (Pa) bulk modulus of elasticity; Gc () GayLussac number (p. 14); Ho () Hooke number (p. 138); Du () Dulong number (p. 191).
It expresses the elasticity effect in the viscoelastic fluid flow process. It depends
only on the physical properties of the fluid.
Info: [A26],[B20].
3.3.14 Elasticity Number (3.) Nelast
Nelast 5
cp
βa2
cp (J kg21 K21) specific heat capacity; β (K21) volume thermal expansion
coefficient; a (m s21) sound velocity.
It expresses the elastic-to-inertia forces ratio or, alternatively, the elasticity
effect in fluid flow processes.
Info: [A152],[B20].
3.3.15 Elasto-capillary Number Nec, Γ 0
Nec 5
τ rx σ
5 De Ca 21
ηL
122
Dimensionless Physical Quantities in Science and Engineering
τ rx (s) relaxation time; σ (N m21) surface tension; η (Pa s) dynamic viscosity of viscoelastic fluid; De () Deborah number (1.) (p. 120); Ca () capillary number (1.) (p. 97).
It characterizes the elastic and capillary properties of adhesive and other nonNewtonian fluids in the production of microfibres. Rheology. Microbiology.
Info: [C136].
3.3.16 Ellis Number El
El 5
ηini w
τd
ηini (Pa s) initial limiting value of viscosity as shear nears to zero; w (m s21) flow velocity; τ (Pa) shear stress by condition η 5 ηini/2; d (m) pipe diameter.
It characterizes the non-Newtonian fluid flowing through a pipeline. Rheology.
Info: [A35],[B20].
Samuel Benjamin Ellis (born 1904), American chemist.
3.3.17 Ericksen Number Er
8wLτ M
8 L 2
Er 5
5
De
UR2
U R
w (m s21) velocity of movement of crystal wall; L Δs (m) dislocation; τ M
(s) molecular time; U () dimensionless molecular concentration parameter;
R (m) space position; De () Deborah number (1.) (p. 120).
This number expresses the ratio of the viscous moment of rheological fluid flow
to the moment of elastic flow and, alternatively, it quantifies the hydrodynamic and
elastic couplings. For example, it describes the Poiseuille flow of nematic liquid
crystals with diverse degrees of space orientation. Here, it expresses the intensity
of the narrow extent of nematic potential, this intensity being related to the broad
extent of the elastic potential.
Info: [C51].
3.3.18 Fourier Rheology Number Forh
Forh 5
στ
ηL
σ (N m21) surface tension; τ (s) natural process time; η (Pa s) liquid
dynamic viscosity; L (m) characteristic length.
It expresses the dimensionless rheological process time expressed by means of
the elastic and capillary properties of fluid. Rheology. Microrheology.
Info: [C136].
Jean Baptiste Joseph Fourier (p. 175).
Fluid Mechanics
123
3.3.19 Fourier Viscoelastic Relaxation Number Forx
Forx Zhrx 5
vτ rx
L2
ν (m2 s21) kinematic viscosity; τ rx (s) relaxation time; L (m) characteristic
length; Zhrx () Zhukovsky viscoelastic number (p. 133).
It expresses the elastic-to-inertia forces ratio and, alternatively, the relaxation-todiffusion time ratio. In the dimensionless form, it characterizes the stroke flow of viscoelastic fluids. In addition, it characterizes the stroke wave propagation time and
viscoelastic vibration of mechanical stress. Sometimes, it is called the Zhukovsky viscoelastic number Zhrx (p. 133) or the viscoelastic number (1.) Nη (p. 131).
Info: [A23].
Jean Baptiste Joseph Fourier (see above).
3.3.20 Galilei Number Ga
Ga1 5
gL3
5 Re2 Fr 21
v2
Ga2 5
Ga1
βΔT
ð2Þ;
ð1Þ;
Ga3 5
R
Ar
ΔR
ð3Þ
g (m s22) gravitational acceleration; L (m) characteristic length; ν (m2 s21) kinematic viscosity; β (K21) volume thermal expansion coefficient; ΔT (K) temperature difference; R, ΔR (kg m23) liquid density and their difference in various
places of non-isothermic flow; Re () Reynolds number (p. 81); Fr () Froude
number (1.) (p. 62); Ar () Archimedes hydrodynamic number (p. 53).
This number characterizes the molecular friction and gravity forces in flowing
fluid. In expression (2), it expresses the mutual action of the molecular friction
force and the buoyancy force in a non-isothermal process. In expression (3), it is
about the mutual action of the molecular friction force and the Archimedes force
determined by different densities of the fluid in a system. Rheology.
Info: [B17],[B20].
Galileo Galilei (15.2.15648.1.1642), Italian mathematician,
astronomer and physicist, originally doctor.
In addition to extensive astronomical observations with a
telescope he had improved, he was also engaged in the
dynamics of bodies. For example, he discovered inertia, the
law of superposition of velocities, the free fall law and
the pendulum swing time dependence on its length. With his
telescope, he was the first to discover the craters on the
Moon, solar spots, the four moons of Jupiter and the Milky
Way as an aggregation of remote stars.
124
Dimensionless Physical Quantities in Science and Engineering
3.3.21 Graetz Rheological Number Gzr
Gzr 5
Rcp wL2n
λL
R (kg m23) density; cp (J kg21 K21) specific heat capacity; w (m s21) reference velocity; L, Ln (m) length of heat transfer in front and vertical flow;
λ (W m21 K21) thermal conductivity.
It expresses the ratio of heat transferred by convection to that transferred by conduction. It characterizes the mutual ratio of the heat transfer by convection to the
transfer by conduction in rheological systems such as molten polymers.
Info: [A29],[B17].
Leo Graetz (p. 193).
3.3.22 Heat Build-up Number Gn
See Brinkman rheological number Br (p. 118).
3.3.23 Hedstrőm Number (1.) Hd1
Hd1 5
σ k L2
5 Re Bm
Rv2
σk (Pa) yield limit; L (m) characteristic length; R (kg m23) fluid density;
ν (m2 s21) kinematic viscosity of plastic; Re () Reynolds number (p. 81);
Bm () Bingham number (p. 118).
It characterizes non-Newtonian fluid flow (the creep of Bingham plastics). See
the Bingham number Bm (p. 118) and the plastic deformation K (p. 128).
Info: [A26],[A29],[B17].
Bengt Olaf-Arvid Hedstrőm (born 1926), Swedish chemist.
3.3.24 Hedstrőm Number (2.) Hd2
Hd2 5
σE L
ηw
σE (Pa) elastic limit stress; L (m) characteristic length; η (Pa s) dynamic
viscosity in the plastic state; w (m s21) mean velocity.
It relates to Bingham plastics flow and corresponds to the Bingham number Bm
(p. 118).
Info: [A26].
Bengt Olaf-Arvid Hedstrőm (see above).
Fluid Mechanics
125
3.3.25 Ilyushin Number Il
Il 5
4 Lτ
4 τ
5 Re
3 ηw
3 Rw2
L (m) characteristic length, channel diameter; τ (Pa) maximum dynamic slip
stress; η (Pa s) dynamic viscosity; w (m s21) mean flow velocity; R (kg m23) density; Re () Reynolds number (p. 81).
It characterizes the viscoplastic fluid flow under the action of a pressure gradient
in a ring canal consisting of two immovable cylinders. With Il-0, a Newtonian
fluid flows.
Info: [A29].
Alexey Antonovich Ilyushin (born 1911), Russian engineer.
3.3.26 MackleySherman Number MS
MS 5
β
5 tgϑ
w
β (m s21) mass transfer coefficient of particle; w (m s21) velocity of incident
particle; ϑ () particle impact angle.
It expresses the ratio of the particle mass transfer velocity to its velocity of
impact on a diaphragm surface which equals the tangent of the particle impact
angle. The Stanton mass number Stm (p. 265) is another analogue of the number
MS for a cluster of particles.
Malcolm Robert Mackley (born 1947), English chemical
engineer.
He is engaged in the flow of rheological fluids and its
industrial applications in the polymerization process in the
food industry. Especially, his research concerns the oscillation
mixing of liquids and the cool processing thereof in production
lines. It includes the drawing-through of molten polyethylene
and other industrial and nutritional materials. In his work, he
uses both direct experiment and numerical modelling.
3.3.27 Magnetic and Thermal Energy Ratio λ
λ5
Emg
Emg
μ μm2
5 0 3 ;
5
Et
kT
16πd kT
where
m5
4 3
πd H
3
Emg (J) magnetic energy; Et (J) thermal energy; k (J K21) Boltzmann constant; T (K) absolute temperature; μ0 (H m21) vacuum magnetic permeability;
126
Dimensionless Physical Quantities in Science and Engineering
μ (H m21) dissolvent magnetic permeability; d (m) particle diameter; m
(A m2) magnetic moment; H (A m21) magnetization.
It describes the microstructure development in magnetorheological suspensions,
which represent artificial suspensions of magnetized particles immersed in nonmagnetic fluid. Together with the Mason number Mas (p. 126), it represents the
basic criterion in magnetorheology. Rheology.
Info: [B74].
3.3.28 Mach Viscoelastic Number Ma
w
Ma 5 rffiffiffiffiffiffiffiffiffi
η
Rτ rx
w (m s21) flow velocity; η (Pa s) dynamic viscosity; R (kg m23) fluid density; τ rx (s) time of stress relaxation.
It expresses the viscoelastic fluid flow rate at the inlet of a pipeline or canal. It
is important in the extruding or injecting of plastics, particularly with geometrically
simple input parts when complex singularities arise in the fluid flow. Fluid mechanics. Rheology.
Info: [B34].
Ernst Mach (p. 73).
3.3.29 Mason Number Mas
Mas 5
122 ηω
μμ0 H 2
η (Pa s) solvent dynamic viscosity; ω (s21) rotating frequency; μ (H m21) solvent magnetic permeability; μ0 (H m21) vacuum magnetic permeability;
H (A m21) particle magnetization.
This number expresses the viscous-to-magnetic forces ratio. It characterizes the
ability of particles suspended in a non-magnetic fluid to be magnetized (the diameter 110 μm approximately) with respect to a rotating magnetic field under the
influence of modulation according to the Mason number. Magnetorheological processes of sedimentation.
Info: [B74].
3.3.30 Nahme Number Na
w2 η
Na 5
;
λΔTrh
where
@ ln Ah1 21
ΔTrh 5 2
@T
Tini
Fluid Mechanics
127
w (m s21) characteristic velocity; η (Pa s) dynamic viscosity; λ (W m21 K21) thermal conductivity; ΔTrh (K) temperature difference in the rheological process;
T (K) temperature; Ah1 () Arrhenius number (1.) energetic (p. 34).
It characterizes the mutual relation between the heat transfer by dissipation and
conduction in non-isothermal rheological systems such as molten polymers. It is a
special case of the Heat build-up number Gn (p. 124) pro ΔT 5 ΔTrh.
Info: [A23].
3.3.31 Nahme Nonisothermal Number Na
Na 5
ηβw2
;
λT
where
_
w 5 Lγ;
β5
T dη
η dT
η (Pa s) dynamic viscosity in the limit of zero shear rate; β () thermal sensitivity of the fluid viscosity; w (m s21) characteristic velocity; λ (W m21 K21) thermal conductivity; T (K) absolute temperature; L (m) characteristic length
(L 5 h); γ_ (s21) shear rate.
It characterizes the non-isothermal instability in the torsion flow of polymeric
fluids and expresses the influence of viscous fluid heating. NaAh0; 1i.
Info: [B98].
3.3.32 NahmeGriffith Number Na
Na 5
ΔTdis
5 Br Ps
ΔTvis
ΔTdis (K) temperature difference developed by viscous dissipation of fluid;
ΔTvis (K) temperature difference developed by temperature-dependent viscosity;
Br () Brinkman number (p. 174); Ps () Pearson number (p. 127).
This number expresses the viscous heating influence in the action of both viscous dissipation and fluid viscosity depending on temperature. NahmeGriffith
number is usually in the range 0 , Na , 200, for 0.1 , Na , 0.5 viscosity is unaffected by temperature. Rheology.
Info: [A9].
3.3.33 Pearson Number Ps
Ps 5
ΔTop
5 αðTS 2 TN Þ
ΔTvis
ΔTop (K) temperature difference imposed by operating conditions; ΔTvis (K) temperature difference required to change the viscosity; TS (K) surface wall
128
Dimensionless Physical Quantities in Science and Engineering
temperature; TN (K) temperature of thermal undisturbed fluid area; α (K21) thermal expansion coefficient; η0, η (Pa s) basic and real dynamic viscosities.
This number expresses the ratio of the operating temperature range to the temperature gradient necessary to change the viscosity. It characterizes the viscosity
temperature dependence in cases when the temperature gradient must be considered
in boundary conditions. For the viscosity exponential dependence on the temperature, it is
η 5 η0 expð2 αΔTop Þ
Rheology.
Info: [A23].
3.3.34 Péclet Rheological Number Perh
Perh 5
L2 γ_
a
L (m) characteristic length, gap width; γ_ (s21) sound velocity; a (m2 s21) thermal diffusivity.
It characterizes the diffusion heat transfer in non-isothermal polymer fluid flow.
PerhBh0; 1i.
Info: [B25].
Jean Claude Eugène Péclet (p. 180).
3.3.35 Plastic Deformation K
See Bingham number Bm (p. 118).
3.3.36 Plastic Viscosity Ratio Pη , k
Pη 5
ηp
ηs 1 ηp
ηs, ηp (Pa s) dynamic viscosities of dissolvent and plastic.
It expresses the total viscosity as the ratio of the Newtonian fluid (solvent) viscosity to the polymeric viscosity. It is used, for example, in drawing or injecting
plastics. Fluid mechanics. Rheology.
Info: [B34].
Fluid Mechanics
129
3.3.37 Plasticity Number Npl
Npl 5
τEL
ηw
τ E (Pa) yield limit; L (m) characteristic length; η (Pa s) dynamic viscosity;
w (m s21) velocity.
It is used in rheology and the creep of plastics. It is an equivalent of the
Bingham number Bm (p. 118) and analogous to the Hedstrőm number (2.) Bm
(p. 124).
Info: [A35].
3.3.38 Prandtl Generalized Number Prgener
Prgener 5
wL 2 2 n n 21 2 1
L kR Þ 1 1 n
ðw
a
w (m s21) flow velocity; L (m) characteristic length; a (m2 s21) thermal diffusivity; n () exponent of non-Newtonian fluid; κ (kg m21 s21) consistency
coefficient of pseudoplastic and dilating fluids; R (kg m23) density.
21
It expresses the relation between the velocity and temperature processes in
flowing non-Newtonian fluids.
Info: [A33].
Ludwig Prandtl (p. 197).
3.3.39 Processability Parameter Np, P
Np 5
βη
5 Sh CaðRe ScÞ 21 5 Stm Ca
σ
β (m s21) mass transfer coefficient; η (Pa s) dynamic viscosity; σ (N m21) surface tension; Sh () Sherwood number (p. 264); Ca () capillary number
(1.) (p. 97); Re () Reynolds number (p. 81); Sc () Schmidt number
(p. 263); Stm () Stanton mass number (p. 265).
In microfibre rheometry, it is applied to predict the fibre shape and to support
the process using adhesive and other non-Newtonian fluids. This parameter serves
to determine the critical time within which the fibre is broken. Above all, this time
depends on the rheological properties of the fluid and the mass transfer as
expressed by this parameter. Rheology. Microrheology.
Info: [C136].
130
Dimensionless Physical Quantities in Science and Engineering
3.3.40 Reynolds Generalized Number for Non-Newtonian Fluids Renon
Renon 5
8Rw2
τ char
ð1Þ;
Renon 5
Rw2
τ char
ð2Þ
R (kg m23) fluid density; w (m s21) fluid velocity; τ char (Pa) characteristic
shear stress depending on Boltzmann constant and absolute temperature.
It is used in form (1) for the flow of non-Newtonian fluids in tubes of circular
cross section. In form (2) to liquid crystals flow, it expresses the inertia-tocharacteristic tensions ratio which originates in liquid crystal flow.
Info: [A29].
Osborne Reynolds (p. 82).
3.3.41 Reynolds Rheological Number (1.) Rerh
Rerh 5
L2 γ_
v0
L (m) characteristic length, gap width; γ_ (s21) shear velocity; ν 0 (m2 s21) kinematic viscosity at zero shear velocity.
It characterizes the convective transfer in the non-isothermic flow of polymer
fluids. Rerh{1. Rheology.
Info: [B98].
Osborne Reynolds (see above).
3.3.42 Stokes Number Stk
Stk1 5
vw
5 Ca Bo 21
gL2
Stk3 5
ωL
v
ð3Þ;
ð1Þ;
Stk4 5
LΔp
ηw
Stk2 5
vτ
5 Sh 21 Re 21
L2
ð2Þ;
ð4Þ
ν (m2 s21) kinematic viscosity; w (m s21) flow velocity; g (m s22) gravitational acceleration; L (m) particle characteristic dimension; τ (s) particle
vibration time in fluid flow; ω (s21) angular frequency of particle; Δp (Pa) pressure difference; η (Pa s) dynamic viscosity; Ca () capillary number (1.)
(p. 97); Bo () Bond number (p. 95); Sh () Strouhal number (p. 87);
Re () Reynolds number (p. 81).
In expression (1), it expresses the viscosity-to-gravity forces ratio. In cases (2)(4),
it expresses the inertia-to-friction forces ratio. Hydromechanics. Dynamics of particles.
Rheology.
Info: [A7],[A29],[A35],[B20].
Fluid Mechanics
131
George Gabriel Stokes (13.8.18191.2.1903), Irish mathematician and physicist.
He was engaged in the mechanics of continua, especially
in hydrodynamics, waves in elastic bodies, acoustics and diffraction. The fundamental continuum mechanics equation is
called the NavierStokes equation after him. In addition to
continuum mechanics, he measured gravitational field
changes on the surface of the earth. He was engaged in vector analysis, and in chemistry and botany as well.
3.3.43 Thermoelastic Rheological Number Nelast
rffiffiffiffiffiffiffiffi
η0 β
Nelast 5 τ rx L
λT
τ rx (s) relaxation time of viscoelastic stresses; L (m) characteristic length; η0
(Pa s) dynamic viscosity at zero shear rate; β () temperature sensibility of fluid
viscosity; λ (W m21 K21) thermal conductivity; T (K) absolute temperature.
It characterizes the non-isentropic flow of polymer fluids. It expresses the ratio
of the heat propagation time to the polymerization time. NelastAh0; 1i.
Info: [B98].
3.3.44 Truncation Number, Shearing Failure Parameter NT
NT 5
ηω
p
η (Pa s) dynamic viscosity; ω (s21) angular speed of shear deformation; p (Pa) pressure.
It is the ratio of the shear stress to the normal pressure. It relates to viscous nonNewtonian fluid flow. Viscoelasticity. See the Hersey number He (p. 165).
Info: [A35].
3.3.45 Viscoelastic Number (1.) Nη, E
Nη 5
τ rx η
RL2
ð1Þ;
Nη 5 M 2 Re 22
ð2Þ
τ rx (s) stress relaxation time; η (Pa s) dynamic viscosity; R (kg m23) density; L (m) characteristic length; M () Mach number (p. 73); Re () Reynolds number (p. 81).
In expression (1) or (2), it expresses the elastic properties of viscoelastic surroundings and their flow. In the case of viscometry, L denotes the capillary tube
132
Dimensionless Physical Quantities in Science and Engineering
diameter. It is also called the Fourier viscoelastic relaxation number Forx (p. 123)
or the Zhukovsky viscoelastic number Zhrx (p. 133).
Info: [A23].
3.3.46 Viscoelastic Number (2.) Nη
Nη 5
G
ηω
G (Pa) shear modulus; η (Pa s) dynamic viscosity; ω (s21) angular speed.
It is the elastic-to-viscous forces ratio. It describes the flow (outflow) of viscous
fluids. Viscometry.
Info: [A24],[B20],[B59].
3.3.47 Viscoelastic Parameter Pη
Pη 5
ηS
η w
5 P
ηP
Lτ char
ηS (Pa s) dynamic viscosity of dissolvent matter; ηP (Pa s) characteristic
dynamic viscosity of the polymer; w (m s21) motion velocity; L (m) wall displacement; τ char (Pa) characteristic stress.
It expresses the ratio of the diluting material viscosity to the characteristic viscosity of the polymer or, alternatively, the ratio of the shear rate to the characteristic tension. It appears in the flow of polymer liquid crystals.22
3.3.48 Weissenberg Generalized Number Wi
Wi 5 τ
pffiffiffi
γ_
τ (s) time scale for viscoelastic stress relaxation; γ_ (s22) shear rate.
It expresses the characteristic material time (relaxation time) and the shear
velocity. It characterizes the velocity and time relations in rheological processes in
viscoelastic shear flow. See also the Deborah generalized number De (p. 119).
Info: [A23],[A35].
Karl Weissenberg (born 1893), German physicist.
3.3.49 Weissenberg Number (1.) Ws
Ws 5
Rw2
5 Re El ð1Þ;
τ
Ws 5
wτ rx
5 Re Forx
L
ð2Þ;
Fluid Mechanics
Ws 5
ðt1 2 t2 Þw
L
Ws 5 Ma2 Re 21
133
ð3Þ;
where
τ 1 t1 τ_ 5 η0 ðε 1 t2 ε_ Þ;
ð4Þ
R (kg m23) fluid density; w (m s21) flow velocity; τ (Pa) shear stress; τ_
(Pa s21) shear stress rate; τ rx (s) relaxation time; L (m) characteristic length,
diameter; t1, t2 (s) time constants; η0 (Pa s) dynamic viscosity; ε () relative elongation; ε_ (s21) relative elongation rate; Re () Reynolds number
(p. 81); El () Ellis number (p. 122); Forx () Fourier viscoelastic relaxation
number (p. 123); Ma () Mach viscoelastic number (p. 126); Re () Reynolds number (p. 81).
It characterizes the influence of the viscoelastic fluid properties on proceeding
processes. In expression (1), it expresses the ratio of the dynamic viscoelastic force
to the viscous force. In expression (2), it expresses the time change of the viscoelastic fluid flow close to a bypassed wall. The expression (3) is analogous to (2),
and expresses more exactly the ratio of viscoelastic force considering the relaxation modulus to the elastic force only. In expression (4), it expresses that the
shear wave propagation rate is analogous to the sound propagation velocity in compressible fluid flow. The Ws is a special case of the Strouhal number Sh. Fluid
mechanics. Rheology.
Info: [A29],[A33],[B20],[B34].
Karl Weissenberg (see above).
3.3.50 Zhukovsky Viscoelastic Number Zhrx
See the Fourier viscoelastic relaxation number Forx (p. 123), or the viscoelastic
number (1.) Nη (p. 131).
Nikolay Yegorovich Zhukovsky (p. 33).
4 Solid Mechanics
I hold a single experiment higher than thousands of opinions originated by
imagination only.
Mikhail Vasilyevich Lomonosov (17111765)
4.1
Linear and Non-Linear Solid Mechanics
In solid mechanics, the dimensionless quantities involve the elastic, viscoelastic
and plastic fields. For the elastic, the linear tension-dependent deformation is valid
and for the viscoelasticplastic sphere, the non-linear behaviour is valid. The relative prolongation, Hooke and Johnson numbers are among the most widespread
similarity criteria.
4.1.1
Angular Displacement Number ND
ND 5
EJk α
qL3
E (Pa) modulus of elasticity; Jk (m4) shear modulus; α (rad m21) angular
displacement; q (N m21) length loading; L (m) characteristic length.
It expresses the angular displacement of an elastically laid beam.
Info: [A24].
4.1.2
Beam Loading N
N5
q
EL
ð1Þ;
N5
qA
E
ð2Þ
q (N m21) length loading; qA (N m22) surface loading; E (Pa) modulus of
elasticity; L (m) characteristic length.
It expresses longitudinal (1) and planar (2) beam loading. Elasticity, statics.
Info: [A24].
Dimensionless Physical Quantities in Science and Engineering. DOI: 10.1016/B978-0-12-416013-2.00004-X
© 2012 Elsevier Inc. All rights reserved.
136
Dimensionless Physical Quantities in Science and Engineering
4.1.3
Crater Scaling Relationship Ncra
Ncra 5
β
hmax
gr
5 αFr 2β 5 α 2
r
wim
hmax (m) maximum crater depth; r (m) radius of the projectile; α, β () empirically or numerically determined constants; g (m s22) gravitational acceleration; wim (m s21) impact velocity; Fr () Froude number (1.).
It expresses the dynamics of projectile penetration into a purely hydrodynamic
target. The data of the constants (α, β), determined by experiment and numerically,
are presented in the literature.
Info: [B122].
4.1.4
Criterion of Stringed Musical Instruments Nsmi
Nsmi 5
RL2 f 2
σ
R (kg m23) string density; L (m) characteristic length of string; f (s21) string
frequency; σ (Pa) mechanical stress of string.
It expresses the dynamic similarity criterion for string instruments provided the
string tension and density are equal for all instruments (violin, viola, violoncello
and double bass). When building these instruments, it has the same value, for
example, Nsmi 5 0.2 usually. The criterion has very deep musical and physical
meaning. Its value is deduced from the instrument prototype and involves all previous experiences. Diverse relative length of the string instruments is 0.9 for the violin, 1 for the viola, 2 for the violoncello and 2.8 for the double bass. With this,
corresponding frequencies are obtained for certain musical tones.
Info: [A24].
4.1.5
Decibel Scales SPL, Lv, La, LF, LP, LI , LE , Lw
p
; where p0 5 20 μPa ð1Þ;
SPL 5 20 log
p0
v
; where v0 5 1 nm s21 ð2Þ;
Lv 5 20 log
v0
a
; where a0 5 1 μm s22 ð3Þ;
La 5 20 log
a0
F
; where F0 5 1 μN ð4Þ;
LF 5 20 log
F0
Solid Mechanics
137
P
; where P0 5 1 pW ð5Þ;
LP 5 20 log
P0
I
LI 5 20 log
; where I0 5 1 pW m22 ð6Þ;
I0
E
; where E0 5 1 pJ ð7Þ;
LE 5 20 log
E0
w
Lw 5 20 log
; where w0 5 1 pJ m23 ð8Þ;
w0
P0 (Pa) reference pressure value; v0 (m s21) reference speed value; a0 (m s22) reference acceleration value; F0 (N) reference force value; P0 (W) reference
power value; I0 (W m22) reference intensity value; E0 (J) reference energy
value; w0 (J m23) reference energy density value.
In the area of dynamic quantities measurement, the decibel scales express the logarithmic dimensionless useful-to-threshold signals ratio denoted hereafter by a zero
index. Usually the quantities are sound, noise, velocity, acceleration, force, power,
intensity, energy and density thereof. Then, the sound (SPL), velocity (Lv), acceleration (La), force (LF), power (LW), intensity (LI), energy (LE) and energy density (Lw)
levels are the corresponding logarithmic scales. The logarithmic scale introduction is
based on the fact that the human ear perceives logarithmically due to which a logarithmic unit can be applied advantageously (the bel or the decibel as the tenth part of
it). Most frequently, the decibel scale is used in acoustics and in measuring instrumental techniques, electronics and other ranges to express the signal-to-noise ratio.
Info: [C34].
4.1.6
Explosion Number Ex
21
E 5 22
Ex 5 rp
τ 5
R
rp (m) blast wave radius; E (J) explosive energy; R (kg m23) density; τ (s) time.
It characterizes the pressure stroke waves in detonations. Special techniques.
Info: [A24], [B20].
4.1.7
Force Numbers NF
Fτ 2
ð1Þ;
mL
F
NF 5
ð2Þ;
qL
NF 5
138
Dimensionless Physical Quantities in Science and Engineering
NF 5
FL2
GJ
ð3Þ;
F (N) force; τ (s) time; m (kg) density; L (m) characteristic length;
q (N m21) length loading; G (Pa) shear modulus; J (m4) square moment of
cross section.
In equation (1), they express the dynamic force in damped mass movement. In
equation (2), they express the static force with longitudinal beam load and in
equation (3), they express the static force with static torsion stress.
Info: [A24].
4.1.8
Fourier Thermoelastic Number Fote
wte τ
;
Fote 5
L
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi
λ 1 2μ
where wte 5
R
wte (m s21) velocity propagation of the thermoelastic deformation; τ (s) time;
L (m) characteristic length; λ, μ (Pa) Lamé functions; R (kg m23) density.
It expresses the ratio of the thermoelastic wave penetration depth to the characteristic system length. In the dimensionless form, it characterizes the thermoelastic
process propagation time in a system.
Info: [A23].
Jean Baptiste Joseph Fourier (p. 175).
4.1.9
Fourier Wave Number Fow,mech
c2 τ 2
Fow;mech 5 2 ;
L
sffiffiffiffi
E
where c 5
R
c (m s21) velocity propagation of the mechanical wave; τ (s) time; L (m) characteristic length; E (Pa) modulus of elasticity; R (kg m23) density of
material.
This number expresses the dimensionless time of the mechanical wave propagation in the material under strokes, for example. Solid phase mechanics. Stroke
phenomena.
Info: [A24].
Jean Baptiste Joseph Fourier (see above).
4.1.10 Hooke Number Ho
Ho Cau 5
Rσ
σF
ð1Þ;
Solid Mechanics
Ho 5
Rw2
E
139
ð2Þ
R (kg m23) density; σ (Pa) stress; σF (Pa) stress from the loading force;
w (m s21) velocity; E (Pa) modulus of elasticity; Cau () Cauchy number
(aeroelasticity parameter) (p. 155).
It is the ratio of the yield point or the breaking strength σ to the tension due to a
loading force (1), which can be the dynamic pressure of the fluid (2). Elasticity
hydrodynamics.
Info: [A23],[A29],[B17].
Robert Hooke (18.7.16353.3.1703), English scientist.
The Hooke’s law is his important discovery, according to
which the body deformation is proportional to the acting
force. This is valid for small deformations to the extent of
the elasticity limit. In the year 1678, he tried to prove that
the Earth’s orbit around the Sun is an ellipse. In the same
year, he discovered that the gravity force drops with the second root of the distance. He informed Newton about this in
writing, but subsequently an authorship conflict arose
between them which led to extreme hostility.
4.1.11 Impact Factor Nimp, I
Nimp 5
Mw2ini
σc d 3
M (kg) mass of a projectile; wini (m s21) impact velocity of a projectile;
σc (Pa) unconfined compressive strength of a concrete target; d (m) diameter
of a projectile.
This factor expresses the ratio of the projectile energy to the inverse deforming
pressure energy in a target. Impact mechanics. Ballistics.
Info: [B66].
4.1.12 Johnson’s Damage Number Dn
Dn 5
Rwini
σk
R (kg m23) material density; wini (m s21) initial impulse velocity; σk (Pa) yield stress of the material.
This number is the measure of the inertia loading force related to the material
resistance against loading. It represents the basic parameter in impact mechanics to
solve the material plastic response especially with dynamic loading. It is the
140
Dimensionless Physical Quantities in Science and Engineering
deformation state measure in the zone where plastic deformation arises abruptly.
The Cauchy number (aeroelasticity parameter) Cau (p. 155) is a similar criterion
for the dynamic elastic response. Non-linear mechanics.
Info: [B127],[B47].
4.1.13 Johnson’s Damage Number Jo
Jo 5
Nimp
w2ini Rt
5
σc
NA
wini (m s21) impact velocity of a projectile; Rt (kg m23) density of a concrete
target; σc (Pa) unconfined compressive strength of a concrete target; Nimp () impact factor (p. 139); NA () slenderness factor (p. 145).
It expresses the degree of the relative damage in the projectile impact on a target. Usually, it is applied to classify the projectile impact. Stroke mechanics.
Ballistics.
Info: [B66].
4.1.14 Loss Coefficient η
Δu
ð1Þ;
2πu
ð σmax
σ2
dε max
where u 5
2E
0
I
Δu 5
σ dε ð3Þ
η5
ð2Þ;
Δu (J m23) change of specific volume energy; u (J m23) specific volume
energy; σmax (Pa) maximal stress; σ (Pa) stress by loading; E (Pa) modulus
of elasticity; ε () relative elongation (p. 144).
It expresses the material dynamic memory function and, alternatively, the degree
to which the material dissipates the energy of vibration. With the elastic load up to
the highest stress σmax, expression (2) is valid for the elastic energy related to the
volume unit. Expression (3) concerns the loading and following unloading in which
the energy dispersion corresponds to the area of the hysteresis stress deformation
loop. Usually, the loss coefficient is time and cycling frequency dependent.
Info: [C65].
4.1.15 Mach Thermoelastic Number Mte
wt
Mte 5
;
wte
a
where wt 5 ;
L
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi
λ 1 2μ
wte 5
R
Solid Mechanics
141
wt, wte (m s21) velocity propagation of thermal and thermoelastic wave in the
system; a (m2.s21) thermal diffusivity; L (m) characteristic length; λ, μ (Pa) Lamé functions; R (kg m23) density.
This number expresses the ratio of the heat propagation rate in elastic surroundings to the propagation rate of mechanical waves (sound rate). It characterizes the
mutual relation between the heat propagation rate and the thermal stress in a thermal system.
Info: [A23].
Ernst Mach (p. 73).
4.1.16 Mass of the Projectile Nm
Nm 5
M
Rd 3
M (kg) mass of a projectile; R (kg m23) material density of a target; d (m) projectile diameter.
In the dimensionless shape, it expresses the ratio of the projectile mass to the
target density. Solid mechanics. Ballistics.
Info: [B22].
4.1.17 Noise Number Noi
1
1
2 S22
Noi 5 Ni Tref
Ni (m) mutually uncorrelated stochastic processes with zero mean value;
Tref (s) dimensionless reference time; S (m2 s) power density spectrum of the
stochastic process.
It characterizes a random process. It appears in a dimensionless mathematical
model to solve the stochastic equations set by means of the continuous hybrid
Monte Carlo method.
Info: [A24],[B36].
4.1.18 Nose Factor Pn
1
1
2
; where Ψ 5
3Ψ 24Ψ2
1
h
Pn 5
; where Ψ 5
2
d
1 1 4Ψ
1
r
Pn 5 1 2 2 ; where Ψ 5
d
8Ψ
Pn 5
R
d
ð1Þ;
ðogive noseÞ
ð2Þ;
ðconical noseÞ
ð3Þ;
ðblunt=spherical noseÞ
142
Dimensionless Physical Quantities in Science and Engineering
Ψ () calibrated radius head (CRH); R (m) calibrated radius or diameter of
the ogive; d (m) projectile diameter; h (m) projectile nose length; r (m) projectile nose radius.
This factor expresses the influence of the projectile nose shape on penetration.
The expression (1) is valid for the calibrated head radius or a broken Gothic arch,
the expression (2) holds for a conical peak, whereas the expression (3) holds for a
spherical top. The projectile nose factor becomes values in the range 0 , Pn # 1.
For a flat top of a projectile, Pn 5 1 is valid. For a semispherical projectile point,
Pn 5 0.5 is valid. Stroke mechanics. Ballistics.
Info: [B66].
4.1.19 Oscillation Amplitude NA
NA 5
EJy
qL4
E (Pa) modulus of elasticity; J (m4) square moment of cross section; y (m) displacement of oscillation; q (N m21) length loading; L (m) characteristic
length.
This number expresses the amplitude of an oscillating beam. Elasticity.
Dynamics.
Info: [A24].
4.1.20 Oscillation Frequency Nf
Nf 5
RL4 f 2
EJ
Nf 5
ω2 L
g
ð2Þ;
Nf 5
ω2 I
GL3
ð3Þ;
ð1Þ;
R (kg m23) density; L (m) characteristic length; f (s21) frequency; E (Pa) modulus of elasticity; J (m4) square moment of cross section; ω (s21) angular
frequency; g (m s22) gravitational acceleration; I (kg m2) moment of inertia;
G (Pa) shear modulus.
In form (1), it expresses the oscillating beam frequency. In form (2), it expresses
the mathematical pendulum oscillation frequency and in form (3), it means the
eigenfrequency of shaft torsion oscillations. Dynamics. Elasticity. Physics.
Info: [A24].
Solid Mechanics
143
4.1.21 Péclet Thermoelastic Number Pete
See Vernotte thermoelastic number Vete (p. 283).
Jean Claude Eugène Péclet (p. 180).
4.1.22 Penetration Depth Hp
Hp 5
h
5 f ðNimp ; NA ; Pn Þ
d
h (m) penetration depth; d (m) diameter of a projectile; Nimp () impact
factor (p. 139); Na () slenderness factor (p. 145); Pn () nose factor.
It expresses the projectile penetration depth of a non-deformable projectile striking a target. It is the function of the impact factor Nimp (p. 139), of the slenderness
factor NA (p. 149) and of the nose factor Pn (p. 141). Stroke mechanics. Ballistics.
Info: [B66].
4.1.23 Poisson’s Ratio ν
ν 52
ε1
E
21
5
ε2
2G
ε1 () relative deformation in the cross section (contraction) by uniaxial stress;
ε2 () relative deformation in axial direction (elongation) by uniaxial stress;
E (Pa) modulus of elasticity; G (Pa) shear modulus.
It characterizes the relative deformation in single axis stress. For perfect noncompressible materials, it would be ν 5 0.5 in theory; for most technical materials
it is νAh0; 0:5i; for cork it is ν-0; for most steels it is ν 0.3; for rubber it is
almost 0.5. For the Poisson constant, it is μ 5 ν21. Mechanics of solid bodies.
Elasticity.
Info: [A23],[A29].
Siméon Denis Poisson (21.6.178125.4.1840), French mathematician and physicist.
He was one of the founders of mathematical physics and
one of Laplace’s students. Especially, he was engaged in
mathematical analysis, variation calculus, probability theory
and hydromechanics. By extending and generalizing the
Laplace equation with source members, he formed the
Poisson equation describing steady-state fields with internal
sources and sinks.
144
Dimensionless Physical Quantities in Science and Engineering
4.1.24 Ratio of the Plug Mass to the Projectile Mass NΔm
NΔm 5
πd2 RH
4M
d (m) projectile diameter; R (kg m23) material density of a plate; H (m) penetration depth of a projectile; M (kg) projectile mass.
It expresses the ratio of the material mass loss of a thin plate, after a projectile
impact, to the projectile mass. Solid mechanics. Ballistics.
Info: [B22].
4.1.25 Relative Elongation ε
ε5
σ
ΔL
F
5
5
E
L
EA
σ (Pa) stress; E (Pa) modulus of elasticity; ΔL (m) absolute elongation;
L (m) characteristic length; F (N) force; A (m2) cross-sectional area.
It expresses the relative elongation of a body.
Info: [A24].
4.1.26 Relative Stiffness Ns
Ns 5
ks y
F
ks (kg s22) stiffness of elastic element, spring constant; y (m) compression of
elastic component; F (N) force.
In the dynamics of mechanical sets, it expresses the ratio of elastic element compression to the acting force.
4.1.27 Relative Volume Change PV
PV 5
ΔV
V0
ΔV (m3) volume change; V0 (m3) initial volume.
It expresses the ratio of the volume change to the original volume before deformation. Elasticity, mechanics.
Info: [A24].
Solid Mechanics
145
4.1.28 Sachs Number Sa
rffiffiffiffiffi
3 Pa
Sa 5 r
E
r (m) blast wave radius (distance from explosion place to reference one); pa (Pa) atmospheric pressure; E (J) explosive energy.
It expresses the superficial explosion propagation. See the analogous explosion
number Ex (p. 137).
Info: [A24].
4.1.29 Slenderness Factor NA
NA 5
M
Rchar d 3
M (kg) mass of a projectile; Rchar(kg m23) characteristic density of environment between projectile and target; d (m) projectile diameter.
It expresses the ratio of the pressure cross section of a projectile to the characteristic density zone for an unlimited target. It represents the projectile slenderness
for a solid characteristic density of the environment between a projectile and a target. Stroke mechanics. Ballistics.
Info: [B66].
4.1.30 Statical Moment Number NMS
NMS 5
M
EL3
ð1Þ;
NMS 5
M
GL3
ð2Þ;
NMS 5
M
qL2
ð3Þ
M (N m) moment of force; E (Pa) modulus of elasticity; G (Pa) shear modulus; L (m) characteristic length; q (N m21) length loading.
In form (1), it expresses the moment for the static tensile or pressure stress; in
form (2), it expresses the torsion stress. The bending moment of a beam, laid on a
common elastic base and loaded longitudinally, is given by form (3).
Info: [A24].
4.1.31 Structural Property Number NS
NS 5
σL
E
146
Dimensionless Physical Quantities in Science and Engineering
σ (N m23) volume loading; L (m) characteristic length; E (Pa) modulus of
elasticity.
This number expresses the mass-to-stiffness ratio of a structure. In a complex
mechanical system, it expresses the resulting value of its partial subsystems.
4.1.32 Thermoelastic Coupling Number (1.) N1
N1 5
ð3λ 1 2μÞα
Eα
3Kα
5
5
cR
ð1 2 2νÞcR
cR
λ, μ (Pa) Lamé functions; α (K21) linear thermal expansion coefficient;
cR (J m23 K21) specific heat capacity; E (Pa) modulus of elasticity; K (Pa) volume modulus of elasticity; ν () Poisson’s ratio (p. 143).
This number expresses the ratio of the specific thermal stress to the specific volume thermal capacity. It characterizes the relation between the thermoelastic stress
and the thermal capacity in a compound thermal and thermal stress process. In a
non-compound process, α 5 0.
Info: [A23].
4.1.33 Thermoelastic Coupling Number (2.) (Gay-Lussac Thermoelastic
Number) N2, Gcte
N2 Gcte 5
ð3λ 1 2μÞ
αTref 5 βTref
ðλ 1 2μÞ
λ, μ (Pa) Lamé functions; α (K21) linear thermal expansion coefficient;
Tref (K) reference temperature; β (K21) volume thermal expansion coefficient.
In an elastic environment, it expresses the ratio of the thermal deformation propagation velocity to the electromagnetic waves propagation velocity. It characterizes
the velocity relation between thermal and mechanical deformation energy propagation. With α 5 0, the thermal and deformation processes do not depend on each
other. It enables the solution of temperature rise connected to the body deformation
in tasks on thermal energy dissipation.
Info: [A23].
Joseph Louis Gay-Lussac (p. 15).
4.1.34 Thermoelastic Coupling Number Resultant Nυ
Nυ 5 N1 N2 5
ð3λ 1 2μÞ2 α1 α2 Tref
Eα1 βTref
3Kα1 βTref
5
5
ðλ 1 2μÞcR
ð1 2 2νÞcR
cR
Solid Mechanics
147
λ, μ (Pa) Lamé functions; α1, α2 (K21) linear thermal expansion coefficient
in system 1 and system 2; Tref (K) reference temperature; cR (J m23 K21) specific volume heat; E (Pa) modulus of elasticity; β (K21) volume thermal
expansion coefficient; K (Pa) volume modulus of elasticity; ν () Poisson’s
ratio (p. 143); N1 () thermoelastic coupling number (1.) (p. 146); N2 () thermoelastic coupling number (2.) (p. 146).
It characterizes a compound thermal and thermal stress process in linear systems. The independence of both processes is expressed by the value of Kυ, because
α1 5 0 or the reference temperature Tref 5 0.
Info: [A23].
4.1.35 Thermoelastic Deformation Nε
Nε 5
ε
αTref
Nε 5
ε
ð1 1 νÞαTref
ð2Þ;
Nε 5
ð1 2 νÞε
ð1 1 νÞαTref
ð3Þ
ð1Þ;
α (K21) linear thermal expansion coefficient; Tref (K) reference temperature;
ν () Poisson’s ratio (p. 143); ε () relative elongation (p. 144);
This quantity expresses the relative thermoplastic deformation with single axis (1),
planar (2) and space (3) thermal stresses.
Info: [A23].
4.1.36 Thermoelastic Displacement Nu
u
ð1Þ;
αTref L
u
Nu 5
αLTref ð1 1 νÞ
Nu 5
Nu 5
uð1 2 νÞ
ð1 1 νÞαTref L
ð2Þ;
ð3Þ
u (m) displacement; α (K21) linear thermal expansion coefficient; Tref (K) reference temperature; L (m) characteristic length; ν () Poisson’s ratio (p. 143).
It characterizes the dimensionless thermoelastic displacement with single axis (1),
planar (2) and space (3) thermal stresses.
Info: [A23].
148
Dimensionless Physical Quantities in Science and Engineering
4.1.37 Thermoelastic Displacement Potential NΦu
NΦu 5
Φ
ð1 2 νÞΦ
ðλ 1 2μÞΦ
5
5
βTref L2
ð1 1 νÞαTref L2
ð3λ 1 2μÞαTref L2
Φ (m2) thermoelastic potential of the displacement; β (K21) volume thermal
expansion coefficient; Tref (K) reference temperature; L (m) characteristic
length; α (K21) linear thermal expansion coefficient; λ, μ (Pa) Lamé functions; ν () Poisson’s ratio (p. 143).
It characterizes the dimensionless thermoelastic displacement potential.
Info: [A23].
4.1.38 Thermoelastic Potential NΦ
NΦu 5
Φ
uL
NΦε 5
Φ
εL2
ð2Þ;
NΦσ 5
GΦ
σL2
ð3Þ
ð1Þ;
Φ (m2) thermoelastic potential of the displacement; u (m) displacement; L (m) characteristic length; G (Pa) shear modulus; σ (Pa) stress; ε () relative elongation (p. 144).
It characterizes the dimensionless thermoelastic potential related to displacement (1),
deformation (2) or stress (3).
Info: [A23].
4.1.39 Thermoelastic Stress Function NF
NF 5
ð1 2 νÞF
EαTref L2
F (N) thermoelastic stress function; E (Pa) modulus of elasticity; α (K21) linear thermal expansion coefficient; Tref (K) reference temperature; L (m) characteristic length; ν () Poisson’s ratio (p. 143).
In the dimensionless form, it expresses the thermoelastic stress function.
Info: [A23].
Solid Mechanics
149
4.1.40 Thermoelastic Stress Number Nσ
Nσ 5
σ
αETref
Nσ 5
σð1 2 νÞ
αETref
ð2Þ;
Nσ 5
σk
αEΔTls
ð3Þ
ð1Þ;
σ (Pa) stress; α (K21) linear thermal expansion coefficient; E (Pa) modulus
of elasticity; Tref (K) reference temperature; σk (Pa) yield limit; ΔTls (K) temperature difference between liquidus and solidus; ν () Poisson’s ratio
(p. 143).
In the dimensionless form, this number characterizes the thermoelastic stress for
planar (1) and space (2) linear stresses. In expression (3), it is about the criterion of
internal residual thermoelastic stress.
Info: [A23].
4.1.41 Viscous Damping Coefficient Nd, .
kd
Nd 5 pffiffiffiffiffiffiffiffi
ks m
kd (kg s21) coefficient of viscous damping; ks (kg s22) stiffness of elastic component, spring constant; m (kg) mass.
In the dynamics of mechanical sets, it expresses the ratio of the viscous damping
force to the elastic element stiffness.
4.2
Fracture Mechanics and Micromechanics
In fracture mechanics, the physical similarity criteria are focused to prevent structure defects from originating, especially those which could cause cracks and material fractures. They characterize the material resistance. This relates to the
elastoplastic and/or fully plastic material loading conditions. Well-known dimensionless numbers include the Fick number, resistance against crack generation,
fracture strength and the crack propagation number.
150
Dimensionless Physical Quantities in Science and Engineering
4.2.1
Crack Filling Number N
N5
pm τ
5 EuFoPrRe2
νR
pm (Pa) metallostatic pressure; τ (s) local solidification time; ν (m2 s21) kinematic viscosity; R (kg m23) density; Eu () Euler number (1.) (p. 61); Fo
() Fourier number (p. 175); Pr () Prandtl number (p. 197); Re ()
Reynolds number (p. 81).
It characterizes the possibility to fill the crack with residual interdendrite molten
material or alternatively to liquefy the existing crack. It is the fracture mechanics
criterion.
Info: [A23].
4.2.2
Crack Propagation Number, Ice Number NCR
Rw2
NCR 5
E
2 rffiffiffiffiffiffi
EL
;
R
where R 5
πσ2 a
E
R (kg m23) density; w (m s21) crack propagation velocity; E (Pa) modulus
of elasticity; L (m) characteristic length; R (kg s21) mechanical resistance;
σ (Pa) stress; a (m) half-length of a crack.
It is the extension of the Cauchy number (aeroelasticity parameter) (p. 155) in
the fracture mechanics. It enables one to follow the evolution of crack propagation
and the influence of resistance on crack freezing.
Anthony G. Atkins, American engineer.
Robert M. Caddell, American engineer.
4.2.3
Energy Release Number Nσ,res
Nσ;res 5 NF
qEf
;
σ2res h
where Ef 5
E
ð1 2 ν 2 Þ
NF () acting force on the notch forehead; q (N m21) length loading; Ef (Pa) plane elasticity modulus of layer and substrate; σres (Pa) residual stress; h (m)
distance in the layer thickness; E (Pa) modulus of elasticity; v () Poisson’s
ratio (p. 143).
It expresses the inner-to-outer stress ratio created due to loosening in the location of a longitudinal notch, crack or cutting in a surface layer.
Info: [B50].
Solid Mechanics
4.2.4
151
Fick Number Fi
Fi FoD21 5
L2
Dτ
L (m) characteristic length; D (m2 s21) diffusivity; τ (s) time; FoD () Fourier mass number (p. 249).
It characterizes the diffusion intensity of the elements in the solid phase in relation to the distance of primary or secondary dendritic axes and to the local solidification time. With increasing Fi, the mixing off of the elements during
solidification rises, the solidification interval increases and the trend to cracks
between dendrites increases (e.g. in steel especially). Of the metallurgic factors, the
steel temperature magnifies the value of the Fi. By refining the dendrite structure,
for example, by modifying the steel at the end of the reduction time in a furnace or
in a pan in tapping, the unfavourable influence of the number Fi can be limited
substantially. It is a criterion of fracture mechanics.
Info: [A23].
Adolf Eugen Fick (3.9.182921.8.1901), German
physiologist.
After an initial study of mathematics, he devoted himself
primarily to physiology. He formulated the diffusion laws
and an equation for neutral particles (1855). With this he laid
an important base for subsequent research of diffusion processes, which led later to contact lens development, among
others. In addition, he was engaged in studying the activity
of the sense organs and in examining electric signals under
the excitation of nerves.
4.2.5
Fracture Strength Nf
σfrac
Nf 5
5 60
σth
rffiffiffiffiffiffi
σ
Eh
σfrac (Pa) fraction stress of material; σth (Pa) theoretical stress of material
(σth 5 E/30); σ (N m21) surface stress; E (Pa) modulus of elasticity; h (m) thickness of material.
This number characterizes fraction stress in nanomaterials. Fracture mechanics
and nanomaterials.
Info: [B38].
4.2.6
Interval of Solidification N
N5
ΔTls
Tc
ΔTls (K) temperature interval of solidification; Tc (K) casting temperature.
152
Dimensionless Physical Quantities in Science and Engineering
It expresses the relation of the thermal solidification interval to the pouring temperature. It characterizes the material trend to cracks between the dendrites during
solidification. With optimal pouring temperature, the tendency increases with the
solidification interval. Then, the crack tendency increases with the increasing criterion value. It is a fracture mechanics criterion.
Info: [A23].
4.2.7
Metallostatic Pressure N
N5
pm
5 EuFr
gRL
pm (Pa) metallostatic pressure; g (m s22) gravitational acceleration;
R (kg m23) density; L (m) characteristic length; Eu () Euler number (1.)
(p. 61); Fr () Froude number (1.) (p. 62).
It characterizes the ratio of the metallostatic pressure in a critical place of a casting to its characteristic wall thickness. The criterion application is joined with the
building of risers on castings. The greater its value, the less the probability of a
crack originating. For certain characteristic casting thicknesses, the tendency to create interdendrite cracks can be suppressed by increasing the metallostatic height
(e.g. riser heightening, riser filling or overpressure riser application). It is a fracture
mechanics criterion.
Info: [A23].
4.2.8
Microcrack Initial Number Ncrini
Ncrini 5
Fini Ef
σ2 A
Fini (N) acting dynamic force; Ef (Pa) elasticity modulus of thin layer; σ (Pa) tension stress in thin layer; A (m2) opening surface in the crack size of surface
layer.
This number characterizes the crack evolution in a thin surface layer or on an
elastic substrate or a viscous sublayer. It expresses the relation between an outer
initializing force acting on the layer and an internal stress force in the layer. For
example, it relates to electronic parts such as solar cells and flat panel display technology. Microelectronics in macroelectronic systems. Fracture mechanics.
Info: [B110].
4.2.9
Microcrack Steady Number Ncrst
Ncrst 5
Fst Ef
σ2 A
Solid Mechanics
153
Fst (N) steady dynamic force; Ef (Pa) elasticity modulus of thin layer; σ (Pa) tension stress in thin layer; A (m2) opening surface in the crack size of surface layer.
It characterizes a steady crack in a thin surface layer or on an elastic substrate
or a viscous sublayer. It expresses the relation between an outer steady-state force
acting on a layer and an internal stress force in a layer. Microelectronics in macroelectronic systems. Fracture mechanics.
Info: [B110].
4.2.10 Relative Fracture Strain εfrac
rffiffiffiffiffiffi
Es
εfrac 5 2
Eh
Es (J m22) density of surface energy; E (Pa) modulus of elasticity; h (m) depth of the crack opening.
It characterizes the fracture deformation in nanomaterials. Fracture mechanics.
Nanotechnology.
Info: [B38].
4.2.11 Relative Strain Index Pε
Pε 5
εyield
εfrac
εyield () yield strain; εfrac () fracture strain.
This index expresses the relation of a relative yield point deformation to its
material destruction value. Fracture mechanics.
4.2.12 Resistivity Against Crack Generation (1.) N
N5
pm L
5 EuFrReSh 21
νRgτ
pm (Pa) metallostatic pressure; L (m) critical defect size; ν (m2 s21) kinematic
viscosity; R (kg m23) density; g (m s22) gravitational acceleration; τ (s) solidification time; Eu () Euler number (1.) (p. 61); Fr () Froude number
(1.) (p. 62); Re () Reynolds number (p. 81); Sh () Strouhal number (p. 87).
It characterizes the resistivity against crack generation in a solidifying casting. It
is a fracture thermomechanics criterion.
Info: [A23].
154
Dimensionless Physical Quantities in Science and Engineering
4.2.13 Resistivity Against Crack Generation (2.) N
N5
σ
5 LeFrðFoWeScÞ 21
νRgτ
σ (Pa) surface stress; ν (m2 s21) kinematic viscosity; R (kg m23) density;
g (m s22) gravitational acceleration; τ (s) solidification time; Le () Lewis
number (p. 254); Fr () Froude number (1.) (p. 62); Fo () Fourier number
(p. 175); We () Weber number (p. 389); Sc () Schmidt number (p. 263).
It characterizes the resistivity against crack generation as a function of the
molten stuff surface energy, viscosity and solidification time. It is a fracture thermomechanics criterion.
Info: [A23].
4.2.14 Strain at Fracture εfrac
εfrac 5
3fr sin α
L2
f (m) deflection of the lateral twig at breakage; r (m) radius of the lateral
twig, distance from the neutral plane; α () angle just before breakage; L (m) distance from the twig base to the point where the force acts.
It expresses the deformation at the point where the fracture arises. Fracture
mechanics.
4.2.15 Stress Index Pσ
Pσ 5
σyield
σfrac
σyield (Pa) yield stress; σfrac (Pa) fraction stress.
It expresses the ratio of the yield point stress to the material destruction stress.
Fracture mechanics.
4.3
Aeroelasticity
In aeroelasticity, the dimensionless quantities express the influence of the interaction between inertia, elasticity and aerodynamic forces. In static aeroelasticity, the
dimensionless quantity relates to the problems of divergence and regressive control
of a bypassed surface. In dynamic aeroelasticity, the dimensionless quantity relates
to the problems of fluttering and the dynamic response of mechanical parts which
are bypassed. The aeroelasticity parameter, Cauchy’s, Connor’s, Frueh’s, Regier’s
Solid Mechanics
155
and other numbers are among the well-known dimensionless quantities in
aeroelasticity.
4.3.1
Aeroelasticity Parameter Ae
See the Cauchy number (aeroelasticity parameter) Cau (p. 155).
Info: [A29].
4.3.2
Aeroelasticity Stiffness NAE
NAE 5
2E
Rw2
E (Pa) modulus of elasticity; R (kg m23) density; w (m s21) flow velocity.
It expresses the ratio of the structure stiffness to the aerodynamic force. It is the
inverse value of the aeroelasticity parameter AE (p. 155). Aeroelasticity.
4.3.3
Cauchy Number (Aeroelasticity Parameter) Cau
Cau M 2 Ho 5
Cau 5
R w2
E
R w2
K
ð1Þ;
ð2Þ
R (kg m23) density; w (m s21) velocity; K (Pa) volume modulus of elasticity; E (Pa) modulus of elasticity; M () Mach number (p. 73); Ho () Hooke number (p. 138).
This parameter expresses the ratio of the inertia force to the material compressibility force. It characterizes the compressible fluid flow (1) and the dynamic material strain (2) by inertia forces. It is often called the aeroelasticity parameter as
well. Aerodynamics, aeroelasticity, dynamics.
Info: [A23],[A43],[B127],[C37].
Augustin Louis Cauchy (21.8.178923.5.1857), French
mathematician.
He was known for his precision and consistency in mathematics. He introduced many concepts such as the determinant, limit, continuity and convergence. He founded complex
analysis and deduced the CauchyRiemann conditions with
Riemann. He was very prolific, publishing nearly 800 works.
156
Dimensionless Physical Quantities in Science and Engineering
4.3.4
Connors Number Con
w
Con 5
5K
fL
sffiffiffiffiffiffiffiffi
mδ
RL2
w (m s21) mean velocity of two-phase flow; f (s21) oscillation frequency;
L (m) characteristic length, pipe diameter; K () instability factor;
m (kg m21) length density of pipe material; δ () damping decrement;
R (kg m23) fluid density.
It relates to the fluid elastic vibration characteristics in the two-phase flow of
fluids. For example, it relates to tubes exposed to a dynamic two-phase flow of air
and water in heat exchangers. The number Con serves to determine the critical
flow rate and is based on the mean flow rate, mean fluid density and damping in a
two-phase flow. Aeroelasticity, two-phase flow, heat exchangers.
Info: [G28].
H. J. Connors, American scientific researcher.
Together with P.M. Moretti, Connors is among the significant personalities in
the aeroelasticity field, in dynamic induced oscillation and fluttering.
Peter M. Moretti, American scientist.
His scientific work is very wide-ranging and involves the
dynamics of mechanical systems, especially those of induced
vibrations excited by fluid streaming. Primarily, this relates
to the fluid stream dynamic acting on the nest of tubes in
heat exchangers, further about the instability solution in
transversal streaming through these nests and the eigenfrequency determination and damping of nests. He was also
engaged in research related to thermal stratification in lakes,
in supersonic streaming and heat transfer problems.
4.3.5
Flutter Number F
F5
weq
M
5
Rg
wR
weq (m s21) equivalent air velocity near the sea level; wR (m s21) Regier surface velocity index (flutter parameter); M () Mach number (p. 73); Rg () Regier number (p. 157).
High velocities aerodynamics. Aeroelasticity. Flutter.
Info: [B94].
4.3.6
Frequency Parameter Pf
Pf 5
ωL
5 2πSh
w
Solid Mechanics
157
ω (s21) angular velocity; L (m) characteristic length (e.g. channel width or distance from a wall); w (m s21) flow velocity; Sh () Strouhal number (p. 87).
It characterizes the unsteady flow in bundles, fluidization and other dynamic
processes acting on mechanical systems, as an example.
Info: [A29],[A35].
4.3.7
Frueh Number Fh
Lω1
Fh 5
a
rffiffiffiffiffiffi
Nm
cL
L (m) characteristic length, (e.g. half-length of wing chord); ω1 (s21) first harmonic frequency of torsional vibration; a (m s21) sound velocity; cL () uplift
curve inclination; m (kg m21) mass to the wing length; R (kg m23) air density;
Nm () mass ratio (p. 75).
Aeroelasticity, transonic flutter of wings.
Info: [A24].
Frank J. Frueh, American aerodynamic engineer.
4.3.8
Mass Ratio Nm
Nm 5
m
πRL3
m (kg) mass of body; R (kg m23) fluid density; L (m) characteristic length
of a body.
Mechanical systems dynamics. Aircraft fluttering and stability. Aeroelasticity.
Info: [A24].
4.3.9
Regier Number Rg
Lω
Rg 5
a
sffiffiffiffiffiffiffiffiffiffi
Rl
πRL2
L (m) characteristic length; ω (s21) angular velocity; a (m s21) sound velocity;
Rl (kg m23) fluid density; R (kg m23) material density.
It expresses aeroelastic relations, transonic flutter of wings and blades of flow
machines. It is a modification of the Strouhal number Sh (p. 87). Aeroelasticity.
Info: [A24].
158
Dimensionless Physical Quantities in Science and Engineering
4.3.10 Regier Surface Number Rgs
Rgs 5
wR
a
wR (m s21) Regier surface velocity index (flutter parameter); a (m s21) sound
velocity.
It is a simpler expression of the Regier number Rg (p. 157) and represents the
elastic-to-aerodynamic forces ratio at sea level. High velocity aerodynamics.
Aeroelasticity. Flutter.
Info: [A24],[B94].
4.4
Tribology
The tribologic dimensionless quantities involve especially the subjects of friction,
lubrication and wearing. Usually, they express the interaction of surfaces in their
relative movement. For example, they relate to the friction and lubrication of bearings and the friction and wear of tools in machining. The dimensionless quantities
especially apply to new micro- and nanotechnologies, in which the tribological processes represent a basic problem. Among the well-known tribologic dimensionless
quantities are Gűmbel’s, Hersey’s and Sommerfeld’s numbers, as well as wear
intensity and friction factor.
4.4.1
Adhesion Parameter Nadh,Θ
3
Nadh 5
Eh2
1
ra2 Δσ
;
where Δσ 5 σ1 1 σ2 2 σ1;2
E (Pa) modulus of elasticity; h (m) standard deflection of peak heights; ra (m) curve radius of surface roughness; Δσ (N m21) Dupré adhesion, adhesion surface
stress; σ1, σ2, σ1,2 (N m21) surface stress of two spheres and their boundary line.
It expresses the ratio of the elastic energy to the adhesion work, provided a contact has occurred. With Nadh c1; there is a partial contact in which the elastic material has contact only on the highest roughness peaks. A full contact occurs for
Nadh {1: The parameter is based on the Gauss transversal roughness distribution on
the elastic material surface. Micro and nanotechnology. Tribology.
Info: [B126].
4.4.2
Coefficient of Bearing Friction f
f1 5 2π2
ηn r
p c
ð1Þ
Solid Mechanics
f2 5
ηn
p
159
ð2Þ
η (Pa s) dynamic viscosity; n (s21) rotational speed; p (Pa) pressure induced
by weight or loading force on bearing surface (2rl ); r (m) radius of shaft journal;
l (m) length of shaft journal; c (m) angular clearance.
Expression (1) serves to estimate the bearing friction or the energy loss in a
bearing approximately. For stable lubrication (boundary, hydrodynamic or mixed),
relation (2) is valid. For fully hydrodynamic lubrication in a plain bearing, it is
ηn
$ 1:7 3 1026
p
The course of the friction coefficient dependence on the Sommerfeld number Sm
(p. 385), expressed with the Stribeck curve, can be replaced by three zones boundary, mixed and hydrodynamic lubrication in each of which the effective
coefficient value can be determined.
Sm{1
fef 5 fs
ðboundary lubricationÞ
Sm 1
fef 5 fs ð1 2 SmÞ 1 SmLh
Smc1
fef 5 SmLh ðhydrodynamic lubricationÞ
ðmixed lubricationÞ
Tribology, plain bearings.
4.4.3
Coefficient of Kinetic Friction fk,μk
fk 5
F
Fn
fk # fs
ð1Þ;
ð2Þ
F (N) friction force; Fn (N) normal force pressing friction surfaces together;
fs () coefficient of static friction (p. 160).
It expresses the friction in movement. In contrast to the coefficient of static friction (p. 160), its size depends on the surface roughness and conditions thereof. In
general, expression (2) holds for the rate range of cm s21 to m s21, and it is supposed to approximate a constant. For comparison, for an iron casting, for example,
the coefficient is fs 5 1.1, fk 5 0.15; for steel/bronze, it is fs 5 0.51, fk 5 0.44; for
steel/steel, it is fs 5 0.74, fk 5 0.57. Mechanics. Tribology.
160
Dimensionless Physical Quantities in Science and Engineering
4.4.4
Coefficient of Static Friction fs,μs
fs 5
Ff
Fn
Ff (N) friction force; Fn (N) normal force pressing standing friction surfaces
together.
This coefficient expresses the friction of solid bodies at a standstill. It depends
on the surface roughness and is greater than the coefficient of kinetic friction fk
(p. 157), generally ( fk , fs ). The rolling coefficient fr belongs here, which occurs
in rolling operations, such as wheel contact with the rolling surface of vehicles.
It expresses the friction of solid bodies at a standstill. Its size does not depend
on the contact surface area. The static friction force is a reaction the size of which
depends on the size of some outer forces tending to shift the surfaces mutually.
The static friction force rises to the maximum value (1), with which the object
starts to move. The static friction coefficient is equal to or greater than the kinetic
friction coefficient (2). For example, for a few materials and dry friction the values
are as follows: for wood/wood fs A h0.25; 0.5i, for steel/steel fs A h0.5; 0.8i, for
aluminium/steel fs A h0.6; 0.8i, for glass/glass fs A h0.9; 1.0i. The oxidation surface cover strongly influences this coefficient, for example, for clean surfaces of
steel/steel fs 5 0.78; for an oxidized surface fs 5 0.27; and for a sulfidated surface
fs 5 0.39. A survey of the static and sliding friction coefficient values is given in
[C103] for various combinations of materials. Mechanics. Tribology.
Info: [C57].
4.4.5
Critical Frictional Temperature Θf
Θf 5
η2 ws ut
L2 σ2f
ð1Þ;
Θf 5
η2 ws ut
L2 σ2m
ð2Þ
η (Pa s) dynamic viscosity at the oil temperature in input to the sliding part;
ws (m s21) sliding velocity; ut (m s21) peripheral velocity; L (m) characteristic length (disc radius); σf (Pa) contact stress considering the surface roughness; σm (Pa) maximum contact stress according to Hertz definition at arbitrary
loading and without seizing.
It characterizes the heating and seizing of discs in rolling movement and lubrication. The expressions (1) and (2) are linearly dependent, approximately.
Info: [B16].
Solid Mechanics
4.4.6
161
Dimensionless Film Pressure N P ; p
p
πbLp
NP 5
5
;
pHZ
2F
rffiffiffiffiffiffiffiffiffiffiffi
FE
2F
where pHZ 5
5
2πLR πbL
p (Pa) oil-film pressure; pHZ (Pa) maximal dry (Hertzian) friction in contact;
b (m) half width of dry contact; L (m) characteristic length (bearing width);
F (N) force loading of bearing; E (Pa) reduced modulus of elasticity; R (m) reduced curve radius.
It expresses the dimensionless oil-film pressure.
Info: [B117].
4.4.7
h5
Dimensionless Film Thickness h
πLEh
;
4F
where E 5 2
1 2 ν 21
1 2 ν 22
1
E1
E2
L (m) characteristic length (bearing width); E (Pa) reduced modulus of elasticity; h (m) film thickness; F (N) force loading of bearing; E1, E2 (Pa) elasticity modulus of bearing parts; ν 1, ν 2 () Poisson’s ratio (p. 143) of bearing
parts.
The dimensionless oil-film thickness. Tribology, plain bearings.
Info: [B117].
4.4.8
Drag Friction Factor fD
fD 5
Ff 1 Fd
Ap Rw2N
1
2
Ff (N) friction force; Fd (N) drag force; Ap (m2) cross-section area;
R (kg m23) fluid density; wN (m s21) free flow velocity.
This factor characterizes friction under the action of dynamic drag and friction
forces. Fluid mechanics. Tribology.13
4.4.9
Duty Cycle Group NC
3
3
24η0 π2 ELR 2
NC 5
F
TE 2
η0 (Pa s) dynamic viscosity at normal conditions; T (s) period of load cycle;
E (Pa) reduced modulus of elasticity; L (m) characteristic length (bearing
width); R (m) reduced curve radius; F (N) force bearing load.
162
Dimensionless Physical Quantities in Science and Engineering
It expresses the influence of the bearing duty cycle on the thickness and pressure
of an oil film in a bearing. The elastohydrodynamic lubrication is considered.
Tribology, lubrication, plain bearings.
Info: [B117].
4.4.10 Dynamic Force NFd
NFd 5
Rw2 E
σ2P
R (kg m23) density; w (m s21) sliding velocity; E (Pa) modulus of elasticity;
σP (Pa) tensile strength.
It expresses the ratio of the dynamic loading force to the material strength force.
Together with the static force NFs (p. 170), it represents two basic parameters influencing the coefficient size of the friction between a rotating wheel and a support
(e.g. road or rail).14
4.4.11 Energetic Wear Intensity NE
NE 5
ΔV
λ
FL
ΔV (m3) volume of wearout layer; F (N) friction force; L (m) length of
sliding; λ (J m23) volume density of deformation energy.
It expresses the ratio of the energy consumed for wearing to the friction energy.
Tribology, material wearing.
4.4.12 Euler Number (2.) Eu2
Eu2 5 2
dp d
5 2fF
dl Rw2
dp
(Pa m21) friction pressure gradient; d (m) pipe diameter; R (kg m23) fluid
dl
density; w (m s21) flow velocity; fF () Fanning friction number (p. 163).
This number expresses the fluid friction if flowing through a pipeline.
Info: [A35].
Leonhard Euler (p. 61).
4.4.13 Fanning Number Fa
Fa 5
2τ
R w2
τ (Pa) shear stress; R (kg m23) fluid density; w (m s21) flow velocity.
Solid Mechanics
163
It expresses the ratio of the shear stress to the fluid dynamic pressure. It characterizes the friction loss for flow around bodies or in a pipeline. See also the
Fanning friction number fF (p. 163).
Info: [B20].
John Thomas Fanning (18371911), American engineer.
4.4.14 Fanning Friction Number fF, cF
fF 5
2Δp DH
fD
5
4
Rw2 L
Δp (Pa) pressure difference; R (kg m23) fluid density; w (m s21) flow velocity; DH (m) hydraulic diameter; L (m) characteristic pipe length; fD () Darcy friction number (p. 119).
This number characterizes the friction in turbulent isothermic flow in a bare
pipeline. In the range of the Reynolds number (p. 81) Re A h5 3 103; 200 3 103i,
the value fF is given by the relation fF 5 0.046 Re20.2.
Info: [A2],[B17].
John Thomas Fanning (see above).
4.4.15 Flexible Number NF, F
F5
1
3
1
R
4
Ered α ηus
Ered (Pa) reduced modulus of elasticity; R (m) effective radius curve;
α (m2 N21) piezoviscosity coefficient; η (Pa s) input dynamic viscosity;
us (m s21) sum of tangential surface velocities.
It influences the minimum thickness of a bearing film. Elastohydrodynamic
lubrication of a bearing.
4.4.16 Friction Factor f
f 52
21
dp
1
DH Rw2
dx
2
dp
(Pa m21) pressure drop per unit length; DH 5 4 SO21 (m) hydraulic diamdx
eter; S (m2) surface area; O (m) perimeter; R (kg m23) fluid density;
w (m s21) fluid density.
2
164
Dimensionless Physical Quantities in Science and Engineering
It represents the general definition of the friction factor. The Fanning friction
number fF (p. 163) and the Darcy friction number fD (p. 119) are practical expressions thereof.
Info: [C29].
4.4.17 Gűmbel Number (1.) Gu
Gu 5
Fh2
2ηuR2
F (N m21) force on length of bearing; h (m) angular clearance; η (Pa s) dynamic viscosity; u (m s21) peripheral velocity; R (m) shaft radius.
It expresses the ratio of the bearing loading force to the dynamic friction force.
Tribology, lubrication, bearings.
Info: [A29].
Ludwig Karl Friedrich Gűmbel (18741923), German naval engineer.
4.4.18 Gűmbel Number (2.) Gu
Gu 5
ηωD2
F
η (Pa s) dynamic viscosity; ω (s21) angular velocity; D (m) diameter of
bearing journal; F (N) loading force.
It expresses the ratio of the dynamic friction force to the loading force of a plain
bearing. Tribology, lubrication, bearings.
Info: [A29].
Ludwig Karl Friedrich Gűmbel (see above).
4.4.19 Harrison Number Hr
Hr 5
6ηuL
pa δ2
η (Pa s) dynamic viscosity; u (m s21) peripheral velocity; L (m) length of
bearing pad in direction of motion; pa (Pa) ambient of inflow pressure; δ (m) bearing film thickness by output.
It expresses the ratio of the dynamic friction force to the static pressure force in
loading a bearing. Tribology. Lubrication. Bearings.
Info: [A29].
William John Harrison (18841969), English mathematician.
Solid Mechanics
165
4.4.20 Hersey Number He
He 5
FL
ηa
He 5
F
ηuL
ð1Þ;
ð2Þ;
Fl (N m21) length force loading (bearing); η (Pa s) dynamic viscosity;
a (m s21) sound space; F (N) load force; u (m s21) peripheral velocity;
L (m) bearing length.
It expresses the loading-to-viscosity forces ratio and characterizes the force relation in tribology (e.g. for bearing loading and lubrication).
Info: [A29],[B20].
Mayo Dyer Hersey (born 1886), American engineer.
4.4.21 Hydrodynamic Load Group NH
NH 5 3π2
η0 u ELR 2
ER
F
η0 (Pa s) dynamic viscosity at normal conditions; u (m s21) peripheral velocity
E (Pa) reduced modulus of elasticity; R (m) reduced radius curve; L (m) characteristic length (bearing width); F (N) bearing load.
It expresses the hydrodynamic bearing load influence on the thickness and pressure of the oil film in a bearing. Elastohydrodynamic lubrication. Tribology, lubrication, plain bearings.
Info: [B117].
4.4.22 Mass Wear Intensity Nm
Nm 5
Δm
LAR
Δm (kg) mass of wearout layer; L (m) length of sliding; A (m2) nominal
area; R (kg m23) material density.
It represents the mass wear of friction surfaces. Tribology, material wearing.
4.4.23 Material Abradability NA
NA 5
ΔV
kA Q A
5
f
f ΔmL
166
Dimensionless Physical Quantities in Science and Engineering
ΔV (m3) volume wearout of material by abrasion; f () coefficient of sliding
friction; Δm (kg) volume wearout of material by sliding; L (m) length of
sliding; kA (m2 kg1) abrasivity factor; QA (kg m22) surface loading limit.
It expresses the ratio of the volume decrease due to abrasion, to the mass loss
due to sliding friction. Tribology, material wear.
4.4.24 Material Wear Resistance NR
NR 5 NA21 5
f
kA QA
f () coefficient of sliding friction; kA (m2.kg21) abrasivity factor;
QA (kg m22) surface loading limit.; NA () material abradability (p. 165).
It expresses the material wear resistance. It is the inverse value of the material
abradability NA (p. 165). Tribology, material wear.
Info: [B92].
4.4.25 Mechanical Severity of Contact MS
pffiffiffi
ð1 1 10 fs Þσmax d
MS 5
K
σmax (Pa) maximum occurring Hertzian stress18; d (m) crack length;
K (m21/2 kg s22) fracture toughness of the material; fs () coefficient of static
friction (p. 160).
It expresses the contact mechanical resistivity under force loading. Tribology.
Info: [B75].
4.4.26 Minimal Film Thickness NH,H
HH 5 hmin
1
α2 η2 u2s R
hmin (m) minimal film thickness; α (m2 N21) piezoviscosity coefficient;
η (Pa s) input dynamic viscosity; us (m s21) the sum of tangential surfaces
velocities; R (m) effective curve radius.
It expresses the minimum thickness of a lubricating film. In general, it increases
with increasing flexibility, though this is not valid in some applications.
Elastohydrodynamic lubrication, bearings.
Info: [B31].
Solid Mechanics
167
4.4.27 Moody Friction Factor fM
fM 5 fD 5 4fF
FD () Darcy friction number (p. 119); fF () Fanning friction number
(p. 169).
It corresponds to the Darcy friction number and fourfold of the Fanning friction
number (p. 169).
4.4.28 Piezoviscous Number NL , L
3
ηus Ered
NL 5 α
R
1
4
α (m2 N21) piezoviscosity coefficient; η (Pa s) dynamic viscosity; us (m s21) the sum of tangential surface velocities; Ered (Pa) reduced modulus of elasticity;
R (m) effective curve radius.
This number expresses the influence of the lubricating film physical properties,
such as viscosity, velocity, shape geometry and elasticity in the sliding lubrication
process. Electrodynamic lubrication, plain bearings.
Info: [B77].
4.4.29 Reduced Frequency Nf,red
Nf ;red 5
ωL
c
ω (s21) angular frequency; L (m) characteristic length; c (m s21) undisturbed
sound velocity.
It is the measure for the ratio of the gap width to the wavelength of sound propagation. Tribology, bearings.
4.4.30 Relative Pipe Roughness εrel
εrel 5
ε
D
ε (m) mean height, roughness height of inner pipe surface; D (m) inner pipe
diameter.
It is analogous to the roughness factor fr (p. 168). It serves to determine the friction factor. Usually, this is determined from the Moody diagram, which represents
168
Dimensionless Physical Quantities in Science and Engineering
its dependence on the Reynolds number Re (p. 81) and the relative pipeline roughness. For example, for Re 5 3 3 104 and εrel 5 1022, the pipeline fracture coefficient is f 5 0.02.
4.4.31 Relative Roughness εrel
εrel 5
Ag
;
Arough
where εrel Ah0; 1i
Ag (m2) geometric sample surface; Arough (m2) surface determined by
roughness of the sample.
It expresses the relative planar roughness of a specimen. Tribology.
4.4.32 Reynolds Friction Number Rer,δ1
Reτ 5
wτ δ
;
ν
where wτ 5 w0
rffiffiffiffi rffiffiffiffiffiffi
cf
τw
5
2
R
wτ (m s21) friction velocity; δ (m) boundary layer thickness; ν (m2 s21) kinematic viscosity; w0 (m s21) reference velocity of the undisturbed flow in
pipes and channels; cf () friction coefficient; τ w (Pa) shear stress on circumfluenced wall.
It characterizes the viscous fluid flow in a boundary layer close to walls, in a
pipeline and canals. Tribology.
Osborne Reynolds (p. 82).
4.4.33 Roughness Factor, Roughness Ratio fr
fr 5
ε
L
fr 5
Ar
Ag
ð1Þ;
ð2Þ
ε (m) height of roughness; L d (m) characteristic length, pipe diameter;
Ar (m2) real surface (interface) area ; Ag (m2) geometric surface (interface) area.
It expresses the ratio of the real interface height to the geometric one (1) or
the ratio of the real surface area (of the interface) to the geometric surface (2).
In the case of a circular cross-section pipeline, the diameter is the characteristic
dimension; the equivalent diameter is that for the non-circular one. It characterizes
the roughness influence of a bypassed surface on the friction of fluids.
Info: [A29].
Solid Mechanics
169
4.4.34 Shear Wave Number Nws
rffiffiffiffi
ω
Nws 5 L
ν
L (m) characteristic length (thickness); ω (s21) angular frequency; ν (m2 s21) kinematic viscosity.
It expresses the inertia-to-friction forces ratio. For low values of the number, the
viscosity influence predominates. On the contrary, for high values of the number,
the inertia influence dominates. From the physical point of view, it represents the
ratio between the gap width and the boundary layer thickness. Tribology, bearings.
4.4.35 Skin Friction Coefficient, Wall Shear Stress Coefficient Cf
Cf 5
τw
1
2
Rw
N
2
0:664
Cf 5 pffiffiffiffiffiffiffi
Rex
ð1Þ;
where τ w 5 η
dw y50 :
dy
ð2Þ
pffiffiffiffiffiffiffi
Cf 5 0:0576 5 Rex ð3Þ
τ w (Pa) shear stress on the wall ( y 5 0); R (kg m21) fluid density; wN (m s21) free fluid flow; η (Pa s) dynamic viscosity; y (m) coordinate perpendicular to
the wall surface; Rex () Reynolds number.
It expresses the dynamic friction resistance originating in viscous fluid flow
around a fixed wall. The expression (2) is valid for laminar flow along a flat plate
(Blasius’ boundary layer) and the expression (3) holds for turbulent flow.
Tribology. Aerohydrodynamics.
Info: [A21].
4.4.36 Squeeze Number Sq
12ηω L 2
Sq 5
pa
δ0
η (Pa s) dynamic viscosity; ω (s21) oscillation frequency; pa (Pa) ambient
pressure; L (m) characteristic length, radius; δ0 (m) initial gap thickness.
It characterizes the damping of a thin compressible film squeezed between a
fixed and movable oscillating plate. It expresses the friction and damping conditions in bearings and lubrication. Tribology.
Info: [A26],[B20].
170
Dimensionless Physical Quantities in Science and Engineering
4.4.37 Stanton Pannell Friction Factor fSP
fSP 5
1
Fa
2
Fa () Fanning number (p. 162).
It corresponds to half of the Fanning number (p. 162).
Info: [C49].
Thomas Edward Stanton (p. 201).
4.4.38 Static Force NFs
NFs 5
FE
σ2P
F (N) static force; E (Pa) modulus of elasticity; σP (Pa) tensile strength.
It expresses the ratio of the static loading force to the material strength force.
Together with the dynamic force NFd (p. 162), it represents two basic parameters
acting on the coefficient size of the friction between a rotating wheel and a support
(road, rail, etc.).
4.4.39 Thermal Severity of Contacts TS
TS 5
fs Fu
ΔTs bλef
F (N) contact load; u (m s21) velocity; ΔTs (K) temperature difference; b (m) half width of the elliptic contact in the direction of sliding; λef (W m21 K21) effective
thermal conductivity; fs () coefficient of static friction (p. 160).
It characterizes the thermal resistivity of a shock loaded contact. Tribology.
Info: [B75].
4.4.40 Torque Frictional Coefficient CM
CM 5
2Mf
r 3 ω2 AR
Mf (N m) friction moment; r (m) characteristic radius; ω (s21) angular
velocity; A (m2) surface; R (kg m23) fluid density.
It expresses the ratio of a friction moment in rotation to the moment due to
dynamic force. Hydromechanics.
Info: [A8].
Solid Mechanics
171
4.4.41 Transfer Number (4.) Friction Nf
Nf 5
τwL
wN η
τ w (Pa) shear stress on circumfluenced wall; L (m) characteristic length
(e.g. radius of pipe, wall and curvature); wN (m s21) free flow fluid velocity;
η (Pa s) dynamic viscosity.
This number expresses the friction-to-viscous forces ratio or dimensionless surface friction alternatively. Laminar flow in a boundary layer.
4.4.42 Wear Intensity PW
Pw 5
h
L
Pw 5
ΔV
LA
ð1Þ;
ð2Þ
h (m) thickness of wearout layer; L (m) length of sliding; ΔV (m3) volume
of wear layer; A (m2) nominal area.
In expression (1), it expresses linear wear intensity. In expression (2), it
expresses volume intensity. Tribology, material wearing.
5 Thermomechanics
The theory becomes the more attractive the simpler are its premises and the
expressive is the heterogeneity of phenomena it involves and the wider is the area
of applicability thereof.
Albert Einstein (18791955)
5.1
Heat Conduction
In a thermal system, the heat conduction manifests itself as a thermal field. The
thermal field can be steady or unsteady, linear or non-linear, with inner sources
or without them, with movable or stationary borders, with basic, composed or
combined boundary conditions, or with a phase conversion. Conjugate processes
of heat and mass transfers, heat and thermally induced stress, as well as nonequilibrium heat propagation processes belong to this area. Dimensionless quantities also correspond to this classification. Among them are the following numbers:
the Biot, Brinkmann, Fourier, Kirpichev, Pe´clet, Pomerantsev and Stefan two-phase
conduction number.
5.1.1
Biot Number Bi
Bi 5
αL
λ
ð1Þ;
Bi 5 αð2πf λcRÞ 21=2
ð2Þ
α (W m22 K21) heat transfer coefficient; L (m) characteristic length; λ
(W m21 K21) thermal conductivity; f (Hz) frequency; c (J kg21 K21) specific heat capacity; R (kg m23) density; V (m3), S (m2) body volume and surface; TS, TP (K) temperature of the body and environment.
This number expresses the ratio of the heat flow transferred by convection on a
body surface to the heat flow transferred by conduction in a body. It characterizes
the third-type boundary condition. The equation Bi 5 L (λα21)21 expresses the ratio
of a characteristic body length to an equivalent environmental thickness adhering
to the surface. With Bi $ 100 in the heat transfer on the surface, the thermal resistance is slight in comparison to the heat transfer by conduction. The temperature
Dimensionless Physical Quantities in Science and Engineering. DOI: 10.1016/B978-0-12-416013-2.00005-1
© 2012 Elsevier Inc. All rights reserved.
174
Dimensionless Physical Quantities in Science and Engineering
field is characterized by considerable non-uniformity. On the contrary, with
Bi # 0.1, difference (TP 2 TS) is great on the body surface and slight inside the
body and the field is uniform. With the general shape of body, the ratio L 5 VS21 is
inserted for the characteristic length. With periodically changing heat transfer, unit
amplitude and frequency, the modified Biot number is used in expression (2).
Info: [A23],[A29],[A31],[A35],[B20].
Jean-Baptiste Biot (21.4.17743.2.1862), French physicist.
He was engaged in a wide range of physics especially
in optics specifically in polarized light, changes in polarization and optical gas refractions. He also studied heat, conduction analysis and the external convection influence on
conduction. He was noted for requiring precision in mathematical formulations and experiments. With Gay-Lussac, he
collaborated to design the first balloon for scientific purposes
(1804) and discovered the BiotSavart law (1870) with
Felix Savart.
5.1.2
Brinkmann Number Br
Br 5
ηw2
ηw2
5
λΔT
λðTS 2 TN Þ
η (Pa s) dynamic viscosity; w (m s21) flow velocity; λ (W m21 K21) thermal
conductivity; ΔT (K) temperature difference; TS (K) surface temperature;
TN (K) temperature in the thermal undisturbed fluid area.
It expresses the ratio of the heat arising due to viscous friction of a fluid to the
heat transferred by molecular conduction. It characterizes the heat conduction in
viscous fluid flow. For high fluid viscosity values and low thermal conductivity
values (e.g. molten polymers), the value is Brc1.
Info: [A23],[A29],[A31],[B20].
Henri Coenraad Brinkmann, German physicist.
5.1.3
Clausius Number Cl
Cl 5
w3 LR
λΔT
w (m s21) velocity of fluid or solid body relative to undisturbed fluid; L (m) characteristic length; R (kg m23) density; λ (W m21 K21) thermal conductivity;
ΔT (K) temperature difference.
Thermomechanics
175
It is the ratio of the kinetic energy propagation rate to the heat propagation rate
by conduction. It characterizes the heat conduction in forced fluid flow or in a
moving environment generally. The characteristic length can be the canal width,
distance from the wall or fluid layer thickness.
Info: [A23],[A29],[A31],[A35],[B20].
Rudolf Julius Emanuel Clausius (2.1.182224.10.1888),
German mathematician and physicist.
He worked in the field of thermodynamics and formulated
the first and second thermodynamic theorems. The first is
about the equivalence of heat and work. By introducing the
entropy concept, he discovered that entropy can never drop
during physical processes, but can only stay constant in
reversal processes, as expressed by the second thermodynamic theorem. Together with Maxwell, he formulated the
kinetic theory of gases.
5.1.4
Fourier Number Fo
Fo 5
aτ
L2
Fo 5
λLτ
cRL3
ð1Þ;
ð2Þ
a (m2 s21) thermal diffusivity; τ (s) time; L (m) characteristic length;
λ (W m21 K21) thermal conductivity; c (J kg21 K21) specific heat capacity;
R (kg m23) density.
It expresses the ratio of the time of a proceeding thermal process to that of the
molecular diffusion of the heat. In this time, expressed in a dimensionless way, the
relations between the thermal field change rate, physical parameters and thermal
system dimensions appear. In form (2), it expresses the ratio of the heat transferred
by conduction in a system to the heat accumulated in the system.
Info: [A23],[A31],[A19],[A35],[A43],[B20].
Jean Baptiste Joseph Fourier (21.3.176816.5.1830),
French mathematician and physicist.
With his work The´orie Analytique de la Chaleur
(Analytic Theory of Heat) of 1822, he created the foundations for the mathematical heat conduction theory. First, he
formulated the important property of mathematical physics
equations: their dimensional homogeneity. His basic contribution was very important for the theory of trigonometric
arrays and that of real variable functions.
176
Dimensionless Physical Quantities in Science and Engineering
5.1.5
Heat Capacities Relation K
K5
cR
ðcRÞref
cR, (cR)ref (J m23 K21) specific volume heat capacity and its reference value.
It is the parametric criterion expressing the dimensionless material thermal
capacity of a body. It can be used in non-linear heat conduction. Usually cR 5
c(T)R(T) is valid.
Info: [A23].
5.1.6
Heat Diffusivities Relation K
K5
a
aref
a, aref (m2 s21) thermal diffusivity and its reference value.
It is the parametric criterion expressing the dimensionless material thermal diffusivity of a body. It can be used in non-linear heat conduction. Usually a 5 a(T) is
valid.
Info: [A23].
5.1.7
Irregularity of Temperature Field Ψ
Ψ5
Tp 2 Ts
Tp 2 Tmed
Tp (K) temperature of external environment; Ts (K) wall surface temperature;
Tmed (K) mean integral volume body temperature.
It expresses the ratio of the environment and surface temperature difference of a
body to the difference between the environment temperature and the mean integral
volume temperature. It characterizes the temperature field non-uniformity in a
body.
Info: [A23].
5.1.8
Kirpichev Heat Number Ki
Ki PoA 5
Ki 5 Bi
qA L
λΔT
TS 2 TP
ΔT
ð1Þ;
ð2Þ
Thermomechanics
177
qA (W m22) surface heat flux density; L (m) characteristic length;
λ (W m21 K21) thermal conductivity; ΔT (K) reference temperature difference or temperature; TS, TP (K) body surface and environment temperatures;
PoA () Pomerantsev heat number (p. 181); Bi () Biot number (p. 173).
This number expresses the ratio of the thermal heat flow surface density to the
density of the thermal heat flow conducted in a body. It characterizes the secondkind boundary condition or, alternatively, a planar heat source.
Info: [A4],[A23],[A26],[A29],[A31],[A20],[B20].
Mikhail Viktorovich Kirpichev (18791955), Russian
engineer and physicist.
Using thermal similarity theory, he laid the foundations of
systematic modelling research for heat transfer, especially in
liquid metals, and elaborated the regular mode theory in thermal systems with internal heat sources. He was engaged in
research of the heat transfer in boiling and condensation.
Essentially, he interpreted similarity theory as theory of an
experiment. His book Teorija podobija i rnodelirovanije
(The Theory of Similarity and Modeling, 1956) has substantial significance.
5.1.9
Kondratiev Number Kd
Kd 5
mL2
a
Kd 5 BiΨ 5
ð1Þ;
α
LΨ
λ
ð2Þ
m (s21) unit speed of cooling; L 5 VS 2 1 (m) characteristic length dimension;
a (m2 s21) thermal diffusivity; α (W m22 K21) heat transfer coefficient;
λ (W m21 K21) thermal conductivity; Bi () Biot number (p. 173); Ψ () irregularity of temperature field (p. 176).
It characterizes the regular state of the heat transfer between a fluid and a body
in which the temperature space distribution in the body preserves the thermal similarity in time. In this state, the temperature is delayed regularly at an arbitrary point
of the body against the external environment temperature. It expresses the intensity
of mutual action of the body surface with the surrounding environment and, further,
the non-uniformity of the body temperature field in the regular state. With uniform
temperature distribution (Bi-0, for Bi , 0.1, essentially), Ψ 5 1 and Kd 5 Bi are
valid. On the contrary, with the greatest thermal field non-uniformity, Ψ 5 0 holds
(Bi-N, for Bi . 100, essentially), and then Kd is constant. The number Kd
involves the zone from zero to a value given by the body shape. It is often called
the thermal inertia criterion as well.
Info: [A23],[A29],[A35].
Georgy Mikhailovich Kondratiev, Russian engineer.
178
Dimensionless Physical Quantities in Science and Engineering
5.1.10 Local Temperature Θ
Θ5
T 2 Tmin
Tmax 2 Tmin
T, Tmin, Tmax (K) local, minimal and maximal temperatures.
It expresses the ratio of the difference between the local and minimal temperature to that between the maximal and minimal temperatures. It is the parametric criterion expressing the dimensionless temperature in a system. Usually, it represents
the dependent variable in the solution of thermal tasks. ΘAh0; 1i.
Info: [A23].
5.1.11 Maxwell Number Ml
Ml 5
λ
5 kPr 21
ηcV
λ (W m21 K21) thermal conductivity; η (Pa s) dynamic viscosity;
cV (J kg21 K21) specific heat capacity; κ () specific heat ratio (p. 28);
Pr () Prandtl number (p. 197).
It expresses the relation between gas thermal conductivity and gas viscosity.
It is a modification of the Prandtl number Pr (p. 197).
Info: [A23].
James Clerk Maxwell (p. 197).
5.1.12 Mikheyev Number Mi
Mi 5
ατ
5 BiFo
cRL
Mi 5
α2 τ
5 Bi2 Fo ð2Þ
λcR
ð1Þ;
α (W m22 K21) heat transfer coefficient; τ (s) time; c (J kg21 K21) specific
heat capacity; R (kg m23) density; L (m) characteristic length; λ (W m21 K21) thermal conductivity; Bi () Biot number (p. 173); Fo () Fourier number
(p. 175).
This number characterizes the dependence of the mean integral quantities in
space and time on a body shape. It expresses the ratio
pffiffiof
ffi the heat passed by the
body to the volume heat. For finite volume bodies L 5 S is valid, and for longitudinally unlimited bodies it is L 0: In equation (2), it expresses
the ratio of the
pffiffiffiffiffiffiffiffi
heat transferred by convection to the body thermal activity ε 5 λcR: It is applied
in cases in which L 5 λα 2 1 is a characteristic length dimension.
Thermomechanics
179
Info: [A23].
Mikhail Alexandrovich Mikheyev (19021970), Russian engineer.
5.1.13 Number of Phase Changes Nph, K
Nph 5
1
5 Sf 21
cp ΔT
Nph 5
rRw
5 Nu Pe 21
q
Nph 5 2
ð1Þ;
rRν
5 Nu Pe 21
qL
ð2Þ;
ð3Þ
l (J kg21) specific latent heat of phase change; cp (J kg21 K21) specific heat
capacity; ΔT (K) temperature difference; r (J kg21) reaction heat of chemical
change; R (kg m23) fluid density; w (m s21) flow velocity; q (W m22) surface heat flux; ν (m2 s21) kinematic viscosity; Sf () Stefan two-phase conduction number (p. 183); Nu () Nusselt number (p. 196); Pe () Péclet heat
number (p. 180).
It expresses phase conversions. It is the inverse value of the Stefan two-phase
conduction number Sf (p. 183). Expression (1) is the ratio of the thermal flow necessary to the phase conversion in a material, to the overheating or subcooling heat
of one of the phases. Expressions (2) and (3) represent the number Nph with phase
transformations in flowing fluid. Expression (2) represents the ratio of the phase
conversion rate to the flow rate. Expression (3) is the ratio of an inertia force in
flowing fluid, originating due to the phase conversion, to the inner friction force.
Therefore, it represents a specific form of the Reynolds number Re (p. 81). It is
analogous to the Jakob evaporation number Jak (p. 220).
5.1.14 Ostrogradsky Number Os
See Pomerantsev heat number PoV (p. 181) and Damkőhler number (4.) (2. heat)
Da4 (p. 37).
Mikhail Vasilyevich Ostrogradsky (24.9.18011.1.1862),
Russian mathematician and physicist.
He was the first mathematician to publish the proof of
the divergence theorem. In the year 1822 he went to Paris,
where he worked on heat theory and on hydrodynamic problems. He applied the divergence theorem as a tool to convert the volume integral to the planar one. He wrote many
articles on partial differential equations, algebra, elasticity,
hydromechanics, heat and electricity. He is considered the
founder of the Russian school of theoretical mechanics.
180
Dimensionless Physical Quantities in Science and Engineering
5.1.15 Péclet Heat Number Pe
Pe 5
wL
5 RePr
a
w (m s21) motion velocity; L (m) characteristic length; a (m2 s21) thermal
diffusivity; Re () Reynolds number (p. 81); Pr () Prandtl number (p. 197).
This number is the criterion for the mutual action of the convective and molecular heat transfers in flowing fluid. It expresses the ratio of the convective thermal
flow transferred by the fluid to that transferred by conduction. It characterizes the
thermal process with consideration of the movement of the environment and physical properties. It can be understood as the ratio of the enthalpy changes of the fluid
flowing in the axial direction to the heat penetrating into the flow by conduction in
the normal direction. With the Pe number increasing, the heat conduction portion
decreases and the convective heat portion grows. Thermal motion processes.
Info: [A4],[A23],[A35],[A43],[B20].
Jean Claude Eugène Péclet (10.2.17936.12.1857),
French physicist.
He was engaged in research on heat transfer by flow and
conduction especially, and in radiation and mass transfer as
well. His publications are well known for his clear and distinct opinions and well-executed experiments. His principal
work is the Traite´ de la Chaleur Conside´rée dans ses
Applications (Essay on Heat and its Applications in Crafts
and Manufacture, 1829).
5.1.16 Péclet Non-Linear Number of Solidification Pe
Pe 5
ωL dh
1
;
λðTÞ dΘ TS 2 TC
where Θ 5
T 2 TC
TS 2 TC
w (m s21) interface motion velocity; L (m) characteristic length; λ
(W m21 K21) thermal conductivity; h (J m23) specific volume enthalpy of
solid; TS, TC, T (K) solidifying interface, cooler and local temperatures.
It characterizes the motion of the isothermic solidliquid interface, for example,
in continuous casting and in solidification of castings.
Info: [A23].
Jean Claude Eugène Péclet (see above).
5.1.17 Phase Change of Enthalpy N
N5
h 2 hs
lsl
ð1Þ;
Thermomechanics
181
N5
cðTs 2 Tsl Þ
lsl
ð2Þ;
N5
cðTi 2 Tsl Þ
lsl
ð3Þ
h (J kg21) specific enthalpy; hs (J kg21) specific enthalpy of solid phase; lsl
(J kg21) specific latent heat of melting; c (J kg21 K21) specific heat capacity;
Ts (K) solid phase temperature; Tsl (K) temperature of solidification or melting; Ti (K) initial (casting) temperature of overheated melting.
In the dimensionless form, it characterizes phase conversion enthalpy. It is analogous to the Stefan two-phase conduction number Sf (p. 183).
Info: [A23].
5.1.18 Pomerantsev Heat Number, Heat Generation Term Po, G
PoV Os 5
PoA 5 Ki 5
qV L2
λΔTr
ð1Þ;
qA L
λΔTr
PoL 5
qL
λΔTr
PoB 5
qB
λLΔTr
ð2Þ;
ð3Þ;
ð4Þ
qV (W m23), qA (W m22), qL (W m21), qB (W) volume, area, linear and point
density of heat flux (source); L (m) characteristic length; λ (W m21 K21) thermal conductivity; ΔTr (K) characteristic temperature difference; Os () Ostrogradsky number (p. 179).
It characterizes volume (1), area (2), linear (3) or point (4) heat sources in a system. Sometimes in expression (1), it is called the Ostrogradsky number Os (p. 179),
whereas it is called the Kirpichev number Ki (p. 176) in expression (2).
Info: [A4],[A23],[A26],[A35],[A42],[B20].
Alexey Alexandrovich Pomerantsev, Russian engineer.
5.1.19 Predvoditelev Number Pd
Pd 5
dΘP
dFo max
ð1Þ;
182
Dimensionless Physical Quantities in Science and Engineering
Pd 5
bL2
aTP;i
ð2Þ;
Pd 5
b2 L2
a
ð3Þ;
Pd 5 2πfL2 a 21
ð4Þ
ΘP () dimensionless environment temperature; b (K s21) constant; L (m) characteristic length; a (m2 s21) thermal diffusivity; TP,i (K) initial temperature; b2 (s21) constant; f (Hz) oscillation frequency of external temperature;
Fo () Fourier number (p. 175).
This number characterizes the temperature time change of the outer environment
or of internal heat sources. For a linear change, TP 5 TP,i 1 bτ holds for expression
(2), and for T(τ) 5 T(N) 2 (T(N) 2 Ti) exp (2b2τ), expression (3) is valid. For
the thermal flow density changing exponentially from an internal source, q(τ) 5 qi
exp (2b2τ) holds, when the constant b2 expresses the maximum change rate of the
thermal source. Expression (4) holds for the environmental temperature T(τ)
Ti 5 Tm cos (2πfτ), which changes in a harmonic way.
Info: [A4],[A23],[A26],[B20].
Alexander Savviich Predvoditelev (18911973), Russian physicist.
5.1.20 StefanFourier Number N
N5
cðTls 2 Ts Þ aτ
5 Sf Fo
lls
L2
ð1Þ;
N5
cðTs 2 Tsl Þ aτ
5 Sf Fo
lsl
L2
ð2Þ
c (J kg21 K21) specific heat capacity at solidification or melting temperature;
Tls, Tsl (K) temperature of solidification or melting; Ts (K) surface temperature
of solidification body (ingot); lls, lsl (J kg21) specific latent heat of solidification
or melting; a (m2 s21) thermal diffusivity; τ (s) time; L (m) characteristic
length; Sf () Stefan two-phase conduction number (p. 183); Fo () Fourier
number (p. 175).
In the dimensionless form, it characterizes the time of the thermal two-phase
process of solidification (1) or melting (2).
Info: [A123].
Josef Stefan (see below).
Jean Baptiste Joseph Fourier (p. 175).
Thermomechanics
183
5.1.21 Stefan Two-Phase Conduction Number Sf
Sf 5
cðTls 2 Ts Þ
lls
ð1Þ;
Sf 5
cðTs 2 Tsl Þ
lsl
ð2Þ
c (J kg21 K21) specific heat capacity at solidification or melting temperature; Tls,
Tsl (K) temperature of solidification or melting; Ts (K) surface temperature of
solidification body (ingot); lls, lsl (J kg21) specific latent heat of solidification or
melting.
It expresses the ratio of the specific enthalpy of an ingot to the specific latent
heat of solidification. It characterizes the sensitivity to the thermal process latent
energy of solidification (1) or melting (2) in a solidusliquidus system under the
condition of linear heat conduction, neglecting the difference of densities of both
phases and considering the phase conversion at an equal temperature.
Info: [A23].
Josef Stefan (p. 214).
5.1.22 Thermal Conductivity Relation Λ
Λ5
λ
λref
λ, λref (W m21 K21) thermal conductivity and its reference value.
It is the parametric criterion expressing the dimensionless thermal conductivity
of a material. It can be used in non-linear heat conduction.
Info: [A23].
5.1.23 Two-Phase Heat Conduction K
K5
λl ðT1 2 Ts Þ
λs ðTs 2 T2 Þ
λl, λs (W m21 K21) thermal conductivity of liquid and solid phases; T1, T2, Ts
(K) input liquid casting, output and solid phase temperatures.
It expresses the thermal flows ratio in liquid and solid phases. It characterizes
the heat conduction process in a two-phase environment during solidification or
melting.
Info: [A23].
184
Dimensionless Physical Quantities in Science and Engineering
5.2
Free Convection
In free (natural) convection, the dimensionless quantities express the spontaneous
heat flow in fluids or gases due to the thermal difference caused by the difference
of fluid densities. The force fields setting the fluid in motion originate directly in
the fluid. Free convection occurs in the atmosphere, in oceans and near the surface
of heated or cooled bodies, for example. For free convection, the Rayleigh,
Grashof, Marangoni, Archimedes, Grigull and Schwarzschild numbers are
characteristic.
5.2.1 Archimedes Thermodynamic Number Ar
Ar 5
ΔT gL
5 ΔΘFr21
T 0 w2
Ar2 5
ð1Þ;
gL
L3 gβΔT ν 2
βΔT
5
5 GrRe22
w2 L2
w2
ν2
ð2Þ
ΔT (K) temperature difference; T0 (K) steady-state temperature; g (m s22) gravitational acceleration; L (m) characteristic length; w (m s21) flow velocity;
β (K21) thermal volume expansion coefficient; ν (m2 s21) kinematic viscosity;
Θ () relative temperature; Fr () Froude number (1.) (p. 62); Gr () Grashof heat number (p. 185); Re () Reynolds number.
This number characterizes the free and forced convection caused by the temperature gradient and the fluid flow rate. In equation (2), it is often called the buoyancy parameter NB (p. 56).
Info: [A4],[A12],[A29],[B20].
Archimedes of Syracuse (p. 54).
5.2.2
Bejan Pressure Number Be
Be 5
ΔpL2
ηa
Δp (Pa) pressure difference in channel; L (m) characteristic length of circumfluenced channel; η (Pa s) dynamic viscosity; a (m2 s21) thermal diffusivity.
It expresses the pressure gradient in a flow-through canal with consideration of
the heat diffusion. It has an analogous role as the Rayleigh number (2.) (heat instability) Ra (p. 187) in natural convection. Natural convection.
Info: [C23].
Adrian Bejan (p. 5).
Thermomechanics
5.2.3
185
Crispation Number Cr
Cr 5
ηa
σL
η (Pa s) dynamic viscosity; a (m2 s21) thermal diffusivity; σ (N m21) surface
tension; L (m) characteristic length, layer thickness.
This number expresses the ratio of a viscous force and diffusive heat action to a
surface tension force. It characterizes the influence of convective flows in heat and
momentum transfers.
Info: [A26],[A29].
5.2.4
Grashof Heat Number Gr
Gr 5
L3 gΔR L3 gβΔT
5
5 βΔTGa
Rν 2
ν2
Gr 5
L3 gΔT
ν2T
ð2Þ;
ω2 L
g
ð3Þ
Grrot 5 Gr ð1Þ;
L (m) characteristic length dimension; g (m s22) gravitational acceleration;
ΔR (kg m23) change of fluid density due to temperature change ΔT; R (kg m23) density; ν (m2 s21) kinematic viscosity; β (K21) volume thermal expansion coefficient; T (K) temperature; ω (Hz) angular frequency; Ga () Galilei number
(p. 123); Gr () Grashof heat number defined in expression (1).
It expresses the buoyancy-to-viscous forces ratio and its action on a fluid. It
characterizes the free non-isothermal convection of the fluid due to the density difference caused by the temperature gradient in the fluid. It is often used in expression (2) for an ideal gas. In form (3), it is used for convective transfer in rotating
canals of which the axis is parallel with the rotation axis. With this modified Gr
number, the centrifugal force influence on the heat transfer is considered. Usually,
the Coriolis force influence is negligible.
Info: [A23],[A26],[A43],[B20].
Franz Grashof (11.7.182626.10.1893), German engineer.
Within the framework of applied mechanics and general
machinery, he was engaged in research on material stiffness,
hydraulics and heat theory. As for heat transfer, he was
engaged in the problem of free convection caused by the
density difference due to the temperature gradient in a fluid.
186
Dimensionless Physical Quantities in Science and Engineering
5.2.5
Grashof Modified Number Grmod
Grmod 5
gR2 βL4
λη2
g (m s22) gravitational acceleration; R (kg m23) liquid density; β (K21) volume thermal expansion coefficient; L (m) characteristic length; λ (W m21 K21) thermal conductivity; η (Pa s) dynamic viscosity.
It is a modification of the Grashof heat number Gr (p. 185). It expresses the natural
convective transfer from a heated or cooled wall into the surrounding viscous fluid.
Info: [B60].
Franz Grashof (see above).
5.2.6
Grigull Number Gg
Gg 5
βgqA τ 2
ηcp
β (K21) volume thermal expansion coefficient; g (m s22) gravitational acceleration; qA (W m22) heat flux density; τ (s) time; η (Pa s) dynamic viscosity;
cp (J kg21 K21) specific heat capacity.
It characterizes the conditions for the origin of the development of free convection in a horizontal layer of a fluid or gas which is heated from below.
Info: [A23],[A33].
Ulrich Grigull (12.3.191220.10.2003), German engineer.
He was engaged in research on thermodynamics and heat
transfer, and in research on turbulent film condensation. He
published the book Grundgesetze der Wärmeűbertragung
(Basic Laws of Heat Transfer). His research included the
investigation of water and water steam properties, thermal
physical properties such as viscosity under high pressure
and temperature and determination of critical water steam
properties.
5.2.7
Marangoni Number Mr
Mr 5
dσ ΔTL
Δσ ΔT L2
5
dT ηa
ΔT ΔL ηa
σ (N m21) surface tension; ΔT (K) temperature difference between upper surface
layer and lower one; L (m) characteristic length (layer thickness); η (Pa s) dynamic viscosity; a (m2 s21) thermal diffusivity.
It is the criterion for mutual influence of capillary forces and molecular friction
forces. It characterizes the thermal capillary phenomena on a fluid-free surface and
Thermomechanics
187
the surface tension influence on the origin of the free convection in a thin fluid
layer. The Mrcrit 5 80 is the critical value leading to instability.
Info: [A4],[A29],[A35],[B20],[C7].
Carlo Guiseppe Matteo Marangoni (18401925), Italian physicist.
5.2.8
Rayleigh Number (2.) (Heat Instability) Ra2
Ra2 5
gL3 ΔTβ
gL3 ΔR
5
5 GrPr
νa
ηa
g (m s22) gravitational acceleration; L (m) characteristic length; ΔT (K) temperature difference; β (K21) volume thermal expansion coefficient;
ν (m2 s21) kinematic viscosity; a (m2 s21) thermal diffusivity; ΔR (kg m23) density difference; η (Pa s) dynamic viscosity; Gr () Grashof heat number
(p. 185); Pr () Prandtl number (p. 197).
It characterizes the free convection heat transfer along a heat-exchanging surface. It expresses the buoyancy-to-diffusion ratio or, alternatively, the free convection thermal instability in fluids. In a closed space with Ra , 103, the heat is
transferred between warmer and cooler walls by conduction only.
Info: [A4],[A23],[A29],[A43],[B20].
Lord Rayleigh, John William Strutt (12.11.1842
30.6.1919), British physicist. Nobel Prize in Physics, 1904.
He was engaged in research on optics, acoustics, electromagnetism and molecular light dispersion. He discovered
one of the radiation laws for perfect black body radiation in
the long-wave zone. He intensively studied Maxwell’s electromagnetism theory, both theoretically and experimentally.
He researched the density of rare gases and discovered
argon.
5.2.9
Rayleigh Number (3.) Ra3
Ra3 5
qA L5 gβ
5 GrNuPrX 21
νλax
qA (W m22) heat flux density; L (m) characteristic length, pipe diameter;
g (m s22) gravitational acceleration; β (K21) volume thermal expansion coefficient; ν (m2 s21) kinematic viscosity; λ (W m21 K21) thermal conductivity;
a (m2 s21) thermal diffusivity; x (m) distance from inlet to vertical tube;
Gr () Grashof heat number (p. 185); Nu () Nusselt number (p. 196);
Pr () Prandtl number (p. 197); X () geometrical coordinates (p. 15).
188
Dimensionless Physical Quantities in Science and Engineering
It characterizes the combined free and forced convections in vertical tubes.
Info: [A4],[A23].
Lord Rayleigh (see above).
5.2.10 Rayleigh Number (4.) Ra4
Ra4 5
qV L5 gβ
5 GrPrPoV
vλa
qV (W m23) volume density of heat flux; L (m) characteristic length;
g (m s22) gravitational acceleration; β (K21) volume thermal expansion coefficient; v (m2 s21) kinematic viscosity; λ (W m21 K21) thermal conductivity;
a (m2 s21) thermal diffusivity; Gr () Grashof heat number (p. 185); Pr () Prandtl number (p. 197); PoV () Pomerantsev heat number (p. 181).
It characterizes the free convection in a horizontal fluid layer with a constant
internal volume heat source.
Info: [A4],[A23],[A29],[A35].
Lord Rayleigh (see above).
5.2.11 Schwarzschild Number Sch
Sch 5
21
gT @v
dT
cp v @T p dl
g (m s22) gravitational acceleration; T (K) temperature; cp (J kg21 K21) specific
heat capacity; v (m3 kg21) specific volume; l (m) length.
This number characterizes the stability conditions for a fluid mechanical balance
with its longitudinally changing temperature or the condition under which no thermal convection occurs in the fluid. In a fluid, over the horizontal surface of a wall,
of which the temperature changes perpendicularly to the surface up to the free fluid
level, the following condition is valid to cause convection
dT gT dv
.
dl c v dT
p
p
Info : [A23],[A33].
Thermomechanics
189
Karl Schwarzschild (9.10.187311.5.1916), German
astronomer, physicist and mathematician.
He was focused on electrodynamics, optics and the radiation of stars related to stellar atmosphere research. He
applied photography to measure the instability of stars and
investigated internal star structures and star gas equilibrium.
He devoted himself also to relativity theory, and the
Schwarzschild radius, which represents the critical gravitational solid-body radius at which a body becomes a black
hole according to general relativity theory, is named after
him.
5.2.12 Thompson Number Th
See Marangoni number Mr (p. 186).
Info: [A29].
5.3
Forced Convection
The dimensionless quantities expressing forced convection heat transfer are among
the most widely used. Forced convection is characterized by a flow with mutual
action of inertia and viscous forces. The predominance of the viscous flow characterizes a laminar flow. On the contrary, turbulent flow arises under the action of
external influences and its development depends on the viscous-to-inertia forces
ratio exclusively. Among the numerous dimensionless quantities, the Reynolds,
Prandtl, Colburn, Dulong, Eckert, Nusselt, Fliegner, Graetz, Stanton and other
numbers are especially important.
5.3.1
Bansen Heat Number Ba
Ba 5
T1 5 T2
T 2 Ts
T1, T2 (K) input and output temperatures in channel; T, Ts (K) fluid flow and
channel wall temperatures.
This number expresses the ratio of the temperature gradient between a channel
inlet and outlet to the temperature gradient between the fluid and the channel wall.
It characterizes the convection heat transfer intensity for fluid flowing in channels.
Info: [A14],[A23],[A29].
R.V. Bansen.
190
Dimensionless Physical Quantities in Science and Engineering
5.3.2
Boundary Layer Number N
N5
τsL
wN η
τ s (N m21) shear surface stress; L (m) characteristic length; wN (m s21) flow velocity far from boundary layer; η (Pa s21) dynamic viscosity.
It characterizes the surface friction in flow in a laminar boundary layer. It is
analogous to the Bingham number Bm (p. 118).
Info: [A23].
5.3.3
Colburn Number JQ, Jh
JQ Jh 5
2
2
1
α
cp η 3
5 StPr 3 5 NuRe21 Pr2 3
Rcp w λ
α (W m22 K21) heat transfer coefficient; R (kg m23) density; cp (J kg21 K21) specific heat capacity; w (m s21) fluid flow velocity; η (Pa s) dynamic viscosity;
λ (W m21 K21) thermal conductivity; St () Stanton number (p. 201);
Pr () Prandtl number (p. 197); Nu () Nusselt number (p. 196); Re () Reynolds number (p. 81); Jh () J-heat transfer factor (p. 195).
It characterizes the heat transfer in forced and free flows of a viscous fluid. The
Colburn number is identical with the J-heat transfer factor (p. 195).
Info: [A23],[A29].
Allan Philip Colburn (8.6.19041955), American chemical engineer.
He was engaged in water steam condensation from a saturated air flow, which was his main interest during his
entire life. It was due to him that the common foundations
of momentum, heat and mass transfer, following from thermodynamic principles, were formulated in the US for the
first time.
5.3.4
Crispation Group Ncr
Ncr 5
ηa
σL
η (Pa s) dynamic viscosity; a (m2 s21) thermal diffusivity; σ (kg s22) undisturbed surface tension; L (m) characteristic length (thin layer thickness).
It expresses the possibility of fluid crispation (tearing) originating in convective
flow and heat transfer.
Info: [A23],[A35].
Thermomechanics
5.3.5
191
Dulong Number Du
Du rh21 5
w2
cp ΔT
w (m s21) flow velocity; cp (J kg21 K21) specific heat capacity; ΔT (K) temperature difference.
This number expresses the ratio of the flowing fluid kinetic energy to the
enthalpy change in adiabatic compression. It characterizes the convection heat
transfer between a wall and fast flowing compressible gas with viscous energy dissipation and adiabatic compression. It represents the volume density of the heat
flow that arises and the heating of the bypassed surface. It corresponds to the
Eckert number Ec (p. 191). Its inverse value corresponds to the restitution enthalpy
coefficient rh (p. 198).
Info: [A23],[A29],[B20].
Pierre Louis Dulong (12.2.178519.7.1838), French physicist and chemist.
Above all, he is well known for the DulongPetit law
(1819), which joins the specific thermal capacity of metals
with their molar mass. It is a classic relation that expresses
the specific thermal crystal capacity dependence on the grid
oscillating frequency. Despite its simplicity, this relation provides a very good result for solid materials with relatively
simple crystal structure in the high-temperature range.
5.3.6
Eckert Number Ec
Ec 5
2
wN
5 2Rf
cp ΔT
2
Ec 5 ðκ 21ÞMN
ð1Þ;
TN
ΔTad
Tad;N 2 TN
52
52
ΔT
ΔT
TN 2 T
ð2Þ
wN (m s21) fluid flow velocity far from body; cp (J kg21 K21) specific heat
capacity of fluid; ΔT (K) temperature difference; T, Tad, TN (K) static, adiabatic and far from body temperatures; k () specific heat ratio (p. 28); Rf () temperature recovery factor (p. 202).
It expresses the ratio of kinetic energy to a thermal energy change. Ec1 is a
special case of the Dulong number Du (p. 191) for w 5 wN. Ec2 is valid for
a compressible fluid.
Info: [A23],[A26],[A29],[A35].
192
Dimensionless Physical Quantities in Science and Engineering
Ernst Rudolf Georg Eckert (13.9.19048.7.2004),
American engineer of Czech-German origin.
He researched a wide range of heat transfer problems,
from cryogenic temperatures to plasma. His research
involves basic studies on humidity migration in porous materials and on related problems of gas turbine cooling. His
principal works are Introduction to the Transfer of Heat and
Mass (1959) and Analysis of Heat and Mass Transfer (1972).
5.3.7
Fliegner Number Fl
Fl 5
Rw pffiffiffiffiffiffiffiffiffiffi
cp Tst ð1Þ;
pst
Fl 5
Rw pffiffiffiffiffiffiffiffi
cp T
p
Fl 5
pffiffiffiffiffiffiffiffi
Rw
cp T ð3Þ;
2
ðps 1 Rw Þ
ð2Þ;
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffi
Q m cp T
kM
ðk 21ÞM 2
p
ffiffiffiffiffiffiffiffiffi
ffi
5
1
1
Fl 5
Aðps 1 Rw2 Þ
2
k 21
ð4Þ
R (kg m23) density; w (m s21) flow velocity; pst (Pa) stagnation absolute
pressure of gas flow; cp (J kg21 K21) specific heat capacity; Tst (K) stagnation
temperature; p (Pa) static absolute pressure; T (K) absolute temperature;
ps (Pa) static pressure; Qm (kg s21) mass flux; A (m2) clear area; k () specific heat ratio (p. 28); M () Mach number (p. 73).
It characterizes the coupling between the fluid flow and the heat transfer. It
expresses the ratio of the heat related to the velocity unit to the fluid momentum
flow. In expression (1), it is the stagnation number; in expression (2), it is the static
number; and in expression (3), it is the impulse number. Generally, the Fl 5 f(M)
expressed in form (4) holds for the compressible fluid flow.
Info: [A23],[A29],[A35],[B20].
Albert Friedrich Fliegner (18421928), Swiss engineer.
5.3.8
Frőssling Heat Number Fs
Fs 5
Nu 22
1
1
Re2 Pr 3
ð1Þ;
Thermomechanics
Fs 5
Nu
1
193
ð2Þ
Re2
Nu () Nusselt number (p. 196); Re () Reynolds number (p. 81); Pr () Prandtl number (p. 197).
It expresses the influence of the fluid molecular properties and character of the
flow in a boundary layer on the average value of the specific thermal transference.
Expression (1) holds for turbulent flow around a cylinder or a ball. Expression (2)
is valid for an isothermic plate with laminar flow in a boundary layer.
Info: [A23],[A29],[A35].
Karl Gustav Frőssling (born 1913), Swedish engineer.
5.3.9
Graetz Number Gz
Gz 5
Qm cp
cp RwA
5
λL
λL
ð1Þ;
Gz 5
wD2
D
5 RePr
aL
L
ð2Þ
Qm (kg s21) mass flux; cp (J kg21 K21) specific heat capacity; λ (W m21 K21) thermal conductivity; L (m) characteristic length; ρ (kg m23) density; w (m s21)
velocity; A (m2) area; D (m) pipe diameter; a (m2 s21) thermal diffusivity;
Re () Reynolds number (p. 81); Pr () Prandtl number (p. 197).
This number expresses the ratio of the fluid thermal capacity to its thermal conductivity. It characterizes the heat transfer in the laminar flow of fluids. The relation (2) is valid for a circular pipeline. A large Gz number represents a
predominating convection influence. A small Gz number indicates to a predominating heat conduction influence. In the last case, the temperature does not change
along the path and the thermal field is fully developed. The Gz number is used
mainly in the American science texts, in which it often replaces the Pe´clet heat
number Pe (p. 180).
Info: [A23],[A29],[A35],[B20].
Leo Graetz (26.9.185612.11.1941), German physicist.
His scientific work was focused on heat transfer by conduction and radiation, and on the problems of friction and
elasticity. For a long time, he was devoted to problems of
electromagnetic waves and cathode radiation. His publications Electricity and Magnetism and Electricity and its
Applications are especially significant.
194
Dimensionless Physical Quantities in Science and Engineering
5.3.10 Heat Non-Stationarity Number (1.) NT
NT 5
@TS
L2
@τ ðTS 2 Tmed Þa
TS (K) wall temperature; τ (s) time; L (m) characteristic length; Tmed (K) mean fluid temperature; a (m2 s21) thermal diffusivity.
This number characterizes the influence of the time-dependent temperature of a
wall on the non-stationary heat transfer of single-phase fluids in passageways.
Info: [A23].
5.3.11 Heat Non-Stationarity Number (2.) NZ
NZ 5
@TS
L
@z TS 2 Tmed
Ts (K) wall temperature; z (m) axis along the pipeline; L (m) characteristic
length; Tmed (K) mean fluid temperature.
It characterizes the influence of the temperature gradient change in a wall on the
heat transfer of single-phase fluids in passageways.
Info: [A23].
5.3.12 Heat Source Effect Number F
F5
DqV L
2Rcp νΔT
D (m) tube diameter; qV (W m23) volume density of heat flux; L (m) characteristic length; R (kg m23) density; cp (J kg21 K21) specific heat capacity;
ν (m2 s21) kinematic viscosity; ΔT (K) temperature difference.
It characterizes the heat source influence on the heat transfer by convection in
wholly developed laminar fluid flow from below upwards in inclined tubes with
uniformly distributed inner sources and uniform thermal flow on the walls. With
increasing parameter F, the Nusselt number Nu (p. 196) drops. With F 5 const, the
heat transfer intensity depends on the Rayleigh number (2.) (heat instability) Ra2.
With large Ra2 (p. 187) numbers and arbitrary F, the tube orientation influences
the heat transfer intensity.
Info: [A23].
5.3.13 Heat Transfer Efficiency N
N5
αLRw2
5 BiEu21
λΔp
Thermomechanics
195
α (W m22 K21) heat transfer coefficient; L (m) characteristic length;
R (kg m23) density; w (m s21) velocity; λ (W m21 K21) thermal conductivity; Δp (Pa) pressure difference; Bi () Biot number (p. 173); Eu () Euler number (1.) (p. 61).
It characterizes the external heat transfer intensity related to system hydraulic
resistance.
Info: [A23],[A33].
5.3.14 Hydrodynamic Non-Stationarity Number NG
NG 5
@Qm L2
@τ νQm
Qm (kg s21) mass flux; τ (s) time; L (m) characteristic length; ν (m2 s21) kinematic viscosity.
It expresses time change in the passage of a mass which influences the nonstationary heat transfer of single-phase fluid.
Info: [A23].
5.3.15 J-Heat Transfer Factor Jh
2
α
cp η 3
Jh JQ 5
c p qm λ
α (W m22 K21) heat transfer coefficient; cp (J kg21 K21) specific heat capacity; qm (kg m22 s21) mass flux density; η (Pa s) dynamic viscosity;
λ (W m21 K21) thermal conductivity; JQ () Colburn number (p. 190).
Generally, it characterizes the heat transfer in free and forced flows. It is identical with the Colburn number JQ (p. 190).
Info: [A35],[B20].
5.3.16 Leibenzon Number (1.) Lb
Lb 5 Nu
Pr 1 2
Pr
Nu () Nusselt number (p. 196); Pr () Prandtl number (p. 197).
It is an extended modification of the Nusselt number Nu (p. 196), with which
convective and diffuse heat transfers are considered.
Info: [A23],[A33].
Leonid Samuilovich Leibenzon (18791951), Russian physicist and
mechanist.
196
Dimensionless Physical Quantities in Science and Engineering
5.3.17 Margoulis Number Mg
Mg 5 St 21 5
cp Rw RePr
5
α
Nu
cp (J kg21 K21) specific heat capacity; R (kg m23) fluid density; w (m s21) fluid flow velocity; α (W m22 K21) heat transfer coefficient; St () Stanton
number (p. 201); Re () Reynolds number (p. 81); Pr () Prandtl number (p.
197); Nu () Nusselt number (p. 196).
It expresses forced convection. It is also called the thermal resistance number.
Info: [A23],[A35].
Wladimir Margoulis (born 1886).
5.3.18 Nusselt Number Nu
Nu 5
αL
5 ReStPr 5 PeSt
λ
α (W m22 K21) heat transfer coefficient; L (m) characteristic length;
λ (W m21 K21) thermal conductivity; Re () Reynolds number (p. 81); St ()
Stanton number (p. 201); Pr () Prandtl number (p. 197); Pe () Péclet
heat number (p. 180).
It expresses the ratio of the total heat transfer in a system to the heat transfer by
conduction. In characterizes the heat transfer by convection between a fluid and the
environment close to it or, alternatively, the connection between the heat transfer
intensity and the temperature field in a flow boundary layer. It expresses the
dimensionless thermal transference. The physical significance is based on the idea
of a fluid boundary layer in which the heat is transferred by conduction. If it is not
so, the criterion loses its significance. In the expression α (λ/L)21, it expresses the
ratio of the heat transfer intensity to heat conduction intensity in a boundary layer.
In the expression L (λ/α)21, it represents the ratio of the characteristic length to the
boundary layer thickness. In the expression L /λ (1/α)21, it expresses the ratio of
thermal resistances by conduction to those by convection in a boundary layer.
Info: [A4],[A29],[A35],[A43],[B20].
Ernst Kraft Wilhelm Nusselt (25.11.18821.9.1957),
German engineer.
In 1915, he published an important work on basic heat
transfer laws. In this work, he first suggested the dimensionless thermal similarity criteria. He was engaged in research
on the analogy between the transfer of heat and mass, and
convective heat transition. He formulated the basic theory of
heat regenerators. He was also engaged in work related to
thermal processes in membrane steam condensation and in
powder fuel combustion.
Thermomechanics
197
5.3.19 Prandtl Number Pr
Pr 5
ηcp
ν
5 5 PeRe 21
λ
a
η (Pa s) dynamic viscosity; cp (J kg21 K21) specific heat capacity;
λ (W m21 K21) thermal conductivity; ν (m2 s21) kinematic viscosity; a (m2 s21) thermal diffusivity; δ, δt (m) thicknesses of thermodynamic and thermal boundary
layer; Pe () Péclet heat number (p. 180); Re () Reynolds number (p. 81).
This number expresses the ratio of the momentum diffusivity (viscosity) to the
thermal diffusivity. It characterizes the physical properties of a fluid with convective and diffusive heat transfers. It describes, for example, the phenomena connected with the energy transfer in a boundary layer. It expresses the degree of
similarity between velocity and diffusive thermal fields or, alternatively, between
hydrodynamic and thermal boundary layers. With Pr 5 1 and grad p 5 0, the thermal and hydrodynamic fields are similar. For example, if diverse molten materials
have equal Prandtl numbers, they have similar velocity and temperature fields in
crystallization. The following are valid:
Pr{1
Pr61
Prc1
δ{δt ;
δ 5 δt ;
δcδt :
With small Pr numbers (Pr , 1), the molecular heat transfer by conduction
predominates over that by convection. With Pr . 1, it is the opposite case.
For example, for air and many other gases, the Pr value is about 0.7, for
coolants PrAh2; 10i, for machine oils PrAh102; 4 3 104i, and for mercury it is
approximately 15 3 1023.
Info: [A23],[A29],[A43],[B20].
Ludwig Prandtl (4.2.187515.8.1953), German physicist.
His principal works relate to the hydrodynamic and thermal boundary layers, turbulent boundary layer analysis and,
especially, heat transfer. He published (1910) his elaboration of Reynolds’ original concept (1874) of the analogy
between turbulent momentum and heat transfer. He applied
modelling methods extensively, above all the hydraulic
analogy to visualize flow.
5.3.20 Prandtl Resultant (Effective, Total) Number Pref
Pref 5
εM 1 ν
εT 1 a
198
Dimensionless Physical Quantities in Science and Engineering
εM (m2 s21) vortex momentum diffusivity; εT (m2 s21) vortex thermal diffusivity; ν (m2 s21) kinematic viscosity; a (m2 s21) thermal diffusivity.
This number expresses the ratio of the resulting momentum diffusivity to the
resulting thermal diffusivity in mixed laminar and turbulent flow. It characterizes
the influence of the physical properties of a fluid on convective heat transfer and
momentum transfer under turbulent flow conditions.
Info: [A23],[A29],[A33].
Ludwig Prandtl (see above).
5.3.21 Prandtl Turbulent Number Prtur
Prtur Scv 5
εM
L
5
εT
LT
εM (m2 s21) vortex momentum diffusivity; εT (m2 s21) vortex thermal diffusivity; L, LT (m) characteristic lengths for momentum and heat transfer by mixing;
Scv () Schmidt turbulent number (p. 264).
It expresses the ratio of turbulent momentum diffusivity to turbulent thermal diffusivity. It characterizes the influence of turbulent flow and turbulence in heat
transfer.
Info: [A23],[A29],[A33].
Ludwig Prandtl (see above).
5.3.22 Rayleigh Number (1.) Ra1
rffiffiffiffiffiffi
RL
Ra1 5 w
σ
w (m s21) flow velocity; R (kg m23) liquid density; L (m) characteristic
length; σ (N m21) surface tension.
It characterizes the condition of fluid jet decomposition. It is equivalent to the
Weber number (2.) We2 (p. 91) and it is often called the Rayleigh parameter.
Info: [A23].
Lord Rayleigh (p. 187).
5.3.23 Restitution Enthalpy Coefficient rh
rh 5
ha 2 hl
hst 2 hl
ð1Þ;
Thermomechanics
rh 5
cp ðTaS 2 Tl Þ
w2
199
ð2Þ
ha (J) specific adiabatic enthalpy of high-speed fluid flow on the wall; hl (J) specific enthalpy of fluid flow; hst (J) specific enthalpy of retarded flow;
cp (J kg21 K21) specific heat capacity; TaS (K) adiabatic temperature of circumfluenced surface; Tl (K) static temperature of fluid flow; w (m s21) flow velocity.
This coefficient expresses the ratio of the fluid flow adiabatic enthalpy on a wall
to the sum of the total flow enthalpy and the flow kinetic energy or, alternatively,
the ratio of the real conversion temperature to the theoretical conversion temperature. It characterizes the change of the specific fluid flow enthalpy when a high
velocity flow bypasses a wall; see the Eckert number Ec (p. 191) and the Dulong
number Du (p. 191). It expresses the ratio of the real flow heating due to conversion to the ideal gas heating. It is used for the convective heat transfer in compressible fluid flow.
Info: [A23].
5.3.24 Restitution Temperature Coefficient r
r5
Taw 2 Tl
Tst 2 Tl
Taw (K) adiabatic wall temperature; Tst (K) stagnation temperature of fluid
flow; Tl (K) static fluid temperature.
It expresses the ratio of the heating of a real wall due to flowing fluid kinetic
energy to the highest theoretical heating of the wall. It characterizes the heating
degree of a wall surface due to flowing fluid kinetic energy in relation to the highest theoretical heating.
Info: [A23].
5.3.25 Shukhov Number Su
Su 5
πkdL
Qcp R
k (W m22 K21) heat passage coefficient; d (m) pipe diameter; L (m) characteristic length; Q (m3 s21) volume flow; cp (J kg21 K21) specific heat capacity;
R (kg m23) density.
It characterizes the influence of the radial and axial temperature gradients, in
a heated or cooled fluid flowing through a pipeline, on the hydraulic resistance
(pressure loss) in a considered pipeline section.
Info: [A23],[A33].
Vladimir Grigoryevich Shukhov (18531939), Russian engineer.
200
Dimensionless Physical Quantities in Science and Engineering
5.3.26 Spalding Function Sp
@Θ
Sp 5 2
@u1 u150
T 2 TN
where Θ 5
;
Tw 2 TN
rffiffiffi
R
u 5w
τ
1
T (K) local temperature; TN (K) free stream temperature; Tw (K) wall temperature; w (m s21) velocity; τ (Pa) shear stress; R (kg m23) density;
u1 () Prandtl velocity ratio (p. 80).
This function characterizes the dimensionless temperature gradient in a wall
with forced convection.
Info: [A23].
Dudley Brian Spalding (see above).
5.3.27 Spalding Number (1.) Sp
Sp
0 12 1
1
2
2
αν
αν @τ A
2
pffiffiffiffiffiffiffiffiffiffi 5
5
5 StPrNe
λ
R
wλ 0:5fF
5 NuRe
21
2
Ne
1
2
5 NuPe
21
2
PrNe
1
2
α (W m22 K21) heat transfer coefficient; ν (m2 s21) kinematic viscosity;
w (m s21) velocity; λ (W m21 K21) thermal conductivity; fF () Fanning
friction number (p. 163); St () Stanton number (p. 201); Pr () Prandtl number (p. 197); Ne () Newton number (p. 75); Nu () Nusselt number (p. 196);
Re () Reynolds number (p. 81); Pe () Péclet heat number (p. 180).
It characterizes the convective heat transfer intensity in a fully developed boundary
layer.
Info: [A23].
Brian Spalding (born 1923), English engineer.
Initially, he devoted himself to the liquid fluid combustion problem and to thermodynamics. He published a whole
range of original scientific works concerning this sphere.
He has also worked on the transfer of heat and mass,
boundary layer theory and the fluid dynamics simulations,
making use of the rapid evolution of calculation techniques
in recent years.
Thermomechanics
201
5.3.28 Stanton Number St
St Mg 21 5
α
Nu
5
5 NuFo
Rcp w RePr
St 5
αS
qmV Vcp
St 5
qA
Rwcp ðTr 2 Ts Þ
ð1Þ;
ð2Þ;
ð3Þ
α (W m22 K21) heat transfer coefficient; R (kg m23) density; cp (J kg21 K21) specific heat capacity; w (m s21) fluid velocity; S (m2) surface area;
qmV (kg m23) volume density of mass flux; V (m3) volume; qA (W m22) surface heat flux density; Tr (K) reference temperature; TS (K) wall temperature; Nu () Nusselt number (p. 196); Re () Reynolds number (p. 81);
Pr () Prandtl number (p. 197); Fo () Fourier number (p. 175).
It expresses the ratio of the heat transferred in a system by convection to the
thermal capacity of heat-bearing surroundings. It characterizes the forced convection heat transfer. In form (2), it is used to express the convective heat transfer in
burning chambers and in steam boiler spaces. In form (3), it expresses the heat
flow through a laminar boundary layer when solid bodies are passed by with fluid.
In form (1), it is also called the Margoulis number Mg (p. 196).
Info: [A4],[A21],[A43],[B11],[B20].
Thomas Edward Stanton (12.12.18651931), English
engineer.
His principal interest was the flowing of fluids and the
friction problem, with its related heat transfer. From 1902
to 1907, he was engaged in wide research on the influence
of wine on structures such as bridges and roofs. After the
first flight of the Wright brothers in an aircraft in 1908, he
devoted himself enthusiastically to airplane and airship
design, and above all to the heat transfer in air-cooled
engines.
5.3.29 Temperature Factor ΘS
ΘS 5
TS
Tr
ð1Þ;
ΘS 5
TS
TaS
ð2Þ
202
Dimensionless Physical Quantities in Science and Engineering
TS (K) surface temperatures of body wall; Tr (K) characteristic temperature of
fluid flow; TaS (K) adiabatic temperature of body surface.
It expresses the ratio of the absolute surface temperature of a body wall to the
characteristic absolute temperature of a flowing fluid. It characterizes the influence
of the flowing fluid temperature on a body. Instead of the relative temperature Tr,
sometimes the adiabatic temperature of a solid, ideally insulated and non-radiating
wall is considered, which is passed by a fluid with heat development caused by
energy dissipation. With outer bypassing of a body, usually the outer flow temperature is chosen as the characteristic temperature Tr. The mean flow temperature is
chosen as the characteristic temperature in flow in tubes and canals.
Info: [A23].
5.3.30 Temperature Recovery Factor Rf
Rf 5
2cp ΔT
5 2Ec 21
w2
cp (J kg21 K21) specific heat, capacity of fluid flow; ΔT (K) difference
between static temperature of flowing gas and adiabatic temperature of circumfluenced wall; w (m s21) fluid velocity; Ec () Eckert number (p. 191).
For an ideal gas, this factor expresses the ratio of the real temperature of a
bypassed wall to the theoretical temperature.
Info: [A15].
5.3.31 Thermal Coupling Number K, χ
K 5 0:5
λl 1
Re2 ;
λs
KAh0; NÞ
λl, λs (W m21 K21) specific thermal conductivity of fluid and solid; Re ()
Reynolds number (p. 81).
This number characterizes the influence of thermally physical properties on the
convective heat transfer in compound tasks. It occurs in compressible fluid flow
when considering the mechanical energy dissipation and heat propagation in a solid
body. Usually, its value is approximately K 5 1. The case of K-N (λs-0) occurs
when a perfect thermal insulator is inserted into the fluid flow.
Info: [A23].
5.3.32 Heat Transfer Number Nq
Nq 5
q
Rw3 L2
Thermomechanics
203
q (W) heat flux; R (kg m23) density; w (m s21) flow velocity; L (m) characteristic length.
It expresses the ratio of the heat transferred in a system to the kinetic energy of
the flowing environment. Heat transfer in flowing fluid.
Info: [A23],[A29],[A33].
5.3.33 Unsteady Heat Transfer Number N
N5
Nu
Nu0
Nu () Nusselt unsteady number, see Nusselt number (p. 196); Nu0 () Nusselt quasi-stationary number, see Nusselt number (p. 196).
It expresses the ratio of unsteady heat transfer to quasi-stationary heat transfer.
It characterizes the dimensionless transfer in unsteady flow.
Info: [A23].
5.4
Radiation
Here radiation denotes thermal radiation, of which the main aspect is electromagnetic radiation. Besides conduction and convection, it describes the third kind of
heat transfer, which is of use with high surface temperatures especially. Among the
basic dimensionless quantities are the Boltzmann and Stefan numbers. The former
expresses the relation of the heat transferred by convection and radiation, whereas
the latter expresses that transferred by radiation and conduction. The Bansen,
Bouguer, Nusselt radiation, Pe´clet radiation and Schuster numbers are also among
the dimensionless values, and so are the radiation pressure and radiation viscosity
numbers.
5.4.1
Bansen Radiation Number Ba
Ba 5
αR S
Qm cp
αR (W m22 K21) radiation heat transfer coefficient; S (m2) surface area of
channel wall; Qm (kg s21) mass flux; cp (J kg21 K21) specific heat capacity.
It expresses the ratio of the heat transferred by radiation to the fluid thermal
capacity.
Info: [A4],[A35],[B20].
R.V. Bansen.
204
Dimensionless Physical Quantities in Science and Engineering
5.4.2
Biot Radiation Number BiR
See the Stefan number Sf (p. 213).
Jean-Baptiste Biot (p. 174).
5.4.3
Boltzmann Number Bo
Bo Tg 5
cp Rw
εσ0 T 3
ð1Þ;
Bo 5 4NRePrτ 21 5 4NPeτ 21
Bo 5
3
τBo1
16
Bo 5
Bo1
2τ
Bo 5
Rwðh1 2 h2 Þ
εσ0 T 4
Bo 5
εσ0 SR T 3
Q m cp
ð2Þ;
ð3Þ;
ð4Þ;
ð5Þ;
ð6Þ
cp (J kg2 1 K2 1) specific heat capacity; R (kg m2 3) flue gases density;
w (m s21) flue gases velocity; σ0 (W m22 K24) StefanBoltzmann constant;
T (K) flue gas thermodynamic temperature; τ () optical thickness, ratio of
characteristic length to mean free path; h1, h2 (J kg21) specific enthalpy of flue
gases; SR (m2) radiation surface; Qm (kg s21) mass flux; ε () emissivity
(p. 206); Tg () Thring radiation number (p. 214); N () radiation number
(2.) (p. 211); Re () Reynolds number (p. 81); Pr () Prandtl number
(p. 197); Pe () Péclet heat number (p. 180).
This number expresses the ratio of the heat transferred by forced convection to
that transferred by radiation. It characterizes the relation between the convective
energy transfer (in the flow direction) and the heat radiation transfer. It expresses
also the ratio of the absorbing surroundings’ enthalpy to the thermal flow radiated
by a surface which limits the surroundings. In heat transfer by radiation, it has similar significance to the Pe´clet heat number Pe (p. 180) in heat conduction.
Characteristic optical thickness is important for determining the relative ratio of the
heat radiation transfer to the convective transfer for optically thick (τc1) or
optically thin (τ{1) layers. This ratio is expressed by the modified Bo number (3)
for an optically thick layer or by (4) for a thin one. Sometimes, the modified
enthalpy expression (5) is used. For radiation in combustion spaces, expression (6)
is applied.
Info: [A23],[B20].
Thermomechanics
205
Ludwig Boltzmann (20.2.18445.10.1906), Austrian
physicist.
He founded gas kinetic theory and statistical physics. He
was engaged in entropy and probability. He deduced one of
the radiation laws now known as the Boltzmann law. His
works on electromagnetism and thermodynamics are especially important.
5.4.4
Bouguer Number (1.) Bu1
Bu1 5
3Ri lR
2Rm d
Ri (kg m23) mass concentration of particles in reference gas volume; lR (m) mean length of radiation path; Rm (kg m23) density of mass particles; d (m) diameter of mass particles.
It relates to fluid flow with mass particles. It characterizes the heat transfer by
radiation into mass particles in flowing gas.
Info: [A23],[A29],[B20].
Pierre Bouguer (16.2.169815.8.1758), French mathematician, astronomer and physicist.
He was engaged in research on earth density, photometry, astronomical photometry and general light propagation
problems. He is often considered the founder of photometry. The Bouguer law expresses the relation between radiation energy absorption and the absorbing surroundings.
5.4.5
Bouguer Number (2.) Bu2
Bu2 5 κL
κ (m21) attenuation coefficient of environment including absorption and dispersion coefficients; L (m) reference length.
It expresses the ratio of the characteristic relative length of a ray path to the free
ray path caused by absorption and radiation scatter. It characterizes the degree of
attenuation (absorbance) of the surroundings in heat radiation transfer in a gas. It is
analogous to the Knudsen number (1.) Kn (p. 69) in gas dynamics.
Info: [A23],[A29].
Pierre Bouguer (see above).
206
Dimensionless Physical Quantities in Science and Engineering
5.4.6
ε5
Emissivity ε
εgb
5 1 2 r;
εbb
where εAh0; 1i
εgb () grey body absorptivity; εbb () ideal black body absorptivity; r () surface reflectance; Q (W m22) heat flux transferred by radiation;
σ0 (W m22 K24) StefanBoltzmann constant; T1, T2 (K) temperature of radiating and absorbing surfaces.
It expresses the relative radiativity (absorbance) of the surface of grey bodies. It
is a non-linear function of the surface temperature and depends on the radiation
wave length. For the amount of heat transferred by radiation, the
StefanBoltzmann law is valid.
Q 5 εσ0 ðT14 2 T24 Þ
Thermomechanics. Radiation. Contactless measurement of surface temperatures.
Info: [C4].
5.4.7
Fourier Radiation Number FoR
FoR 5 aκ2 τ
a (m2 s21) thermal diffusivity; κ (m21) attenuation coefficient; τ (s) time.
It expresses the ratio of the radiation heat transferred in a system of molecular
diffusion to the radiation heat absorbed in the system. In the dimensionless form, it
characterizes the time of non-stationary heat transfer by radiation. It determines the
relation between the velocity due to temperature field radiation and the physical
parameters of the system.
Info: [A23].
Jean Baptiste Joseph Fourier (p. 175).
5.4.8
Hottel Number Hot
Hot 5
α
εσ0 T 3
α (W m22 K21) convective heat transfer coefficient; σ0 (W m22 K24) StefanBoltzmann constant; T (K) absolute temperature; ε () emissivity
(p. 206).
It expresses the ratio of heat flows by convection to those by radiation. Heat
transfer by radiation. Thermomechanics. High-temperature devices. Industrial
furnaces.
Info: [A2].
H.C. Hottel.
Thermomechanics
5.4.9
207
Kirpichev Radiation Number KiR
KiR N21 5
σT 3
5 SfBu21
2
λκ
σ (W m22 K24) absorptance (absorption capacity, radiating capacity); T (K) temperature; λ (W m21 K21) thermal conductivity; κ (m21) attenuation coefficient; N () radiation number (2.) (p. 211); Sf () Stefan number (p. 213);
Bu2 () Bouguer number (2.) (p. 205).
It expresses the ratio of the heat transferred by radiation to that transferred by
molecular conduction in a turbulent flow. It characterizes the heat transfer by radiation in an environment with attenuation effect. Its inverse value is called the radiation number (2.) N (p. 211).
Info: [A23],[A33].
Mikhail Viktorovich Kirpichev (p. 177).
5.4.10 Local Radiation Heat Transfer ξ
ξ5
εσ0 T 3 κx
κx
Ei2 Re
5 KReEc21 5 Kη 2
5
Rcp wr
Bo
Bo Ec
σ0 (W m22 K24) StefanBoltzmann constant; T (K) temperature; κ (m21) attenuation coefficient; x (m) coordinate; R (kg m23) density; cp (J kg21 K21) specific heat capacity; wr (m s21) reference velocity; ε () emissivity (p. 206);
Bo () Boltzmann number (p. 204); K () radiation high-temperature number
(p. 210); Re () local Reynolds number, see Reynolds number (p. 81); Ec () Eckert number (p. 191); Kη () radiation viscosity (p. 212); Ei () Einstein
number (p. 317).
This transfer expresses the ratio of the heat transferred by radiation in absorbing
surroundings to that transferred by convection. It characterizes the influence of the
radiation and convective heat transfer on the local heat transfer in high-temperature
processes, where the hydrodynamic and radiation gas viscosities and gas flow rate
are considered. The thermal radiation dissipation by gas molecules is presumed to
be negligible.
Info: [A23],[A33].
5.4.11 Nusselt Radiation Number NuR
NuR 5
qA κL
ðT 2 TS ÞσT 3
qA (W m22) heat flux density; κ (m21) attenuation coefficient; L (m) characteristic length; T, TS (K) radiating environment and wall surface temperature;
σ (W m22 K24) absorptance (absorption capacity, radiating capacity).
208
Dimensionless Physical Quantities in Science and Engineering
It characterizes the radiation heat transfer intensity of the radiating turbulent
flow into a canal wall. It expresses the size of the thermal impact on the interface
between the radiating gas and a wall.
Info: [A23],[A33].
Ernst Kraft Wilhelm Nusselt (p. 196).
5.4.12 Optical Thickness of Heat Boundary Layer τ
τ 5 kδ
τ 5 5κ
ð1Þ;
rffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffi
νx
5 5 PrNξ ð2Þ
wr
κ (m21) attenuation coefficient; δ (m) thickness of thermal boundary layer;
ν (m2 s21) kinematic viscosity; x (m) coordinate; wr (m s21) reference
velocity; Pr () Prandtl number (p. 197); N () radiation number (2.)
(p. 211); ξ () local radiation heat transfer (p. 207).
It expresses the ratio of the thickness of the thinned thermal boundary layer to
that of the unthinned thermal boundary layer. In the dimensionless form, it characterizes
theffiffiffiffiffiffiexpression
pffiffiffiffiffiffi the optical thickness of the thermal boundary layer. With p
Nξc1; it is about an optically thick boundary layer, but with Nξ {1; it is
about an optically thin one. The size of the τ depends strongly on pressure. It drops
with increasing pressure and vice versa.
Info: [A23].
5.4.13 Péclet Radiation Number PeR
PeR 5
wRcp
κL
εσ0 T 3
w (m s21) velocity; R (kg m23) density; cp (J kg21 K21) specific heat capacity;
σ0 (W m22 K4) StefanBoltzmann constant; T (K) temperature; κ (m21) attenuation coefficient; L (m) characteristic length; ε () emissivity (p. 206).
It characterizes the relation between the convective and radiation heat transfers
in the turbulent flow of a radiating media.
Info: [A23],[A33].
Jean Claude Eugène Péclet (p. 180).
5.4.14 Pomerantsev Radiation Number PoR
PoR 5
qRV
κσT 4
ð1Þ;
Thermomechanics
PoR 5
qV L
Rwcp T
209
ð2Þ
qRV (W m23) volume density of radiation heat flux; κ (m21) attenuation coefficient; σ (W m22 K24) absorptance (absorption capacity, radiating capacity);
T (K) temperature; qV (W m23) volume density of heat flux; L (m) characteristic length; R (kg m23) density; w (m s21) velocity; cp (J kg21 K21) specific heat capacity.
This number expresses the ratio of the internal volume heat source to the heat
transferred by radiation. It characterizes the internal volume heat source in radiation heat transfer.
Info: [A23].
Alexey Alexandrovich Pomerantsev, Russian engineer.
5.4.15 Radiation Heat Flow (1.) QR1
QR1 5
qRA
qRA
5
4
σT
pR
qRA (W m22) radiation heat flux density; σ (W m22 K24) absorptance (absorption capacity, radiating capacity); T (K) temperature of outer environment;
pR (W m22) radiation pressure.
It expresses the ratio of the heat flow transferred by radiation to that absorbed
by a surface. It characterizes the resulting radiation heat flow in the heat transfer in
a system.
Info: [A23].
5.4.16 Radiation Heat Flow (2.) QR2
QR2 5
DR σT 3
D R aR T 3
5
λw
λ
DR (m2 s21) radiation diffusion coefficient; σ (W m22 K24) fluid absorptance
(absorption capacity, radiating capacity); T (K) temperature; λ (W m21 K21) thermal conductivity; w (m s21) flow velocity; aR (J m23 K24) radiation
constant.
It expresses the ratio of the heat transferred by flowing fluid radiation to that
transferred by conduction. It characterizes the heat transfer in various high-temperature heat-exchanging devices.
Info: [A23].
210
Dimensionless Physical Quantities in Science and Engineering
5.4.17 Radiation Heating Up (1.) N1
N1 5
σT 3 τ
cRL
σ (W m22 K24) absorptance (absorption capacity, radiating capacity); T (K) temperature; τ (s) time; cR (J m23 K21) specific volume heat; L 5 h (m) depth of warming through.
It expresses the ratio of the heat, carried to a body by radiation, to the thermal
content of the body. It characterizes the radiation heat transfer on a body surface.
Info: [A23].
5.4.18 Radiation Heating Up (2.) N2
N2 5
ðσT 3 Þ2 τ
λcR
σ (W m22 K24) absorptance (absorption capacity, radiating capacity); T (K) flue gases temperature; τ (s) time; λ (W m21 K21) thermal conductivity;
cR (J m23 K21) specific volume heat.
It characterizes the velocity of the radiation heating of bodies, where not the
geometrical body size but other thermal parameters act strongly. These parameters
are expressed by the body thermal conductivity and the boundary condition of surface heat transfer.
Info: [A23].
5.4.19 Radiation High-Temperature Number K
0
12
0
12
σT 3 ηκ ηc2 κ @ σT 3 A
η @ σT 3 A
K5 2 2 5
5
R cp
σT 4 Rcp c
ηR Rcp c
0 12 0
0 12
12
η @wA @ σT 3 A
Ei
5
5 Kη @ A
ηR c
Bo
Rcp w
σ (W m22 K24) absorptance (absorption capacity, radiating capacity); T (K) temperature; η, ηR (Pa s) dynamic (hydrodynamic) viscosity, radiation dynamic
viscosity; κ (m21) attenuation coefficient; R (kg m23) density; cp (J kg21 K21) specific heat capacity; c (m s21) light velocity; w (m s21) flow velocity;
Kη () radiation viscosity (p. 212); Ei () Einstein number (p. 317); Bo () Boltzmann number (p. 204).
Thermomechanics
211
This criterion expresses the ratio of the energy transferred by hydrodynamic and
radiation viscosities to the total energy transferred by radiation and convection. It
characterizes the influence of the hydrodynamic and radiation viscosities and the
flow rate on the resulting energy transfer by radiation and convection in hightemperature processes.
Info: [A23].
5.4.20 Radiation Number (1.) K
K5
λE
5 WeHo 21 Sf 21
εσ0 σT 3
λ (W m21 K21) thermal conductivity; E (Pa) modulus of elasticity;
σ0 (W m22 K24) StefanBoltzmann constant; σ (N m21) surface tension;
T (K) temperature; ε () emissivity (p. 206); We () Weber number
(p. 91); Ho () Hooke number (p. 138); Sf () Stefan number (p. 213).
It characterizes the radiation heat transfer (Sf ) under conditions of inertia force
and surface tension (We) and elasticity (Ho) acting on the flowing fluid.
Info: [A23].
5.4.21 Radiation Number (2.) N
N5
λκ
σT 3
λ (W m21 K21) thermal conductivity; k (m21) attenuation coefficient;
σ (W m22 K24) absorptance (absorption capacity, radiating capacity); T (K) temperature.
It expresses the ratio of heat transferred by conduction to that transferred by
radiation. It characterizes the radiation heat transfer in attenuating the surroundings.
It is analogous to the Planck number (p. 24).
Info: [A23],[A29].
5.4.22 Radiation Parameter Φ
Φ5
εσ0 T 3 Lh
λ
ð1Þ;
Φ5
f εσ0 T 3 Lh
λ
ð2Þ
σ0 (W m22 K24) StefanBoltzmann constant; T (K) gas temperature;
Lh 5 AO21(m) hydraulic diameter; A (m2) channel cross-section surface area;
212
Dimensionless Physical Quantities in Science and Engineering
O (m) channel perimeter with fluid contact; λ (W m21 K21) thermal conductivity; ε () emissivity of inner channel wall (p. 206); f () function of mean
surface wall temperature.
This parameter expresses the ratio of the heat transferred by radiation in a passageway to that transferred by conduction in a channel wall. It characterizes the
relation between the radiation and conduction heat transfers in passageways.
Alternatively, it expresses the influence of the radiation on the convective transfer.
Info: [A23],[A29].
5.4.23 Radiation Pressure Rp
Rp 5
σT 4
cp
ð1Þ;
Rp 5
ηR ck
p
ð2Þ
σ (W m22 K24) absorptance (absorption capacity, radiating capacity); T (K) temperature; c (m s21) light velocity; p (Pa) gas pressure; ηR (Pa s) radiation dynamic viscosity; k (m21) attenuation coefficient.
It expresses the ratio of the radiation pressure to the static pressure of a gas. It
characterizes the relation between the radiation heat transfer energy and the gas
pressure energy. The value of the number Rp is very small and so is the significance of the radiation pressure if the temperature T is not greater than approximately 105 K or if the pressure p is not too low. In most technical applications, the
Rp number can be neglected. However, it is very important, for example, in hightemperature plasma physics.
Info: [A23],[B20].
5.4.24 Radiation Viscosity Kη
Kη 5
η
ηc2 κ
5
ηR
σT 4
η, ηR (Pa s) dynamic (hydrodynamic) viscosity, radiation dynamic viscosity;
c (m s21) light velocity; κ (m21) attenuation coefficient; σ (W m22 K24) absorptance (absorption capacity, radiating capacity); T (K) temperature.
It characterizes the relation between hydrodynamic and radiation dynamic
viscosities.
Info: [A23].
Thermomechanics
213
5.4.25 Schuster Number Sch
Sch 5
κp
κc
κp (m21) attenuation coefficient by absorption of radiation; κc (m21) attenuation coefficient by absorption and dispersion on particles in gas flow.
This number expresses the ratio of the radiation attenuation by dissipation to the
total radiation attenuation by dissipation and absorption. It characterizes the relation between absorbing and dissipating properties of particles floating in a radiating
gas flow. It states the dispersion portion of the total attenuation of radiation due to
surroundings.
Info: [A23],[A33].
Sir Franz Arthur Friedrich Schuster (18511934),
British physicist of German origin.
He is well known for his work in spectroscopy, in which
he is a renowned authority. In addition, he was engaged in
research on electricity behaviour in gases, terrestrial magnetism, radiometry, calorimetry and the mathematical periodicity theory. He wrote the book An Introduction to the
Theory of Optics (1904).
5.4.26 Stark Number Sk
See the Stefan number Sf (p. 213).
5.4.27 Stefan Number Sf
Sf Sk 5
εσ0 T 3 L
λ
σ0 (W m22 K24) StefanBoltzmann constant; T (K) temperature; L (m) characteristic length; λ (W m21 K21) thermal conductivity; Sk () Stark number (p. 213); ε () emissivity of inner channel wall (p. 206).
It expresses the ratio of the heat transferred by radiation to that transferred by
conduction. It characterizes the coupling between the thermal field in a solid body
and the heat transfer by radiation on the body surface. It is analogous to the Biot
number Bi (p. 173) and therefore is called the Biot radiation number BiR (p. 204)
sometimes. It is often called the Stark number Sk (p. 213).
Info: [A4],[A23],[B20].
214
Dimensionless Physical Quantities in Science and Engineering
Josef Stefan (24.3.18357.1.1893), SlovenianAustrian
mathematician and physicist.
He was engaged in research on the kinetic theory of
gases, thermodynamics and heat transfer by radiation. In
the year 1891, he published his famous work on iceberg
melting in polar seas, in which he laid the foundation for
the non-linear heat conduction theory with phase change:
the eponymous Stefan problem. Contrary to Kirchhoff,
Stefan showed that radiated energy exists in all wavelengths
and is governed by the 4th power of the absolute
temperature.
5.4.28 Thring Radiation Number Tg
See the Boltzmann number Bo (p. 204).
Meredith Wooldridge Thring (17.12.191515.9.2006), English engineer.
5.5
Boiling
Boiling represents a phase conversion with fast fluid evaporation within the entire
fluid mass. It passes through three stages involving the initial creation of vapour
kernels, a transient zone and a film boiling. The complicated phase conversion
process is described by a lot of dimensionless quantities. Among the basic ones are
the following: the Bond, Archimedes, Jakob, Kutateladze, LockhartMartinelli,
Stanton, Sterman and other numbers, including classic criteria modified for boiling.
5.5.1
Archimedes Steam Number Ar
1
3
R2l 2 σ2
Rv 21
12
Ar 5
Rl
η2l g
Rl, Rv (kg m23) liquid and vapour density; σ (N m21) surface tension; ηl (Pa s) liquid dynamic viscosity; g (m s22) gravitational acceleration.
It expresses the ratio of the buoyancy force (the Archimedes force) of the vapour
phase in a non-isotropic fluid to the molecular friction force in the fluid. It characterizes the mutual action of capillary, viscous and gravity forces in a vapourfluid
mixture. It influences the origin and development of gravity capillary waves in a
viscous fluid. It is a special modification of the Galilei steam number (p. 217).
Heat transfer in boiling.
Archimedes of Syracuse (p. 54).
Thermomechanics
5.5.2
215
Beginning of Boiling Instability N
N5
ðTS 2 Tv Þllv Rl L
Tv σ
TS (K) wall temperature before rising of vapour bubbles (initial maximal superheating of liquid); Tv (K) vapour temperature; llv (J kg21) specific latent heat
of vaporization; Rl (kg m23) density of liquid; L (m) characteristic length;
σ (N m21) surface tension.
It characterizes the origin of the instability of a fluid as it begins boiling.
Info: [A23].
5.5.3
Boiling Number Boi
Boi 5
qA
llv Rl w
Boi 5
L2 qA
qA
5
Qm llv
qm llv
ð1Þ;
ð2Þ
qA (W m22) area heat flux density; llv (J kg21) specific latent heat of vaporization; Rl (kg m23) liquid density; w (m s21) flow velocity; L (m) characteristic
length; Qm (kg s21) mass flux; qm (kg s21 m22) area mass flow density.
It expresses the ratio of the specific vapour heat flow perpendicular to a
bypassed wall to the total heat flow parallel with the wall. It characterizes the process of creating vapour kernels in fluid boiling and the process of heat transfer
intensification. For flow through a pipeline, with the internal diameter d, L 5 d
holds.
Info: [A23],[B56].
5.5.4
Boiling Stability K
K5
qcrit
1
1
llv R2v ½σgðRl 2 Rv Þ4
qcrit (W m22) surface density of critical heat flux; llv (J kg21) specific latent
heat of vaporization; Rv, Rl (kg m23) density of vapour and liquid; σ (N m21) surface tension; g (m s22) gravitational acceleration.
It expresses the ratio of the evaporation rate on a heated surface to the mean
velocity of vapour bubble growth. It characterizes the degree of the dynamic action
of the evaporation process on the vapourfluid system stability in boiling.
Sometimes, it is called the criterion of the heat transfer hydrodynamic crisis in
boiling.
Info: [A23].
216
Dimensionless Physical Quantities in Science and Engineering
5.5.5
Convection Number Ncon
1 2 x 0:9 Rg 0:5
Ncon 5
Rl
x
x () mass ratio of vapour fraction; Rg, Rl (kg m23) density of gas (vapour)
and liquid.
Essentially, it represents the modified Martinelli parameter X (p. 106) used for
vapour bubble flow in boiling fluid.
Info: [A23],[B56].
5.5.6
Frequency Evolution of Steam Bubbles N
N5
Rl cl σ
Rl
Rv dllv pβ Rl 2 Rv
Rl, Rv (kg m23) liquid and vapour; cl (J kg21 K21) liquid and vapour density;
σ (N m21) surface tension; d (m) characteristic dimension of vapour bubble;
llv (J kg21) specific latent heat of vaporization; p (Pa) pressure; β (K21) thermal expansivity of vapour.
It expresses the ratio of the surface stress force to the thermogravitational force.
It characterizes the evolution frequency of vapour bubbles in boiling fluid.
Info: [A23].
5.5.7
Froude Boiling Number Frb
2 ðTS 2 Tv Þcl Rl 4
Frb 5 3
ðπal Þ2
llv Rv
R g
w2
Rv 21
12
Frb2 5
gR
Rl
ð1Þ;
ð2Þ
R (m) radius of vapour bubble or drop; g (m s22) gravitational acceleration;
TS, Tv (K) temperature of heated wall and vapour; cl (J kg21 K21) specific
heat capacity of liquid; Rl, Rv (kg m23) density of liquid and vapour; llv (J kg21) specific latent heat of vaporization; al (m2 s21) thermal diffusivity of liquid;
w (m s21) motion velocity.
This number expresses the inertia-to-buoyancy forces ratio. It characterizes the
influence of force relations on the origin and evolution of the vapour bubbles in
boiling. In form (1), it expresses the condition for bubbles originating; in form (2),
it expresses the condition for vapour bubble motion. The vapour bubble radius can
Thermomechanics
217
be determined in a linear scale R 5 V1/3, where V denotes the bubble volume, or an
effective radius R 5 (3V/4πn)1/3 can be used.
Info: [A23].
William Froude. (p. 63)
5.5.8
Froude Heat Number Frt
Frt 5
gL3
a2
g (m s22) gravitational acceleration; L (m) characteristic length; a (m2 s21) thermal diffusivity.
It expresses the gravitation-to-thermodiffusion forces ratio. It characterizes the
mean intensity of the heat transfer under forced convection and fluid boiling in a
tube.
Info: [A23].
William Froude (see above).
5.5.9
Galilei Steam Number Ga
Ga 5
3
2
g
σ
2
ν l gðRl 2 Rv Þ
g (m s22) gravitational acceleration; ν l (m2 s21) kinematic viscosity of liquid;
σ (N m21) surface tension; Rl, Rv (kg m23) density of liquid and vapour.
It characterizes the heat transfer in closed two-phase thermosiphon systems
(a fluid boiling in an evaporator, vapour motion in a transfer part, condensation,
condensate return to the evaporator), all under the influence of gravity and inertia
forces.
Info: [A23].
Galileo Galilei (p. 123).
5.5.10 Heat Activity K
K5
ðRcλÞl
ðRcλÞw
R (kg m23) density; c (J kg21 K21) specific heat capacity; λ (W m22 K21) thermal conductivity; subscripts: l liquid; w channel wall.
This activity expresses the ratio of the fluid thermal activity to the canal wall
thermal activity. It characterizes the influence of material thermal properties on the
non-stationary heat transfer in fluid boiling. Together with the off liquid heating Ψ,
218
Dimensionless Physical Quantities in Science and Engineering
(p. 222) it expresses the temperature of the film boiling crisis. In bubble boiling, it
is of use similarly as the characteristic roughness dimension.
Info: [A23].
5.5.11 Impulse Heating NP
2 1
k
qA R v
NP 5
Rb
Rl llv Rl
qA (W m22) heat flux density; Rl, Rv (kg m23) density of liquid and vapour;
R (kg m23) density of generated vapour nuclei; llv (J kg21) specific latent heat
of vaporization; b () mean value of coefficient depending on liquid superheating; k () boiling constant.
This quantity expresses the conditions to bring a fluid to boiling with impulse
heating when the vapour origin is caused spontaneously by originating randomcharacter kernels. For k . 1, the temperature of intensive kernel nuclei origin is
reached. For kc1, the vapour phase nuclei are created in a fluctuating way even
for small masses of the fluid, and the created vapour conversion centres do not
play any role. With k 5 3 in an experiment, the growth of bubbles is delayed by
inertia forces (Rayleigh’s case). With k 5 0.5, the growth rate is limited by the heat
supply. Heat transfer in boiling.
Info: [A33].
5.5.12 Jakob Evaporation Speed Number Ja
Ja To 5
qA
llv Rv df
qA (W m22) heat flux density; llv (J kg21) specific latent heat of vaporization;
Rv (kg m23) density of saturated vapour; d (m) diameter of vapour bubble;
f (Hz) generation frequency of vapour bubbles; To () Tolubinsky number
(p. 226).
It expresses the ratio of the vapour phase generation (bubbles) rate, on a heated
surface, to the mean growth rate of vapour bubbles. It characterizes the heat transfer intensity in boiling. It is the equivalent of the modified Pe´clet boiling speed
number Pewb (p. 223). It expresses also the ratio of the maximum vapour bubble
diameter to the overheated fluid film thickness. Sometimes, it is called the
Tolubinsky number To (p. 226).
Info: [A23].
Max Jakob (see below).
Thermomechanics
219
5.5.13 Jakob Number Ja1
Ja1 5
c1 ΔT Rl
llv Rv
cl (J kg21 K21) specific heat capacity of liquid; ΔT 5 TlTn (K) liquid super
heating; Tl, Tn (K) temperature of liquid and saturated vapour; llv (J kg21) specific latent heat of vaporization; Rl, Rv (kg m23) liquid and vapour density.
This number expresses the ratio of the heat supplied to overheat the unit volume
of a fluid to the specific volume latent heat of evaporation. It characterizes the heat
transfer intensity in fluid boiling. It depends on the fluid pressure and overheating.
It drops with the pressure increasing because the vapour density increases, and vice
versa. With the pressure greater than the atmospheric pressure (Ja1 # 20), the
vapour bubbles grow in size due to the heat supplied from a heated surface over
the adhering fluid layer. With low pressures (Ja1 $ 20), the heat that led to a
vapour bubble is transferred from the overheated fluid on the inter-phase surface.
To determine the vapour bubble diameter, the formula is in the first case
R5
pffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffi
2βJa1 aτ
and in the second case
pffiffiffiffiffi
R 5 2γJa1 aτ
and generally
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffi
R 5 γJa1 1 γ 2 Ja21 1 2βJa1
aτ
The last expression is valid for Ja1 5 0.11000, which corresponds to pressures
of 0.0110 MPa. The factors β and γ depend on geometric parameters. For example, for the wettability angle ϑ 5 4090 , it is γ 5 0.10.49, β66.
Info: [A23],[B20],[B56].
Max Jakob (20.7.18791955), American physicist of
German origin.
After the year 1910, he was engaged in research on thermodynamics and heat transfer and carried out a great deal
of work in this area. His work concerned, especially, steam
and air properties under high pressure, devices to measure
thermal conductivity, explanations of the boiling mechanism and condensation, and flow in pipelines and channels.
In this sphere, he published approximately 500 works.
220
Dimensionless Physical Quantities in Science and Engineering
5.5.14 Kutateladze Evaporation Number Ku
Ku 5
llv
cl ΔT
Ku 5
llv Rl w
5 Ku PeNu 21
qA
ð2Þ;
Ku 5
llv Rl ν l
5 Ku PrNu 21
qA L
ð3Þ;
Ku4 5 Ja121
Rl
Rv
ð1Þ;
ð4Þ
llv (J kg21) specific latent heat of vaporization; cl (J kg21 K21) specific heat
capacity of liquid; ΔT 5 Tv 2 Ts (K) difference of vapour and wall temperatures;
Rl, Rv (kg m23) density of liquid and vapour; w (m s21) liquid velocity;
qA (W m22) heat flux density; vl (m2 s21) kinematic viscosity of liquid; L (m) characteristic length; Ku () Kutateladze evaporation number defined by the relation (1); Pe () Péclet heat number (p. 180); Nu () Nusselt number (p. 196);
Pr () Prandtl number (p. 197); Ja1 () Jakob number (p. 219).
It expresses the ratio of the specific evaporation heat to the heat content increase
in the overheating of one of the phases. In form (2), it expresses the ratio of the
phase conversion rate to the phase flow rate. In form (3), it expresses the ratio of
flow inertia forces originating due to the phase conversion process to viscous
forces. It is a special expression of the Reynolds number Re (p. 81). In form (1), it
is also called the Trawton number Tr (p. 232), and in form (4) it is the Jakob
vapour number.
Info: [A23].
Samson Semenovich Kutateladze (p. 237).
5.5.15 Local Boiling Stability K
Rv w2v
ffi
K 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
σgðRl 2 Rv Þ
ð1Þ;
Rl w2l
ffi
K 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
σgðRl 2 Rv Þ
ð2Þ
Rv, Rl (kg m23) density of vapour and liquid; wv, wl (m s21) velocity of vapour
and liquid; σ (N m21) surface tension; g (m s22) gravitational acceleration.
It characterizes the local vapourfluid set stability under dynamic and capillary
gravitational mutual action. Equation (1) expresses the approximate transition of the
non-viscous fluid bubble boiling to the film boiling, and equation (2) expresses the
Thermomechanics
221
effect of non-viscous fluid being driven out from the microporous surface in the
barbotage.
Info: [A23].
5.5.16 LockhartMartinelli Number (1.) LM1, Lp
LM1 5
Rv σL
η2v
Rv (kg m23) vapour density; σ (N m21) surface tension; L (m) characteristic
length; ηv (Pa s) dynamic viscosity of vapour.
It expresses the ratio of the fluid surface stress force to the vapour viscosity
force. It characterizes the influence of hydrodynamic forces on the heat transfer in
film boiling, for example, of cryogen fluids in a pipeline.
Info: [A23].
5.5.17 LockhartMartinelli Number (2.) LM2, Xtt
LM2 5
12x
x
0:9 0:5 0:1
Rv
ηl
Rl
ηv
x () relative mass flux of vapour fraction; Rv, Rl (kg m23) density of vapour
and liquid; ηv, ηl (Pa s) dynamic viscosity of vapour and liquid.
It characterizes the convective fluid boiling in vertical tubes and constant crosssection canals. Together with the boiling number Boi (p. 215), it determines the
heat transfer. The formula
(
C3 )C4
α
1
5 C1 Boi 1 C2
αl
Xtt
is valid, where C1 5 6.7 3 103; C2 5 3.5 3 1024; C3 5 0.67; C4 5 1; αl is the heat
transfer coefficient on the fluid side of a channel; α 5 αk 1 αv, where αk is the convection part of heat transfer and αv is the part corresponding to the formation of
vapour nuclei.
Info: [A23].
5.5.18 Number of Steam Nucleus N
dp
N5
dT
d2 ðRl 2 Rv ÞqA
λl σRl
Tn
222
Dimensionless Physical Quantities in Science and Engineering
p (Pa) pressure; T, Tn (K) temperature and saturation temperature; d (m) characteristic dimension of vapour bubble; Rl, Rv (kg m23) density of liquid and
vapour; qA (W m22) area heat flux density; λl (W m21 K21) thermal conductivity of liquid; σ (N m21) surface tension.
It characterizes the number of vapour nuclei originating in the vapour creation
in fluid.
Info: [A23].
5.5.19 Nusselt Boiling Number Nub
Nub 5
αd
α
5
λl
λl
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
σ
gðRl 2 Rv Þ
α (W m22 K21) heat transfer coefficient; d (m) characteristic dimension of
vapour bubble; λl (W m21 K21) thermal conductivity of liquid; σ (N m21) surface tension; g (m s22) gravitational acceleration; Rl, Rv (kg m23) density
of liquid and vapour.
This number characterizes the convective heat transfer in the heating and boiling
of fluids.
Info: [A23].
Ernst Kraft Wilhelm Nusselt (p. 196).
5.5.20 Off Liquid Heating Ψ
Ψ5
cl ðTn 2 Tl Þ
llv
cl (J kg21 K21) specific heat capacity of liquid; Tn, Tl (K) temperature of saturation and liquid; llv (J kg21) specific latent heat of vaporization.
It expresses the ratio of the specific heat required to finish the fluid heating to
the boiling point to the specific latent heat of evaporation. It characterizes the
degree of insufficient fluid heating in boiling. With increasing Ψ, the effect of heat
flow intensification is reduced.
Info: [A23].
5.5.21 Péclet Boiling Number Peb
qA cl
Peb 5
llv λl
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
σ
gðRl 2 Rv Þ
qA (W m22) area heat flux density; cl (J kg21 K21) specific heat capacity of
liquid; λl (W m21 K21) thermal conductivity of liquid; llv (J kg21) specific
Thermomechanics
223
latent heat of vaporization; σ (N m21) surface tension; g (m s22) gravitational
acceleration; Rl, Rv (kg m23) density of liquid and vapour.
It characterizes the heat transfer by convection and conduction in a fluid boundary layer in boiling.
Info: [A23].
Jean Claude Eugène Péclet (p. 180).
5.5.22 Péclet Boiling Speed Number Pewb
Pewb
fLRv cl d
wRv cl
5
λl
λl
Pewb Sh21 5
df
wt
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
σ
gðRl 2 Rv Þ
ð1Þ;
ð2Þ
f (Hz) frequency of rising bubbles; L (m) characteristic length; Rv, Rl (kg m23) density of vapour and liquid; cl (J kg21 K21) specific heat capacity of liquid;
λl (W m21 K21) thermal conductivity of liquid; d (m) characteristic dimension of
vapour bubble; w 5 qA (llvRv)21 (m s21) boiling velocity; qA (W m22) heat flux
density; llv (J kg21) specific latent heat of vaporization; σ (N m21) surface tension; g (m s22) gravitational acceleration; wt (m s21) heat diffusion velocity;
Sh () Strouhal number defined by the relation (3) (p. 87).
It expresses the ratio of the velocity of vapour bubbles originating in boiling to
the heat diffusion rate in the fluid. It characterizes the rate at which vapour bubbles
originate in boiling. In equation (2), it represents the inverse value of the Strouhal
number Sh defined by equation (3) (p. 87).
Info: [A23].
Jean Claude Eugène Péclet (see above).
5.5.23 Rayleigh Steam Number Ra
Ra 5 GrPr 5
Ra 5
gL3 β l ΔT
ν l al
gL3crit Rl 2 Rv
ν v av Rv
ð1Þ;
ð2Þ
1
2
σ
where Lcrit 5 2π
gðRl 2 Rv Þ
g (m s22) gravitational acceleration; L (m) characteristic length; β l (K21) volume thermal expansion coefficient of liquid; ΔT (K) temperature difference;
ν l, ν v (m2 s21) kinematic viscosity of liquid and vapour; al, av (m2 s21) thermal
224
Dimensionless Physical Quantities in Science and Engineering
diffusivity of liquid and vapour; Rl, Rv (kg m23) density of liquid and vapour; σ
(N m21) surface tension; Lcrit (m) critical wavelength; Gr () Grashof heat
number (p. 185); Pr () Prandtl number (p. 197).
It characterizes the heat transfer by free convection in boiling and thermal instability in fluid. In the basic form (1), it is valid for fluid heating, and in the modified
form (2), it holds for the boiling and resulting instability of a fluid. In the latter
case, the critical wavelength is taken as the characteristic length.
Info: [A23],[A29].
Lord Rayleigh, John William Strutt (p. 187).
5.5.24 Reynolds Boiling Number Reb
qA
Reb 5
llv Rv ν l
Reb 5
qA d
llv Rv ν l
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
σ
R
5 Nub Pr 21 Ku 21 l
Rv
gðRl 2 Rv Þ
ð1Þ;
ð2Þ
qA (W m22) heat flux density; llv (J kg21) specific latent heat of vaporization;
Rv, Rl (kg m23) density of vapour and liquid; ν l (m2 s21) kinematic
viscosity of liquid; σ (N m21) surface tension; g (m s22) gravitational acceleration; d (m) characteristic tube dimension; Nub () Nusselt boiling number
(p. 222); Pr () Prandtl number (p. 197); Ku () Kutateladze evaporation
number (p. 220).
It expresses the ratio of the inertia force arising in flowing fluid with phase conversion to the internal friction force. It characterizes the heating and boiling of a
fluid. It is the hydrodynamic boiling criterion. In equation (2), it expresses the ratio
of the pipe surface heat loading to the specific latent evaporating heat. It is the
measure for pipe wall thermal loading in fluid boiling.
Info: [A33].
Osborne Reynolds (p. 82).
5.5.25 Reynolds Boiling Speed Number Rewb
Rewb 5
d π 3
d Rv fn
ν l Rl 6
d (m) characteristic dimension of vapour bubble; ν l (m2 s21) kinematic viscosity of liquid; Rl, Rv (kg m23) density of liquid and vapour; f (Hz) generation
frequency of vapour bubbles; n () number of vapour nuclei passing through
surface unit.
It characterizes the fluid boiling rate.
Thermomechanics
225
Info: [A23].
Osborne Reynolds (see above).
5.5.26 Reynolds Bubbling Evaporation Number Rebb
Rebb 5
Rl wd
wd
5
ηl
νl
Rl (kg m23) liquid density; ν l (m2 s21) kinematic viscosity of liquid; ηl (Pa s) dynamic viscosity of liquid; w (m s21) propagation velocity of vapour bubble;
d (m) characteristic dimension of vapour bubble.
It expresses the ratio of the vapour bubble inertia force to the fluid friction force.
It characterizes the hydrodynamic relations in fluid bubble boiling.
Info: [A23].
Osborne Reynolds (see above).
5.5.27 Stanton Boiling Number Stb
Stb 5
αllv
qA c l
Stb 5
qA
Rv cl wl ðTS 2 Tn Þ
ð1Þ;
ð2Þ
α (W m22 K21) heat transfer coefficient; llv (J kg21) specific latent heat of
vaporization; qA (W m22) surface density of heat flux; cl (J kg21 K21) specific
heat capacity of liquid; Rv (kg m23) vapour density; wl (m s21) liquid motion
velocity; TS, Tn (K) wall and saturation temperatures.
This number characterizes the convective heat transfer in fluid boiling. In equation (2), it is valid for the shooting process4 in film boiling.
Info: [A23],[A29].
Thomas Edward Stanton (p. 201).
5.5.28 Steam Pressure Np
p
Np 5
σ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
σ
gðRl 2 Rv Þ
p (Pa) pressure; σ (N m21) surface tension; g (m s22) gravitational acceleration; Rl, Rv (kg m23) density of liquid and vapour.
It characterizes the pressure in heat transfer, for example, in closed two-phase
thermosiphon systems in which the boiling and condensation processes proceed
under the action of gravitation and inertia forces.
Info: [A23],[A29].
226
Dimensionless Physical Quantities in Science and Engineering
5.5.29 Steam Superheating Number Ψ
Ψ5
cpv ðTv 2 Tn Þ
llv
cpv (J kg21 K21) specific heat capacity at constant pressure; Tv (K) vapour
temperature; Tn (K) saturation temperature; llv (J kg21) specific latent heat of
vaporization.
It expresses the superheated vapour specific heat to the specific latent heat of
evaporation. With Ψ-0, the heat flow from a wall causes vapour generation and
superheating.
Info: [A23].
5.5.30 Sterman Number Sn
Sn 5
llv
cpv Tn
llv (J kg21) specific latent heat of vaporization; cpv (J kg21 K21) specific heat
capacity of saturated vapour; Tn (K) temperature of saturated vapour.
It expresses the ratio of the specific evaporation heat to the specific heat of saturated steam. It characterizes the developed bubble boiling.
Info: [A23].
5.5.31 Tolubinsky Number To
See the Jakob evaporation speed number Ja (p. 218).
5.5.32 Wallis Number Ws
Ws 5
1
1 3
½d gðRl 2 Rv ÞRl 2
η
η (Pa s) dynamic viscosity of liquid; d (m) tube diameter; g (m s22) gravitational acceleration; Rl, Rv (kg m23) density of liquid and vapour or gas.
It characterizes the fluid viscosity influence on stroke flow in boiling.
Info: [A23].
5.5.33 Weber Boiling Number Web
Web 5
Rv ðw2v 2 w2l ÞL
σ
Thermomechanics
227
Rv (kg m23) vapour density; wv, wl (m s21) flow velocity of vapour and liquid;
L (m) characteristic length dimension (wall thickness, boundary layer thickness,
drop or bubble diameter); σ (N m21) surface tension.
It expresses the inertia-to-capillary forces ratio. It characterizes the hydrodynamic forces influence on the process of bubbles originating in fluid boiling.
Info: [A23].
Ernst Heinrich Weber (p. 92).
Wilhelm Eduard Weber.
5.6
Evaporation
Evaporation is the phase conversion in which a fluid is converted to vapour on its
surface. The evaporation occurs under an arbitrary fluid temperature. The higher
the temperature is, the faster the evaporation. Further, the evaporation is influenced
by the fluid’s properties, by gas pressure and motion over the fluid and by the
evaporating area size. In addition to several evaporation numbers, some of the fundamental dimensionless quantities are the Borishanskyi, Bulygin, Gukhman, Jakob
evaporation, Lomonosov, Richman, Spalding and Trawton numbers.
5.6.1
Borishansky Number Bs
p
Bs 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
σðRl 2 Rv Þg
p (Pa) pressure; σ (N m21) surface tension; Rl, Rv (kg m23) density of liquid
and vapour; g (m s22) gravitational acceleration.
This number expresses the ratio of the absolute pressure in the system to the
pressure gradient on the interface of phases. It characterizes the pressure conditions
in fluid evaporation.
Info: [A23].
Veniamin Mironovich Borishansky (born 1917), Russian engineer.
5.6.2
Bulygin Number Bu
Bu 5
llv cmp p
cl ðT 2 Ti Þ
llv (J kg21) specific latent heat of evaporation; cmp (Pa21) pressure mass capacity of vapour; p (Pa) pressure; cl (J kg21 K21) specific heat capacity of liquid;
T, Ti (K) material and initial temperatures.
It expresses the ratio of the evaporation heat to the fluid heat (the heat necessary
to bring the fluid to boiling). It characterizes the evaporation heat transfer and the
228
Dimensionless Physical Quantities in Science and Engineering
body accumulating ability in high-intensity heat transfer. The influence of the heat
which is lost during evaporation in molecular transfer is considered.
Info: [A23],[A29].
N.P. Bulygin.
5.6.3
Drag Evaporation Coefficient CD,ev
CD;ev 5
FD
1
2
2Rc w
FD (N) drag force acting on particle; Rc (kg m23) density of cold liquid;
w (m s21) velocity difference between particle and cold liquid.
It expresses the resistance of the vapour layer arising on the surface of a hot particle during its penetration into cold fluid. It is analogous to the Dalton number Dal
(p. 394). Evaporation.
Info: [B15].
5.6.4
Euler Modified Number Eumod
Eumod 5
HL Rl g
w2G Rg
HL (m) pressure height of liquid; Rl (kg m23) liquid density; Rg (kg m23) vapour density; g (m s22) gravitational acceleration; WG (m s21) vaporization
velocity from free surface.
It expresses the vapour flow across a mass transfer.
Info: [A29].
Leonhard Euler (p. 61).
5.6.5
Evaporation Flow Number Ev
Ev 5
wev Dp
η
wev (kg m22 s21) vaporization velocity; Dp (m) particle diameter; η (Pa s) dynamic viscosity.
It expresses the dynamic-to-viscous forces ratio. It characterizes the vapour flow
inside a vapour film, with the vapour being induced during cold fluid evaporation
at the interface between the cold fluid and the vapour. Evaporation.
Info: [A23].
Thermomechanics
5.6.6
229
Evaporation Number (1.) Nev
Nev 5
w2
llv
w (m s21) flow velocity; llv (J kg21) specific latent heat of vaporization.
It characterizes the evaporation process. It expresses the ratio of the flow kinetic
energy to the fluid latent heat.
Info: [A23].
5.6.7
Evaporation Number (2.) Nev
Nev 5
cl
5 Nev
GcDu 21
βllv
cl (J kg21 K21) specific heat capacity of liquid; β (K21) volume thermal
() expansion coefficient; llv (J kg21) specific latent heat of evaporation; Nev
evaporation number (1.) (p. 229); Gc () Gay-Lussac number (p. 14); Du () Dulong number (p. 191).
It characterizes the evaporation process. It expresses the ratio of the heat necessary to heat the fluid to the evaporation temperature to the specific latent heat of
evaporation.
Info: [A23].
5.6.8
Evaporation Number (3.) Elasticity Nev
Nev 5
K
a2
5
5 Nev
Cau
llv
llv Rl
K (Pa) bulk modulus; llv (J kg21) specific latent heat of vaporization;
() evaporation
Rl (kg m23) liquid density; a (m s21) sound velocity; Nev
number (1.) (p. 229); Cau () Cauchy number (aeroelasticity parameter)
(p. 155).
It characterizes the fluid evaporation process. It expresses the ratio of the energy
of forces acting with fluid evaporation to the specific latent evaporation heat. It is
called the elasticity evaporation number as well.
Info: [A23].
5.6.9
Gravitational Evaporation Number Ngv
Ngv 5
D3p R2c g
η2c
Dp (m) particle diameter; Rc (kg m23) density of cooling agent; g (m s22) gravitational acceleration; ηc (Pa s) dynamic viscosity of cooling agent.
230
Dimensionless Physical Quantities in Science and Engineering
It describes the particles in a cooling system under the influence of gravitation.
It expresses the gravity-to-viscosity forces ratio. Together with the evaporation
flow number Ev (p. 228) and the Reynolds number Re (p. 81), it is among the fundamental criteria influencing the force acting on a cooled particle under laminar
flow conditions during film evaporation.6
5.6.10 Gukhman Evaporation Number Gu
Gu 5
tg 2 ts
Tg
tg, ts (K) hot gas temperature, moist material surface (liquid level) temperature;
Tg (K) thermodynamic gas temperature.
With evaporation from a free fluid surface, it expresses the ratio of the outer
mass transfer intensity (moisture), expressed by the volume fluid evaporation intensity (moisture) in the air boundary layer, to the outer heat transfer intensity. It is
the thermodynamic criterion of evaporation in isobaric, adiabatic conditions. It is
used to describe the evaporation from a fluid-free surface or moist material drying.
With non-adiabatic evaporation, an additional criterion Θ 5 T/Tl must be introduced to express the thermal conditions’ influence if the fluid surface temperature
Tl differs from the adiabatic air saturation temperature close to the moist body surface. In moist material drying, it is called the Gukhman drying number Gu (p. 251).
Info: [A4],[A23].
Alexander Adolfovich Gukhman, Russian chemical engineer.
5.6.11 Jakob Evaporation Number Jak
Jak 5
llv
cp ΔT
llv (J kg21) specific latent heat of vaporization; cp (J kg21 K21) specific heat
capacity; ΔT (K) temperature difference.
It expresses the ratio of the specific latent heat of evaporation to the fluid heat
capacity. It is analogous to the number of phase changes N (p. 179).
Info: [B20].
Max Jakob (p. 219).
5.6.12 Leidenfrost Number Lei
Lei 5
Bi
Pe2
Bi () Biot number (p. 173); Pe () Péclet heat number (p. 180).
Thermomechanics
231
It characterizes the heat transfer by evaporation with hot wall cooling by a thin
fluid film flowing over its surface. The hardening of bodies in fluids and other
technological processes are examples. The Leidenfrost phenomenon.
Info: [B30].
Johann Gottlob Leidenfrost (27.11.17152.12.1794),
German physicist and chemist.
Above all, his basic contribution consisted in the discovery of the Leidenfrost phenomenon. With this, in the vicinity of a hot body being much hotter than the boiling
temperature, which is the so-called Leidenfrost point, the
fluid forms a thin lagging steam layer. This steam layer
forms a thermal barrier which prevents the fluid from getting into fast boiling. Therefore, it takes much longer for
the fluid to evaporate. With the temperature under the
Leidenfrost point, this thermal barrier does not exist, due to
which the fluid evaporates swiftly.
5.6.13 Lomonosov Number Lo
Lo 5
gΔRL
5 ArRe22
Rw2
g (m s22) gravitational acceleration; ΔR (kg m23) vapour and air density difference; L (m) characteristic length; R (kg m23) air density; w (m s21) airflow velocity; Ar () Archimedes hydrodynamic number (p. 53); Re () Reynolds number (p. 81).
It expresses the ratio of the buoyancy force (Archimedes force) to the inertia
force. It acts in processes of non-isothermal water evaporation from surfaces of
vessels and tanks into flowing air.
Info: [A23],[A33].
Mikhail Vasilyevich Lomonosov (19.11.171115.4.1765),
Russian scientist, encyclopedist, writer and poet.
He was a founder of both Russian science and culture.
He studied physics, chemistry, metallurgy, history and arts,
and he was interested in other fields as well. By creating
the mechanical heat theory, expressing and proving the law
of mass conservation, he laid the first foundations for general heat theory. He showed that heat is a form of movement of particles. He wrote a whole range of works,
including the first history of Russia.
232
Dimensionless Physical Quantities in Science and Engineering
5.6.14 Richman Number Ri
Ri 5
qm L wL
5
Rν
ν
qm (kg s21 m22) mass flow density; L (m) characteristic length; R (kg m23)
liquid density; ν (m2 s21) kinematic viscosity; w (m s21) flow velocity.
It expresses the ratio of inertia forces to viscous forces which act in condensate
evaporation. It characterizes the influence of hydrodynamic conditions on the evaporation process. Sometimes, it is called the transversal flow criterion for parallel
running heat and mass transfers during evaporation or vapour condensation in a
vapourgas mixture. It can be called the Reynolds mass number Rem (p. 262).
Info: [A33].
Georg Vilhelm Richman (17111753), Russian physicist.
5.6.15 Spalding Number (2.) Sp, B’
Φ 21
Sp 5 cp ΔT llv 2
Qm
cp (J kg21 K21) specific heat capacity of surroundings; ΔT (K) surroundings
and liquid level temperature difference; llv (J kg21) specific latent heat of vaporization; Φ (W) radiation heat flux; Qm (kg s21) mass flux.
This number expresses the ratio of the received heat to the latent heat of an
evaporating material. It characterizes the heat transfer in the evaporation of fluid
drops.
Info: [A23].
Brian Spalding (born 1923), English engineer (p. 200).
5.6.16 Trawton Number Tr
Tr Ku 5
llv
ΔTcp
llv (J kg21) specific latent heat of vaporization; ΔT (K) temperature difference; cp (J kg21 K21) specific heat capacity; Ku () Kutateladze evaporation
number defined by equation (1) (p. 220).
It expresses the ratio of the specific latent evaporation heat to that delivered to
heat a fluid volume unit. It characterizes the fluid evaporation process with
the temperature corresponding to normal conditions. It is a special case of the
Kutateladze evaporation number Ku defined by relation (1) (p. 220).
Info: [A23].
Frederic Thomas Trawton (18631922), Irish physicist.
Thermomechanics
5.7
233
Condensation
For condensation, several dimensionless quantities are used, to represent the gas
or vapour transformation of the fluid. For example, there is a set of quantitative
condensation criteria expressing energetic, force and other influences which
are connected with condensation. Other dimensionless quantities are the following:
the Archimedes, Galilei, Kapica, Grigul, Kirbride, Kutateladze, Laplace,
McAdams, Nusselt, Prandtl, Reynolds, Sherwood, Stanton and Weber numbers. In
addition, some dimensionless quantities are used for such things as various binary
mixtures and thermocapillary processes.
5.7.1
Archimedes Condensation Number Ark
Ark 5
gL3
Rv
Rv
5
Ga
1
2
1
2
k
Rl
Rl
ν 2l
g (m s22) gravitational acceleration; L (m) characteristic length; ν l (m2 s21) kinematic viscosity of liquid; Rv, Rl (kg m21) density of vapour and liquid; Gak
() Galilei condensation number (p. 235).
This number expresses the ratio of the product of the static buoyancy, arising
from the difference of densities, and the inertia force to the square of the friction
force. It characterizes the influence of the free two-phase vapourcondensate environment on the heat transfer during condensation.
Info: [A23].
Archimedes of Syracuse (p. 54).
5.7.2
Condensation Number (2.) K
K5
L3 R2 glvl
ληΔT
L (m) characteristic length; R (kg m23) condensation liquid density; g (m s22) gravitational acceleration; lvl (J kg21) specific latent heat of condensation;
λ (W m21 K21) thermal conductivity of condensation liquid; η (Pa s) dynamic
viscosity of condensation liquid; ΔT (K) temperature difference during
condensation.
It characterizes the condensation process on a vertical wall.
Info: [A23].
5.7.3
Condensation Number of Binary Steam Mixtures Kb
Kb 5 2σ
@σ
ðlvl η2l Þ 21
@cm
234
Dimensionless Physical Quantities in Science and Engineering
σ (N m21) surface tension; cm () mass fraction; lvl (J kg21) specific latent
heat of condensation; ηl (Pa s) dynamic viscosity of liquid.
It characterizes the influence of a vapour binary mixture on drop condensation.
The surface stress is presumed to be determined by the condensation liquid first.
Info: [A23].
5.7.4
Condensation Number of Flowing Steam Kw
Kw 5
2w2v Rv
2
ðgν Þ3 Rl
l
wv (m s21) vapour velocity; g (m s22) gravitational acceleration; ν l (m2 s21) kinematic viscosity of liquid; Rl, Rv (kg m23) density of liquid and vapour.
It characterizes the influence of clean vapour velocity on the heat transfer during
drop condensation on a vertical surface. With it, the mutual vapour friction force
on the condensate film surface, the gravity force and the molecular film friction are
considered.
Info: [A23].
5.7.5
Condensation Number of Thermocapillary Movement Kk
Kk 5 β σ ΔTcrit
where β σ 5
σRcrit
5 β σ ΔTcrit Lak
Rl ν 2l
1 @σ
;
σ @T
Rcrit 5
2σTn
Rl lvl ΔTcrit
β σ (K21) temperature coefficient of surface tension; ΔTcrit (K) critical temperature difference; σ (N m21) liquid surface tension; Rcrit (m) critical radius
of condensation nucleus; Rl (kg m23) liquid density; ν l (m2 s21) kinematic
viscosity of liquid; T (K) temperature; Tn (K) saturation temperature; lvl
(J kg21) specific latent heat of condensation; Lak () Laplace condensation
number (p. 238).
It expresses the thermocapillary-to-viscosity forces ratio. It characterizes the
influence of thermocapillary motion in immovable, clean vapour condensation.
Info: [A23].
5.7.6
Condensation Number of Thermocapillary Separation Kδ
Kδ 5 β p ΔT
E
Rl ν 2l Rcrit
;
where Rcrit 5
2σTn
Rl lvl ΔTcrit
Thermomechanics
235
β p (K21) thermal coefficient of displaced pressure; ΔT (K) temperature difference between saturation temperature and surface wall temperature; E (J) pressure
energy of separation; Rl (kg m23) liquid density; ν l (m2 s21) kinematic viscosity of liquid; σ (N m21) surface tension; Tn (K) saturation temperature;
lvl (J kg21) specific latent heat of condensation; ΔTcrit (K) critical temperature
difference.
It expresses the ratio of mutually acting forces between condensate molecules
and those of molecularly active surface material on a condensation surface. It characterizes the influence of the wall surface material on the drop condensation of
immobile clean vapour.
Info: [A23].
5.7.7
ε5
Degree of Vapour Condensation ε
Qm1 2 Qm2
Qm1
Qm1, Qm2 (kg s21) vapour mass fluxes on input and output of vapour condenser.
It expresses the ratio of the vapour mass flow on a condenser inlet to that on its
outlet. It characterizes the condensation efficiency of the condenser.
Info: [A23].
5.7.8
Galilei Condensation Number Gak
Gak 5
gL3
ν 2l
ð1Þ;
Gak 5
gx3
ν 2l
ð2Þ
g (m s22) gravitational acceleration; L (m) characteristic length (wall height);
ν l (m2 s21) kinematic viscosity of liquid; x (m) distance in direction of condensation liquid layer generation.
It expresses the ratio of the molecular friction force during condensation to the
gravitation force. It characterizes the influence of friction and gravitation on the
heat transfer in condensation. Expression (1) holds for developed condensation, and
expression (2) is valid for a developing condensation layer.
Info: [A23].
Galileo Galilei (p. 123).
236
Dimensionless Physical Quantities in Science and Engineering
5.7.9
Grigull Condensation Number Ggk, Z
1
1
g 3 λl ΔTL
λl ΔT
Ggk 5 2
5 Ga3k
lvl ηl
lvl ηl
νl
g (m s22) gravitational acceleration; ν l (m2 s21) kinematic viscosity of condensate liquid; λl (W m21 K21) thermal conductivity of condensate liquid
ΔT 5 TnTS (K) temperature difference during condensation; Tn (K) saturation
temperature; TS (K) condensation wall surface temperature; L (m) characteristic
length; lvl (J kg21) specific latent heat of condensation; ηl (Pa s) dynamic
viscosity of condensation liquid; Gak () Galilei condensation number (p. 235).
It characterizes the condensate film flow. It represents the relative effective
length of the condensate flow-down surface. The condensate film flow is laminar
in horizontal tubes with Ggk , 3900 and in vertical ones with Ggk , 2300. A turbulent flow occurs with Ggk . 3900 in horizontal tubes and with Ggk . 2300 in the
case of vertical tubes.
Info: [A23].
Ulrich Grigull (p. 186).
5.7.10 Kapica Number Ka
Ka 5
σ3
gR3l ν 4l
σ (N m21) surface tension; g (m s22) gravitational acceleration; Rl (kg m23) liquid density; ν l (m2 s21) kinematic viscosity of liquid.
This number expresses the ratio of the surface stress force to the product of the
volume force and the viscosity force. It characterizes fluid film thermal conductivity increase due to wave periodic flow on a vertical wall.
Info: [A23].
Peter Leonidovich Kapica (8.7.18948.4.1984), Russian
physicist. Nobel Prize in Physics, 1978.
He was engaged in research on magnetism under low
temperatures and helium superfluidity problems. He
designed several unique devices during his stay in England,
where he collaborated with Ernst Rutherford, and in the former Soviet Union as well. His research on microwave generators led him to the study of controlled nuclear reaction
starting in the 1960s.
Thermomechanics
237
5.7.11 Kirbride Condensation Number Kb
1
1
2
α ν 2l 3
Kb 5
5 NuGak 3
λl g
α (W m22 K21) heat transfer coefficient; λl (W m21 K21) thermal conductivity
of condensation liquid; ν l (m2 s21) kinematic viscosity of condensation liquid;
g (m s22) gravitational acceleration; Nu () Nusselt number (p. 196); Gak ()
Galilei condensation number (p. 235).
It characterizes the heat transfer in membrane clean vapour condensation and
turbulent condensate flow. It is often called the condensation number (1.).
Info: [A23].
5.7.12 Kutateladze Condensation Number Ku
Kup 5
lvl
cpv ðTv 2 Tl Þ
ð1Þ;
Kuk 5
lvl
cpl ðTl 2 TS Þ
ð2Þ
lvl (J kg21) specific latent heat of condensation; cpv, cpl (J kg21 K21) specific
heat capacity of vapour and condensation liquid; Tv (K) vapour temperature;
Tl (K) mean temperature of condensation liquid; TS (K) external surface wall
temperature.
This number expresses the ratio of the condensation specific latent heat to the
specific heat cooling of one phase. In equation (1), it characterizes the vapour condensation process, and in equation (2), it characterizes the condensate subcooling.
Therefore, it is called the vapour (1) or condensate (2) Kutateladze number.
Info: [A23].
Samson Semenovich Kutateladze (18.7.191420.3.1986),
Russian engineer and physicist.
In the year 1936, he formulated the basic similarity conditions for heat-exchange processes with the change of state
phase of material. His experimental work with the heat transfer in liquid metals in nuclear engineering is extensive.
He formulated the basic ideas of the hydrodynamic theory of
the boiling crisis. His books Osnovy teorii teploobmena
(Fundamentals of Heat Exchange Theory, 1979) and Analiz
podobija v teplofizike (Similarity Analysis in Thermophysics,
1982) are significant in the evolution of the field.
238
Dimensionless Physical Quantities in Science and Engineering
5.7.13 Laplace Condensation Number Lak
Lak 5
ΔpL
σ
ð1Þ;
Lak 5
Rl σL
η2l
ð2Þ
Δp (Pa) pressure difference across condensation liquid layer; L (m) thickness
of condensation layer; σ (N m21) surface tension; Rl (kg m23) density of
condensation liquid; ηl (Pa s) dynamic viscosity of condensation liquid.
It expresses the ratio of the pressure force to the surface stress force. It characterizes the force relations in the membrane or drop condensation of vapour.
Info: [A23].
Pierre-Simon Laplace (p. 71).
5.7.14 McAdams Number Mc
Mc 5
α4 Lηl ΔT
λ3l R2l glvl
α (W m22 K21) heat transfer coefficient; L (m) characteristic length; ηl (Pa s) dynamic viscosity of condensation liquid layer; λl (W m21 K21) thermal conductivity of condensation liquid layer; Rl (kg m23) density of condensation liquid;
ΔT (K) mean temperature difference between vapour and wall g (m s22) gravitational acceleration; lvl (J kg21) specific latent heat of condensation.
It characterizes the heat transfer in fluid membrane condensation. It is constant
for a specific surface orientation.
Info: [A23].
William Henry McAdams (15.3.18922.5.1975), American
chemical engineer.
He was engaged in the heat transfer, distillation and flowing of viscous fluids. His publication Heat Transmission is
widely known. In addition, he edited many of technical
papers. His work on the Manhattan project and his original
work on nuclear submarines and jet aircraft are well known.
5.7.15 Middle Mass Temperature Θ
Θ5
T 2 T1
Tn 2 T1
Thermomechanics
239
T (K) mean temperature in condenser; T1 (K) temperature in input in condenser; Tn (K) temperature of saturated vapour.
It characterizes the mean mass temperature of the dispersed water flow in a condenser (water and water vapour).
Info: [A23].
5.7.16 Nusselt Condensation Number (1.) Nukl
Nukl 5
L3 R2l glvl
5 PrKuGak
λl ηl ΔT
Nukl 5
qA L
λl ΔT
ð1Þ;
ð2Þ
L (m) characteristic length; Rl (kg m23) condensation liquid density;
λl (W m21 K21) thermal conductivity of condensation liquid; ηl (Pa s) dynamic viscosity of condensation liquid; g (m s22) gravitation acceleration;
lvl (J kg21) specific latent heat of condensation; ΔT (K) temperature difference
during condensation; qA (W m22) surface heat flux density; Pr () Prandtl
number (p. 197); Ku () Kutateladze evaporation number (p. 220); Gak () Galilei condensation number (p. 235).
It characterizes the influence of molecular friction and gravity acting on the heat
transfer in membrane condensation.
Info: [A23].
Ernst Kraft Wilhelm Nusselt (p. 196).
5.7.17 Nusselt Condensation Number (2.) Nuk2
1 2 1
1
λl R2l g 3
Rl g 3
3
Nuk2 5
5
5
Ga
k
α η2l
η2l
λl (W m21 K21) thermal conductivity of condensation liquid; Rl (kg m23) density of condensation liquid; ηl (Pa s) dynamic viscosity of condensation liquid;
g (m s22) gravitational acceleration; α (W m22 K21) heat transfer coefficient;
δ (m) thickness of condensation liquid layer; Gak () Galilei condensation
number (p. 235).
It characterizes the heat transfer in membrane condensation on a vertical wall
under the mutual influence of friction force, gravity and molecular friction in
flowing-down condensate film.
Info: [A23].
Ernst Kraft Wilhelm Nusselt (see above).
240
Dimensionless Physical Quantities in Science and Engineering
5.7.18 Phase Transformation Number N
N5
lvl
cl ðTn 2 T1 Þ
lvl (J kg21) specific latent heat of condensation; cl (J kg21 K21) specific heat
capacity of condensation liquid; Tn (K) saturation temperature of vapour; T1 (K) steam condenser input temperature.
It expresses the ratio of the specific latent condensation heat to that withdrawn
from the saturated vapour by a unit quantity of the cooling water. It is among the
set of quantitative condensation criteria F, G, J, K, L, M, R (p. 240).
Info: [A23].
5.7.19 Prandtl Condensation Number Prk
Prk 5
νl
5 PeRe21
al
ν l (m2 s21) kinematic viscosity of condensation liquid; al (m2 s21) thermal diffusivity of condensation liquid; Pe () Péclet heat number (p. 180); Re () Reynolds number (p. 81).
It characterizes the relation between the temperature and velocity fields in condensation or, alternatively, between the thickness of the temperature and hydrodynamic boundary layers in gradientless bypassing of solid bodies. In liquid metals, it
is often Prk{1, which can be of use both in drop condensation (mercury) and
membrane condensation (sodium, potassium).
Info: [A23].
Ludwig Prandtl (p. 197).
5.7.20 Proportion of Vapour Content x
x5
Qmv
Qmv 1 Qml
Qmv, Qml (kg s21) vapour and condensation liquid mass fluxes.
It expresses the ratio of the vapour mass flow to the mass flow of a
vapourcondensate mixture. It characterizes the relative vapour content in the mixture, for example, at the inlet and outlet of a condenser tube.
Info: [A23].
5.7.21 Quantitative Condensation Criteria F, G, J, K, L, M, R
F5
gdηl lvl
5 Fr21 PrN
2 λ ΔT
wN
l
ð1Þ;
Thermomechanics
241
rffiffiffiffiffiffiffiffiffi
ΔTλl Rl ηl
5 Pr21 N 21 M 5 RM
G5
ηl lvl
Rv ηv
sffiffiffiffiffiffiffiffiffiffi
1
glvl Rl ηl d
J5
5 Re2 ðFrLÞ21 ð3Þ;
3
qA
wN
qA
K5
lvl
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
d
5 MLRe2 2
wN Rv ηv
ð2Þ;
ð4Þ;
qA d
ð5Þ;
ηl lvl
rffiffiffiffiffiffiffiffiffi
Rl ηl
ð6Þ;
M5
Rv ηv
L5
R 5 Pr 21 N 21
ð7Þ
g (m s22) gravitation acceleration; d (m) outer tube diameter; ηl, ηv (Pa s) dynamic viscosity of condensation liquid and vapour; lvl (J kg21) specific latent
heat of condensation; wN (m s21) velocity of vapour free flow; λl, λv
(W m21 K21) thermal conductivity of condensation liquid and vapour; ΔT (K) temperature difference; Rl, Rv (kg m23) density of condensation liquid and
vapour; qA (W m22) surface density of heat flux; Fr () Froude number (1.)
(p. 62); Pr () Prandtl number (p. 197); N () phase change (p. 180); Re ()
Reynolds number (p. 81).
These criteria characterize the energetic, force and other quantitative relations
under certain conditions, which are, for example, constant density of the throughwall heat flow (criteria J, K, L), constant wall temperature (criteria F, G) and physical parameters of phases (M). The phase change N (p. 180) is also among the quantitative criteria.
Info: [A23].
5.7.22 Reynolds Number of Film Condensation Rebk
Rebk 5
αΔTL
cl ΔT 21
11ϕ
lvl ηl
lvl
Rebk 5
Qm
2πrηl
ð1Þ;
ð2Þ
α (W m22 K21) heat transfer coefficient; ΔT (K) temperature difference during condensation; L (m) characteristic length; lvl (J kg21) specific latent
heat of condensation; ηl (Pa s) dynamic viscosity of condensation liquid;
242
Dimensionless Physical Quantities in Science and Engineering
cl (J kg21 K21) specific heat capacity of condensation liquid; ϕ () relative
subcooling of condensation liquid; Qm (kg s21) mass flux in condensation layer;
r (m) inner tube radius.
It expresses the ratio of the condensate film inertia force to the molecular friction in a condensate film. It characterizes the hydrodynamic force relations in a
thin condensate film. In equation (2), it is used with partial condensation in tubes.
Info: [A23].
Osborne Reynolds (p. 82).
5.7.23 Reynolds Number of Two-Phase Condensation Rek
wN L λl ðTn 2 TS Þ
5
νv
lvl ηl
wv x
ð2Þ
Rep 5
νv
Rek 5
ð1Þ;
wN (m s21) vapour free flow velocity; L (m) characteristic length; ν v (m2 s21) kinematic viscosity of vapour; λl (W m21 K21) thermal conductivity of condensation liquid; ηl (Pa s) dynamic viscosity of condensation liquid; Tn (K) temperature
of saturated vapour; TS (K) temperature of wall surface; lvl (J kg21) specific latent
heat of condensation; wv (m s21) local vapour velocity; x (m) distance in direction
of condensation liquid layer generation.
It expresses the ratio of the flowing fluid inertia force to the molecular flowing
fluid friction force. It characterizes the hydrodynamic force relations in vapour condensation. In expression (1), it represents a resultant criterion, and in expression
(2), it represents a local criterion.
Info: [A23].
Osborne Reynolds (see above).
5.7.24 Sherwood Condensation Number Shk
L3 gRl lvl
Shk 5
4ν l λl ΔT
1
4
L (m) characteristic length; g (m s22) gravitational acceleration; Rl (kg m23) density of condensation liquid; ν l (m2 s21) kinematic viscosity of condensation
liquid; λl (W m21 K21) thermal conductivity of condensation liquid; lvl (J kg21) specific latent heat of condensation; ΔT (K) temperature difference between
vapour and wall.
It characterizes the influence of molecular friction and gravity on the heat transfer in vapour membrane condensation on a vertical wall.
Info: [A23].
Thomas Kilgore Sherwood (p. 265).
Thermomechanics
243
5.7.25 Stanton Condensation Number Stk
Stk 5
α
5 NuPe21
R l cl w
Stk 5
αlvl
qA c l
ð1Þ;
ð2Þ
α (W m22 K21) heat transfer coefficient; Rl (kg m23) density of condensation
layer; cl (J kg21 K21) specific heat capacity of condensation layer; w (m s21) velocity growth of condensation film; lvl (J kg21) specific latent heat of condensation; qA (W m22) surface heat flux density; Nu () Nusselt number (p. 196); Pe
() Péclet heat number (p. 180).
It expresses the ratio of the condensate temperature change along a wall to the
temperature drop between the wall and the condensate. It characterizes the convective heat transfer process in drop and membrane condensations. Its significance is
similar to that of the Nusselt number Nu (p. 196).
Info: [A23].
Thomas Edward Stanton (p. 201).
5.7.26 Weber Condensation Number Wek
Wek 5
ðRl 2 Rv ÞgL2
σ
Wek 5
Rl gL2
σ
ð1Þ;
ð2Þ
Rl, Rv (kg m23) density of liquid and vapour; g (m s22) gravitational acceleration; L (m) characteristic length; σ (N m21) surface tension.
It expresses the ratio of the gravity force to the surface stress force. It characterizes the force relations in water condensation which appear, for example, in the
formation of the arising condensate drop. Sometimes, it is used in form (2), in the
description of the surface distillation process.
Info: [A23].
Ernst Heinrich Weber (p. 92).
Wilhelm Eduard Weber.
5.8
Heat and Mass Transfer
Compound heat and mass transfer is characterized by a mathematical model in the
expression of second-order partial equations set with composed boundary conditions. A great number of dimensional quantities are used to describe heat and mass
244
Dimensionless Physical Quantities in Science and Engineering
transfer. Among the most important are the Bulygin, Fedorov, Fourier, Kirpichev,
Kossovitch, Lykov and Posnov numbers, and also many mostly modified, dimensionless quantities for heat or mass transfer, such as the Sherwood, Schmidt and
Stanton numbers.
5.8.1
Absorption Number Ab
rffiffiffiffiffiffiffi
x
Ab 5 β
ð1Þ;
Dw
rffiffiffiffiffiffiffiffiffi
xL
Ab 5 β
ð2Þ
Dw1
β (m s21) absorption coefficient of liquid transfer; x (m) surface length with
liquid layer; D (m2 s21) gas diffusivity in liquid layer; w (m s21) mean liquid
velocity in the layer; L (m) layer thickness; w1 (m2 s21) volume velocity.
In the dimensionless form, it expresses the mass transfer coefficient, for example, in gas absorption on wet walls of chambers. In expression (2), the fluid film
thickness is considered.
Info: [A23].
5.8.2
Biot Mass Number Bim
Bim 5
αm L βL
5
λm
am
αm (kg m22 s21) specific mass transfer; L (m) characteristic length;
λm (kg m21 K21) specific mass transfer; β (m s21) mass transfer coefficient;
am (m2 s21) mass diffusivity.
This number expresses the ratio of the mass transfer on a fluidbody interface
to the mass transfer inside the L-thick wall of the body. It characterizes the relation
between the mass transfer on the outer surface of the body by convection and that
by conduction inside the body. The third-type boundary condition is used in solving
heat transfer tasks, for example, in the drying or wetting of various materials.
Info: [A23].
Jean-Baptiste Biot (p. 174).
5.8.3
Bodenstein number Bd
Bd Pem 5
w L
D
w (m s21); L (m) characteristic axial length; D (m2 s21) effective axial diffusivity; Pem () Péclet mass number (p. 258).
Thermomechanics
245
It expresses the ratio of the convective heat transfer rate to the diffusion rate. It
characterizes the diffusion process in chemical reactors. In substance, it is about
expressing the degree of the approximation of a system being examined to idealized models of reacting flows. With Bd-N, it is about total displacement of a
system and with Bd-0, it is about total mixing. For real diffusion and combined
models, N , Bd . 0 is valid. It is a special case of the Pe´clet mass number Pem
(p. 258) for mass transfer. It describes the diffusion in granulated material layers in
forced convection.
Info: [A23],[A33],[B20].
Max Ernst August Bodenstein (15.7.18713.9.1942),
German physical chemist.
In the year 1916, he explained the velocity of certain
chemical reactions with the chain reaction concept. He was
convinced that in those cases the process starts by random
collision of two molecules of different chemicals. This collision generates heat, causing another couple of molecules
to react and generate further heat, and so on. In later years,
this chain reaction concept was applied to designing the
atomic reactor and the atomic bomb.
5.8.4
Bulygin Number Bu
Bu 2
llv δp Δp
5 RmKo
cl ΔT
llv (J kg21) specific latent heat of evaporation; δp (Pa21) Soret pressure coefficient; p (Pa) pressure difference; cl (J kg21 K21) specific heat capacity of liquid;
ΔT (K) temperature difference; Rm () Ramzin number (p. 261);
Ko () Kossovitch number (p. 253).
It expresses the ratio of the heat consumed to evaporate a portion of the moisture in a molecular vapourgas mixture transfer to that consumed to heat the wet
material to the boiling point. It characterizes the phase conversion in high-temperature mass transfer especially. It is about a filtering vapour transfer in wet material
and the wet material’s ability to accumulate heat, connected with this transfer.
With it, the heat which is lost to change the moisture to vapour participating in the
molecular transfer is considered.8
Info: [A23],[A33].
N.P. Bulygin.
5.8.5
Colburn Mass Number Jm, Com
Jm Sc Prm 5
ν
Sh
ffiffiffiffiffi
5 p
D Re 3 Sc
246
Dimensionless Physical Quantities in Science and Engineering
ν (m2 s21) kinematic viscosity; D (m2 s21) diffusion coefficient; Sc () Schmidt
number (p. 263); Prm () Prandtl mass number (p. 260); Sh () Sherwood
number (p. 264); Re () Reynolds number (p. 81).
This number characterizes the mass transfer in forced viscous fluid flow. It is
also called the Schmidt number Sc (p. 263) or the Prandtl mass number Prm
(p. 260). Heat and mass transfers.
Info: [A20].
Allan Philip Colburn (p. 190).
5.8.6
Cooling of Porous Bodies Np
Np 5
qm c p L
λð1 2 μÞ
qm (kg s21 m22) mass flux density; cp (J kg21 K21) specific heat capacity;
L (m) characteristic length; λ (W m21 K21) thermal conductivity of porous
wall; μ () porosity (p. 24).
It expresses the ratio of heat transferred by convection inside a porous wall to
that transferred by conduction with an equal temperature gradient.
Info: [A23].
5.8.7
Coupling Energy Number E
E5
l
llv
l (J kg21) specific bond heat; llv (J kg21) specific latent heat of evaporation.
It expresses the ratio of the heat consumed to overcome the moisture-material
coupling to the specific evaporation heat. It characterizes the energy consumed to
release the moisture in hygroscopic material drying.
Info: [A23],[A33].
5.8.8
Drew Number Dr
Dr 5
ðMA 2 MB ÞxA 1 MB Mv
ln
xA 2 xAS ðMB 2 MA Þ MS
MA, MB (kg mol21) molar mass (gram molecule) of components A, B; xA ()
molar fraction of A in diffusion flow; xAS () molar fraction of component A in
the wall; Mv, MS (kg mol21) molar mass (gram molecule) of the mixture in
vapour and wall.
Thermomechanics
247
It characterizes the mass transfer intensity in a boundary layer with said intensity causing the change in the velocity profile in binary systems. For these systems,
it also represents the drag coefficient CD (p. 60).
Info: [A33].
5.8.9
Dufour Number Du
Du 5
r
ðN1 N 21 Þ0
r
ðp1 p21 Þ0
5
Le
Le
cp ðN1 N 21 ÞðN2 N 21 Þ cp ðp1 p 21 Þðp2 p21 Þ
Du 5
qD
cp T
ð1Þ;
ð2Þ
r (J kg21 K21) specific gas constant; cp (J kg21 K21) specific heat capacity;
N1, N2 () molecule number of components 1 and 2 in volume of binary gas
mixture, N 5 N1 1 N2; p1, p2 (Pa) partial pressures, p 5 p1 1 p2; qD (J kg21) heat transferred by diffusion in gas mixture; T (K) gas mixture temperature;
Le () Lewis number (p. 254).
It characterizes the ratio of the diffuse heat and mass transfers, in a binary mixture of gases under isotropic conditions, to the enthalpy of the unit mixture mass,
provided the linear diffusion rate equals that of conduction. Thermodiffusion.
Info: [A23],[A33].
Louis Dufour (18321892), Swiss physicist.
5.8.10 Fedorov Number (1.) Fe
Fe 5
εδt llv
5 εKoPn
cp
δt (K21) Soret thermal coefficient; llv (J kg21) specific latent heat of evaporation; cp (J kg21 K21) specific heat capacity of damp material; ε () liquid
phase conversion in vapour (p. 255); Ko () Kossovitch number (p. 253);
Pn () Posnov diffusion number (p. 259).
This number expresses the ratio of the heat necessary to evaporate the moisture
in a material internally to the average specific thermal capacity of the wet material.
It characterizes the heat and mass transfer process, especially with respect to the
drying of wet material, and defines the transition between quiet and boiling fluid
layers. The criterion does not depend on the choice of heat and mass transfer potentials, but only on ε, δt coefficients and the thermodynamic characteristics. It is
applied as a generalized variable in the analytical solution of material drying
problems.9
Info: [A23],[A33].
Igor Mikhailovich Fedorov, Russian physicist.
248
Dimensionless Physical Quantities in Science and Engineering
5.8.11 Fedorov Number (2.) Fe
Fe 5 δt β 5 Ntr Pn
δt (K21) Soret thermal coefficient; β (K) Dufour coefficient; Ntr () transfer
number (p. 268); Pn () Posnov diffusion number (p. 259).
It characterizes the transfer processes. It is analogous to the Posnov diffusion
number Pn (p. 259) for mass transfer.
Info: [A23].
Igor Mikhailovich Fedorov (see above).
5.8.12 Fedorov Number (3.) Fe
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
!ffi rffiffiffiffiffiffiffiffiffi
u
2
u
3 4gRg
R
3 4
s
Ar
Fe 5 dekv t 2
21 5
3η
Rg
3
dekv (m) equivalent particle diameter; g (m s22) gravitational acceleration;
Rg, Rs (kg m23) density of gas and particles; η (Pa s) dynamic viscosity;
Ar () Archimedes hydrodynamic number (p. 53).
It characterizes the transition from the immobile layer to the uplifting one in
fluidizing processes. The criterion is analogous to the Archimedes hydrodynamic
number Ar (p. 53).
Info: [A23],[A33].
Igor Mikhailovich Fedorov (see above).
5.8.13 Fourier Filtration Number Fop
Fop 5
ap τ
5 FoLuf ;
L2
where ap 5
kh
μ
ap (m2 s21) pressure filtration velocity; τ (s) time;
length; k (m s21) filtration coefficient; h (m) mean
porosity (p. 24); Fo () Fourier number (p. 175); Luf
number (p. 256).
It characterizes the time of non-stationary filtration of
material.
Info: [A23].
Jean Baptiste Joseph Fourier (p. 175).
L (m) characteristic
static height; μ () () Lykov filtration
moisture in the porous
Thermomechanics
249
5.8.14 Fourier Mass Number FoD, Fom
FoD 5
Dτ
L2
Fom 5
am τ
5 FoLu
L2
ð1Þ;
ð2Þ
D (m2 s21) diffusivity; τ (s) time; L (m) characteristic length; am (m2 s21) mass diffusivity; Fo () Fourier number (p. 175); Lu () Lykov number
(p. 256).
This number expresses the ratio of the intensity of the volume mass diffusion
change to that of the time change of moisture content in a material. In a dimensionless expression, it characterizes the non-stationary mass (moisture) transfer. It
determines the relation between the change rate of the mass transfer potential field
and the physical properties and size of the system in molecular material transfer in
a non-stationary state.
Info: [A23],[A33].
Jean Baptiste Joseph Fourier (see above).
5.8.15 Frőssling Mass Number Fsm
Fsm 5
Sh 22
1
1
Re2 Sc3
Sh () Sherwood number (p. 264); Re () Reynolds number (p. 81);
Sc () Schmidt number (p. 263).
It characterizes the convective mass transfer for a ball located in flowing fluid.
Info: [A23].
Karl Gustav Frőssling (born 1913), Swedish engineer.
5.8.16 Geometric Number Ge
Ge 5
h
H
h (m) bundle perimeter; H (m) bundle height.
It expresses a dimensionless bundle height. It is used to determine the mass
transfer in bundles, for example, in drying wood.
250
Dimensionless Physical Quantities in Science and Engineering
5.8.17 Graetz Mass Number Gzm
Gzm 5 Rem Prm 5 Pem
L
dH
Rem () Reynolds mass number (p. 262); Prm () Prandtl mass number (p.
260); Pem () Péclet mass number (p. 258); L (m) characteristic length; dH
(m) hydraulic diameter.
It expresses the ratio of the convective mass transfer to the diffusion heating or
cooling in fluid flow inside closed pipelines and other systems.
Info: [A23].
Leo Graetz (p. 193).
5.8.18 Granulation Number Au
Au 5
lls
llv Δu
lls (J kg21) specific heat of crystallization; llv (J kg21) specific latent heat of
evaporation; Δu () moisture difference.
It expresses the ratio of the heat arising in crystallization and drying to the total
heat necessary to evaporate moisture from a dried solution. With it, the influence
of the phase conversion (crystallization) of the solution (molten material) layer on
the moisture evaporation intensity on the granule surface is considered. It determines the heat flow direction among the granules (grains) and bypassing gas. With
the granulation in a fluidization layer, it determines the boundary between the granulation zones in drying (Au , 1) and cooling of molten material (Au . 1).
Info: [A23],[A33].
5.8.19 Grashof Mass Number Grm
gL3 β c
gL3 MN TS
Grm 5
ðRs 2 RÞ 5 2
21
MTP
ν 2l
νl
Grm 5
gL3 ΔR
ν 2l R
ð1Þ;
ð2Þ
g (m s22) gravitational acceleration; L (m) characteristic length; β c (m3 kg21) volume expansion coefficient inducing solution; ν 1 (m2 s21) kinematic viscosity of
fluid; Rs (kg m23) mass concentration of dissolved matter at surface; R (kg m23) mass concentration of dissolved matter; M, MN (kg mol21) molar mass of
steamgas mixture at liquid surface and in sufficient distance from surface; TS, TP
(K) liquid surface temperature and external environment temperature.
Thermomechanics
251
It expresses the ratio of the product of inertia and buoyancy forces to the square
of a viscous force. It characterizes the mass transfer by natural convection in a
non-isotropic vapourgas mixture, with the transfer being caused by the temperature difference and vapour concentration in the air of the vapourgas mixture.
Info: [A23].
Franz Grashof (p. 185).
5.8.20 Gukhman Drying Number Gu
See the Gukhman evaporation number Gu (p. 230).
Alexander Adolfovich Gukhman, Russian chemical engineer.
5.8.21 Heating Up of Moist Particles Nt, Nm
rffiffiffiffiffiffiffiffiffiffi
bw
L ð1Þ;
Nt 5
a
rffiffiffiffiffiffiffi
kL2
ð2Þ
Nm 5
a
w (m s21) motion velocity of material in dryer; a (m2 s21) thermal diffusivity;
L 5 Vd S21 (m) ratio of absolute dry body volume to its surface; Vd (m3) volume of absolute dry body; S (m2) surface of absolute dry body; k (s21) dry
coefficient; b () logarithmical decrement (p. 18).
It characterizes the heating intensity of dispersive spherical particles in gas flow
drying with convective and radiation heat supply of constant value and temperature
changing T 5 To exp (28wτ). It also characterizes material motion velocity in a
drying stove w(Nt) and evaporation rate Nm.
Info: [A33].
5.8.22 Internal Heat and Mass Transfer N
N5
Bi
εKo
Bi () Biot number (p. 173); ε () liquid phase conversion in vapour
(p. 255); Ko () Kossovitch number (p. 253).
It expresses the ratio of the heat supplied to a body material to the heat consumed in internal mixture evaporation in the material.
Info: [A23],[A33].
252
Dimensionless Physical Quantities in Science and Engineering
5.8.23 J-Mass Factor Jm
Jm 5
2
2
1
2
βR ν 3 β ν 3
β 2
5
5 Num Re 21 Prm 3 5 Sc3
qm D
w D
w
β (m s21) mass transfer coefficient; R (kg m23) density; qm (J kg21 m22) mass flux density; ν (m2 s21) kinematic viscosity; D (m2 s21) diffusivity;
w (m s21) flow velocity; Num () Nusselt mass number (p. 258); Re () Reynolds number (p. 81); Prm () Prandtl mass number (p. 260); Sc ()
Schmidt number (p. 263).
It characterizes the convective transfer of mass, heat and momentum.
Info: [A23].
5.8.24 Jermakov Number Er
Er 5
kqS
Rl
k (m s21) energy absorbing coefficient by freeze-drying; qS (W m22) surface
heat flux density; R (kg m23) density of liquid or vapour; l (J kg21) specific
latent heat.
This number characterizes the sublimation drying process in electromagnetic
heating with a constant drying rate. It is the measure of the ratio of the heat amount
delivered to the drying material in a certain time to that consumed to evaporate
unit volume of drying material in the same amount of time.
Info: [A33].
E.A. Jermakova.
5.8.25 Kirpichev Mass Number Kim
Kim 5
qm L
qm L
5
λmt Δϑ amt RΔuΔϑ
Kim 5 Bim
ϑS 2 ϑP
Δϑ
ð1Þ;
ð2Þ
qm (kg s21 m22) mass flux density; L (m) characteristic length; λmt
(kg m21 s21 K21) thermal mass conductivity; Δϑ (K) potential difference of
mass transfer; amt (m2 s21 K21) thermal mass diffusivity; R (kg m23) density;
Δu () moisture content difference; ϑS, ϑP (K) mass transfer potential of wall
and environment; Bim () Biot mass number (p. 244).
It characterizes the relation between outer and inner mass flow intensities. In
mass transfer tasks, it represents the second-type boundary condition.
Thermomechanics
253
Info: [A23].
Mikhail Viktorovich Kirpichev (p. 177).
5.8.26 Kossovitch Number Ko
llv cm
llv Δu
5
cp ΔT
cp ΔT
Ko 5
llv (J kg21) specific latent heat of evaporation; cm () mass fraction;
cp (J kg21 K21) specific heat capacity of damp material; ΔT (K) temperature
difference; Δu () dampness difference.
It is the phase conversion criterion. It expresses the ratio of the evaporation heat
for moisture evaporation to the heat for wet body heating. It characterizes the convective heat transfer in evaporation. It represents the ratio of the specific heat consumed to evaporate the whole amount of moisture to that consumed by wet
material when heating to maximum temperature.
Info: [A23].
5.8.27 Lagrange Number (1.) Lg1
Lg1 5
D 1 εD
D
D (m2 s21) molecular diffusivity; εD (m2 s21) vortex mass diffusivity.
It expresses the ratio of the sum of molecular and swirl mass transfer intensities
to the molecular heat transfer intensity. It characterizes the mass transfer in a turbulent flow.
Info: [A23].
Joseph-Louis Lagrange (p. 70).
5.8.28 Lebedev Number Lb
Lb 5
μbðTP 2 TÞ
Rs cmp p
Lb 5
TR
TP
Lb 5
umed
ucrit
ð1Þ;
ð2Þ;
ð3Þ
μ () porosity (p. 24); b 5 @ðmv Vv21 Þ[email protected]ðkg m23 K21 Þ temperature gradient
of vapour concentration; mv (kg) vapour mass; Vv (m3) vapour volume;
254
Dimensionless Physical Quantities in Science and Engineering
T, TP (K) initial temperature of material and environment; Rs (kg m 2 3) density of solid phase; cmp (kg kg21 Pa 2 1) pressure mass capacity of vapour
(vapour mass per unit mass of dry gas and per unit of pressure difference); p (Pa) pressure; TR (K) temperature of radiating source; umed () mean moisture
content in a body; ucrit () critical moisture content.
This number expresses the ratio of the vapour flow by molecular expansion to
the total macroscopic vapour flow from porous material. It characterizes thermal
processes in drying intensification. In expression (2), it represents the volume evaporation of dispersed fluid in a boundary layer. It determines the heat transfer
increase due to boundary layer thickness reduction with the temperature increase of
a radiating source. In expression (3), it represents the ratio of a material’s mean
moisture with drying rate reduction to a material’s mean critical moisture. With it,
the heat transfer coefficient reduction due to material moisture reduction is considered. Sometimes expression (2) is called the Gukhman modified criterion.
Info: [A23].
Peter Nikolaevich Lebedev (8.3.186614.3.1912),
Russian physicist.
He founded the Russian school of physicists. He was the
first to discover and investigate mm-range electromagnetic
waves (1895) and addressed himself to the nature of the terrestrial magnetic field. He was engaged in the action of
light on solid substances (1900) and gases (1908). He confirmed the electromagnetic theory of light from the quantitative point of view. In his last years, he devoted himself to
the problem of ultrasound.
5.8.29 Lewis Number Le
Le 5
λ
a
5 5 ScPr21 5 Prm Pr21
Rcp D D
Le 5
D
a
ð1Þ;
ð2Þ
λ (W m21 K21) thermal conductivity; R (kg m23) density; cp (J kg21 K21) specific heat capacity; D (m2 s21) molecular diffusivity; a (m2 s21) thermal
diffusivity; Sc () Schmidt number (p. 263); Prm () Prandtl mass number
(p. 260); Pr () Prandtl number (p. 197).
It expresses the thermal-to-molecular diffusivities ratio. It characterizes the
mutual relation of heat and mass transfers in various materials. It is the ratio of
chemical potential field change to a thermal field in the flow of gas or fluid mixtures. For gases, it is LeAh0.8; 1.2i, for fluids LeAh70; 100i. In expression (2), it is
usually called the LewisSemenov number, or sometimes only the Lewis number.
Info: [A23].
Thermomechanics
255
Warren Kendall Lewis (21.8.18829.3.1975), American
chemical engineer.
He is called the father of modern chemical engineering.
He was engaged in researching distillation and evaporation,
petroleum flow modelling, predicting the lifetime of petroleum deposits and developing methods for exploiting them.
His work concerning powder fluidization and the control of
its movement in combustion chambers, and in heat and catalytic cracking, was very significant.
Nikolay Nikolayevich Semyonov (p. 48).
5.8.30 Lewis Turbulence Number Letur
Letur 5
R cp εD
lD
εD
5
5
λT
lT
εT
R (kg m23) density; cp (J kg21 K21) specific heat capacity; εD, εT (m2 s21) vortex mass and thermal diffusivity; λT (W m21 K21) vortex thermal diffusivity;
lD, lT (m) length of mixing for mass and heat.
This number expresses the ratio of the molecular mass transfer intensity of a
given component in a mixture to the turbulent heat transfer intensity. It characterizes the kinematic properties in heat and mass transfers in gas mixture flow. It
expresses the relation between thermal and concentration fields.
Info: [A23].
Warren Kendall Lewis (see above).
5.8.31 Liquid Phase Conversion in Vapour ε
ε5
duf
du
ε5
qmv
qmv 1 qml
ð1Þ;
ð2Þ
0#ε#1
uf () body moisture content with phase change; u () body moisture content;
qmv, qml (kg s21 m22) mass flux density of vapour and liquid.
It expresses the ratio of the relative moisture content change by evaporation to
the total moisture content change in a given place of a body under conditions of
non-stationary moisture transfer in drying. It characterizes the vapour amount, diffusing in capillary porous material, with respect to fluid and vapour flow without
256
Dimensionless Physical Quantities in Science and Engineering
considering the convective transfer. For the moisture content change in an arbitrary
place, the following limits are valid:
ε 5 0 the moisture is changed by fluid transfer only,
ε 5 1 the moisture is changed only by evaporation.
In hygroscopic materials drying, it characterizes, in equation (2), the ratio of the
absolute value of the resulting vapour mass flow density to the sum of densities of
the vapour and fluid mass flows in non-stationary moisture transfer.
Info : [A23],[A33].
5.8.32 Lukomsky Number Lk
Lk Lu21 5
a
am
a (m2 s21) thermal diffusivity; am (m2 s21) mass diffusivity; Lu () Lykov
number (p. 256).
It characterizes combined heat and mass transfer. It is analogous to the Lykov
number Lu (p. 256).
Info: [A29].
5.8.33 Lykov Filtration Number Luf
Luf 5
ap
a
ap, a (m2 s21) pressure filtration and thermal diffusivity.
It expresses the pressure to temperature fields propagation intensities ratio. It
characterizes the pressure to thermal diffusion ratio in heat and mass transfers.
Usually, it is Lufc1, but it is 1001000 most frequently.
Info: [A23].
Alexey Vasilievich Lykov (see below).
5.8.34 Lykov Number Lu
Lu 5
am
a
am, a (m2 s21) mass and thermal diffusivity.
This number expresses the ratio of the mass to temperature field propagation
intensities. It characterizes the relation between the temperature field propagation
and the mass (moisture) field. It expresses the mutual heat-to-mass transfer ratio in
a capillary porous material and in dispersive materials. It is the degree of the potential molecular field relaxation of the materials. The difference between the Lykov
Thermomechanics
257
and Lewis number Le (p. 254) is analogous to that between the Biot Bi (p. 174) and
Nusselt number Nu (p. 196), for example, as the intensities of corresponding phenomena ratio.
Info: [A23].
Alexey Vasilievich Lykov (20.9.191028.6.1974), Russian
physicist.
In the year 1935, he discovered the thermodiffusion of
the humidity in capillary porous material. It was called the
Lykov effect. He was engaged in solving complicated problems of heat and mass transfer in capillary porous material
and dispersal environments under acting of phase and
chemical conversions. Of his many monographs, the
Teorija suški (Theory of Drying, 1951) and Teorija teploprovodnosti (Theory of Thermal Conductivity, 1969) are his
most important works.
5.8.35 Margoulis Mass Number Mgm
See the Stanton mass number Stm (p. 256).
Wladimir Margoulis (born 1886).
5.8.36 Merkel Number Me
Me 5
qm S
Qm
qm (kg s21 m22) mass flux density; S (m2) total surface of water level that is
in contact with air; Qm (kg s21) mass flux.
It expresses the ratio of water mass, transferred in cooling, to dry gas mass. It
characterizes the mass transfer in basin coolers and cooling towers.
Info: [A23],[B11].
Friedrich Merkel (born 1892), German engineer.
5.8.37 Mikhailov Number Mi
Mi 5 δp
dp
5 BuRb
du
δp (Pa21) Soret pressure coefficient; p (Pa) pressure; u () moisture content; Bu () Bulygin number (p. 227); Rb () Rebinder number (p. 262).
It characterizes the filtering mass transfer kinetics in intense heat and mass
transfer in drying. It expresses the degree to which pores are filled by a fluid at a
258
Dimensionless Physical Quantities in Science and Engineering
given temperature or, alternatively, the total pressure change of the vapourgas
mixture or a fluid inside a material.
5.8.38 Miniovich Number Mn
Mn 5
Sr
Vp
S (m2) particle surface, V (m3) particle volume, r (m) pore radius; p() porosity (p. 24).
In heat and mass transfer processes in a very small area of a layer of material, it
expresses the layer geometry influence on temperature and moisture fields in the
layer. Drying of porous materials.
Info: [A33].
J.M. Miniovich.
5.8.39 Nusselt Mass Number Num
See the Sherwood number Sh (p. 264).
Ernst Kraft Wilhelm Nusselt (p. 196).
5.8.40 Péclet Mass and Vortex Number Pemu
Pemu 5
wL
εD
w (m s21) velocity; L (m) characteristic length; εD (m2 s21) vortex mass
diffusivity.
It expresses the ratio of the convective mass propagation to the propagation rate
caused by molecular vortex diffusion. It characterizes the convective-vortex diffusive mass transfer in flowing fluid.
Info: [A23].
Jean Claude Eugène Péclet (p. 180)
5.8.41 Péclet Mass Number Pem
Pem 5
wL
5 RePrm 5 ReSc
D
ð1Þ;
Pem 5
J
kBD
ð2Þ
w (m s21) velocity; L (m) characteristic length; D (m2 s21) diffusivity;
J (kg s21) convective mass flux through the membrane; kBD (kg s21) total
Thermomechanics
259
coefficient of back diffusion; Re () Reynolds number (p. 81); Prm () Prandtl mass number (p. 260); Sc () Schmidt number (p. 263).
In expression (1), it expresses the ratio of the volume mass transfer rate,
with forced convection, to the diffusive mass transfer rate. It characterizes the
convectivediffusive mass transfer in flowing fluid. It is also called the Reynolds
mass number Rem (p. 262).
For diaphragm systems, the dimensionless mass transfer parameter (2) is used,
which expresses the ratio of convective flow through a diaphragm to a phenomenon
which causes return diffusion expressed by the total coefficient kBD.
Info: [A23].
Jean Claude Eugène Péclet (see above).
5.8.42 Pomerantsev Mass Number Pom
Pom 5
qmV L2
am RΔu
qmV (kg s21 m23) volume density of mass flux; L (m) characteristic length;
am (m2 s21) mass diffusivity; R (kg m23) material density; Δu () difference of moisture content.
This number expresses the ratio of unit mass development by an internal source
to the maximum possible unit mass amount transferred by conduction. It characterizes an internal mass source in a dimensionless way.
Info: [A23].
Alexey Alexandrovich Pomerantsev, Russian engineer.
5.8.43 Posnov Diffusion Number Pn
δt ΔT
amt RΔT
amt RΔT
5
ð1Þ;
5
Δu
amt RΔu
λmt Δϑ
" #
@ϑ
cmt @T
ΔT
@ ϑ
u
Pn 5
5 Δϑ
ð2Þ
T
ðcmt Þmed Δϑ
@ ΔT
Pn 5
u
δt (K21) Soret thermal coefficient; ΔT (K) temperature difference; Δu () difference of moisture content; amt (m2 s21 K21) thermal mass diffusivity; R (kg m23) density of material; am (m2 s21) mass diffusivity; λmt (kg m21 s21 K21) thermal
mass conductivity; Δϑ (K) difference of mass transfer potential; cmt (K21) thermal
mass capacity.
It expresses the ratio of the thermodiffusive moisture transfer intensity to the diffusive moisture transfer intensity. It characterizes the internal moisture transfer by
conduction under non-isentropic conditions. The relative non-uniformity of a moisture field in a material is caused by heat and mass transfer in a steady state. With
260
Dimensionless Physical Quantities in Science and Engineering
small Δu or linear dependence of u(ϑ), the Pn number becomes a dynamic transfer
parameter a relative temperature product of the mass transfer potential expressed
by expression (2). In this form, it expresses the ratio of the potential change of
mass (moisture content) to the relative temperature change with a given mass steadystate content in a body. It is analogous to the Fedorov number (2.) Fe (p. 248).
Info: [A23].
5.8.44 Posnov Filtration Number Pnp
Pnp 5
δp Δp
λmp Δp
5
Δu
Ram Δu
δp (Pa21) Soret pressure coefficient; Δp (Pa) pressure difference; Δu () difference of moisture content; λmp (kg m21 s21 Pa21) pressure mass conductivity; R (kg m23) density; am (m2 s21) mass diffusivity.
It expresses the ratio of the filtration to diffusive moisture transfer intensities. It
characterizes the internal moisture transfer in material during filtration. The relative
moisture field non-uniformity in the material is caused by a pressure gradient
(filtration potential difference).
Info: [A23].
5.8.45 Prandtl Mass Number Prm
See the Schmidt number Sc (p. 263).
Ludwig Prandtl (p. 197)
5.8.46 Predvoditelev Mass Number Pdm
21
dðqm qmi
Þ
Pdm 5 2
dFo
max
Pdm 5
kL2
am
ð1Þ;
ð2Þ
qm, qmi (kg s21 m23, or kg s21 m22) volume and surface mass flow intensity and
its initial value; k (s21) drying coefficient; L (m) characteristic length;
am (m2 s21) mass diffusivity; Fo () Fourier number (p. 175).
It expresses the ratio of the material concentration change rate in the surroundings to that in a body. It characterizes the mass flow density time change, for example, that of an internal source (qm qmV) or of the second-type boundary condition
(qm qmA). In tasks involving wet material heating in constant temperature surroundings, when the surface moisture evaporation occurs in the second drying
Thermomechanics
261
period according to the law qm 5 qmi exp (2 kτ), expression (2) is valid, and the
first period of drying Pdm 5 0 is valid.
Info: [A23].
Alexander Savviich Predvoditelev (18911973), Russian physicist.
5.8.47 Psychrometric Relation K
K5
αcm
qm c
α (W m22 K21) heat transfer coefficient; cm () mass ratio; qm (kg s21 m22) mass flux density; c (J kg21 K21) specific heat capacity.
It expresses the ratio of the heat transferred by convection to that transferred by
the mass transfer. It characterizes wet air as a heat transfer means and a drying
environment simultaneously. Wet and dry thermometer well.
Info: [A23].
5.8.48 Ramzin Number Rm
Rm 5
δp Δp δp Δp
5 BuKo21
5
Δu
δt Δϑ
δp (Pa21) Soret pressure coefficient; Δp (Pa) pressure difference; Δu () difference of moisture content; δt (K21) Soret thermal coefficient; Δϑ (K) difference of moisture transfer potential; Bu () Bulygin number (p. 227); Ko () Kossovitch number (p. 253).
In molecular transfer, it characterizes the vapour content described as a mass
content of the material which takes part in the molecular transfer. See the Posnov
filtration number Pnp (p. 260).
Info: [A23].
5.8.49 Rayleigh Mass Number Ram
Ram 5
XgL3 Δc gL3 ΔR
5
5 GrSh
νD R
νD
X () dimensionless diffusion number; g (m s22) gravitational acceleration;
L (m) characteristic length; Δc () difference of initial and final mass ratio of
chemical reaction products; ν (m2 s21) kinematic viscosity; D (m2 s21) diffusivity; ΔR (kg m23) density difference; R (kg m23) density; Gr () Grashof
heat number (p. 185); Sh () Sherwood number (p. 264).
It characterizes the natural convection influence on the mass transfer in fluids,
for example, the concentration and temperature propagation influence on the
262
Dimensionless Physical Quantities in Science and Engineering
thermodiffusion separation of liquid mixtures in a fluid thermodiffusion separating
column.
Info: [A23].
Lord Rayleigh (p. 187).
5.8.50 Rebinder Number Rb
Rb 5
cb
dTmed c
5
5 BKo 21
dumed llv
llv
c (J kg21 K21) specific heat capacity; b 5 dTmed/dumed (K) change of mean
integral temperature to change of mean integral moisture content ratio; llv (J kg21) specific latent heat of vaporization; B 5 ΔubΔT21 () dimensionless drying
coefficient; Ko () Kossovitch number (p. 253).
This number expresses the ratio of the heat consumed in material heating to that
consumed in moisture evaporating during an infinitely short time interval. It relates
to the kinematic characteristics of integral properties of heat and mass transfer. It
expresses the characteristics of local changes of temperature and moisture content.
Info: [A23].
Peter Aleksandrovich Rebinder (21.9.189812.7.1972),
Russian physical chemist.
He was engaged in the problems of the origin, stability,
surface phenomena and structure formation in disperse systems. In addition, he explained the idea of the molecular
mechanism of surface active substances and their applications in diverse technological processes. In the year 1928,
he clarified the influence of absorption in reducing the
rigidity of solid bodies.
5.8.51 Reynolds Mass Number Rem
See the Pe´clet mass number Pem (p. 258), Richman number Ri (p. 232).
Osborne Reynolds (p. 82).
5.8.52 Romankov Number Ro
Ro 5
Tg 2 Ts
Tg
Ro 5
Tg
Ts
ð2Þ
ð1Þ;
Thermomechanics
263
Tg (K) hot gas temperature used for drying; Ts (K) temperature of dried-up
material.
This number expresses the ratio of the temperature gradient between gas and
material in drying to the heated gas temperature at the drying stove outlet. In drying, it characterizes the potential wet gas possibilities to heat the dried material.
Sometimes, it is used in the form of equation (2).
Info: [A23].
Peter Grigorievich Romankov (17.1.19041.10.1990),
Russian physical chemist.
He was engaged in the analysis of kinetic laws and the
generalization of the methods of similarity theory and
dimensional analysis. He applied the results on heat and
mass transfer processes and hydromechanical ones in
chemical technology. His principal work is the monograph
Gidravličeskije processy v chimičeskoj technologii
(Hydraulic Processes in Chemical Engineering). He was
also engaged in fluid mixing, filtration, centrifugal separation, cleaning of gas, drying, absorption, extraction and
distillation.
5.8.53 Schmidt Effective Number Scef
Scef 5
εM 1 ν
εD 1 D
εM, εD (m2 s21) vortex momentum diffusivity and vortex mass diffusivity; ν
(m2 s21) kinematic viscosity; D (m2 s21) diffusivity.
It expresses the ratio of the total momentum diffusivity to the total mass diffusivity in the mass transfer in combined laminar and turbulent flow. It characterizes
the mass transfer in combined turbulent and laminar flow.
Info: [A23].
Ernst Schmidt (see below).
5.8.54 Schmidt Number Sc
Sc JQ Prm 5
ν
D
ν (m2 s21) kinematic viscosity; D (m2 s21) diffusivity; JQ () Colburn
number (p. 190); Prm () Prandtl mass number (p. 260).
This number expresses the ratio of the kinematic viscosity, or momentum transfer by internal friction, to the molecular diffusivity. It characterizes the relation
between the material and momentum transfers in mass transfer. It provides the
264
Dimensionless Physical Quantities in Science and Engineering
similarity of velocity and concentration fields in mass transfer. For example, molten materials with an equal Schmidt number have similar velocity and concentration
fields. Higher Sc number values characterize slower mass exchange and higher
values of dividing coefficients. This leads to higher mixing and a tendency to crack
in a solidified casting. The criterion was first introduced by Schmidt in 1929. It
is also called the Colburn number Jq (p. 190) or the Prandtl mass number Prm
(p. 260).
Info: [A23],[A43].
Ernst Schmidt (11.2.189222.1.1975), German engineer.
He was a pioneer in engineering thermodynamics, especially in heat and mass transfer. Initially, he was engaged in
measuring the radiation properties of solid materials, which
was the reason to design an effective thermal shield of aluminium foil. In addition, he was engaged in solving nonstationary thermal fields by a graphic difference method, in
determining the local heat transfer coefficient and in analogy between heat and mass transfer.
5.8.55 Schmidt Turbulent Number Scv
Scv 5
εM
εD
εM, εD (m2 s21) vortex momentum and mass diffusivity.
It expresses the ratio of the vortex momentum diffusivity to the vortex mass diffusivity. It characterizes the mass transfer in a turbulent flow.
Info: [A23].
Ernst Schmidt (see above).
5.8.56 Sherwood Number Sh
Sh Num Tam 5
βL
D
β (m s21) mass transfer coefficient; L (m) characteristic length; D (m2 s21) molecular diffusivity; Num () Nusselt mass number (p. 258); Tam () Taylor
mass number (p. 266).
It expresses the ratio of the heat transfer to the molecular diffusion. It characterizes the mass transfer intensity at the interface of phases. It is also called the
Nusselt mass number Num (p. 258) or the Taylor mass number Tam (p. 266).
Info: [A23].
Thermomechanics
265
Thomas Kilgore Sherwood (25.7.190314.1.1976),
American chemical engineer.
He was engaged primarily in research on mass transfer
and its interactions with flow, chemical reactions and industrial applications in which these processes play a principal
role. After having published Absorption and Extraction
(1937), which was the first, sui generis in this field, he
became a world famous personality.
5.8.57 Soret Thermodiffusion Number So
So 5 Le
ðN2 N 21 Þini
ðN1 N 21 ÞðN2 N 21 Þ
N1, N2 () molecule number of components 1 and 2 in binary gas mixture,
N 5 N1 1 N2; Le () Lewis number (p. 254).
It characterizes the thermodiffusion effect in mass transfer. In binary gas mixtures with equal initial content of components, it equals half of the thermodiffusion
constant. It expresses the coupling between heat and mass transfers. It represents
the dimensionless thermodiffusion coefficient.
Info: [A23].
Charles Soret (18541904), Swiss physicist.
5.8.58 Spalding Transfer Number Spp
ΦR
Spp 5 cp ΔT llv 2
Qm
21
cp (J kg21 K21) specific heat, capacity; ΔT (K) temperature difference;
llv (J kg21) specific latent heat of vaporization; ΦR (W) radiation heat flux;
Qm (kg s21) mass flux.
It expresses the ratio of the thermal energy change to the latent evaporation heat
of material. It characterizes heat and mass transfers under conditions of thermal
radiation and phase conversion.
Info: [A23].
Brian Spalding (p. 200).
5.8.59 Stanton Mass Number Stm
Stm Mg 5
β
5 ShRe 21 Sc 21
w
ð1Þ;
266
Dimensionless Physical Quantities in Science and Engineering
Stm 5
qm
1
5
Rw αmp w
ð2Þ
β (m s21) mass transfer coefficient; w (m s21) flow velocity; qm (kg s21 m22) mass flux density; R (kg m23) fluid density; αmp (kg m22 s21 Pa21) pressure
specific mass transfer; Mg () Margoulis number (p. 196); Sh () Sherwood
number (p. 264); Re () Reynolds number (p. 81); Sc () Schmidt number
(p. 263).
It expresses the ratio of the mass transfer perpendicular to a solid phase surface
to the mass transfer by flowing in parallel with the solid phase surface. It characterizes the mass transfer at the interface between the solid and fluid phases. It is
called the Margoulis number Mg (p. 196) as well.
Info: [A23].
Thomas Edward Stanton (p. 201).
5.8.60 Stefan Parameter Sf
Sf 5
p
p 2 ppP
ln
ppS 2 ppP p 2 ppS
p (Pa) local absolute static pressure; ppS (Pa) partial vapour pressure above
the surface of evaporation or condensation; ppP (Pa) partial vapour pressure in
environment.
It expresses the influence of the transversal material flow on the size of the
mass transfer coefficient in evaporation or vapour condensation from a vapourgas
mixture.
Info: [A23].
Josef Stefan (p. 214).
5.8.61 Taylor Mass Number Tam
See the Sherwood number Sh (p. 264).
Geoffrey Ingram Taylor (p. 89).
5.8.62 Temkin Heat and Mass Transfer Number Di
Di 5
qm llv L
5 KoLuKim
λΔT
qm (kg s21 m22) density of mass flux; llv (J kg21) specific latent heat of evaporation; L (m) characteristic length; λ (W m21 K21) thermal conductivity of
material; ΔT (K) initial and final temperature difference; Ko () Kossovitch
Thermomechanics
267
number (p. 253); Lu () Lykov number (p. 256); Kim () Kirpichev mass
number (p. 252).
This number expresses the ratio of the external heat transition intensity, caused
by the external mass transition, to the internal heat transfer intensity by conduction
in wet material. It characterizes the mean thermal gradient value in wet material.
Info: [A23].
A.G. Temkin.
5.8.63 Temperature Criterion N
N5
Ta 2 Ts
Ta 2 Tad
Ta (K) temperature of moist air; Ts (K) material temperature after drying-up;
Tad air temperature in a state of adiabatic saturation (temperature of a wet
thermometer).
It characterizes the potential possibilities of wet air as a heating material and
drying agent as well.
Info: [A23].
5.8.64 Thiele Modulus (1.) Th, mT
Th Da4 5
euL2
λT
e (m2 s22) specific energy; u (kg m23 s21) chemical reaction rate; L (m) characteristic length; λ (W m21 K21) thermal conductivity; T (K) temperature;
Da4 () Damkőhler number (4.) (p. 37).
It expresses the ratio of the surface reaction rate to the thermal diffusion rate. It
characterizes the non-isothermal heat and mass transfer in physical, chemical and
biotechnical processes. Catalysis. Diffusion in porous catalysers.
Info: [A29].
Ernest William Thiele (8.12.189529.11.1993), American
chemical engineer.
Initially, he was engaged in work related to carbon reactions and evolved an idea leading to the McCabeThiele
method of fractional column design. During the war, he
participated in the atomic energy development program and
was engaged in heavy water extraction. After the war
(1948), he worked on a project to exploit the nuclear drive
in aeronautics. Then he was engaged in refining processes
and the distillation of hydrocarbon mixtures. In connection
with this, he solved the heat and mass transfer problems.
268
Dimensionless Physical Quantities in Science and Engineering
5.8.65 Transfer Number Ntr
Ntr 5
β Δn
ΔT
ð1Þ;
Ntr 5
β Δu
ΔT
ð2Þ
β (K) Dufour coefficient; Δn () difference of mass concentration; ΔT (K) temperature difference; Δu () difference of relative moisture content.
In equation (1), it expresses the general transfer phenomena which represent the
mass concentration change due to a temperature change, and in equation (2) it
expresses a moisture change. Thermomechanics. Heat and mass transfer.
Info: [A23].
5.8.66 Transfer Number (1.) Thermodynamic Ntr , ε
Ntr 5 β
c
r
β (K21) volume thermal expansion coefficient; c (J kg21 K21) specific heat
capacity; r (J kg21) specific heat of evaporation.
It characterizes the thermodynamic heat and mass transfer processes. It is analogous
to the Posnov diffusion number Pn (p. 259) and the Fedorov number (2.) Fe (p. 248).
Info: [A23].
5.8.67 Transfer Number (2.) Concentration Ntr
Ntr 5 δt
Δcm
ΔT
δt (K21) Soret thermal coefficient; Δcm () difference of relative substance
concentration; ΔT (K) temperature difference.
It characterizes the mass concentration process in heat and mass transfer.
Info: [A23].
5.8.68 Transfer Number (3.) Diffusion Ntr , ε
Ntr 5
Dx
5 Ab2 Sh22
Qδ
Thermomechanics
269
D (m2 s21) molecular diffusivity; x (m) wetted length; Q (m3 s21) volume
flow; δ () relative film thickness; Ab () absorption number (p. 244);
Sh () Sherwood number (p. 264).
It characterizes dimensionless diffusivity. It expresses the gas absorption on wet
chamber walls. Mass transfer.
Info: [A23].
5.9
Non-Equilibrium Thermomechanics
The dimensionless quantities for non-equilibrium thermomechanics involve the
processes of wave heat propagation and thermal stress in material and parallel
heat and mass propagation. The wave propagation is closely related to thermal
strokes and the action of intensive heat sources on the material. A survey of the
wave propagation theory is given in [A46]. Parallel heat and mass propagation is
applied, for example, in the transfer in porous and capillary materials and is related
to new technologies and materials especially. Most of the dimensionless quantities
are more complicated modifications of basic criteria and represent the influence of
the heat and mass propagation velocities, relaxation action, thermalization and
internal heat transfer. The following numbers are important: the Fourier, Biot,
Nusselt, Pe´clet, Vernotte, Sparrow and others. In this section, the more detailed
explanation of some dimensionless quantities relates, above all, to their specific
significance in non-equilibrium thermomechanics and in the actual complexity of
wave and wave-diffusion processes.
5.9.1
Biot Volumetric Number of Internal Heat Transfer BiV
BiV 5
αV L2
λ
BiV 5
τ 2 α2V
5 Fo2 ðBiV Þ2 Pc
c1 c2
ð1Þ;
ð2Þ
αV (W m23 K21) volume coefficient of internal heat transfer; L (m) characteristic length; λ (W m21 K21) thermal conductivity; τ (s) time; c1, c2
(J m23 K21) specific volume heat capacity of material components; Fo () Fourier number (p. 175); BiV () Biot volumetric number of internal heat transfer defined by the relation (1); Pc () relative parallel heat capacity (p. 279).
The Biot volumetric number of internal heat transfer expresses the ratio of the
internal inter-component conductivity (heat transfer between the material components) and the standard volume heat conductivity.
270
Dimensionless Physical Quantities in Science and Engineering
With great internal heat transfer, the heat propagates as in homogeneous material and only one diffusive heat conduction equation can be used for the
description.
With the internal heat transfer comparable to the material thermal conductivity,
the heat propagates in parallel within the framework of material components and
heat exchange occurs among them. To describe it mathematically, one equation of
parallel heat propagation must be used or, alternatively, two diffusive equations
with mutual heat transfer coupling.
With less intensive internal heat transfer, the heat propagates in the material in
parallel within the framework of the components, between which thermal interaction scarcely occurs. Then, the mathematical description of the heat propagation
consists of two independent equations for the heat conduction in material components. If the thermal conductivities of the material components do not differ too
much, only one single equation can be used for heat conduction, as in a homogeneous material.
In contrast to the standard expression (1), with the characteristic dimension
being replaced by the product of the thermal process time and the equivalent parallel heat propagation rate, the Biot volumetric number of internal heat transfer can
be defined as in expression (2).
Info: [A17],[B45].
Jean-Baptiste Biot (p. 174).
5.9.2
Cattaneo Number Cat
Cat 5
τλw2
ν 2 Rcp
τ (s) čas; λ (W m21 K21) thermal conductivity of fluid flow; w (m s21) flow velocity; ν (m2 s21) kinematic viscosity; R (kg m23) fluid density;
cp (J kg21 K21) specific heat capacity.
It expresses the existence and propagation of a discontinuity in the velocity gradient and temperature of a flowing fluid along a wall, heated with an impulse
method or randomly by an intensive heat source. Non-equilibrium
thermomechanics.
Info: [B93].
5.9.3
Coupling Factor Ncoup
Ncoup 5
GL2
λref
G (W m23 K211) electronphonon coupling factor; L (m) characteristic
length; λref (W m21 K21) thermal conductivity.
Thermomechanics
271
It represents the ratio of the square of the convective electron heat transfer to
the heat transferred by conduction. It expresses an energetic coupling between electron gas and a grid. Wave heat propagation. Non-equilibrium thermomechanics.
Physical technology.
Info: [A70],[B63].
5.9.4
Dominance Number K
a @2 T @T 21
K5 2 2
c @t
@t
a (m2 s) thermal diffusivity; c (m s21) velocity propagation of thermal wave;
T (K) temperature; t (s) time; τ (s) relaxation time ðτ 5 a=c2 Þ:
The wave character of heat transfer predominates for Kc1; i.e. (1) if the critical
frequency c2/a is small (it is the inverse value of the relaxation time τ), (2) if the
process time t is very short or (3) if the heating or cooling rate (caused by the heat
flow) is high. The first condition involves the solid material properties only,
whereas the second and third depend also on the combined influence of the geometric arrangement and system thermal load. This criterion is more general than
the usual wave propagation one.
Info: [A46].
Robert Tzou (born 1955), American physical engineer.
He is engaged in unbalanced macro- and microscopic
heat propagation, in shock heat processes and their modelling. Furthermore, he was engaged in the diffusion anomaly
in amorphous surroundings, control of fast shock thermal
processes, thermomechanical properties of composites, thermomechanical interactions in fractures and high-velocity
penetration through surroundings. His monographs Macroto Microscale Heat Transfer (1997) and Ultrafast Heating
and Thermomechanical Coupling Induced by Femtosecond
Lasers (2007) are especially significant.
5.9.5
Fourier Mass Relaxation Number Form
Form 5
am τ rx
L2
am (m2 s21) mass diffusivity; τ rx (s) relaxation time of material; L (m) characteristic length.
It characterizes the relaxation time of a non-stationary mass transfer.
Info: [A23].
Jean Baptiste Joseph Fourier (p. 175).
272
Dimensionless Physical Quantities in Science and Engineering
5.9.6
Fourier Porous Number (1.) Fop
Fop 5
λekv τ t
cf
5 Sp21
2
cL
c
where τ t Rh c f
;
α
c 5 εcf 1ð1 2 εÞcs ;
ε5
Vf
V
λekv (W m21 K21) equivalent thermal conductivity; τt (s) time delay of temperature; c, cf, cs (J m23 K21) specific volume heat capacity: final, fluid and
solid material; L (m) thickness of porous layer; Rh (m) hydraulic diameter;
α (W m22 K21) internal heat transfer; ε () volume ratio; Vf, V (m3) fluid
volume, total volume; Sp () Sparrow number (p. 281).
This number characterizes the dimensionless time in heat transfer by fluid flowing through porous material. It expresses the influence of the non-equilibrium heat
propagation.
Info: [B76].
Jean Baptiste Joseph Fourier (see above).
5.9.7
Fourier Porous Number (2.) Fop
Fop 5
c2f w2 τ
λekv τ cf wL
5
5 Fop Pe2p
cλekv
cL2 λekv
cf (J m23 K21) fluid specific volume heat capacity; w (m s21) constant mean
velocity; τ (s) time; c (J m23 K21) final specific volume heat capacity; λekv
(W m21 K21) equivalent thermal conductivity; L (m) porous layer thickness;
τ q, τ t (s) time delay of heat flux and temperature; Fop () Fourier porous
number (1.) (p. 272); Pep () Péclet porous number (p. 279).
It characterizes the dimensionless time in forced flow and heat transfer in a
porous material. The equilibrium state, constant velocity and times τ q 5 τ t are
considered.
Info: [B79].
Jean Baptiste Joseph Fourier (see above).
5.9.8
Fourier Relaxation Heat Number Forx
Forx 5
aτ rx
5 ShPe21
L2
a (m2 s21) thermal diffusivity; τ rx (s) relaxation time; L (m) characteristic
length; Sh () Strouhal number (p. 87); Pe () Péclet heat number (p. 180).
Thermomechanics
273
It characterizes the inertia of temperature equalizing in a thermal system. It
represents the part of the Fourier number Fo (p. 175) corresponding to the thermal
field relaxation time in a system. The relaxation time is that during which all parameters of the considered thermal system are equalized.
The influence of the Fourier relaxation heat number can be expressed, analogously to the molecular heat and mass transfers, in three bands: wave conduction
for Forx . 102; wave-diffusive conduction for 1024 , Forx , 102; and diffusive conduction for Forx , 1024. Non-equilibrium thermomechanics. Physical technology.
Info: [A23],[B45].
Jean Baptiste Joseph Fourier (see above).
5.9.9
Fourier Relaxation Parallel Number Foq
aτ q
5 BiV21 P1c ð1Þ;
L2
1 λ1 1 λ2 1 1
1 21
1
Foq 5 2
L c1 1 c2 αV c1
c2
Foq 5
ð2Þ
a (m2 s21) thermal diffusivity; τ q (s) relaxation time of parallel propagation;
L (m) characteristic length; λ1, λ2 (W m21 K21) thermal conductivity of material components; c1, c2 (J m23 K21) specific volume heat capacity of material
components; αV (W m23 K21) volume coefficient of internal heat transfer;
BiV () Biot volumetric number of internal heat transfer (p. 269); Pc () relative parallel heat capacity (p. 279).
It determines the timescale for material transition from the non-equilibrium
state, which is caused by different temperatures of its individual components in one
place. The velocity, with which the material reaches equilibrium by the action of
internal heat transfer, depends on the relaxation time value of parallel heat propagation, where the equilibrium is expressed by equal temperatures of all transition
components. The relaxation time value is defined as the product of the material
thermal capacity and the internal transfer flow of unit energy. The relaxation time
represents the effective time during which the whole material thermal content could
be transported among its components. The Fourier relaxation parallel number
expresses the ratio of the series and parallel material thermal capacities and the
inner heat transfer (1). In the case of two-component material, it can be expressed
as in expression (2).
Info: [A17].
Jean Baptiste Joseph Fourier (see above).
5.9.10 Fourier Relaxation Wave Number Foq
Foq 5
2
aτ q
l
5
2
L
L
274
Dimensionless Physical Quantities in Science and Engineering
a (m2 s21) thermal diffusivity; τ q (s) relaxation time of wave heat propagation; L (m) characteristic length; l (m) mean free path of heat carrier.
The relaxation time of heat wave propagation describes the effect of inelastic
collisions with which the thermal wave energy converts to the system internal
energy and is damped in passing through the material. The Fourier relaxation
wave number equals the second power of the ratio of the mean free path and the
characteristic system dimension.
By making use of the analogy with low pressure gas flow, the influence of Foq
on the propagation character can be determined. With Foq . 102, almost no collisions occur in the considered material volume, there is no diffusion, and the energy
carriers transfer the heat freely directly inside the material. With 1024 , Foq , 102,
a transient state exists. With the Foq value decreasing, the heat transfer wave character weakens steadily because of the effect of collisions, damping the wave propagation, increases, and, therefore, the propagation character changes to diffusive
propagation. For Foq , 1024, the collisions break the propagation of heat and distribute it in all directions, resulting in the known character of diffusive propagation.
Info: [A17],[B45].
Jean Baptiste Joseph Fourier (see above).
5.9.11 Fourier Thermalization parallel Number (1.) FoPT
aP τ T
5 BiV21 Pλ21 ð1Þ;
L2
1
λ1 λ2
1 λ1 c2 1 λ2 c1
FoPT 5 2
λ1 1 λ2
L λ1 c2 1 λ2 c1 αV
FoPT 5
ð2Þ
aP (m2 s21) parallel thermal diffusivity; τ T (s) thermalization time of parallel
heat propagation; L (m) characteristic length; λ1, λ2 (W m21 K21) thermal
conductivity of material components; c1, c2 (J m23 K21) specific volume heat
capacity of material components; αV (W m23 K21) volume coefficient of internal
heat transfer; BiV () Biot volumetric number of internal heat transfer (p. 269);
Pλ () relative parallel thermal conductivity (p. 280).
It expresses the ratio of series and parallel thermal material conductivities and
internal heat transfer (1). In the case of two-component material, it can be
expressed as in equation (2).
Info: [A17],[B45].
Jean Baptiste Joseph Fourier (see above).
5.9.12 Fourier Thermalization Parallel Number (2.) FoT
FoT1 5
aτ T
5 BiV21 Pλ21 P121 5 FoPT Pa21
L2
ð1Þ;
Thermomechanics
1 λ1 1 λ2 1 λ1 c2 1 λ2 c1
FoT 5 2
λ1 1 λ2
L c2 1 c1 αV
275
ð2Þ
a (m2 s21) thermal diffusivity; τ T (s) thermalization time of parallel heat propagation; L (m) characteristic length; λ1, λ2 (W m21 K21) thermal conductivity
of material components; c1, c2 (J m23 K21) specific volume heat capacity of
material components; αV (W m23 K21) volume coefficient of internal heat transfer;
BiV () Biot volumetric number of internal heat transfer (p. 269); Pλ () relative parallel thermal conductivity (p. 280); Pa () relative parallel thermal diffusivity (p. 280); FoPT () Fourier thermalization parallel number (1.) (p. 274).
It expresses the relation between internal heat transfer, series and parallel thermal material conductivities and diffusivity (1). It is a modification of the Fourier
thermalization parallel number (1.) (p. 274) by the mutual ratio of the series and
parallel material thermal diffusivities. In the case of two-component material, it
can be expressed as in equation (2).
Info: [A17],[B45].
Jean Baptiste Joseph Fourier (see above).
5.9.13 Fourier Thermalization Wave Number FoT
FoT 5
vτ q vτ T
aτ T
5 2
L L
L
v (m s21) heat carrier velocity propagation; τ q (s) relaxation time of wave
heat propagation; τ T (s) thermalization time of wave heat propagation; L (m) characteristic length; a (m2 s21) thermal diffusivity.
The thermalization time of the wave heat propagation describes the effect of
elastic collisions with which the thermal wave momentum is not lost, but the velocity distribution of heat carriers is formed. The Fourier thermalization wave number
expresses the product of the ratio of the path covered by a heat carrier during the
relaxation time to the characteristic length of the observed material zone and
the ratio of the path covered by a heat carrier during the thermalization time to the
characteristic length of the observed material zone.
Info: [A17],[B45].
Jean Baptiste Joseph Fourier (see above).
5.9.14 Fourier Wave Number Fo
Fo 5
vτ vτ q
L L
v (m s21) heat carrier velocity propagation; τ (s) time; τ q (s) relaxation
time of wave heat propagation; L (m) characteristic length of a system.
276
Dimensionless Physical Quantities in Science and Engineering
It characterizes the dimensionless time of the wave heat propagation process. It
is the product of the ratio of the path covered by a heat carrier during the time τ to
the characteristic system length and the ratio of the path covered by a heat carrier
during the relaxation time to the characteristic system length.
Info: [A17],[B45].
Jean Baptiste Joseph Fourier (see above).
5.9.15 Heat Transfer in Micropores K τ
Kτ 5
τ ekv
ð1 2 εÞcs
5
τt
c
where τ t Rh c f
;
α
τ ekv 5
ð1 2 εÞcs τ t
;
c
ε5
Vf
V
τ ekv (s) equivalent time; τ t (s) time delay, thermalization time of non-equilibrium heat propagation; ε () volume ratio; Vf, V (m3) fluid volume and total
volume; c, cs, cf (J m23 K21) specific volume heat capacity: total, solid and fluid;
Rh (m) hydraulic radius of nano or microchannels; α (W m22 K21) heat transfer coefficient.
It limits the character of non-equilibrium heat propagation in fluid flow through
micropores.
5.9.16 Knudsen Phonon Number Knph
pffiffiffiffiffiffiffiffiffiffi
3aτ d pffiffiffiffiffi
5 3τ ;
Knph 5
h
where τ d 5
h2
a
a (m2 s21) thermal diffusivity; τ d (s) diffusion time; h (m) thickness; τ (s) time constant of CattaneoVernotte.
It characterizes heat wave propagation and determines when the classic
CattaneoVernotte mathematical model can be used. This model cannot be used in
cases in which it exceeds the limits of KnphAh0.35; 0.5i.
Info: [B35].
Martin Hans Christian Knudsen (p. 420).
5.9.17 Mikheyev Volumetric Number of Internal Heat Transfer MiV
See the non-equilibrium parallel heat transfer number Knep (p. 277).
Mikhail Alexandrovich Mikheyev (19021970), Russian engineer.
Thermomechanics
277
5.9.18 Number of Non-Equilibrium Heat Transfer K τ
Kτ 5
τT
FoT
5
τq
Foq
τ T (s) thermalization time of non-equilibrium (wave or parallel) heat propagation; τ q (s) relaxation time of non-equilibrium (wave or parallel) heat propagation; FoT () Fourier thermalization wave number (p. 275) or Fourier
thermalization parallel number (2.) (p. 274); Foq () Fourier relaxation wave
number (p. 273) or Fourier relaxation parallel number (p. 273).
This number defines the physically allowed zone of wave and parallel non-equilibrium heat propagation. The character of non-equilibrium heat propagation is
determined by the thermalization to relaxation time ratio of material. The value
Kτ 5 1 represents equilibrium diffusion propagation. With Kτ , 1, it expresses the
non-equilibrium heat wave propagation which changes to diffusion with increasing
time. For Kτ . 1, it expresses the non-equilibrium parallel heat propagation.
Info: [A17],[B45].
5.9.19 Number of Non-Equilibrium Parallel Heat Transfer Knep
Knep 5
Fo
ταV
5
5 FoBiV Pc
cS
Foq
τ (s) time; αV (W m23 K21) volume coefficient of internal heat transfer;
cS (J m23 K21) series specific volume heat capacity; Fo () Fourier number
(p. 175); Foq () Fourier relaxation parallel number (p. 273); BiV () Biot
volumetric number of internal heat transfer (p. 269); Pc () relative parallel heat
capacity (p. 279).
This criterion is defined as the ratio of the Fourier number (p. 175) to the
Fourier relaxation parallel number (p. 273). It determines the time zone of
the action of parallel non-equilibrium heat propagation. It expresses the ratio of the
heat transferred inside a material between its components to the total thermal content of a body. The Biot volumetric number of internal heat transfer BiV (p. 269)
and the relative parallel heat capacity Pc (p. 279) decide whether in the time Fo,
since the origin of the non-equilibrium state, the material is already in an equilibrium state or still in a non-equilibrium state. The criterion is also called the
Mikheyev volumetric number of internal heat transfer MiV (p. 276).
With Knep , 1022, the internal energy exchange between material components
with unequal temperatures is not able to manifest itself in the short time, since the
origin of the non-equilibrium state with the action of a heat source. In this phase,
the thermal process can be described by a set of independent diffusive heat conduction equations for individual material components. If 1022 , Knep , 101, the energy
transfer appears between the components inside the material. With increasing time,
the temperatures of individual components come near to the equilibrium values.
278
Dimensionless Physical Quantities in Science and Engineering
The heat propagation process must be expressed by the parallel heat propagation
equation or by the set of diffusive equations with mutual coupling. For Knep . 101,
sufficient time has elapsed already since the origin of the non-equilibrium state that
the material may get, due to internal redistribution of energy, into the equilibrium
state with equal temperature of all of its components. Then, the heat propagation
can be expressed as a rule by one diffusion heat propagation equation for the material as a whole.
Info: [A17],[B45].
5.9.20 Number of Non-Equilibrium State of Matter Knsm
Knsm 5
Fo
Forx
Fo () Fourier number Fo (p. 175); Forx () Fourier relaxation heat number
(p. 272).
It expresses the time zone of non-equilibrium wave heat conduction. It is the criterion which enables the division of the wave-diffusive heat conduction process
into three time zones: the non-equilibrium heat conduction with Knsm , 1021; the
transient zone with 1021 , Knsm , 101; and the equilibrium diffusive heat conduction with Knsm . 101. Non-equilibrium thermomechanics. Physical technology.
Info: [A17],[B45].
5.9.21 Number of Non-Equilibrium Wave Heat Transfer Knev
Knev 5
Fo
Foq
Fo () Fourier wave number (p. 275); Foq () Fourier relaxation wave
number (p. 273).
This criterion is defined as the ratio of the Fourier wave number (p. 275) to the
Fourier relaxation wave number (p. 273). It determines the time of acting of the
wave non-equilibrium heat propagation.
With Knev , 1022, the heat carriers forming the thermal wave propagate through
a material without collisions. In the short time since the origin of non-equilibrium
state due to a heat source, no collision process creating material thermal resistance
occurred yet. In this phase, the heat propagation process can be described by a
wave propagation equation. If 1022 , Knev , 101, the wave propagation damping
appears with increasing time and the material gets into the equilibrium state gradually. The heat propagation process is expressed by a wave-diffusion equation. With
Knev . 101, sufficient time has elapsed since the origin of the non-equilibrium state
that the wave propagation character may be damped. Then, the heat propagation
can be expressed by a heat conduction diffusion equation.
Info: [A17], [B45].
Thermomechanics
279
5.9.22 Nusselt Porous Number Nup
Nup 5
αRp
λf
α (W m22 K21) heat transfer coefficient; Rp (m) hydraulic radius expressed as
pore volumes ratio to their surface (ΔVp/ΔAp); λf (W m21 K21) thermal conductivity of fluid.
This number characterizes the convection heat transfer in porous material. It is
applied especially if the heat conduction predominates, due to which its size is constant. It occurs in cases where the Rayleigh pore flow number Rapor (p. 111) is
small and the natural convection in pores predominates in the heat conduction.
Equilibrium and non-equilibrium heat transfers. Two-phase flow.
Info: [B55].
Ernst Kraft Wilhelm Nusselt (p. 196).
5.9.23 Péclet Porous Number Pep
Pep 5
cf wL
λekv
cf (J m23 K21) specific volume heat capacity of fluid; w (m s21) mean flow
velocity; L (m) characteristic length, porous layer thickness; λekv (W m21 K21) equivalent thermal conductivity; Sp () Sparrow number (p. 281); Pe () Péclet heat number (p. 180).
It characterizes the velocity of fluid flowing through porous material. The variable velocity frequency has a strong influence on the heat transfer. The ratio
SpPe21 shows approximately whether there is a local equilibrium state in the
porous material.
Info: [B76].
Jean Claude Eugène Péclet (p. 180).
5.9.24 Relative Parallel Heat Capacity Pc
Pc 5
X X1
cp
5
ci
cs
ci
P
cp (J m23 K21) parallel
ðcp 5
ci Þ; cs ðJ m 23 K 21 Þ series specific
heat capacity
P 1 21
volume heat capacity cs 5
; ci ðJ m 23 K 21 Þ specific volume heat
ci
capacity of ith material component.
It expresses the parallel-to-series thermal capacities ratio of a material, with said
capacities being determined by the heat capacities of individual material components.
Info: [A17],[B45].
280
Dimensionless Physical Quantities in Science and Engineering
5.9.25 Relative Parallel Heat Transfer Rate Pv
Pv 5
1
wt
1
Pv 5
wt
rffiffiffiffiffi
a
ð1Þ;
τq
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
λ1 1 λ2
αV
c1 c2
ð2Þ
wt (m s21) heat propagation velocity; a (m2 s21) thermal diffusivity; τ q (s) relaxation time of parallel heat propagation; αV (W m23 K21) volume coefficient
of internal heat transfer; λ1, λ2 (W m21 K21) thermal conductivity of material
components; c1, c2 (J m23 K21) specific volume heat capacity of material
components.
It expresses the ratio of the apparent parallel heat transfer rate to the heat propagation rate in a material (1). In the case of a two-component material, it can be
expressed as in equation (2).
Info: [A17],[B45].
5.9.26 Relative Parallel Thermal Conductivity Pλ
Pλ 5
X X1
λp
5
λi
λs
λi
P
λp (W m21 K21) parallel thermal conductivity ðλp 5P λi Þ;
21 21
λs (W m K ) series thermal conductivity ðλs 5 ð λ1i Þ21 Þ;
λi (W m21 K21) thermal conductivity of ith material component.
It expresses the ratio of the parallel thermal material conductivity to the series
conductivity, with said conductivities being determined by the thermal conductivities of individual material components.
Info: [A17],[B45].
5.9.27 Relative Parallel Thermal Diffusivity Pa
Pa 5
ap
a
a (m2 s21) thermal diffusivity; ap (m2 s21) parallel thermal diffusivity.
It expresses the character of non-equilibrium heat propagation by means of the
parallel-to-standard heat diffusivities ratio.
The relative parallel diffusivity has two limits. In the case Pa-0, i.e. ap-0, the
heat propagates in material mostly by mediation of one kind of heat carrier. With
Pa-1, i.e. ap-a, the heat propagates nearly exclusively by a parallel mechanism.
Thermomechanics
281
In materials with two kinds of heat carriers, the thermal capacity of one of the
carriers is often many times greater than that of the second carrier and, on the contrary, the thermal conductivity of the first carrier is substantially less than that of
the second one.
Info: [A17],[B45].
5.9.28 Relaxation Time Nτ,rx
Nτ;rx 5
τ rx w2
5 CatPr
ν
τ rx (s) relaxation time; w (m s21) fluid flow velocity; ν (m2 s21) kinematic
viscosity; Cat () Cattaneo number (p. 270); Pr () Prandtl number (p. 197).
In non-equilibrium thermodynamics, this number expresses the relaxation time in dimensionless form in flowing fluid along a plate wall, heated by impulses or
strokes. It is used, for example, in observing the discontinuities of velocity gradients
and temperatures.
Info: [B77].
5.9.29 Sound Speed Number K
K5
vτ 2
q
L
v (m s21) adiabatic sound velocity in liquid; τ q (s) relaxation time; L (m) characteristic length.
It expresses the ratio of the product of the relaxation time characterizing the
hyperbolic fluid behaviour and the in-fluid propagation rate in the fluid to the characteristic dimension of the system.
Info: [B43].
5.9.30 Sparrow Numbers Sp
2
αL2
λf
L
Sp 5
5 Nu
λekv Rh
λekv
Rh
α (W m22 K21) equivalent heat transfer coefficient; L (m) characteristic
length, porous layer thickness; λekv (W m21 K21) effective thermal conductivity;
λf (W m21 K21) fluid thermal conductivity; Rh 5 ΔVp =ΔAp ðmÞ hydraulic
radius; ΔVp (m3) pore volume; ΔAp (m2) pore surface; Nu () Nusselt
number (p. 196); Pe () Péclet heat number (p. 180).
282
Dimensionless Physical Quantities in Science and Engineering
It characterizes the non-equilibrium heat transfer in porous material. Essentially,
it expresses the extension of the Nusselt number Nu (p. 196) for equilibrium heat
propagation to non-equilibrium heat propagation, which is of use in the flow in
porous materials. Usually λf , λekv holds. The number Sp enables the estimation of
the time delay due to the process non-equilibrium. The fraction SpPe21 enables
estimating local equilibrium in fluid flow through porous material.
Info: [B76].
5.9.31 Thermal Wave Speed Number K
K5
Lv
a
K5
Ll
1
5 pffiffiffiffiffiffiffiffi
aτ q
Foq
ð1Þ;
ð2Þ
L (m) characteristic length; v (m s21) velocity of the heat wave propagation;
a (m2 s21) thermal diffusivity; l (m) mean free path; τ q (s) relaxation time;
Foq () Fourier relaxation wave number (p. 273).
It expresses the ratio of the thermal wave propagation rate to the characteristic
system dimension and the thermal diffusivity of the material or surroundings. With
the propagation rate expressed by the mean free path and the wave propagation
relaxation time v 5 1=τ q ; the criterion is simplified to equation (2). Then, it
expresses the inverse root of the Fourier relaxation wave number Foq (p. 273).
Info: [B24].
5.9.32 Vernotte Heat Number Ve
Ve 5 Pe21 5
Ve 5
a
wtr L
a
wt L
ð1Þ;
ð2Þ;
rffiffiffiffi
a
where wtr 5
τr
a (m2 s21) thermal diffusivity; wt (m s21) velocity of heat propagation;
wtr (m s21) relaxation velocity of heat propagation; L (m) characteristic
length; τ r (s) relaxation time; Pe () Péclet heat number (p. 180).
It characterizes the influence of the finite heat propagation rate in a body. Wave
heat transfer in stroke phenomena. Non-equilibrium thermomechanics. The action
of intensive heat sources on material. New technologies.
Info: [A23].
Pierre Vernotte (born 18.5.1898), French physicist.
Thermomechanics
283
5.9.33 Vernotte Thermoelastic Number Vete
Vete 5 Pete21
a
;
5
wte L
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi
λ 1 2μ
where wte 5
R
a (m2 s21) thermal diffusivity; L (m) characteristic length; wte (m s21) velocity of thermoelastic deformation propagation; λ, μ (Pa) Lamé functions;
R (kg m23) density; Pete () Péclet thermoelastic number (p. 143).
It characterizes the wave propagation rate of the thermoelastic stress or deformation in heat systems with stroke phenomena. Its inverse value is called the Pe´clet
thermoelastic number Pete (p. 143). Thermomechanics of stroke thermal processes.
New technologies.
Info: [A23].
Pierre Vernotte (see above).
5.9.34 Wave Penetration Depth (1.) H
Fo
H 5 pffiffiffiffiffiffiffiffiffi
Forx
ð1Þ;
Fo
ΘðFoÞ 5 Θð0Þexp 2
2Forx
ð2Þ
Fo () Fourier number Fo (p. 175); Forx () Fourier relaxation heat number
(p. 272).
It characterizes a material zone in which the heat wave propagation influence
predominates. In the time Fo, it expresses the heat wave penetration depth
into material with wave-diffusive heat propagation and impulse action of surface
thermal flow. The front of the arising thermal wave is damped gradually (2).
Non-equilibrium thermomechanics. Physical technology.
Info: [A17],[B63].
5.9.35 Wave Penetration Depth (2.) Effective Hef
pffiffiffiffiffiffiffiffiffi
Hef 5 2 Forx
Forx () Fourier relaxation heat number (p. 272).
It expresses the heat wave penetration depth into a material with which the
wave temperature drops e times in passing through the material. It also confirms
that the penetration depth is proportional to the mean free path of the heat carriers.
Non-equilibrium thermomechanics. Physical technology.
Info: [A17],[B63].
6 Electromagnetism
Finally, some observations of the momentum transfer from the Sun to the Planets
have been carried out and have fundamental significance for the theory. The significance of magneto-hydrodynamic waves is remarkable from this aspect.
Hannes Olof Gösta Alfvén (19081955)
Electromagnetism represents the fundamental interaction between electricity and
magnetism. The electric dimensionless quantities involve electric fields and circuits, electronics, illumination, electrical energy and a wide range of applications
in various fields. The magnetic dimensionless quantities express, above all, basic
magnetic forces caused by electric charge movement, propagation of electromagnetic energy and waves, and other transfer phenomena, such as plasma physics and
magneto-hydrodynamics. The magneto-hydrodynamic dimensionless quantities
express the propagation of magneto-hydrodynamic waves, dynamic pressure, force
relations and energy conversion.
6.1
Electricity
A relatively large number of dimensionless quantities involve both electric circuits
and fields, and are as diverse as electrothermal, ionization and corona diffusion
processes; natural and forced electrothermal convection; heat transfer intensification with dielectric fluid condensation; condensation in an electric field; condensate
film stabilization; mass transfer in electrochemical processes; the motion of
charged particles in flowing gas; and many other applications. Among them, the
dimensionless criteria for diverse boundary conditions and internal sources express
the inhomogeneity and non-linearity of the environment, charge density, electric
field intensity, corona phenomenon, earthing and puncture. Of the many dimensionless numbers, the following are especially important: the electromechanical and
electronic numbers, the Fourier electrical number, the electric field intensity number, the Joule electrothermal number, the corona number, the Kronig number, the
Ohm number and many equivalents of basic quantities.
Dimensionless Physical Quantities in Science and Engineering. DOI: 10.1016/B978-0-12-416013-2.00006-3
© 2012 Elsevier Inc. All rights reserved.
286
Dimensionless Physical Quantities in Science and Engineering
6.1.1
Biot Electrical Number Bie
Bie 5
L
Rα γS
L(m) characteristic length; Rα ðΩÞ electrical resistance equivalent to convective
thermal resistance; γ (S m21) specific electrical conductance; S (m2) surface
area in accordance with resistance Rα.
It characterizes the third-type boundary condition of an electric system. It is
analogous to the thermal Biot number Bi (p. 173).
Info: [A23].
Jean-Baptiste Biot (p. 174).
6.1.2
Breakdown Coefficient N
N5
Iρ
U
5
L2 Ep
LEp
I (A) electric current; ρ (Ω m) specific electrical resistance; L (m) characteristic
length; Ep (V m21) earth breakdown gradient; U (V) spark zone surface voltage.
This coefficient expresses the dimensionless value of the earth breakdown gradient. Earthing.
Info: [A24].
6.1.3
Characteristic Electrical Number Nche
2
dε L ΔTE2
Nche 5 ρ
η2
dT
ρ (kg m23) density; ε (F m21) permittivity; T (K) temperature; L (m) characteristic length; ΔT (K) temperature difference; E (V m21) electric field
intensity; η (Pa s) dynamic viscosity.
It characterizes the effect of the temperature change of environmental dielectric
parameters with the electrostatic field in relation to the heat transfer from a heated
surface. Sometimes, it is incorrectly called the Kronig number Kr (p. 299).
Info: [A23].
6.1.4
Charge Density Number NQ
NQ 5
wLρe
H
w (m s21) velocity; L (m) characteristic length; ρe (C m23) volume charge
density; H (A m21) magnetic field intensity.
Electromagnetism
287
It expresses the ratio of the electric charge flow intensity to the magnetic field
intensity.
Info: [C173].
6.1.5
ClausiusMossotti relation K
KðωÞ 5
ε1 2 ε2
;
ε1 1 2ε2
where ε 5 ε 2
jγ
ω
ε1 ; ε2 (F m21) permittivity of two different environments; γ (S m21) specific
electrical conductance; ω (s21) angular frequency of electrical field; j (2) imaginary unit.
This relation is used in electrokinetics. The real part determines the coefficient
of the electromagnetic force acting on a particle. The imaginary part expresses the
electrorotating moment. The particle size and the electric field are other influences.
Electricity. Electrokinetics.
Info: [C20].
Rudolf Julius Emanuel Clausius (p. 175).
Ottaviano-Fabrizio Mossotti
6.1.6
Communication Coefficient N
N 5 ω ε0 β AR
ω (s21) angular frequency; ε0 (F m21) vacuum permittivity; β (m21) wave
number (β 5 2πλ21); A (m2) cross section area of electron flux; R (Ω) resistance.
It expresses the relative efficiency of the electron flow and the electric field
wave. Communications engineering.
Info: [A24].
6.1.7
Condensate Film Stability (1.) N
N5
εE2 L2
ρl ν 2l
ε (F m21) permittivity; E (V m21) electric field intensity; L (m) characteristic
length; ρl (kg m23) condensation layer density; ν l (m2 s21) condensation layer
kinematic viscosity.
It expresses the Coulomb-to-viscosity forces ratio. It characterizes the degree of
the stability damage of a condensate film and its deformation in an electric field.
This appears as formation of Coulombs and abruptly formed condensate drops on
288
Dimensionless Physical Quantities in Science and Engineering
the surface due to longitudinally nonuniformly distributed pressure (εE2/2). With
sufficiently great E, the film is damaged and the condensate drops are sprayed,
with the electric force overcoming the viscosity force.
Info: [A23].
6.1.8
Condensate Film Stability (2.) N
N5
εE2 L
σ
ε (F m21) permittivity; E (V m21) electric field intensity; L (m) characteristic length; σ (N m21) surface tension.
It expresses the ratio of the Coulomb force to the surface stress force. It characterizes the stability degree of a condensate film and its deformation in an electric
field in relation to the surface stress force.
Info: [A23].
6.1.9
Condensation Electrostatic Number N
N5
ðεgrad E2 Þl δl
gρL
ð1Þ;
N5
ðεE2 Þυ
gρL
ð2Þ
ε (F m21) permittivity; E (V m21) electric field intensity; δl (m) condensate
film thickness; g (m s22) gravitational acceleration; ρ (kg m23) condensate
density; L (m) characteristic length; index y: l condensate, υ vapour.
This number expresses the ratio of the gravity force to the ponderomotive force
of an electric field. It characterizes the heat transfer intensity with dielectric fluid
condensation on an outer tube surface (L 5 D/2) in an inhomogeneous electrostatic
field. In expression (1), it represents the volume ponderomotive force; in expression (2), it represents the surface force. The surface electrostatic force, acting on
the interface of phases, has an especially strong influence on heat transfer
intensification.
Info: [A23].
6.1.10 Corona Number NK
pffiffiffiffiffiffiffiffi
r0 ε 0 ω
NK 5 pffiffiffiffiffi
τk
r0 (m) wire radius; ε0 (F m21) vacuum permittivity; ω (s21) angular frequency; τ (A s m21) linear density of electric charge; k (A s2 kg21) ion
mobility.
It expresses the corona phenomenon.
Info: [A24].
Electromagnetism
289
6.1.11 Current Source Number NJ
J0 R Δ x
U
NJ 5
J0 (A m21) linear density of electric current; R (Ω) resistance; Δx (m) length of element; U (V) voltage.
It concerns the current source related to 1 m of the conductor length. It determines, for example, the electric loss in conductors. Electric machines and devices.
Info: [A24].
6.1.12 Dielectric Constant εr
εr 5
εs
ε0
ð1Þ;
ε 5 ε0 εr
ð2Þ
εs (F m21) static material permittivity; ε0 (F m21) vacuum permittivity
(ε0 5 8,854187 3 10212 F m21); ε (F m21) absolute permittivity.
In the case of a static field, it expresses the insulation properties of a dielectric
material. In the case of an alternating field or electromagnetic oscillation, it
expresses the relation between the vectors of electric induction and electric field
and depends on the frequency. Expression (2) represents the absolute permittivity.
The values of εr are: for air ðεr 61Þ, for glass ðεr 67:6Þ and for water ðεr 680Þ.
Electromagnetism. Electric fields. Physical properties.
Info: [C113].
6.1.13 Diffusion Corona Number N
N5
D
LkE
D (m2 s21) mass diffusion coefficient in electric field; L (m) characteristic
length; k (A s2 kg21) ion mobility; E (V m21) electric field intensity.
It expresses the ratio of the diffusion mass transfer to rectified ion movement. It
characterizes the mass diffusion process in a corona discharge. The diffusion begins
to appear with N $ 1, approximately.
Info: [A23].
6.1.14 Dynamics of Inner Heat Source Number Dy
Dy 5
w2 tδT 2 w1t
;
w1t
where w1t 5
w 1 qV
w2 qV
; w2t 5
cρ
cρ
290
Dimensionless Physical Quantities in Science and Engineering
δT (2) thermogradient coefficient; w1, w2 (K s21) warming velocity;
qV (W m23) volume density of heat flux; c (J kg21 K21) specific heat capacity;
ρ (kg m23) density.
It characterizes the influence of an inner heat source on heat and mass transfers,
when power changes are affected by material temperature changes. When Dy , 0,
the power increase; when Dy . 0, it decreases. In dielectric heating, the internal
volume source qV is known. Induction and dielectric heating. Electric heat.
Info: [A33].
6.1.15 Electric Field Intensity N
E
N5
wμH
ð1Þ;
EL
N5
w
rffiffiffi
γ
ð2Þ
η
E (V m21) electric field intensity; w (m s21) velocity; μ (H m21) permeability;
H (A m21) magnetic field intensity; L (m) characteristic length; γ (S m21) specific electrical conductance; η (Pa s) dynamic viscosity.
It characterizes the ratio between the fed and induced electric field intensities in
moving electrically conductive fluids. In examining the outer electric field influence independently from the induced electric field e.g. with a zero value for the
outer magnetic field it is more suitable to use expression (2). Sometimes it is
called the criterion of loading characteristics MHD canal.
Info: [A23],[A33].
6.1.16 Electrical Circuit Number N
N5
Uτ
R2 C
5
LI
L
ð1Þ;
N5
τ
RC
ð2Þ
U (V) voltage; τ (s) time; L (H) inductance; I (A) electric current; R (Ω) resistance; C (F) capacitance.
Expression (1) describes the transient phenomena in an electric RLC circuit after
a sudden alternating voltage connection. The ratio of time to the electric circuit
time constant is expressed by (2).
Info: [A24].
6.1.17 Electrical Convection Number N
N5
α
grad E2 ρ
η2
α (A2 s4 kg21) molecule polarization coefficient; η (Pa s) dynamic viscosity;
E (V m21) electric field intensity; ρ (kg m23) density.
Electromagnetism
291
It expresses the ratio of the molecular friction force to the electroconvective
one. It characterizes the dielectric fluids flow intensity influenced by the
electric field inhomogeneity. In natural convection, it appears as the Grashof heat
number. (p. 185)
Info: [A23],[A33].
6.1.18 Electrical Field Number NE
NE 5
QNL4
τkU 2
Q (C) electric charge; N (m23 s21) number of ion pairs; L (m) length;
τ (C m21) linear density of an electric charge; k (A s2 kg21) anion mobility;
U (V) voltage.
This number represents an electric field in ionized gas.
Info: [A12].
6.1.19 Electric Instability N2
N2 5 β ε
dT
L
dx
β ε (K21) thermal coefficient of permittivity; dT/dx (K m21) temperature gradient; L (m) characteristic length.
It characterizes the inhomogeneity grade of the electric environment permittivity
in solving problems of equilibrium fluid stability in a horizontal planar condenser.
Info: [A23].
6.1.20 Electric Relaxation Number Nrelax
Nrelax 5
νεE
JA L2
ν (m2 s21) kinematic viscosity; ε (F m21) permittivity; E (V m21) electric
field intensity; JA (A m22) surface density of electric current; L (m) characteristic length.
It expresses the ratio of electric phenomena relaxation in surroundings to the
mechanical relaxation time. It characterizes the relaxation effects related to electric
wind. When the ratio is small, electric relaxation occurs independently of mechanical relaxation. Heat transfer in electrothermal systems.
292
Dimensionless Physical Quantities in Science and Engineering
6.1.21 Electrocalorimetric Effect N
N5
cp ρΔT
εE2
cp (J kg21 K21) specific heat capacity; ρ (kg m23) density; ΔT (K) temperature difference; ε (F m21) permittivity; E (V m21) electric field intensity.
This effect expresses the ratio of the heat obtained by transfer in the unit volume
to the energy density in an electric field. It characterizes the thermal effect of electric current passage through fluids without considering the fluid heating by friction,
polarization energy and energy of electrostatic charges. (K{1).
Info: [A12].
6.1.22 Electromechanical Number Es
Es 5
JA ρL3
k 1 η2
JA (A m22) surface density of electric current at corona discharge in gas;
ρ (kg m23) gas density; L (m) characteristic length; k 1 (A s2 kg21) positive
charge carrier mobility; η (Pa s) dynamic viscosity.
It characterizes the influence of the gas boundary layer damage, by corona electric discharge, on the intensity of the convective heat transfer between an electrode
and gas under conditions of natural or forced convection near the coronizing electrode surface.
Info: [A23],[A33].
6.1.23 Electronic Number N
JA ε0 B
Uε0
ð2Þ;
ð1Þ; N 5 2
2
L ρ
Lρ
JA L
Dε0 B
N5
ð4Þ;
ð3Þ;
N5 2
Dρ
Lρ
N5
JA (A m22) surface density of electric current; ε0 (F m21) vacuum permittivity;
B (T) magnetic induction; L (m) characteristic length; ρ (C m23) volume
density of electric charge; U (V) voltage; D (m2 s21) diffusion coefficient.
In expression (1), it expresses the current, limited by the volume charge, in real
dielectrics and semiconductors. In expression (2), it represents a voltage. In expression (3), it describes the current, limited by the volume charge, when considering
the diffusion of charge carriers in semiconductors and dielectrics with the diffusion
Electromagnetism
293
predominating over drift. In expression (4), the equivalent drift and diffusion influence is considered.
Info: [A24].
6.1.24 Electrothermal Convection Number Net
Net 5
εβ τ E2 L3 grad T
ρνa
ε (F m21) permittivity; β τ 5 τ r21 dτ r =dTðK 21 Þ temperature coefficient of
relaxation time of electrical effects; E (V m21) electric field intensity; T (K) temperature; L (m) characteristic length; ρ (kg m23) density; ν (m2 s21) kinematic viscosity; a (m2 s21) thermal diffusivity; τ r 5 εγ21 (s) relaxation
time of electrical effects.
It characterizes the electrothermal convection in fluids under the action of a
homogeneous or inhomogeneous field. It expresses also the forces of the electrothermal substance acting with the electrothermal convection. With no electric field
acting, it is Net 5 0. The number Net expresses the electrothermal convection suitably, especially with a homogenous field.
Info: [A23].
6.1.25 Electrothermal Convection Parameter N, r
N5
εβ τ E2
εβ ΔV 2
5 Net Ra221 5 τ 3
ρβgL
ρβgL
ε (F m21) permittivity; β τ 5 τ r21 dτ r =dTðK 21 Þ temperature coefficient of electrical effects on relaxation time; E (V m 2 1) electric field intensity; ρ (kg m23) density; β (K21) volume thermal expansion coefficient; g (m s22) gravitational
acceleration; L (m) characteristic length; V ðVÞ electric potential; Net (2) electrothermal convection number (p. 293); Ra2 (2) Rayleigh number (2.)
(p. 187); τ r (s) relaxation time of electrical effects.
This parameter expresses the electrothermal to buoyancy forces ratio. It characterizes the relation of electrothermal convection to free convection.
Info: [A23].
6.1.26 Electrothermal Conversion (1.) N
N5
ρwh
JA E L
294
Dimensionless Physical Quantities in Science and Engineering
ρ (kg m23) density; w (m s21) flow velocity; h (J kg21) specific enthalpy;
JA (A m22) surface density of electric current; E (V m21) electric field intensity; L (m) characteristic length.
It characterizes the process of electric energy conversion to thermal energy,
which proceeds in a gas arc. This criterion is among the fundamental ones of use,
for example, in work related to plasmatrones with heated gas.
Info: [A23].
6.1.27 Electrothermal Conversion (2.) N
N5
ρw3
JA E L
ρ (kg m23) density; w (m s21) velocity; JA (A m22) surface density of electric
current; E (V m21) electric field intensity; L (m) characteristic length.
It characterizes the process of electric energy conversion to kinetic energy of a
rectified flow. The sudden temperature increase of the heated gas and corresponding density drop, with a given passage and cross section of a canal, are connected
to a remarkable velocity increase of the gradual gas flow. The criterion is of use,
for example, in a gas heater solution with supersonic flow in aerodynamic tunnels
if an electric arc is used in an accelerating nozzle.
Info: [A23].
6.1.28 Electrothermal Coupling Number NET
NET 5
JA2 L2
γλΔT
JA (A m22) surface density of electric current; L (m) characteristic length;
γ (S m21) specific electric conductance; λ (W m21 K21) thermal conductivity;
ΔT (K) temperature difference.
It expresses the coupling between electric current and heat conductions. With
this relation as defined by the number, further criteria expressing Joule heat action
can be obtained, for example, in flowing surroundings with physical and chemical
conversions.
Info: [A28].
6.1.29 Electrothermal Power N
N5
γ2 U 4
gρ2 c2 T 2 d3 βΔT
5
γ2 U 4
ηλ
5K1 Gr 21 Pr 21
ηλcT 2 gρ2 cd 3 βΔT
Electromagnetism
295
γ (S m21) specific electrical conductance; U (V) voltage; g (m s22) gravitational acceleration; ρ (kg m23) density; c (J kg21 K21) specific heat capacity; T
(K) local temperature in tank furnace; d (m) electrode diameter; β (K21) volume thermal expansion coefficient; ΔT (K) temperature difference; η (Pa s) dynamic viscosity; λ (W m21 K21) thermal conductivity; K1 (2) criterion K1;
Gr (2) Grashof heat number (p. 185); Pr (2) Prandtl number (p. 197).
It expresses the ratio of electric forces in molten material to mechanical forces
in free moving molten material with thermoelectric convection. It characterizes the
mutual relation among fundamental physical processes in an electric induction furnace and their influence on heat transfer. The influence of the product (Gr Pr)
characterizes mechanical forces distributed in a tank that depends on slag parameters, temperature distribution and geometric dimensions. When Gr Pr . 2 3 107,
the process becomes automatic modelling. The criterion K1 depends on the character of the electric field and inner source distribution.
Info: [A23].
6.1.30 Erlang Unit Eb, P
EbðN; AÞ5
AEbðN 21; AÞ
;
N 1 AEbðN 21; AÞ
AN N
N! N 2 A
Pð . 0Þ5 PN 21
Ax AN N
x 5 0 x! N! N 2 A
Ebð0; AÞ51 ð1Þ;
ð2Þ
N (2) number of sources, for example, phones, servers or circuits; A (2) operation size in dimensionless Erlang units.
The Erlang unit expresses the statistical degree of telecommunications operations used in telephony or internet connections, for example. One Erlang expresses,
for example, one source in uninterrupted operation or two sources at half of their
capacities. The Erlang unit is used to determine whether a telecommunications system is overloaded or, on the contrary, underutilized in other words, whether it
has many or very few utilized sources. It is applied to calculate the grade of service
(GoS) and the quality of service (QoS) as well. There are several diverse Erlang
equations, among which the Erlang B equation (1) and Erlang C equation (2) are
especially important.
Erlang relation B (1) expresses the probability Eb that a system is blocked. This
means that with a request not being realized immediately, it is blocked and cancelled immediately. This relation is used for telephone systems which do not use
the so-called queues.
The Erlang relation C (2) expresses the probability P that a request remains in
the system queue for a certain length of time. In such a way, the quantity P( . 0)
expresses the probability that the response delay to the request is greater than zero,
meaning that the request is not realized immediately. This relation is used for telephone ‘call centre’ systems, in which blocked requests remain in the queue as long
296
Dimensionless Physical Quantities in Science and Engineering
as they can be realized. This system is utilized also in networks transmitting data
sets, for example, on the internet.
In addition to the Erlang unit, the Engset relation is applied to express the statistical degree of a telecommunications operation where there is a narrow relation
between these quantities.
Info: [C52].
Agner Krarup Erlang (1.1.18783.2.1929), Danish mathematician, statistician and engineer.
He was the first to systematically deal with the problems
of telephone networks. In examining the activities of
municipal phone systems, he elaborated a dimensionless
expression which is known as the Erlang unit nowadays. In
particular, it represents the telecommunications criterion.
Even though the Erlang model is simple, the mathematical
foundation of present complex telecommunications networks is substantially based on this principle.
6.1.31 Fourier Electrical Number Foe
Foe 5
τΔx2
τ
5
2
RCL
RCn2
τ (s) time; Δx (m) geometric step; R (Ω) resistance; C (F) capacitance;
L (m) characteristic length; n (2) number of cells.
This is the dimensionless description related to time of electric current propagation through a conductor, e.g. through a geometrically discretized RC model.
Info: [A23].
Jean Baptiste Joseph Fourier (p. 175).
6.1.32 Fourier Relaxation Number of Electrical Field Foe
Foe 5
νeτe
νeε
5 2 ;
2
L
L γ
where τ e 5
ε
γ
ν e (m2 s21) electric viscosity; τ e (s) relaxation time of electric field propagation; L (m) characteristic length; ε (F m21) permittivity; γ (S m21) specific
electrical conductance.
This number expresses the charge relaxation time which influences the electric
and electromagnetic field distribution. Together with the Fourier magnetic number
Fom (p. 308), it expresses the Fourier electromagnetic number Foem (p. 308).
Electric fields.
Jean Baptiste Joseph Fourier (see above).
Electromagnetism
297
6.1.33 Gain
Gain 5 10 log
P2
ðdBÞ
P1
ð1Þ ;
Gain 5 20 log
V2
ðdBÞ ð2Þ
V1
P1, P2 (W) input and output power signal; V1, V2 (V) input and output voltage
signal.
It is used in electronics especially and expresses the output-to-input power (1)
or voltage (2) signals ratio. It is used widely in amplifiers. Electronics,
Telecommunications. Automatic control.
Info: [C62].
6.1.34 Grashof Electrical Number Gre
Gre1 5
F
εE2 L2
5
ρν 2
ρν 2
ð1Þ;
Gre2 Kr1
ð2Þ
F (N) Coulomb force; ρ (kg m23) density; ν (m2 s21) kinematic viscosity;
ε (F m21) permittivity; E (V m21) electric field intensity; L (m) characteristic
length; Kr1 (2) Kronig number (p. 299).
It expresses the electric-to-viscous forces under thermoelectric convection conditions. It characterizes the Coulomb forces influencing the origin of the electric
convection and wind which occurs due to gradual environmental neutrality loss. In
gases, it is about a secondary phenomenon arising with corona discharge. The force
F expresses the force close to the coronizing electrode. However, the number Gre
does not express the field inhomogeneity as a necessary condition for electric convection generation. The Grashof electrical number Gre2 is known as the Kronig
number Kr1 (p. 299) as well.
Info: [A23].
Franz Grashof (p. 185).
6.1.35 Grashof Electrical Relative Number Grrel
Grrel 5
εE2
5 Gre1 Gr 21
LgΔρ
ε (F m21) permittivity; E (V m21) electric field intensity; L (m) characteristic
length; g (m s22) gravitational acceleration; Δρ density difference; Gre1 (2) Grashof electrical number (p. 297); Gr (2) Grashof heat number (p. 185).
It expresses the ratio of a free electrothermic convection force to a free thermal
convection force. It characterizes the ponderomotive force influence on the heat
transfer under the action of electric convection.
Info: [A23].
Franz Grashof (see above).
298
Dimensionless Physical Quantities in Science and Engineering
6.1.36 Ground Parameter N
N5
UL
Iρ
ð1Þ;
N5
ρλΔT
U2
ð2Þ
U (V) voltage; L (m) characteristic length; I (A) electric current; ρ (Ω m) specific electrical resistance; λ (W m21 K21) thermal conductivity; ΔT (K) ground warming around earthed system.
Expression (1) describes the voltage-to-grounding ratio recalculation, and
expression (2) represents the relation between the voltage and the heating of the
earth.
Info: [A24].
6.1.37 Joule Electrothermal Number Jo
Jo 5
λΔT
L 2 JA E
ð1Þ;
Jo 5
λT
λT
5
γU 2
γE2 L2
ð2Þ
λ (W m21 K21) thermal conductivity; ΔT (K) temperature difference; L (m) characteristic length; JA (A m22) surface density of electric current; E (V m21) electric field intensity; T (K) temperature; γ (S m21) specific electric
conductance; U (V) voltage.
This number expresses the ratio of Joule heat to electric energy. Equation (1)
can be expressive for devices with low efficiency and low gradual motion velocity,
for example, the plasma in an electric arc. Equation (2) is called the Joule effect
criterion, which expresses the thermal to electric flows ratio in a fluid. With higher
values, the fluid Joule heating influence can be neglected in comparison to the
heating by simple heat conduction in the fluid.
Info: [A23].
James Prescott Joule (18181889), English physicist.
Above all, he is well known due to his experimental
determination of mechanical heat theory. This was connected with his initial effort to determine the efficiency of
electric motors. He was one of the authors of the energy
conservation law, discoverer of the Joule heat unit and the
co-discoverer of the JouleThomson phenomenon. He
determined the mechanical equivalent of heat created by
friction.
Electromagnetism
299
6.1.38 Kirpichev Electric Number Kie
Kie 5
JA L
γU
JA (A m22) surface density of electric current; L (m) characteristic length;
γ (S m21) specific electrical conductance; U (V) voltage.
It relates to the planar current density, planar source, surface loss and secondtype boundary condition. It is also called the Pomerantsev electrical number PoeA
(p. 302).
Info: [A23].
Mikhail Viktorovich Kirpichev (p. 177).
6.1.39 Kronig Number Kr
Kr1 Gre2 5
Kr2 5
L2 βΔTE2 Nk
2p2
α
1
Mν 2
3kTN
εð3β 1 2β γ ÞE2 L2 ΔT
5 Net Pr 21
ρν 2
ð1Þ;
ð2Þ
L (m) characteristic length; β (K21) volume thermal expansion coefficient;
ΔT 5 TS 2 TN (K) temperature difference small against TN; TS, TN (K) wall
and undisturbed fluid flow temperatures; E (V m21) electric field intensity;
Nk (mol21) Avogadro constant; M (kg mol21) molar mass; ν (m2 s21) kinematic viscosity; α (A2 s4 kg21) gas molecule polarization coefficient;
p (C m) electrical molecular dipole moment; k 5 RNk21 ðJ K 21 Þ Boltzmann
constant; R (J mol21 K21) molar gas constant; ε (F m21) permittivity;
β ε (K21) thermal coefficient of permittivity; β γ (K21) thermal coefficient of
specific electrical conductance; ρ (kg m23) density; Gre2 (2) Grashof electrical
number (p. 297); Net (2) electrothermal convection number (p. 293); Pr (2) Prandtl number (p. 197).
This number expresses the ratio of the product of the inertia and electrostatic
forces to the square of the viscous force. It characterizes the electrostatic field
effect on the heat transfer from a surface heated by convection. It expresses the
force relations caused by the inhomogeneity of electrophysical properties and viscous friction in a fluid. Essentially, it is analogous to the Archimedes thermodynamic number Ar (p. 184). In form (2), it is a modified Kronig electrothermal
convection number. The Kronig number Kr1 is also called the Grashof electrical
number Gre2 (p. 297).
Info: [A23].
Ralph De Laer Kronig (born 1904), German-American physicist.
300
Dimensionless Physical Quantities in Science and Engineering
6.1.40 Kutateladze Electrothermal Number Kue
Kue 5
JA EL
ρwΔh
JA (A m22) surface density of electric current; E (V m21) electric field intensity; L (m) characteristic length; ρ (kg m23) density; w (m s21) fluid velocity;
Δh (J kg21) increment of specific enthalpy.
It characterizes the electrothermal process with an electric discharge in a flowing fluid.
Info: [A23].
Samson Semenovich Kutateladze (18.7.191420.3.1986), Russian engineer
and physicist (p. 327).
6.1.41 Length Current Density K
K5
γE2 d2
cηT
γ (S m21) specific electrical conductance; E (V m21) electric field intensity;
d (m) electrode diameter; c (J kg21 K21) specific heat capacity; η (Pa s21) dynamic viscosity; T (K) temperature.
It expresses the ratio of longitudinal densities of electric and thermal flows. It
characterizes the electrothermal process in diverse electrothermal systems, for
example, in electric furnaces.
Info: [A23].
6.1.42 Motulevitch Number Mo
E2
Mo 5 ρL 2 Δε;
η
2
dε
ΔT
where Δε 5
dT
ρ (kg m23) density; L (m) characteristic length; E (V m21) electric field
intensity; η (Pa s) dynamic viscosity; ε (F m21) permittivity; T (K) temperature.
It expresses the ratio of the electric force caused by dielectric susceptibility
dependence on the environmental density to the viscosity force. It characterizes the
influence of the inhomogeneity, and the thermal system non-uniformity connected
with it, on the electric convection in dielectric fluids. It expresses the influence of
an electric field on the transfer phenomenon.
Info: [A23].
Electromagnetism
301
6.1.43 Nusselt Electrical Number Nue
Nue 5
wL
;
D
where D 5
1 1
ðD 1 D2Þ
2
w (m s21) flow velocity; L (m) characteristic length; D (m2 s21) final ion
diffusivity.
It expresses the ratio of the convective flow to the diffusion electric flow. It
characterizes the electrothermal convection in electrochemical processes.
Sometimes, it is called the Reynolds electrical number (2.) NuEL2 (p. 304).
Info: [A23].
Ernst Kraft Wilhelm Nusselt (p. 196).
6.1.44 Ohm Number Oh
Oh 5
γE
JA
γ (S m21) specific electrical conductance; E (V m21) electric field intensity;
JA (A m22) surface density of electric current.
In the dimensionless state, it characterizes the electric conductance of a system.
Info: [A23].
Georg Simon Ohm (16.3.17896.7.1854), German
physicist.
By experiment, he deduced the Ohm law and introduced
the electrical resistance concept. He was engaged in acoustics as well. He was influenced by French mathematicians
and physicists. On the base of the Fourier heat conduction
law, he modelled the electric current conduction in an electric network and described it mathematically.
6.1.45 Péclet Electrical Number Pee
Pee 5
wLRC
Δx2
w (m s21) velocity; L (m) characteristic length; RC (s) time constant of
electrical RC network; R (Ω) resistance; C (F) capacitance; Δx (m) element
length dimension.
It expresses the relationship among motion velocity of a source, zone boundaries
and conduction in a geometrically discrete RC model.
Info: [A24].
Jean Claude Eugène Péclet (p. 180).
302
Dimensionless Physical Quantities in Science and Engineering
6.1.46 Pomerantsev Electrical Number Poe
PoeV 5
JV L2
γUr
PoeL 5
JL
γUr
ð1Þ;
ð3Þ;
PoeA Kie 5
PoeB 5
I
γUr L
JA L
γUr
ð2Þ;
ð4Þ
JV, JA, JL volume density of electric current (A m23), surface density (A m22)
and linear density (A m21); L (m) characteristic length; γ (S m21) specific
electrical conductance; Ur (V) reference voltage; I (A) electric current; Kie
(2) Kirpichev electrical number (p. 299).
It expresses the ratio of the fed electric flow to that transferred by conduction in
a conductor. It characterizes volume (1), planar (2), longitudinal (3) or spot (4)
inner current sources. It represents thermal loss caused by electric current passage
through conductors.
Info: [A23].
Alexey Alexandrovich Pomerantsev, Russian engineer.
6.1.47 Rayleigh Electrothermal Condensation Number Rael
Rael 5
Feh ρlvl
ηλΔT
Feh (N) equivalent excitation electrodynamic force acting on condensing film;
ρ (kg m23) density; lvl (J kg21) specific latent heat of condensation; η (Pa s) dynamic viscosity; λ (W m21 K21) thermal conductivity; ΔT (K) temperature
difference.
It expresses the ratio of the product of the buoyancy electrohydrodynamic force
and the inertia force to the square of the viscosity force. It characterizes the acting
forces influence on the condensation process in an electric field. Condensation in
the electric field.
Info: [A23].
Lord Rayleigh (p. 187).
6.1.48 Rayleigh Electrothermal Instability Number Rae1
Rae1 5
β 2ε D2 ðrTÞ2 L4
ερνa
β ε (K21) thermal coefficient of permittivity for τ r =τ D {1; τ r (s) relaxation
time of temperature field; τ D (s) characteristic time of electric induction change;
D (C m21) electric induction; T (K) temperature; L (m) characteristic
Electromagnetism
303
length; ε (F m21) permittivity; ρ (kgτ r m23) density; ν (m2 s21) kinematic
viscosity; a (m2 s21) thermal diffusivity.
It expresses the ratio of the product of the electric buoyancy force and the inertia
force to the square of the viscosity force. It characterizes the electrothermal convection in ideal liquid dielectrics. It occurs in the stability description of a balanced
planar horizontal layer in a vertical electric field.
Info: [A23].
Lord Rayleigh (see above).
6.1.49 Rayleigh Electrothermal Number Rae
Rae 5 n
εβ ε L3 E3 rT
ρνa
n 5 0, 1, 2; ε (Fτ r m21) permittivity; β ε (K21) thermal coefficient of permittivity;
L (m) characteristic length; E (Vτ r m21) electric field intensity; T (K) temperature; ρ (kg τ m23) density; ν (m2 s21) kinematic viscosity; a (m2 s21) thermal diffusivity.
This number characterizes the electrothermal convection originating in a fluid
due to field inhomogeneities which are influenced by the geometric shape of
electrodes.
Info: [A23].
Lord Rayleigh (see above).
6.1.50 Reynolds Electrical Number (1.) ReEL1
2 21
wL
γL
wε
ReEL1 5
5 wL
5
ε
νe
γL
ReEL1 5
wε
w
5
ρkL kE
ð1Þ;
ð2Þ
w (m s21) flow velocity; L (m) characteristic length; ν e (m2 s21) electrical
kinematic viscosity; γ (S m21) specific electrical conductance; ε (F m21) permittivity; k (A s2 kg21) charge carrier mobility; E 5 ρL /ε (V m21) electric
field intensity; ρ (C m23) volume density of electric charge.
This number expresses the ratio of the fluid inertia force to the friction electric one
(of atoms, electrons or ions). It characterizes the influence of the inertia force and
that of the electric viscosity with forced electrothermal convection. In equation (2), it
expresses the volume density of charges. Magneto-hydrodynamics.
Info: [A23].
Osborne Reynolds (p. 82).
304
Dimensionless Physical Quantities in Science and Engineering
6.1.51 Reynolds Electrical Number (2.) ReEL2
See the Nusselt electrical number Nue (p. 301).
Osborne Reynolds (see above).
6.1.52 Schmidt Electrodiffusion Number Sc3
Sc3 5
νε
γL2
ν (m2 s21) kinematic viscosity; ε (F m21) permittivity; γ (S m21) specific
electrical conductance; L (m) characteristic length.
It expresses the ratio of the swirl diffusivity of a system to the mass diffusivity
of ions. It characterizes the mass transfer in electrochemical processes.
Info: [A23].
Ernst Schmidt (p. 264).
6.1.53 Senftleben Number Se
Se 5
Nk E2
2 p2
5 KrGr 21
α1
LMg
3 kTN
Nk (mol21) Avogadro constant; E (V m21) electric field intensity; L (m) characteristic length; M (kg mol21) molar mass; g (m s22) gravitational acceleration; α (A2 s4 kg21) gas molecule polarization coefficient; p (C m) electric
molecular dipole moment; k (J K21) Boltzmann constant; TN (K) temperature
of undisturbed fluid flow; Kr (2) Kronig number (p. 299); Gr (2) Grashof
heat number (p. 185).
This number characterizes the electric field influence on the heat transfer of
paraelectric fluids (with constant dipole moment). Essentially, it is another expression of the Kronig number Kr (p. 299).
Info: [A23].
Hermann Senftleben (born 1890), German physicist.
6.1.54 Source of Electrical Heat N
N5
JA2 L
u
5
γρwΔh Δh
JA (A m22) surface density of electric current; L (m) characteristic length; γ
(S m21) specific electrical conductance; ρ (kg m23) density; w (m s21) motion velocity; Δh (J kg21) specific enthalpy change; u (J kg21) specific
inner energy.
Electromagnetism
305
It expresses the ratio of the fed specific inner energy to the specific enthalpy
change of moving surroundings. It characterizes the thermal source caused by electric current passage through moving surroundings.
Info: [A23].
6.1.55 Stefan Electrical Number Sfe
Sf e 5
σ 3
U L
γ
σ (A m22 V24) electric equivalent of surface immisivity or emissivity; γ (S m21) specific electrical conductance; U (V) voltage; L (m) characteristic length.
This number expresses the ratio of the electric energy transmitted by a surface
under a non-linear boundary condition to that transmitted by conduction in a system. It characterizes a non-linear boundary condition, expressed by a power function, in an electric system which is analogous to a thermal one. It is of use, for
example, in the electrothermal modelling of heat transfer by radiation. It is the
physical analogue to the Stefan number Sf (p. 213).
Info: [A23].
Josef Stefan (p. 214).
6.1.56 Temperature Non-Uniformity N
N5
εβ 2ε L2 E2 ðrTÞ2
ρνa
ε (F m21) permittivity; β ε (K21) thermal coefficient of permittivity; L (m) characteristic length; E (V m21) electric field intensity; T (K) temperature;
ρ (kg m23) density; ν (m2 s21) kinematic viscosity; a (m2 s21) thermal
diffusivity.
It characterizes the influence of environmental thermal non-uniformity on an
inhomogeneous electric field and the phenomena it causes. Especially with high
field intensities, it has influence but does not have any substantial influence on
thermal process intensification except in cases of thin fluid layers.
Info: [A23].
6.1.57 Viscous Electrical Number E
E5
rffiffiffiffiffiffiffiffiffiffi 2
ρ ρL Q
2πε0 ηm
306
Dimensionless Physical Quantities in Science and Engineering
ρ (kg m23) density; ε0 (F m21) permittivity; L (m) characteristic length;
Q (C) electric charge; η (Pa s) dynamic viscosity; m (A m2) magnetic
moment.
This number expresses the relation between electrostatic and hydrodynamic
quantities or, alternatively, the motion of charged particles in flowing gas.
Info: [A24].
6.2
Magnetism
In magnetism, the dimensionless quantities are related to magnetic processes and
forces caused by the electric charge motion. Some of them express steady-state
magnetic fields or unsteady electromagnetic ones, propagation of electromagnetic
energy and magnetic waves, thermomagnetic convection and transfer phenomena
in plasma and magnetic turbulence. Among the important numbers are the Ampe`re,
magnetic field, Joule magnetic, Kubo, Lorentz, Maxwell and Stuart, and other numbers. A great number of magnetic dimensionless quantities are related to magnetohydrodynamics.
6.2.1
Ampère Number Am
Am 5
JA L
H
JA (A m22) surface density of electric current; L (m) characteristic length;
H (A m21) magnetic field intensity.
This number expresses the ratio of flow density to magnetic field intensity. It
characterizes the mutual action of the current and the magnetic field in magnetohydrodynamic processes.
Info: [A23].
André-Marie Ampère (20.1.177510.6.1836), French
mathematician and physicist.
He contributed markedly to the electricity and magnetism theory which became the base of the nineteenth century scientific development. He executed many experiments
and, in addition to other things, he demonstrated the origin
of a magnetic field around a conductor through which electric current was flowing. The unit of electric current was
named in his honour.
6.2.2
Electromagnetic Coupling Constant α
α5
e2
1
B
h̄c 137:03599 . . .
Electromagnetism
307
e (C) elementary charge; h̄ (J s) Dirac constant; c (m s21) speed of light in
vacuum.
This constant characterizes an electromagnetic field. It can be utilized to demonstrate the probability that an electron emits or absorbs a photon. It is a very important phenomenon in modern physics and in quantum electrodynamics especially. It
is a non-linear expression. With extremely high energy levels, the effective value
of the electric charge changes the coupling constant value. Electromagnetic fields.
Quantum physics. Astrophysics.
Info: [C23].
6.2.3
Electromagnetic Effect K
K5
μH 2
εE2
μ (H m21) permeability; H (A m21) magnetic field intensity; ε (F m21) permittivity; E (V m21) electric field intensity.
This effect expresses the ratio of the magnetic field energy density to that of the
electric field. It characterizes the degree of the hydrodynamic action of electric and
magnetic fields in fluids. It measures the degree of magnetic effect on electric
convection.
Info: [A23].
6.2.4
Electrothermal Time Nτ
Nτ 5
c
wA
2
5
τ rm
5 γμc2 τ rm ;
τ
where
τ rm 5
ρ
;
γμ2 H 2
τ5
1
γμc2
c (m s21) speed of light; wA (m s21) speed of Alfvén waves; τ rm (s) relaxation time of magnetic field; τ (s) conversion time of electromagnetic field energy
into Joule heat; γ (S m21) specific electrical conductance; μ (H m21) permeability; ρ (kg m23) fluid density; H (A m21) magnetic field intensity.
It expresses the ratio of the lines-of-force relaxation time in an electrically conductive fluid to the conversion time of the electromagnetic field energy to the heat
in Joules. It characterizes time relations involved in the conversion of the electromagnetic field to heat in Joules.
Info: [A23].
308
Dimensionless Physical Quantities in Science and Engineering
6.2.5
Fourier Electromagnetic Number Foem
Foem 5
ν m τ em
;
L2
where τ em 5
pffiffiffiffiffiffiffiffiffiffi
τeτm;
τe 5
ε
;
γ
τ m 5 μγL2
ν m (m2 s21) magnetic viscosity; τem (s) propagation time of electromagnetic
wave; L (m) characteristic length; τ e (s) relaxation time of electrical field
propagation; τ m (s) time of magnetic diffusion; ε (F m21) permittivity;
γ (S m21) specific electrical conductivity; μ (H m21) permeability.
In the dimensionless expression, it characterizes the magnetic field propagation
time, for example, in an electrically conductive fluid to the depth L.
Info: [A23],[A33].
Jean Baptiste Joseph Fourier (p. 175).
6.2.6
Fourier Magnetic Number Fom
Fom 5
νmτm
5 ν m μγ;
L2
where τ m 5 μγL 22
ν m (m2 s21) magnetic viscosity; τ m (s) time of magnetic diffusion; L (m) characteristic length; γ (H m21) magnetic permeability; γ (S m21) specific
electrical conductivity.
It expresses magnetic diffusion time. It points to the electromagnetic effects
influence on electromagnetic wave propagation. Together with the Fourier relaxation number of electrical field Foe (p. 296), it influences the characteristics of an
electromagnetic field. Magnetism.
Info: [A24].
6.2.7
Grashof Magnetic Number Grm
Grm 5 μ0
Grm 5
@M rHL3
@T βρν 2
ð1Þ;
μ0 β M MðTS 2 TN ÞL3 dH
ρν 2
dx
Grm 5 4πγμνGr
ð2Þ;
ð3Þ
μ0 (H m21) vacuum permeability; M (A m21) magnetization; T (K) temperature; H (A m21) magnetic field intensity; L (m) characteristic length;
β (K21) volume thermal expansion coefficient of fluid; ρ (kg m23) density;
ν (m2 s21) kinematic viscosity; βM (K21) volume thermal expansion of
magnetization; TS, TN (K) wall and undisturbed fluid flow temperature;
Electromagnetism
309
x (m) length, coordinate; γ (S m21) specific electrical conductance;
μ (H m21) permeability; Gr (2) Grashof heat number (p. 185).
This number characterizes the heat transfer by free thermomagnetic convection
in paramagnetic fluids with a gravity acceleration which depends on the space
coordinates. It is of use under non-isothermic conditions with the action of thermomagnetophoresis transfer of particles.
Info: [A23],[A29].
Franz Grashof (p. 185).
6.2.8
Interaction Parameter Ninter
See the Stuart number (1.) (magnetic force) St (p. 329).
6.2.9
Joule Magnetic Number Jom
Jom 5
2ρcp ΔT
μH 2
ρ (kg m23) density; cp (J kg21 K21) specific heat capacity; T (K) temperature; μ (H m21) permeability; H (A m21) magnetic field intensity.
It expresses the ratio of the heat in Joules to the total magnetic field energy. It
characterizes the heat transfer in Joules when acted upon by a magnetic field, for
example, in magneto-hydrodynamics and inductive heating.
Info: [A23].
James Prescott Joule (p. 298).
6.2.10 Kubo Number Kub, R
Kub 5
δB Lpar
B0 Lper
δB (T) mean fluctuation of magnetic field; B0 (T) mean value of magnetic
induction; Lpar, Lper (m) parallel and perpendicular correlation lengths in regard
to B0.
The Kubo number is used to classify the transfer phenomena in turbulent magnetic and other systems. With a small Kubo number, the quasi-linear diffuse coefficient is of use, as is the filtration diffusion coefficient with a large Kubo number.
The Kubo number expresses the level of a magnetic flux lines chaos. For Kub{1,
a weak chaos is valid and for Kub . 0.3, the extent of the chaos damages the surfaces which had been closed before. With Kubc1, a complete stochastic behaviour
occurs in which the filtration diffusion coefficient is of use. Transient phenomena
in plasma. Magnetic turbulence. Diffusion.
Info: [B130].
310
Dimensionless Physical Quantities in Science and Engineering
6.2.11 LorentzLorenz Number n
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3Ap
n 11
RT
A (m3 mol21) molar index of refraction; p (Pa) gas pressure; R (J mol21 K21) molar gas constant; T (K) absolute temperature.
This number represents the refraction index of rarefied gas. It expresses its
dependence on absolute temperature, pressure and the refraction index.
Electrodynamics. Magnetism. Electromagnetic radiation.
Info: [C109].
Hendrik Antoon Lorentz (p. 323).
Ludvig Valentine Lorenz.
6.2.12 Magnetic Dimensionless Frequency Nω
Nω 5
τ rx wref
L
τ rx (s) magnetic relaxation time of particles; wref (m s21) reference velocity of
particles; L (m) characteristic length.
It expresses the relation between the relaxation time of particles and the convection time. Usually, Nω {1 holds, which means that the influence of the angular
velocity of fluid particles is not as effective as the convective influence. Magnetohydrodynamics. Magnetic fields. Technology of magnetic separation, emulsification, connection, etc.
Info: [B27].
6.2.13 Magnetic Field Number N
N5
JA L
I
5
H
LH
ð1Þ;
L2 μγ
N 5 L f μγ 5
τ
2
N 5 L2 f 2 με ð2Þ;
ð3Þ;
U
N5
I
rffiffiffi
ε
ð4Þ
μ
JA (A m22) surface density of electric current; L (m) characteristic length;
H (A m21) magnetic field intensity; I (A) electric current; f (s21) frequency;
μ (H m21) permeability; ε (F m21) permittivity; γ (S m21) specific electrical conductance; τ (s) time; U (V) voltage.
In equation (1), it expresses a steady-state magnetic field. In expression (4), it
relates to an unsteady electromagnetic field. Further, expression (2) represents
Electromagnetism
311
electromagnetic energy propagation and expression (3) represents it electromagnetic waves propagation.
Info: [A24].
6.2.14 Magnetic Force Parameter NmgF
See Stuart number (1.) (magnetic force) St (p. 329).
6.2.15 Maxwell Number (1.) Fundamental Mx
Mx 5
ε E2
μH
ε (F m21) permittivity; E (V m21) electric field intensity; μ (H m21) magnetic permeability; H (A m21) magnetic field intensity.
It expresses the electric-to-magnetic field intensity ratio. It is the basic criterion
in the set of dimensionless Maxwell equations. Electromagnetic fields.
Info: [C87].
James Clerk Maxwell (13.6.18315.11.1879), Scottish
theoretical physicist.
He made revolutionary works in electrical engineering,
magnetism and the kinetic theory of gases. He predicted the
existence of electromagnetic waves, which propagate at
the same velocity as light. He created the electromagnetic
theory describing the phenomena of electrodynamics, optics
and thermal radiation. The substance of the Maxwell theory
is expressed by four Maxwell equations describing the time
and space dependence of the electromagnetic field.
6.2.16 Maxwell Number (2.) Electrodynamical Mx
Mx5
τ2
5 Fom
εμL2
τ (s) time; ε (F m21) permittivity; μ (H m21) magnetic permeability; L (m) characteristic length; Fom (2) Fourier magnetic number (p. 308).
This number expresses the dynamics of the electromagnetic field. It occurs in
the system of dimensionless Maxwell equations. Electromagnetic fields.
Info: [C87].
James Clerk Maxwell (see above).
312
Dimensionless Physical Quantities in Science and Engineering
6.2.17 Maxwell Number (3.) Time Mx
Mx 5
ρτ
5 Foem
μL2
ρ (Ω m) specific electrical resistance; τ (s) time; μ (H m21) magnetic permeability; L (m) characteristic length; Foem (2) Fourier electromagnetic number (p. 308).
Essentially, it expresses the dimensionless time of an electromagnetic process. It
is also called the Fourier electromagnetic number (p. 308). Electromagnetic
processes.
Info: [A23],[C87].
James Clerk Maxwell (see above).
6.2.18 Ohmic Heating Oht
Oht 5
λ γ ΔT
JA2 L2
λ (W m21 K21) thermal conductivity; γ (S m21) specific electrical conductance; ΔT 5 T 2 TS (K) temperature difference, difference of mean plasma temperature and temperature of channel wall surface; JA (A m22) surface density of
electric current; L (m) characteristic length.
It expresses the ratio of the heat transferred by conduction to the electric energy
transferred by plasma. It characterizes the influence of the heat led away by conduction on the electric arc motion in a transversal electric field. It expresses the
heat transfer intensity from a plasma into a canal wall with laminar equilibrium
and fully developed flow in a canal spot heated with an arc plasma generator.
Info: [A23].
6.2.19 Péclet Magnetic Rotational Number Pemg
Pemg 5
τB
L2
5
τ rx
Dmg τ rx
τ B (s) Brown relaxation time; τ rx (s) magnetic relaxation time of particles;
L (m) characteristic length; Dmg (m2 s21) magnetic diffusivity.
It expresses the relation between the Brown time and the magnetizing relaxation
time and, alternatively, it defines how fast the magnetic particles can be oriented
according to the local magnetic field. Magnetic field. Magnetic emulsification.
Magnetic separation. Magnetic spraying. Destructive methods.
Info: [B27].
Jean Claude Eugène Péclet (p. 180).
Electromagnetism
313
6.2.20 Pomerantsev Electromagnetic Number Pomg
Pomg 5
Lεqmg
λT
L (m) characteristic length of solid particle; qmg (W m22) surface density of
electromagnetic flux; λ (W m21 K21) thermal conductivity; T (K) source temperature; ε (2) emissivity (absorptivity) (p. 206).
This number characterizes the heating of solid particles in an electromagnetic
field. It is applied, for example, to discover the thermal properties of microscopic
particles by means of the impulse laser heating method. It is analogous to the
Pomerantsev heat number Po (p. 181).
Alexey Alexandrovich Pomerantsev, Russian engineer.
6.2.21 Prandtl Magnetic Number Prm
Pr m 5El 21 5νμγ 5
ν
5Rem Re 21
νm
ν (m2 s21) kinematic viscosity; μ (H m21) permeability; γ (S m21) specific
electrical conductance; νm (m2 s21) magnetic viscosity; El (2) Elsasser number (p. 318); Rem (2) Reynolds magnetic number (p. 328); Re (2) Reynolds
number (p. 81).
It expresses the ratio of viscosity to magnetic diffusion. It characterizes the similarity between the magnetic and the velocity fields of an electrically conductive
fluid flowing in the magnetic field. It is used to express the exciting effect on the
induced magnetic field current.
Info: [A23],[B20].
Ludwig Prandtl (p. 197).
6.2.22 Rayleigh Magnetic Number Ram
Ram 5 γ 2 μ2 L3 gβΔT 5 GrPr 2m
γ (S m21) specific electrical conductance; μ (H m21) permeability; L (m) characteristic length; g (m s22) gravitational acceleration; β (K21) volume
thermal expansion coefficient; ΔT (K) temperature difference; Gr (2) Grashof heat number (p. 185); Prm (2) Prandtl magnetic number (p. 313); Ra2
(2) Rayleigh number (2.) (p. 187).
It characterizes the magnetic convection influence on heat transfer intensification. The influence of the Ram number appears from Ram . 150 Ra2 approximately.
With Ram cRa2 , only the thermomagnetic convention is of use.
Info: [A23].
Lord Rayleigh (p. 187).
314
Dimensionless Physical Quantities in Science and Engineering
6.3
Magneto-hydrodynamics
Magneto-hydrodynamics is focused on the hydrodynamics of electrically conductive fluids on plasma and liquid metals, especially. In this field, the range of dimensionless quantities used is extensive, which corresponds to the significance of
dimensionless quantities of this field and to the wide range of applications in
applied physics, engineering branches, geophysics, astrophysics and other spheres.
Individual dimensionless quantities are focused on dynamic velocity, pressure and
force conditions in MHD, on the continuity of flow of electrically conductive fluid
or plasma and on the level of non-stationarity. The Alfve´n, Cowling, Einstein,
Elsasser, Hartman, Kármán, Mach magnetic, Reynolds and Strouhal numbers are
among the basic dimensionless quantities used within this area.
6.3.1
Alfvén Number Al
1
Al 5 Co2 5 Mm21 5
1
Al 5 Eu2m 5
H
w
wA
B
5 pffiffiffiffiffiffi
w
w μρ
ð1Þ;
where wA 5H
rffiffiffi
μ
;
ρ
rffiffiffi
μ
ð2Þ
ρ
wA (m s21) Alfvén’s wave velocity; w (m s21) velocity; B (T) magnetic
induction; μ (H m21) permeability; ρ (kg m23) density; H (A m21) magnetic field intensity; Co (2) Cowling number (p. 316); Mm (2) Mach
magnetic number (p. 324); Eum (2) Euler magnetic number (p. 319).
This number expresses the magnetic-to-inertia forces ratio or the ratio of the
Alfvén wave propagation rate to the fluid flow propagation rate. It characterizes
the propagation of magneto-hydrodynamic waves (Alfvén waves) along the magnetic field flux lines in a flowing electrically conductive fluid flow (the plasma). It
corresponds to the Euler magnetic number Eum (p. 319) and sometimes is used in
equation (2) to express the magnetic-to-kinetic energy ratio in unit volume. Its
inverse value is called the Mach magnetic number Mm (p. 324) or the Kármán number (2.) Ka (p. 321). Magneto-hydrodynamics.
Info: [A23],[B20].
Hannes Olof Gősta Alfvén (30.5.19082.4.1995),
Swedish astronomer. Nobel Prize in Physics, 1970.
He was engaged in research in electrical engineering,
electronics and astrophysics, and is one of the founders of
modern plasma physics. He published the theory of magnetic storms and of polar glares and laid the foundation of
the Earth magnetosphere theory. His principal works concern theoretical research of magneto-hydrodynamics, for
which he received the Nobel Prize.
Electromagnetism
6.3.2
315
Batchelor Number Bt
Bt5
wγL
c2 ε
w (m s21) velocity; γ (S m21) specific electrical conductance; L (m) characteristic length; c (m s21) speed of light; ε (F m21) permittivity.
It characterizes the dynamic relations in magneto-hydrodynamics.
Info: [A23],[A35]
George Keith Batchelor (8.3.192030.3.2000), Australian
mathematician and physicist.
He worked in hydrodynamics and magneto-hydrodynamics. The focus of his work was primarily in turbulence
research. Of his monographs, Homogenous Turbulence
(1953), Turbulent Diffusion (1956) and An Introduction to
Fluid Dynamics (1957) are the most important.
6.3.3
Brinkmann Modified Number Brmod
L
L
Brmod 5 Br pffiffiffi 5 EcPr pffiffiffi
ζ
ζ
L (m) characteristic length (width) of channel; ζ (m2) permeability of porous
material; Br (2) Brinkmann number (p. 174); Ec (2) Eckert number (p. 191);
Pr (2) Prandtl number (p. 197).
It expresses the ratio of the heat caused by viscous friction of the fluid in microscopic canals of a porous material to the heat transferred by molecular conduction.
Two-phase fluid mechanics. Magneto-hydrodynamics.
Info: [B70]
Henri Coenraad Brinkmann, German physicist.
6.3.4
Chandrasekhar Number Ch
Ch 5
B2 L2
ρνρe
B (T) magnetic induction; L (m) characteristic length; ρ (kg m23) fluid density;
ν (m2 s21) kinematic viscosity; ρe (Ω m) specific electrical resistance.
It expresses the magnetic-to-dispersion forces ratio. Magneto-hydrodynamics.
316
Dimensionless Physical Quantities in Science and Engineering
Subrahmanyan Chandrasekhar (19.10.191021.8.1995),
Indian astrophysicist. Nobel Prize in Physics, 1983.
He was engaged in research in various areas of physics,
in astrophysics especially. Besides the theory of the structure
and dynamic stability of stars, he devoted himself to radiation, the mathematical theory of black holes and general relativity. His name has been given to the Chandrasekhar limit.
According to this, a cooling star, with 1.5 times the Sun’s
mass, cannot resist its own gravitation. He received the
Nobel Prize for his theoretical work about the gravitational
collapse of stars.
6.3.5
Cowling Number, Magnetic Force Number Co
Co 5 Al2 5
w 2
A
w
5
B2
BH
5
μρw2
ρw2
wA (m s21) Alfvén’s wave velocity; w (m s21) velocity; B (T) magnetic
induction; μ (H m21) magnetic permeability; ρ (kg m23) density; H (A m21) magnetic field intensity; Al (2) Alfvén number defined by equation (1) (p. 314).
This number expresses the magnetic-to-inertia forces ratio. It is also called the
magnetic force number. It expresses the second power of the ratio of the Alfvén
wave velocity to the fluid velocity. Magneto-hydrodynamics.
Info: [A29],[A43],[B17]
Thomas George Cowling (17.6.199616.6.1990), English
mathematician and physicist.
He contributed significantly to modern research on stellar energy, especially that of the sun. He played an important role in finding the convective nucleus in stars. Due to
this convection, the sun can behave as a giant dynamo, the
rotation, internal circulation and flow of which generate
electric currents and magnetic fields connected with solar
spots.
6.3.6
Eckert Magnetic Number Ecm
Ecm 5
B2
w2A
5
μρcp ΔT
cp ðT 2 TS Þ
B (T) magnetic induction; μ (H m21) permeability; ρ (kg m23) density;
cp (J kg21 K21) specific heat capacity; ΔT 5 T 2 TS (K) fluid and wall temperatures difference; wA (m s21) Alfvén’s wave velocity.
Electromagnetism
317
It characterizes heat transfer under the conditions of a laminar magneto-hydrodynamic boundary layer caused by a magnetic field of induced currents. The value
of Ecm . 0 corresponds to outer heating and the Ecm , 0 represents fluid flow
cooling.
Info: [A23].
Ernst Rudolf Georg Eckert (p. 192).
6.3.7
Einstein Number Ei
Ei5
w
c
ð1Þ;
Ei R 5
w2
w
5 2
2
c
c γμL
ð2Þ
w (m s21) flow velocity; c (m s21) speed of light; γ (S m21) specific electrical conductance; μ (H m21) permeability; L (m) characteristic length; R (2) relativistic parameter (p. 327).
This number expresses the ratio of the local flow rate of electrically conductive
fluid to the conversion rate of electromagnetic field energy to heat in Joules. In the
form of equation (2), it is also called the velocity criterion or the relativistic parameter R (p. 327).
Info: [A23].
Albert Einstein (14.3.187918.4.1955), American physicist of German origin. Nobel Prize in Physics, 1921.
He is the ingenious physicist who brought to physics the
modern vision of physical reality, particularly with his special and general relativity theories. He also explained
the concept of the photoelectric phenomenon. He received
the Nobel Prize for his photoelectric phenomenon explanation, but not for the theory of general relativity, which was
his main contribution. He spent the last years of his life trying to create a unified theory which could explain all
known forces in nature.
6.3.8
Electric Field Number RE
RE 5
E
wμH
E (V m21) electric field intensity; w (m s21) flow velocity; μ (H m21) permeability; H (A m21) magnetic field intensity.
It characterizes the mutual action of hydrodynamic and electromagnetic fields in
magneto-hydrodynamics.
Info: [A23],[A33].
318
Dimensionless Physical Quantities in Science and Engineering
6.3.9
Electric Field Parameter NE
NE 5
E
E
5
wμH Bw
E (V m21) electric field intensity; w (m s21) velocity; μ (H m21) permeability;
H (A m21) magnetic field intensity; B (T) magnetic induction.
It expresses the ratio of the active electrodynamic force to the induced electrodynamic force. Magneto-hydrodynamics.
Info: [A23].
6.3.10 Elsasser Number El
El 5 ReRem21 5 Pr m21 5
ρ
ηγμ
ρ (kg m23) density; η (Pa s) dynamic viscosity; γ (S m21) specific electrical
conductance; μ (H m21) permeability; Re (2) Reynolds number (p. 81);
Rem (2) Reynolds magnetic number (p. 328); Prm (2) Prandtl magnetic number (p. 313).
Under magneto-hydrodynamic conditions, it expresses the hydrodynamic-tomagnetic field propagation ratio. It characterizes the relation between hydrodynamic and magnetic quantities in the magneto-hydrodynamics field.
Info: [A23],[B20].
Walter Maurice Elsasser (20.3.190414.10.1991),
American theoretical physicist of German origin.
He occupied himself with fundamental problems of
atomic physics, geophysics, radiation heat transfer in the
atmosphere, terrestrial magnetism and in the last years of
his life the theory of organisms. In 1939, he proposed
that the Earth’s rotation generates turbulent currents in both
its liquid nucleus and in its terrestrial magnetism.
6.3.11 Entropy Generation Number NS
NS 5 Sg
L2 T0
λ ðΔTÞ2
Sg (W m23 K21) entropy generation rate; λ (W m21 K21) thermal conductivity; ΔT (K) temperature difference; L (m) characteristic length (width) of
channel; T0 (K) initial temperature.
This number expresses the ratio of the generated entropy rate change to the
characteristic entropy transfer. In magneto-hydrodynamics, it is used to express the
Electromagnetism
319
entropy changes when acted upon by mixed convection and radiation in vertical
porous canals. Magneto-hydrodynamics.
Info: [B70].
6.3.12 Euler Magnetic Number Eum
Eum 5
μH 2
B2
5
ρw2
μρw2
μ (H m21) permeability; H (A m21) magnetic field intensity; ρ (kg m23) density; w (m s21) velocity; B (T) magnetic induction.
In a fluid moving in a magnetic field, it expresses the magnetic-to-dynamic pressures ratio. It characterizes the voltage acting along the magnetic lines of force in
an electrically conductive fluid which moves in a magnetic field. Essentially, it is
the degree of the magnetic-to-kinetic energies ratio for a unit fluid volume.
Sometimes, it is also called the magnetic pressure N (p. 325).
Info: [A23].
Leonhard Euler (p. 61).
6.3.13 Fourier Magnetic Relaxation Number Fomr
Fomr 5
τ rm ν m
5 ReRem21 Ha 22 5 Eum21 Rem22 5 St 21 Rem21 ;
L2
where
τ rm 5
ρ
;
γμ2 H 2
νm 5
1
γμ
τ rm (s) magnetic relaxation time; ν m (m2 s21) magnetic viscosity; L (m) characteristic length; ρ (kg m23) density; γ (S m21) specific electrical
conductance; μ (H m21) permeability; H (A m21) magnetic field intensity;
Re (2) Reynolds number (p. 81); Rem (2) Reynolds magnetic number
(p. 328); Ha (2) Hartmann number (p. 320); Eum (2) Euler magnetic number
(p. 319); St (2) Stuart number (1.) (magnetic force) (p. 329).
In an electrically conducting fluid, moving at the characteristic velocity
we (γμL)21, it expresses the ratio of lines of force relaxation (attenuation) time to
the lines of force passage time through a magnetic field and the conversion of its
energy to electric heat.
Info: [A23].
Jean Baptiste Joseph Fourier (p. 175).
320
Dimensionless Physical Quantities in Science and Engineering
6.3.14 Hall Coefficient CH
CH 5
λ
rL
λ (m) mean free path; rL (m) Larmor radius. This coefficient expresses the
ratio of the gyro-frequency to the frequency of collisions. Magneto-hydrodynamics.
Info: [A29].
Edwin Herbert Hall (see below).
6.3.15 Hall Parameter, Hall Coefficient H, β
H 5 ωτ 5
eBτ
5 KnLr 21
m
ω (s21) resonant frequency of electron; τ (s) mean time of electron relaxation;
e (C) electron charge; B (T) magnetic induction; m (kg) electron mass;
Kn (2) Knudsen number (1.) (p. 69); Lr (2) Larmor number (p. 322).
It represents the ratio of the gyro-frequency to the frequency of collisions. In the
magneto-hydrodynamics, it expresses the ratio of the mean time between collisions
to the time of their circulation around a magnetic flux line.
Info: [B20].
Edwin Herbert Hall (7.11.185520.11.1938), American
physicist.
In the year 1879, he discovered the Hall effect, the substance of which is the transversal electric voltage in a current loaded conductor or in a semiconductor located in an
outer magnetic field. In two-dimensional electron systems,
the quantum Hall effect appears in strong magnetic fields
and under temperatures close to those of liquid helium.
Together with superconductivity, this effect is one of the
most significant manifestations of quantum theory.
6.3.16 Hartmann Number Ha
rffiffiffi
rffiffiffi pffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
γ
γ
5 BL
5 StRe 5 ReEum Rem
Ha 5 μHL
η
η
μ (H m21) permeability; H (A m21) magnetic field intensity; L (m) characteristic length; γ (S m21) specific electrical conductance; η (Pa s) dynamic viscosity; B (T) magnetic induction; St (2) Stuart number (1.) (magnetic force)
(p. 329); Re (2) Reynolds number (p. 81); Eum (2) Euler magnetic number
(p. 319); Rem (2) Reynolds magnetic number (p. 328).
Electromagnetism
321
It is an important criterion of magneto-hydrodynamics. It expresses the ratio of
the induced electrodynamic (magnetic) force to the hydrodynamic force of the viscosity or, alternatively, the ratio of the ponderomotive force (the electromagnetic
volume force by means of which the magnetic field acts on a conductor through
which electric current flows, which causes magnetic pressure) to the molecular friction force. It characterizes the magnetic field influence on the flow of viscous, electrically conducting fluid. With small Ha values, the motion proceeds as if no
magnetic field were acting. With great Ha values, the viscosity forces act only on a
thin layer of the electrically conducting fluid (ionized gas) which adheres closely
to a by-passed wall surface. In other cases, the motion resistance does depend on
the viscosity and is determined completely by electromagnetic volume forces which
are acting on the fluid. With high velocities and turbulent flow, it is more
suitable to use the Stuart number (2.) St (p. 330), expressing the mutual magnetohydrodynamic action, instead of the Ha number.
Info: [A23],[A43],[B20].
Julius Frederik Georg Poul Hartmann (18811951), Danish physicist.
6.3.17 Hydromagnetic Velocity Number Nhm
Nhm 5
w
wmg
w (m s21) fluid flow velocity; wmg (m s21) velocity of magnetic field
propagation.
It expresses the ratio of the fluid flow rate to the magnetic field propagation
rate. It corresponds to the Reynolds magnetic number Rem (p. 328). Magnetohydrodynamics.
6.3.18 Kármán Number (2.) Ka
pffiffiffiffiffiffi
w μρ
w
21
;
Ka 5
5 Al 5
B
wA
rffiffiffi
μ
where wA 5 H
ρ
w (m s21) local fluid velocity; wA (m s21) velocity propagation of Alfvén’s
magnetic wave; μ (H m21) permeability; ρ (kg m23) fluid density; B (T) magnetic induction; H (A m21) magnetic field intensity; Al (2) Alfvén number (p. 314).
It characterizes the influence of the mutual action of hydrodynamic and magnetic fields in magneto-hydrodynamics. It expresses the ratio of the fluid flow rate
to that of the Alfvén waves. It expresses also the kinetic-to-magnetic energies ratio
in a unit volume. Magneto-hydrodynamics.
Info: [A35].
Theodore von Kármán (p. 67).
322
Dimensionless Physical Quantities in Science and Engineering
6.3.19 Larmor Number Lr
Lr 5
rL
L
rL (m) Larmor radius; L (m) characteristic length.
In the magnetic field, it expresses the ratio of the Larmor rotation radius of a
charged mass particle to a characteristic longitudinal dimension. It characterizes
the continuity of the electrically conductive flow. With the Lr negligibly small, the
fluid or plasma flow can be described as a continuous environment exposed to
magnetic field influence. For a large Lr value and the Knudsen number (1.) (p. 69)
Knc1, the fluid flow must be described as the flow of mutually acting discrete
charged particles, exposed to magnetic field influence. Magneto-hydrodynamics.
Info: [A23].
Joseph Larmor (11.7.185719.5.1942), Irish physicist.
He was engaged in researching electricity, dynamics and
thermodynamics. He suggested the physical theory of light,
according to which ether can represent a homogenous fluid
which would be both perfectly incompressible and perfectly
elastic. He developed a theory according to which the spectra could be explained by oscillations of electrons. He
determined the energy released with radiation of an accelerated charge and explained the spectral lines splitting in a
magnetic field. The Larmor radius represents that of the
charged particle movement in a magnetic field.
6.3.20 Lorentz Factor, Lorentz Term γ, Lof
γ5
dτ m
1
5 pffiffiffiffiffiffiffiffiffiffiffiffiffi
dτ
1 2 β2
ð1Þ;
c
γ 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
c 2 u2
ð2Þ;
where
β5
u
c
τ m (s) measured time; τ (s) supposed time; u (m s21) velocity at reference
point where time τ m is measured; c (m s21) speed of light.
It is used in the special relativity theory to simplify the recording of equations
and, especially, to express the related time dilation, length contraction and rest-mass
conversion to the relativistic mass. Physics. Electromagnetism. Electrodynamics.
Magneto-hydrodynamics.
Info: [C79].
Electromagnetism
323
Hendrik Antoon Lorentz (18.7.18534.2.1928), Dutch
physicist. Nobel Prize in Physics, 1902.
He was engaged in optics, electrodynamics and atomistics. He contributed significantly to the general relativity
theory, elaborated by Einstein, specifically by experimentally verifying the special relativity theory (1919). He is
known for his work on the electric and magnetic field influence on electromagnetic radiation. He laid the foundation
for the electron theory of the electric conductivity of substances. He received the Nobel Prize for his work on electromagnetic radiation.
6.3.21 Lundquist Number Ld
3
Ld 5
γHμ2 L
1
ρ2
1
1
5 HaRe2m Re2 2
γ (S m21) specific electrical conductance; H (A m21) magnetic field intensity;
μ (H m21) permeability; L (m) characteristic length (thickness of conductive
fluid); ρ (kg m23) density; Ha (2) Hartmann number (p. 320); Rem (2) Reynolds magnetic number (p. 328); Re (2) Reynolds number (p. 81).
This number characterizes the size of the attached magnetic field influence on
the distribution of the induced magnetic field in the flow of an electrically conductive fluid. It has been used in magneto-hydrodynamics to express unidirectional
Alfvén waves or, alternatively, the waves caused by the magnetic field in the flow
of the electrically conductive fluid.
Info: [A23].
6.3.22 Lykoudis Number Ly
1
Ly 5
γμ2 H 2 L2
1
ðgβΔTÞ2 ρ
1
5 Ha2 Gr 2 2
γ (S m21) specific electrical conductance; μ (H m21) permeability; H (A m21) magnetic field intensity; L (m) characteristic length; g (m s22) gravitational
acceleration; β (K21) volume thermal expansion coefficient; T (K) temperature;
ρ (kg m23) density; Ha (2) Hartmann number (p. 320); Gr (2) Grashof heat
number (p. 185).
It expresses the ratio of the ponderomotive force to the product of buoyancy and
inertia forces. It characterizes the magnetic field influence on the convection of an
electrically conductive fluid in magneto-hydrodynamics.
Info: [A23].
Paul S. Lykoudis (born 1926), American engineer of Greek origin.
324
Dimensionless Physical Quantities in Science and Engineering
6.3.23 Mach Electromagnetic Number Mae
E
we
5 ReEL1 Rem 5
w
Bw
Mae 5
E (V m21) electric field intensity; B (T) magnetic induction; w (m s21) fluid
flow velocity; we (m s21) drift velocity; ReEL1 (2) Reynolds electrical
number (1.) (p. 303); Rem (2) Reynolds magnetic number (p. 328).
It expresses the drift-to-flow rates ratio. Magneto-hydrodynamics.
Info: [A24].
Ernst Mach (p. 73).
6.3.24 Mach Magnetic Number Mm
Mm 5 Al 21 5
Mm 5
w
a 1 wA
pffiffiffiffiffiffi
w ρμ
w
5
wA
B
ð1Þ;
Mm 5
w
1
ða2 1 w2A Þ2
ð2Þ;
ð3Þ
w (m s21) velocity; wA (m s21) Alfvén wave velocity; ρ (kg m23) density;
μ (H m21) permeability; B (T) magnetic induction; a (m s21) sound speed;
Al (2) Alfvén number (p. 314).
This number expresses the ratio of the local flow rate of an electrically conductive fluid to the propagation rate of the Alfvén (magneto-hydrodynamic) wave.
Like the Alfve´n number (p. 314) or the Euler magnetic number Eum (p. 319), it
characterizes the flow rate in magneto-hydrodynamics. Sometimes, expression
(2) or (3) is used. It is also called the Alfve´nMach number.
Info: [A23].
Ernst Mach (p. 73).
6.3.25 Magnetic Fluidization Number Nmgf
Nmgf 5
μs H 2 x
ρs u2f
μs (H m21) powder permeability; H (A m21) magnetic field intensity; x (2) volume fraction of magnetic powder; ρs (kg m23) mean density of fluidized bed;
uf (m s21) surface fluid velocity.
It characterizes the magnetic fluidization of powders in a flowing fluid. It
expresses the magnetic-to-kinetic energy ratio for particles. It is a modification of
the Euler magnetic number Eum (p. 319). Magneto-hydrodynamics.
Info: [B129].
Electromagnetism
325
6.3.26 Magnetic Force Parameter Nmg,F
Nmg;F 5
μHγL
ρw
μ (H m21) permeability; H (A m21) magnetic field intensity; γ (S m21) specific electrical conductance; L (m) characteristic length; ρ (kg m23) density
of environment; w (m s21) flow velocity.
It expresses the magnetic-to-dynamic forces ratio. It is analogous to the magnetic number Nmg (p. 325). Magneto-hydrodynamics.
Info: [A29],[B17].
6.3.27 Magnetic Interaction Nmg
Nmg 5
μ0 H 2 r
2σ
μ0 (H m21) vacuum permeability; H (A m21) magnetic field intensity; r (m) radius; σ (kg s21) liquid surface tension.
It relates to ferrous fluids dynamics. Magneto-hydrodynamics.
Info: [A24],[B20].
6.3.28 Magnetic Number Nmg
sffiffiffiffiffiffi
1
γL
2
5 NmgF
Nmg 5 B
ρw
B (T) magnetic induction; γ (S m21) specific electrical conductance; L (m) characteristic length; ρ (kg m23) density; w (m s21) fluid flow velocity;
NmgF (2) magnetic force parameter (p. 311).
It expresses the magnetic-to-inertia forces ratio. It is analogous to the magnetic
force parameter NmgF (p. 311). Magneto-hydrodynamics.
Info: [C84].
6.3.29 Magnetic Pressure N, S
N5
p
2μ
5 nkT 2
pm
B
ð1Þ;
S5
μH 2
ρw2
ð2Þ
p (Pa) gas pressure; pm (Pa) magnetic pressure; n (m23) numerical
density of particles; k (J K21) Boltzmann constant; T (K) temperature;
326
Dimensionless Physical Quantities in Science and Engineering
μ (H m21) permeability; B (T) magnetic induction; H (A m21) magnetic
field intensity; ρ (kg m23) density; w (m s21) velocity.
It expresses the ratio of the magnetic pressure to that of the gas. It characterizes
the pressure in magneto-hydrodynamics. pm 5 B2/2μ holds for magnetic pressure
and p 5 nkT for gas pressure. It is also called the Euler magnetic number Eum
(p. 319).
Info: [A23],[B20].
6.3.30 MagneticDynamic Number Nmgd
γwB2 L
ρw2
Nmgd 5
γ (S m21) specific electrical conductivity; w (m s21) fluid flow velocity;
B (T) magnetic induction; L (m) characteristic length; ρ (kg m23) fluid
density.
It expresses the magnetic-to-dynamic pressures ratio of the fluid. Magnetohydrodynamics.
Info: [A24],[B20].
6.3.31 Magnetization Parameter Ω
Ω5
ωef
ν ef
ωef (s21) angular frequency; ν ef (s21) effective electron collision.
It represents the local electron magnetization parameter. Magneto-hydrodynamics.
Flow in accelerator canals.
6.3.32 Morozov-Hall Parameter Mor
Mor 5
Ω
Rem
Ω (2) magnetization parameter (p. 329); Rem (2) Reynolds magnetic number
(p. 328).
It characterizes the magneto-hydrodynamic processes for flow in accelerator
canals. Magneto-hydrodynamics.
Info: [C96].
V.A. Morozov, Russian physicist.
Edwin Herbert Hall (p. 320).
Electromagnetism
327
6.3.33 Naze Number Na
Na 5
wA
5 MAl
a
wA (m s21) Alfvén’s wave velocity; a (m s21) sound speed; M (2) Mach
number (p. 73); Al (2) Alfvén number (p. 314).
In an electrically conductive fluid, it expresses the ratio of the propagation rate
of magneto-hydrodynamic waves (the Alfvén waves) to the sound propagation rate
in the fluid. It characterizes the velocity relations in electro-magnetohydrodynamics.
Info: [A23].
Jacqueline Naze (born 1935), French mathematician.
6.3.34 Oseen Magnetic Number Os
Os 5
1
ð1 2 Al2 ÞNmg
2
Al (2) Alfvén number (p. 314); Nmg (2) magnetic number (p. 325), or the
Stuart number (1.) (magnetic force) (p. 329).
It expresses the magnetic-to-inertia forces ratio. It characterizes the magnetohydrodynamic relations in electrically conductive fluid flow in the magnetic field.
Info: [A23].
6.3.35 Porous Media Inertia Coefficient Γ, Npmi
Ck ε2 L
Γ 5 pffiffiffi
ζ
Ck (2) Forchheimer coefficient; ε (2) porosity (p. 24); ζ (m2) permeability
of porous material; L (m) characteristic length (width of channel).
It characterizes the porous environment behaviour with convective and mixed
convective radiation flow and interaction in vertical porous canals. It occurs, for
example, in magneto-hydrodynamics and is accompanied by entropic changes.
Two-phase flow. Magneto-hydrodynamics.
Info: [B70].
6.3.36 Relativistic Parameter R
See the Einstein number Ei defined by relation (2) (p. 317).
328
Dimensionless Physical Quantities in Science and Engineering
6.3.37 Reynolds Magnetic Number Rem
Rem 5 RePr m 5
Rem 5
wL
w
5 γμwL 5
νm
wm
γμEL
5 Rem Mm21
B
ð1Þ;
ð2Þ
w (m s21) velocity; L (m) characteristic length; ν m 5 (μmσ)21 (m2 s21) magnetic viscosity or magnetic diffusivity; γ (S m21) specific electrical conductance; μ (H m21) permeability; wm 5 (γμL)21 (m s21) characteristic velocity;
E (V m21) electric field intensity; B (T) magnetic induction; Re (2) Reynolds
number (p. 81); Prm (2) Prandtl magnetic number (p. 313); Rem (2) Reynolds
magnetic number defined by the equation (1); Mm (2) Mach magnetic number
(p. 324).
This number expresses the ratio of a movable induced magnetic field to an introduced outer magnetic field. It characterizes the influence of the electrically conductive fluid flow rate on the magnetic field distribution in the fluid. Sometimes, it is
also called the velocity number Rw.
It can be expressed as the ratio of the fluid flow rate to the characteristic velocity
wm 5 (γμL)21 of a magnetic field moving over a conductor, or as the ratio of the
flow dimension L to the characteristic length of a magnetic field Lm 5 (γμwm)21
which moves through a conductor. It is the degree of outer magnetic field excitation. For Rem 5 0, the fluid flow does not influence the magnetic flux lines of the
field in which the fluid is moving. The magnetic field of induced currents does not
originate in the fluid. For Rem {1, or w{wm , L{Lm alternatively, the magnetic
field influence can be ignored and it can be assumed that the flow is due to outer
magnetic field influence only, for example, without fluid motion influence on the
magnetic field. The plasma moving in an electromagnetic field or liquid metal in a
pipeline, in canals of magnetic pumps, or in MHD generators, are examples. For
Rem1, the induced magnetic field can influence the outer magnetic field action
considerably. It causes some entraining of a resulting magnetic field by the fluid,
this entraining growing greater with the increasing Rem. For Rem c1, the fluid flow
becomes a good electric conductor and acts on a magnetic field strongly. It is as if
the magnetic flux lines are frozen in the fluid flow and entrained by it. These questions are the subject of magneto-hydrodynamics research in astrophysics.
Info: [A23],[A29],[B20].
Osborne Reynolds (p. 82).
6.3.38 Roberts Number Rob
Rob 5
a
5 Rem Pe 21
νm
a (m2 s21) thermal diffusivity; ν m (m2 s21) magnetic viscosity; Rem (2) Reynolds magnetic number (p. 328); Pe (2) Péclet heat number (p. 180).
Electromagnetism
329
It characterizes the magnetic convection of an electrically conducting fluid in
fast rotating systems under action of a toroidal magnetic field. It expresses the ratio
of induced and outer magnetic fields to convective and molecular heat transfers in
the flowing fluid. It is a very important criterion in examination of the convection
around planets. Magneto-hydrodynamics. Astrophysics.
Info: [B125].
6.3.39 Strouhal Electromagnetic Number Shm
Shm 5
Eτ
we T
5
BL
L
Shm 5
τ
γμL2
ð1Þ;
Shm 5
wB
E
ð2Þ;
ð3Þ
E (V m21) electric field intensity; τ (s) time; B (T) magnetic induction;
L (m) characteristic length; we 5 E/B (m s21) drift velocity; w (m s21) velocity; γ (S m21) specific electrical conductance; μ (H m21) permeability.
It expresses the ratio of the local inertia force to the convective electromagnetic
force. It characterizes the dynamic relations in non-stationary electromagnetic processes. It expresses the degree of the movement non-stationarity in a system.
Info: [A23].
Vincenc Strouhal (p. 87).
6.3.40 Stuart Number (1.) (Magnetic Force) St
St 5
γμ2 H 2 L B2 Lγ
5
5 Ha2 Re21 5 Eum Rem
ρw
ρw
γ (S m21) specific electrical conductance; μ (H m21) permeability; H (A m21) magnetic field intensity; L (m) characteristic length; ρ (kg m23) density;
w (m s21) velocity; B (T) magnetic induction; Ha (2) Hartmann number
(p. 320); Re (2) Reynolds number (p. 81); Eum (2) Euler magnetic number
(p. 319); Rem (2) Reynolds magnetic number (p. 328).
This number expresses the ratio of the ponderomotive force caused by induced
currents to the fluid inertia force. It characterizes the electrically conductive fluid
flow in a magnetic field. This criterion can also be held as the ratio of the persistence time of the fluid in a magnetic field to the relaxation time of magnetic field
flux lines. Sometimes, it is called the magnetic parameter or the parameter of magneto-hydrodynamic mutual interaction (interaction parameter). For St . 1 with
Rem {1, the magnetic forces strongly influence the flow. Magneto-hydrodynamics.
Info: [A23],[A33].
John Trevor Stuart (born 1929), English mathematician.
330
Dimensionless Physical Quantities in Science and Engineering
6.3.41 Stuart Number (2.) (Charge Volume Density) St
St 5
εE2
ρ EL
5 e 2
ρw2
ρw
ε (F m21) permittivity; E (V m21) electric field intensity; ρ (kg m23) fluid
density; w (m s21) flow velocity; ρe (C m23) volume density of an electric
charge; L (m) characteristic length.
It expresses the electrostatic-to-dynamic pressures ratio for a fluid. It is also
called the electrodynamic number Nmgd. It characterizes the electrodynamic process
in magneto-hydrodynamics.
Info: [A23].
John Trevor Stuart (see above).
7 Physical Technology
There is a Plenty of Room at the Bottom
Richard Phillips Feynman (19181988)
7.1
Micro- and Nanotechnology
In micro- and nanotechnology, the dimensionless quantities differ substantially
from those in macroscopically oriented physics and mechanics. Here, specific physical problems of these technologies appear as adhesion, tension, deformation, plasticity, peeling and processes of flow in microchannels and nanopores. The
microfluidic phenomena the flow of micro- and nanoscopic particles, such as
carbon soot, dust and tissue cells are connected with these technologies. The
Knudsen, Mason, microfluidic, Tabor and Zhang-Zhan numbers are among the
most popular dimensionless quantities in micro- and nanotechnology.
Richard Phillips Feynman (11.5.191815.2.1988),
American theoretical physicist. Nobel Prize in Physics, 1965.
The main area of his research was in the field of quantum physics, in which he participated in the origin of quantum electrodynamics. He contributed to defining the
mechanisms of interactions between elementary particles
and elaborated on the technique to describe them, namely
the eponymous Feynman diagrams. He predicted the existence of the internal structures of the proton and the neutron. His works contributed significantly to describing the
microworld we now know. He played an important role in
creating a new sphere: nanotechnology.
7.1.1
Adhesion Parameter Nadh, Θ
Nadh 5
Eh3=2
1=2
ra Δσ
5
Eh 1=2
f ;
Wadh r
where Δσ 5 σ1 1 σ2 2 σ1;2
E (Pa) modulus of elasticity; h (m) standard deviation of the peak heights of
roughness; ra (m) radius of curvature of an asperity; Δσ (N m21) Dupré
Dimensionless Physical Quantities in Science and Engineering. DOI: 10.1016/B978-0-12-416013-2.00007-5
© 2012 Elsevier Inc. All rights reserved.
332
Dimensionless Physical Quantities in Science and Engineering
adhesion, adhesion surface tension; σ1, σ2, σ1,2 (N m21) surface tension between
two spheres; Wadh (J) adhesion work; fr (2) roughness factor.
It expresses the ratio of the elastic energy to the adhesion work, provided there
is a contact. For Nadh c1, there is only a partial contact, where the elastic material
has contact only on peaks of the highest roughness. Full contact occurs with
Nadh {1 only. The parameter is based on the Gauss transversal roughness distribution on an elastic material surface. Micro- and nanotribology.
Info: [B128].
7.1.2
BénardMarangoni Instability Number Mrcrit
Mrcrit 5
σ0 ΔTh
5 80;
ηa
where σ0 5
dσ
dT
σ0 (N m21 K21) rate of change of surface tension with temperature; ΔAT (K) temperature difference between lower level and free liquid level; h (m) depth of
fluid layer; η (Pa s) dynamic viscosity; a (m2 s21) thermal diffusivity;
σ (N m21) surface tension; T (K) temperature.
This number is expressed by the value of the critical Marangoni number Mr
(p. 186), with which instability occurs in a fluid surface layer. This instability manifests itself by loss of contact with the bottom wall, of which the temperature must
be higher than the air temperature above the fluid layer surface. The
BénardMarangoni set of hexagonal cells builds a system in which the fluid flows
upwards through the centre of the cells, and downwards near the walls.
BénardMarangoni instability arises from the temperature nonuniformity in the
fluid and is a certain modification of the classic RayleighBénard instability number, which dates from the beginning of the 20th century. It relates to hydrothermal
flow and surface waves under the action of an intensive heat source (i.e. laser or
electron beam) on the molten material surface. Thermocapillary phenomena.
Microgravitational applications. Microtechnology.
Info: [C7].
Carlo Guiseppe Matteo Marangoni (18401925), Italian physicist.
7.1.3
Darcy Number (3.) Porous Da
Da 5
h2
ξ
h (m) thickness of porous layer; ξ (m2) permeability of porous material.
It characterizes the permeability in porous material and in microchannels. It is
analogous to the Darcy granulation number (2.) Dc (p. 98). Two-phase flow.
Micro- and nanotechnologies. Thin layers.
Info: [B95].
Henry Philibert Gaspard Darcy (p. 98).
Physical Technology
7.1.4
333
Deborah Number (2.) De
De 5 τ rx γ 5
2τ rx wmed
k
De 5 ReNelast
ð2Þ
ð1Þ;
τ rx (s) fluid relaxation time; γ (s21) shear rate; wmed (m s21) mean fluid
flow velocity; k (m) contraction width of the channel; Re (2) Reynolds number (p. 81); Nelast (2) elasticity number (4.) (p. 333).
It characterizes the low viscosity elastic fluid flow in industrially made microchannels with various degrees of narrowing when the fluid contracts and expands.
It is a modification of Deborah number (1.) De (p. 120). It relates to rheologic fluid
flow in technological microchannels with diverse geometric contraction.
Info: [B97].
7.1.5
Dispersion Number Ndis
Ndis 5
2kd
σw R2h
kd (N) dispersion coefficient; σw (N m21) fluid surface tension; Rh (m) hydraulic diameter.
It represents the quantity of the dispersion force in the thin surface layer of a
microchannel. Two-phase flow in microchannels. Micro- and nanotechnology. Thin
layers.
Info: [B95].
7.1.6
Elasticity Number (4.) Nelast , El
Nelast 5
τ rx η
5 DeRe 21 ;
RbDh
where Dh 5
2bh
b1h
τ rx (s) fluid relaxation time; η (Pa s) dynamic viscosity; R (kg m23) fluid
density; b (m) width of contraction; Dh (m) hydraulic diameter; h (m) height of channel; De (2) Deborah number (1.) (p. 120); Re (2) Reynolds
number (p. 81).
It characterizes the non-Newtonian flow of the low viscosity elastic fluid
flow in industrially made microchannels in which the concentration and expansion
processes of the fluid flow occur. It is a modification of elasticity number (1.) El
(p. 121). It relates to the flow of polyethylene and similar rheologic fluids through
technological channels with diverse geometric contraction.
Info: [B97].
334
Dimensionless Physical Quantities in Science and Engineering
7.1.7
Evaporation Momentum Force (1.) Nv
q
Nv 5
qm lv
2
Rl
Rv
q (W m22) heat flux; qm (kg m22 s21) mass flux density; lv (J kg21) latent
heat of vapourization; Rl, Rv (kg m23) density of liquid and vapour, respectively.
It expresses the ratio of the momentum force, during fluid evaporation, to the
inertia force in microchannels. Primarily, it involves the boiling number Boi
(p. 215) and fluid to vapour densities ratio. Boiling. Heat transfer in microchannels.
Microtechnology.
7.1.8
Evaporation Momentum Force (2.) Nv
2
q
L
Nv 5
lv R v σ
q (W m22) heat flux; lv (J kg21) latent heat of vapourization; L (m) characteristic length (bubble diameter); Rv (kg m23) vapour density; σ (N m21) surface tension.
It expresses the ratio of the momentum force, in fluid evaporation, to the surface
strain force with fluid passage and evaporation in microchannels. Heat transfer.
Boiling. Microtechnology.
7.1.9
Force Flow Boiling Number in Microchannels (1.) NF1
NF1 5
qA
qm l v
2
Rl
Rv
qA (W m22) surface heat flux; qm (kg m22 s21) mass flux density; lv (J kg21) specific latent heat of vapourization; Rl, Rv (kg m23) density of liquid and vapour.
It expresses the ratio of the evaporating momentum force to the inertia force in
fluid flow and evaporation in microchannels. For these relations, low values of the
hydraulic diameter and Reynolds numbers ReAh(100; 1000i (p. 81) are characteristic. Two-phase flow. Boiling. Nonequilibrium processes. Micro- and
nanotechnologies.
Info: [B57].
Physical Technology
335
7.1.10 Force Flow Boiling Number in Microchannels (2.) NF2
2
qA
L
NF2 5
lv Rv σ
qA (W m22) surface heat flux; lv (J kg21) specific latent heat of vapourization;
L (m) characteristic length, bubble diameter; Rv (kg m23) vapour density;
σ (N m21) surface tension.
This number expresses the ratio of the evaporating momentum force to the surface strain force in fluid flow and evaporation in microchannels. The expression
does not involve the influence of the contact angle. Two-phase flow. Boiling.
Nonequilibrium processes. Micro- and nanotechnologies.
Info: [B57].
7.1.11 Heat Removal Nrem
Nrem 5
λl ðTsat 2 Tw Þ
Rl ν l llv
λl (W m21 K21) fluid thermal conductivity; Tsat (K) saturation temperature;
Tw (K) wall temperature; Rl (kg m23) liquid density; ν l (m2 s21) kinematic
viscosity; llv (J kg21) latent heat of evaporation.
It expresses the ratio of the heat transfer rate, in terms of the channel length unit,
to the transfer of fluid properties. Two-phase flow. Micro- and nanotechnology.
Info: [B95].
7.1.12 Kandlikar Number (1.) Kan
Kan 5
q 2 R
l
Gr Rg
q (W m22) heat flux; G (kg m22 s21) mass flux density; r (J kg21) specific
heat of evaporation; Rl, Rg (kg m23) fluid and vapour density.
This number expresses the ratio of the evaporation momentum force to the inertia force. Comparing to the boiling number Boi (p. 215) which does not, itself,
represent the real evaporation momentum effect the Kandlikar number involves
the ratio of fluid and vapour densities, too. Among other things, the number Kan
describes the heat transfer mechanism of a boiling fluid flowing through microchannels. Two-phase flow.
Info: [B56].
336
Dimensionless Physical Quantities in Science and Engineering
7.1.13 Kandlikar Number (2.) Kan
Kan 5
q2 L
r Rg σ
q (W m22) heat flux; r (J kg21) specific heat of evaporation; L D (m) characteristic number, hydraulic diameter D; Rg (kg m23) vapour density;
σ (N m21) surface tension.
It expresses the ratio of the evaporation momentum force to the surface stress
force. It describes the heat transfer mechanism of a boiling fluid flowing through
microchannels. Two-phase flow.
Info: [B56].
7.1.14 Knudsen Micro- and Nanometric Number Kn
rffiffiffiffiffi
rffiffiffiffiffiffiffiffiffi
πk M
π η
Kn 5
5
2 Re
2RT RL
R (J mol21 K21) molar gas constant; T (K) absolute fluid temperature; η (Pa s) dynamic viscosity; R (kg m23) fluid density; L (m) characteristic length
(microchannels and nanopores dimension); K (2) specific heat ratio (p. 28);
M (2) Mach number (p. 73); Re (2) Reynolds number (p. 81).
It characterizes the flow in microchannels and nanopores. For KnAh0; 1023i, it
describes continuous flow; for KnAh1023; 1021i it describes shear flow; for
KnAh1021; 101i it describes transient flow; and for KnAh101; Ni it describes free
molecular flow. Micro- and nanomechanics.
Info: [B99].
Martin Hans Christian Knudsen (p. 420).
7.1.15 Mason Dielectrophoretic Number Masdf
Masdf 5
Mf1
8ηw
5
Mfdf1
ε0 εf β 2 dE2
η (Pa s) fluid dynamic viscosity; w (m s21) characteristic velocity; ε0,
εf (F m21) vacuum and fluid permittivity, respectively (ε0 5 8.8542 3
10212 F m21); β (2) ClausiusMossotti factor; d (m) particle diameter;
E (V m21) electric field intensity; Mf1 (2) microfluidic number (p. 338);
Mfdf1 (2) microfluidic dielectrophoretic number (1.) (p. 337).
It expresses the ratio of the viscosity force to the electrostatic interaction force
between particles. It characterizes the microfluidic phenomena in the flow of
micro- and nanoscopic particles (i.e. carbon black, glass dust or tissue cells).
Info: [B55].
Physical Technology
337
7.1.16 Mass Flux Number Nqm
Nqm 5
2 2 c λv
2c
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
RðTw 2 Tl Þ3
pffiffiffi
σw llv 2π
c (2) constant, usually c 5 1; λv (W m21 K21) vapour thermal conductivity;
R (J kg21 K21) gas constant; Tw (K) wall temperature; Tl (K) fluid temperature; σw (N m21) fluid surface tension at the wall; llv (J kg21) latent heat of
vapourization.
It characterizes the mass flow on the interface between phases. It is implied
from the kinetic theory of fluid flow in thin layers and microchannels under phase
conversion conditions.
Info: [B95].
7.1.17 Microchannel Flow Boiling Number Boim, Co
Boim 5
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
σ
gðRl 2 Rg ÞD2h
σ (N m21) surface tension; g (m s22) gravitational acceleration; Rl, Rg
(kg m23) density of liquid, gas or vapour; Dh (m) hydraulic diameter.
When boiling fluid passes through microchannels, it expresses the ratio of the
surface strain force to the buoyancy force acting on a vapour bubble.
Thermomechanics. Boiling. Micro- and nanotechnologies.
Info: [B56].
7.1.18 Microfluidic Dielectrophoretic Number (1.) Mfdf1
Mfdf1 5
3πε0 β 2 d 3 E2
4mw2
ε0 (F m21) vacuum permittivity (ε0 5 8.8542 3 10212 F m21); β (2) positive
or negative direction of dielectrophoretic force; d (m) particle diameter;
E (V m21) electric field intensity; m (kg) particle mass; w (m s21) characteristic velocity of particle.
It expresses the ratio of the electrostatic interaction force, between particles, to
the inertia force. It relates to microfluidic phenomena in the flow of micro- and
nanoscopic particles in an electrical field.
Info: [B55].
338
Dimensionless Physical Quantities in Science and Engineering
7.1.19 Microfluidic Dielectrophoretic Number (2.) Mfdf2
Mfdf2 5
4πε0 εf βd4 E2
;
mw2 L
where β 5
εp 2 εf
εp 1 2εf
ε0 (F m21) vacuum permittivity (ε0 5 8.8542 3 10212 F m21); εf (F m21) fluid
permittivity; β (2) ClausiusMossotti factor; d (m) particle diameter;
E (V m21) electric field intensity; m (kg) particle mass; w (m s21) particle
flow velocity; L (m) characteristic length (electrodes distance).
It expresses the dielectrophoresis and inertia forces ratio in the flow of particles
in an electric field. Microfluidic phenomena.
Info: [B55].
7.1.20 Microfluidic Dielectrophoretic Number (3.) Mfdf3
Mfdf3 5
Mfdf1
3βL
5
Mfdf2
16d
β (2) ClausiusMossotti factor; L (m) characteristic length (electrodes distance); d (m) particle diameter; Mfdf1 (2) microfluidic dielectrophoretic number (1.) (p. 337); Mfdf2 (2) microfluidic dielectrophoretic number (2.) (p. 338).
When the particles flow in an electric field, it characterizes the force relations.
It always applies in cases of important influence of mutual interaction of particles
and the dielectrophoresis force. Microfluidic phenomena.
Info: [B55].
7.1.21 Microfluidic Number Mf1
Mf1 5
6πηd 2
mw
η (Pa s) dynamic viscosity of fluid; d (m) particle diameter; m (kg) particle
mass; w (m s21) characteristic velocity.
It characterizes the behaviour of micro- and nanoscopic particles in a flowing
fluid, for example, that of live cells which are suspended in the fluid. It expresses
the viscous to inertia forces ratio.
Info: [B55].
7.1.22 Mismatch Parameter δ
δ 5 100
sff 2 sss
;
sss
where sfs 5
sff 1 sss
2
Physical Technology
339
sff (m) atom geometric parameter in thin layer; sss (m) atom geometric parameter in substrate; sfs (m) atom geometric parameter on the boundary line of thin
layer and substrate.
It characterizes the growth of thin layers, with deposition processes, and the
structure originating on the interface between a thin layer and the substrate material. It is applied, for example, in the simulation of deposition processes to create
interatomic potentials. Micro- and nanotechnology. Thin layers.
Info: [B84].
7.1.23 Peel Number Np
Np 5
3Es3 h2
2L4 Wadh
E (Pa) modulus of elasticity; s (m) layer thickness; h (m) distance of
unstuck layer from substrate; L (m) characteristic length of crack; Wadh (J) adhesion work.
With the deposited layer deviated from the substrate, it characterizes the relations of adhesion. It expresses the ratio of the elastic deformation energy, stored in
the deformed microstructure memory, to the adhesion work between the microstructure and the substrate. When Np . 1, the elastic deformation stored in the
memory is greater than the work of adhesion and the microstructure will not adhere
to the substrate. When Np # 1, the deformed layer does not have enough energy to
overpower the adhesion.
Info: [B128]
7.1.24 Plastic Adhesion Index Npl.ad.
Npl:ad: 5
π 2 H 4 ra σ
2 W2
8Eekv
a
H (N) material hardness; ra (m) curvature radius of the roughness peaks,
in point as a sphere; σ (m) standard deviation of the roughness height distribution; Eekv (Pa) equivalent modulus of elasticity (see Tabor number Tab)
(p. 343); Wa (J) Dupré adhesion work.
It characterizes the quantitative point of view of the relation between adhesion,
plastic deformation and other corresponding phenomena. Micro- and nanotechnology.
Info: [B100].
7.1.25 Plasticity Index Nplast
Nplast 5
Eekv
H
rffiffiffiffi
σ Eekv pffiffiffi
5
f
H
ra
340
Dimensionless Physical Quantities in Science and Engineering
Eekv (Pa) equivalent modulus of elasticity (see Tabor number Tab) (p. 343);
H (N) material hardness; σ (m) standard deviation of the roughness height distribution; ra (m) curvature radius of the roughness peaks, in point as a sphere;
f (2) relative surface roughness.
It is the plastic deformation criterion of a microscopic contact. For elastic roughness it is Nplast , 0.6; for plastic roughness it is Nplast . 1; and for elastoplastic
roughness it is 0.6 , Nplast , 1. Micro- and nanotechnology.
Info: [B128].
7.1.26 Prediction Number of Adhesion Npred
4 Eσ2
Npred 5 2
3π Wadh L
2=3
E (Pa) equivalent modulus of elasticity; σ2 (m2) quadratic deviation of the
roughness height distribution; Wa (J) adhesion work; L (m) characteristic correlation length.
It characterizes the quantitative predictions of adhesion between deformable
fractal surfaces. It expresses the ratio between the standard deviation of the roughness heights and the maximum height of the roughness peaks before transforming.
Micro- and nanotechnologies.
Info: [B128].
7.1.27 Pulse Electromagnetic Heating of Microparticles Nheat , β
Nhead 5
dεP
2λTref
d (m) microparticle diameter; P (W m22) power density of the electromagnetic field; λ (W m21 K21) thermal conductivity of microparticles; Tref (K) reference temperature; ε (2) emissivity (p. 206).
It characterizes the pulse heating of microscopic particles with a laser, as an
example. Among other things, it is used to determine the thermal, physical properties of the particles. Physical technology. Microtechnology. Thermomechanics.
Electromagnetism.
7.1.28 Pumping Intensity Npi
Npi 5
ðpl 2 pv ÞL2
ν 2l Rl
Physical Technology
341
pl (Pa) fluid pressure; pv (Pa) vapour pressure; L (m) characteristic pressure
length; ν l (m2 s21) kinematic viscosity of liquid; Rl (kg m23) liquid density.
It represents the ratio of the maximum power which can be obtained with the
pressure difference between a fluid and a vapour to the transfer parameters.
Porous flow through microchannels. Two-phase flow. Micro- and nanotechnology.
Thin layers.
Info: [B95].
7.1.29 Pumping Resistance Npr
Npr 5
pRh
4σ
p (Pa) fluid pressure; Rh (m) hydraulic diameter; σ (N m21) fluid surface
tension on the channel wall.
It characterizes the resistance of the two-phase miniature microchannel flow in
porous layers which serve, for example, to cool down the electronic circuits or as
miniature heat exchangers or pumping microcircuits. Two-phase flow. Micro- and
nanotechnology. Thin layers.
Info: [B95].
7.1.30 Response Number Rn
2
Rw2ini L 2
L
Rn 5
5 Dn
τ
H
H
I2
L 2
ð2Þ
Rn 5
2
RτH H
ð1Þ;
R (kg m23) material density; wini (m s21) initial velocity of particles; τ (Pa) shear stress of material; L (m) characteristic length; H (m) thickness of beam
or slab; I (Pa s21) pressure impulse velocity of rectangular shape; Dn (2) Johnson’s damage number.
This number characterizes the dynamic plastic response in a material exposed to
impulse or stroke pressure loading. It especially relates to diverse beams or plates.
However, for the dynamic load of generally shaped bodies, expression (2) is valid.
Info: [B127].
7.1.31 Reynolds Rheological Number (2.) Rerh
Rerh 5
wmed Dh
2Q
5
;
ν0
ðb 1 hÞν 0
where wmed 5
Q
bh ;
Dh 5
2bh
b1h
342
Dimensionless Physical Quantities in Science and Engineering
wmed (m s21) average fluid velocity; Dh (m) hydraulic diameter; ν 0 (m2 s21) kinematic viscosity at zero shear speed; Q (m3 s21) volume flow rate; b (m) width of contraction; h (m) height of the channel.
It characterizes the forced flow of a low viscosity elastic fluid through microchannels with a range of technological cross section narrowing when the fluid is
contracted and expanded. Rheology. Micro- and nanotechnologies.
Osborne Reynolds (p. 82).
7.1.32 Strain Peel Number Npε
"
#
128Es3 h2
4σr L2
256 h 2
Npε 5
11
1
8L4 Wadh
21Es2
2205 s
ð1Þ;
"
#
186Es3 h2
27ð1 2 ν 2 Þσr b2
12 h 2
11
1
Npε 5
ð1 2 ν 2 Þb4 Wadh
310Es2
31 s
40Es3 h2
51ð1 2 ν 2 Þσr R2
Npε 5
11
3ð1 2 ν 2 ÞR4 Wadh
160Es2
ð2Þ;
ð3Þ
E (Pa) modulus of elasticity; s (m) layer thickness; h (m) distance of
unstuck layer from substrate; L (m) length of double-sided, close-set beam;
Wadh (J) adhesion work; σr (Pa) residual stress; ν (2) Poisson’s ratio
(p. 143); b (m) width of beam; R (m) radius of circular plate.
In a dimensionless shape, the number Npε expresses the residual strain in a
beam fixed at both ends (1), in a square plate (2) or in a round plate (3). From these
equations, the maximum dimensions in the microstructure can be determined
(the beam length or fixation, the width or the plate radius), with which the layer
will not be connected with the substrate. It is valid provided the condition Np 5 1
holds, with the inner energy of the elastic deformation corresponding to the
adhesion.
Info: [B127].
7.1.33 Subcooling Number (2.) Nsc
Nsc 5
2ηv λv ðTw 2 Tl Þ
Rh Rv πllv σw
ηv (Pa) vapour dynamic viscosity; λv (W m21 K21) vapour thermal conductivity;
Tw (K) wall temperature; Tl (K) fluid temperature; Rh (m) hydraulic diameter;
Rv (kg m23) vapour density; llv (J kg21) latent heat of vapourization; σw (N m21) fluid surface tension at the wall.
Physical Technology
343
In porous material microchannels, this number expresses the condensation
process and its influence on the heat dissipation with transfer zone reduction and
very high dissipation capability. Heat transfer in microchannels. Condensation and
controlled cooling. Microtechnology.
Info: [B95].
7.1.34 Tabor Number Tab, μ
Tab 5
RWa2
2 ε2
Eekv
1 2 ν2
1=3
;
h5
1=3
RWa2
2
Eekv
1 2 ν2
21
where Eekv
5 E1 1 1 E2 2 ; R (m) equivalent radius of curvature; Wa (J) Dupré adhesion work; Eekv (Pa) equivalent modulus of elasticity; E1, E2 (Pa) modulus of elasticity of two hemispheres; ν 1, ν 2 (2) Poisson’s constant of two
hemispheres; ε (m) intermolecular distance; h (m) depth of penetration.
The Tabor number is the degree of elastic deformation, which depends on the
surface forces’ size. It characterizes the nonequilibrium adhesion mechanism of a
contact in micromechanical systems. Alternatively, it expresses the ratio of the
impression height to the intermolecular distance. Bodies in contact characterized
by a small number, Tab , 0.1, have slight influence on the elastic deformation
and adhere one to the other more easily. For Tab . 5, it is the opposite. With both
materials and quantities Wadh, Eekv and ε being equal, TabBR1=3 holds. Micro- and
nanotribology.
Info: [B128].
7.1.35 Transport Heat Number Nheat
Nheat 5
Rl ν 2l
σL
Rl (kg m23) liquid density; ν l (m2 s21) liquid kinematic viscosity; σ (N m21) surface tension; L (m) characteristic length of channel.
It characterizes the condition for the maximum heat transfer on the microchannel length unit. Two-phase flow. Micro- and nanotechnologies.
Info: [B95].
7.1.36 Zhang-Zhao Number (1.) Zh1
Zh1 5
Wa
σH
Wa (J) Dupré adhesion work; σ (m) standard deviation of the roughness height
distribution; H (N) hardness of material.
344
Dimensionless Physical Quantities in Science and Engineering
It expresses the ratio of the adhesion work to the product of the standard deviation
of the roughness peak heights and the material hardness. The greater this number is,
the more easily the plastic deformation occurs. Micro- and nanotechnologies.
Info: [B90].
7.1.37 Zhang-Zhao Number (2.) Zh2
Zh2 5
d
σ
d (m) difference between mean surface and glare surface during loading; σ (m) standard deviation of the roughness heights.
It expresses the ratio of the difference between the central surface height and the
smooth surface during loading. It affects the plastic deformation adhesion.
7.2
Plasma Physics and Technology
The dissimilarity of plasma in the form of ionized gas from the solid, liquid
or gaseous phases manifests itself in dimensionless quantities as well. The quantities relate to gas particle behaviour; wavelength; frequency and thermal equilibrium
of the plasma; magnetic field influence on the plasma; and plasma pressure observation with regard to magnetic pressure, to magnetohydrodynamic load and to other
quantities. Plasma is associated with various technologies utilizing intensive
sources such as the laser, an electric arc in arc welding, etching of dielectric layers
in electronics with integrated circuits manufacture, thermal loading of the front surface of space bodies in descent into the atmosphere, nuclear fusion research and
other areas of physics and technology. The de Broglie and Larmor numbers, Hall’s
parameter and many other numbers are well known dimensionless quantities in
plasma physics. Systematic using of the dimensionless quantities for plasma technologies has been lacking up to now.
7.2.1
Anisothermal Plasma Number Nta,τ
Nta 5
Te
Ti
Te, Ti (K) kinetic temperature of electrons and ions.
This number expresses a dimensionless parameter characterizing thermal plasma
anisotropy. It appears in plasma density determination and discovering out the ion
collision influence in measurement with the Langmuir cylindrical probe. The
Physical Technology
345
anisothermal plasma number and the Knudsen number for electrons (ions) Kne,i
(p. 349) represent two parameters which presume the Maxwell distribution of electron energy. Plasma physics and technology.
Info: [B88].
Irving Langmuir (31.1.188116.8.1957), American chemist and physicist. Nobel Prize in Chemistry, 1932.
He devoted himself to studying light and contributed significantly to vacuum engineering improvement, which led
to the idea of a high vacuum tube. He was engaged in
research related to glowing filament behaviour in a vacuum
and in the thermal emission problem. He was one of the
first scientists who worked with plasma, which he called
ionized gas. He introduced the concept of electron temperature and the diagnostic method to measure vacuum temperature and density with the Langmuir probe. He devoted also
to atmospheric sciences and meteorology.
7.2.2
Attraction Number Na
Na 5
A
12πηrp2 w
A (J) Hamaker constant; η (Pa s) dynamic viscosity; rp (m) particle radius;
w (m s21) approximate fluid velocity.
It represents the combined influence of the van der Waals attraction forces and
the fluid rate on the deposition velocity with particle capture. Deposition processes.
Physical and chemical filtration in saturated porous materials.
Info: [B48].
7.2.3
de Broglie Thermal Wavelength Λ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h2
Λ5
2πmkT
V
#1
nΛ3
ð2Þ;
V
c1
nΛ3
ð3Þ;
Λ5
ch
pffiffiffi
2kT 3 π
ð1Þ;
ð4Þ
346
Dimensionless Physical Quantities in Science and Engineering
h (J s) Planck constant; m (kg) mass of gas particles; k (J K21) Boltzmann
constant; T (K) absolute temperature; c (m s21) speed of light; V (m3) volume; η (m23) number of particles in volume V.
This number represents roughly the average of the de Broglie wavelength of ideal
gas particles with specific density.
It canffi be held as a mean distance between the parpffiffiffiffiffiffiffiffiffiffi
ticles in the gas, approximately 3 Vn 21 : When the thermal wavelength is much less
than the distance between the particles, the gas behaves as a classic
MaxwellBoltzmann gas. However, when the thermal de Broglie wavelength is
greater by one or more orders than the distance between the particles, the quantum effects become dominant and the gas must be considered a Fermi gas or a Bose
gas, depending on the substance of gas particles. In this type of situation, expression
(2) is valid, and for the preceding situation, the MaxwellBoltzmann expression (3)
holds. For a massless particle, the thermal wavelength can be expressed by expression (4). Plasma physics.
Info: [C133].
Louis-Victor Pierre Raymond, duc de Broglie
(15.8.189219.3.1987), French mathematician and physicist. Nobel Prize in Physics, 1929.
He studied quantum theory and the wave theory of electrons based on Einstein’s and Planck’s works. This led to
the proposal of the wave particle duality theory. According
to it, a substance has properties both of particles and of
waves. The results of de Broglie’s theory of electromagnetic waves later served Schrődinger, Dirac and other scientists in the creation of wave mechanics.
7.2.4
de Broglie Wavelength λ
λ5
h
p
λ5
h
mv
ð1Þ;
ð2Þ
h (J s) Planck constant; p (kg m s21) momentum of particle; m (kg) mass of
particle; v (m s21) speed of particle.
It represents a wavelength as the ratio of the kinetic energy of a particle to its
momentum. In physics, de Broglie’s hypothesis primarily expresses the fact that all
materials have properties of waves. Plasma physics.
Info: [C31].
Louis-Victor Pierre Raymond, duc de Broglie (see previous entry).
Physical Technology
7.2.5
k5
347
de Broglie Wave Number k
2πp p
5
h
h̄
p (kg m s21) momentum; h (kg m s21) Planck constant; h̄ (kg m s21) Planck
constant (h-bar).
This number expresses the ratio of a particle’s momentum to its kinetic energy.
Plasma physics.
Info: [C32].
Louis-Victor Pierre Raymond, duc de Broglie (see previous entry).
7.2.6
Debye Number ND
4
ND 5 πneλ2D ð1Þ;
3
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ε0 kqe22
P
ð2Þ;
λD 5
ne Te21 1 ij j2 nij Ti21
sffiffiffiffiffiffiffiffiffiffiffi
ε0 kTe
λD 5
ð3Þ
ne q2e
ne (m23) electron density; λd (m) Debye length ε0 (F m21) vacuum permittivity; k (J K21) Boltzmann constant; qe (C) electron charge; Te, Ti (K) electron
and ion temperatures, respectively; nij (m23) density of atom particles i with positive ion charge jqe.
This number represents the relation between the Debye length and the probe
radius. It expresses the state of the thermal mode and the density during a plasma’s
existence. For ND 5 1, the state occurs which is denoted as the ratio of the collective behaviour to the independent behaviour of particles. When ND c1, it represents the collective behaviour mode with which the dynamic properties of the
plasma are changing. When ND , 1, it is called the mode of independent particles,
in which the collective phenomenon has slight influence in the plasma. Expressions
(2) and (3) represent the Debye length, which expresses the shielding diameter in
the plasma. Expression (3) is valid only if the ions are much colder than the electrons. The size of λD changes substantially from 10211, for the plasma in the Sun’s
nucleus (ne 5 1032), over 1024 for a gas discharge (ne 5 1016), to values of 105
(ne 5 1) for the intergalactic environment. Plasma physics and technology.
Tokamak. Astrophysics.
Info: [A24], [C33].
348
Dimensionless Physical Quantities in Science and Engineering
Peter Josephus Wilhelmus Debye (24.3.18842.11.1966),
Dutch-American electrical engineer, physicist, chemist.
Nobel Prize in Chemistry, 1936.
He formulated the specific heat theory (1911) and
formed the theory of electrolytic solutions. He engaged in
work related to dipole moments of molecules. He had
remarkable knowledge of the arrangement of atoms and
molecules and of the distances between atoms. Together
with P. Scherrer, he developed (1916) the method to determine the atomic structure of crystals by means of X-rays.
This is called the DebyeScherrer method.
7.2.7
Hall Parameter, Hall Coefficient H, β
H 5 ωτ 5
eBτ
5 KnLr 1
m
ω (s21) resonant frequency of electron; τ (s) average relaxation time of electron; e (C) electron charge; B (T) magnetic induction; m (kg) electron
mass; Kn () Knudsen number (1.) (p. 69); Lr () Larmor number (p. 322).
In magnetohydrodynamics, it expresses the ratio of the mean time between collisions of particles to the time of their circulation around a magnetic flux line. It is
an important criterion for examining three plasma acceleration modes in a magnetic
field. There is no dispersion for βc1 , but it occurs for β 5 1 and is aperiodic for
β{1 . Plasma physics.
Info: [A29].
Edwin Herbert Hall (p. 320).
7.2.8
Ionization Plasma Number Ni
3
2
ni
Ui
27 T
Ni 5
3 3 10
exp 2
nn
ni
kT
ni (m23) number of ionized atoms per unit volume; nn (m23) number of neutral atoms per unit volume; T (K) absolute temperature; Ui (J) ionizing energy;
k (J K21) Boltzmann constant.
It expresses the general approximation of the thermal plasma equilibrium. It predicts which ionization degree can be expected in a gas with thermal equilibrium.
Plasma physics.
Info: [C103].
Physical Technology
7.2.9
349
Knudsen Number for Electrons (Ions) Kne,i
Kne;i 5
λe;i
rp
λe,i (m) electron (ion) mean free path; rp (m) probe radius.
It is the characteristic number for the theory of collisions, as it describes the
mean free path of electrons or ions in relation to the dimension of the applied
probe. For collisionless conditions, Kne,i - N. In other words, with a probe, the
collisions act on the charged particles more intensely as the probe radius increases,
especially under experimental conditions. Plasma physics and technology.
Info: [B88].
Martin Hans Christian Knudsen (p. 420).
7.2.10 Larmor Plasma Number Lrpl
Lrpl 5
rB
v
5
;
rt
rt ω B
where ωB 5 gB
rB (m) Larmor radius; rt (m) toroid radius; v (m s21) particle velocity perpendicular to magnetic field; ωB (s21) Larmor frequency; g (s21 T21) gyromagnetic radius; B (T) magnetic induction.
It expresses the ratio of the Larmor radius to that of a toroid. Alternatively, it is
the ratio of the particle velocity to the magnetic field intensity. Plasma and atomic
physics.
Info: [B86],[C131],[C104].
Joseph Larmor (p. 322).
7.2.11 Magnetization Parameter δ
δ5
R
L
ð1Þ
R (m) Larmor radius, gyroradius; L (m) characteristic length.
It characterizes the magnetic field influence on a plasma. The plasma process or
system is magnetized if its characteristic length is much greater than the gyroradius. When R{L, the charged particles follow the magnetic field lines of trajectories directly. In the opposite case, when RcL; the charged particles strike against a
chamber wall before they can be accelerated due to the magnetic field influence.
Plasma physics.
Info: [C131].
350
Dimensionless Physical Quantities in Science and Engineering
7.2.12 Controlled Thermonuclear Fusion number
β5
2μ0 p
B2
μ0 (H m21) vacuum permeability; p (Pa) plasma pressure; B (T) magnetic
induction.
It plays a key role in controlled thermonuclear fusion. It expresses the plasma to
magnetic pressures ratio. To maintain the plasma, the value must be β , 1 and the
magnetic flow must be higher than the plasma flow. Nevertheless, in reality it is
often β # 0.2, which limits, for example, the pressure p # 107 Pa with BB10 T.
Info: [C131].
7.2.13 Plasma Collision Frequency Nfpl ,γ
w
5 Nπd 2
Nfpl 5 Nπd
ω
2
rffiffiffiffiffiffiffiffiffi
8πT 1
mπ ω
N (m23) number of particles per unit volume m3; d (m) atom diameter;
w (m s21) average atom velocity; ω (s21) characteristic frequency of plasma;
k (J K21) Boltzmann constant; T (K) absolute temperature; m (kg) atom
mass.
This expression represents an estimate based on the gas kinetic theory of
the plasma collision frequency. When Nfpl {ω; then Nfpl 5 0.01. Plasma physics.
Info: [B86],[C104].
7.2.14 Plasma Parameter Λ
Λ 5 4πnλ3D
n (m23) number of particles per unit volume; λD (m) Debye length.
It is among the basic plasma parameters. Usually, it expresses the number of
particles contained in a Debye sphere. When Λ{1, the Debye zone is rarely occupied, which corresponds to a weakly coupled plasma. When Λc1; in the Debye
sphere there is a great concentration of particles to which strongly coupled plasma
corresponds. For strongly coupled plasma, lower temperatures and great densities
of particles are typical, whereas weakly coupled plasma is diffusive with low concentrations of particles and high temperatures. Plasma with a laser ablation and
plasma with an arc discharge are examples of strongly coupled plasma. Some
examples of typical values of a weakly coupled plasma are Λ 5 5 3 108 for nuclear
fusion and Λ 5 4 3 104 for the interstellar environment. Λ 5 2 3 103 is valid for a
chromosphere and Λ 5 3 3 102 for a glow discharge. Plasma physics.
Info: [C131].
Physical Technology
351
7.2.15 Pressure Plasma Number Nβ , β
Nβ 5
2μ0 p
;
B2
where p 5 nkT
μ0 (H m21) vacuum permeability; p (Pa) plasma pressure; B (T) magnetic
induction; n (m23) number of particles per unit volume; k (J K21) Boltzmann
constant; T (K) absolute temperature.
It expresses the ratio of plasma to magnetic pressures or, alternatively, that of
thermal energy density to magnetic energy density. It characterizes the conditions
under which certain limitations are valid to maintain the plasma for a magnetic
field. The value must be less than unity (Nβ , 1). As a matter of fact, the most
suitable value is NβB0.2. The pressure plasma number plays a key role, for example, in controlled thermonuclear fusion. Plasma physics and technology.
Info: [B86],[C131],[C104].
7.2.16 Thrust Coefficient Ct
Ct 5
4π FT
μ0 I 2
μ0 (H m21) vacuum permeability; FT (N) thrust force; I (A) total current
between electrodes.
It characterizes the magnetoplasmadynamic load by the total current.
Magnetodynamics. Electromagnetic plasma accelerators.
Info: [B23].
8 Technology and Mechanical
Engineering
The intuition is the only valuable thing indeed.
Albert Einstein (18791955)
8.1
Technology and Material
In this section, quantities are presented which are applied only in specific areas
of a technology or materials processing. Others are not considered here because
their main application is in fundamental fields, and they are presented, therefore,
in other chapters of this book. In this section, technologies such as heating in
diverse combustion devices, welding, melting, surface treatment and solidification
processes, and other technologies, are discussed. The following numbers are
among the known quantities: the Boltzmann and CrayaCurtet numbers for fireplaces and furnaces, the Christensen, Rosenthal and Rykalin numbers for welding,
the Tikhonov number for thermochemical materials processing, and the Chvorinov
number for solidification time.
8.1.1
Agglomerate Number Ag
sffiffiffiffiffiffiffiffiffiffiffi
5 2
3 D E
Ag 5 Rw2
Γ5
R (kg m23) particle density; w (m s21) particle velocity; D (m) particle
diameter; E (Pa) modulus of elasticity; Γ (J m22) surface energy.
It expresses the degree of contact damage during agglomeration due to the
impact rate, the adhesion energy, and the properties and shape of particles. It is
based on the assumption that the energy used to erode the material is proportional
to the incidental kinetic energy of the agglomerate. The number is a modification
of the Weber agglomerate number Weagl (p. 363).
Dimensionless Physical Quantities in Science and Engineering. DOI: 10.1016/B978-0-12-416013-2.00008-7
© 2012 Elsevier Inc. All rights reserved.
354
Dimensionless Physical Quantities in Science and Engineering
8.1.2
Boltzmann Number of Combustion Chamber Bo
Bo 5
k1 Q m cp
3
σk2 STsp
k1 (2) heat hold coefficient in combustion chamber, comprising heat losses to
ambient; Qm (kg s21) fuel mass flux calculated; cp (J kg21 K21) specific heat
capacity; σ (W m22 K24) absorptance (absorption capacity, radiating capacity);
k2 (2) contamination coefficient of wall surface; S (m2) surface area of
absorption walls; Tsp (K) theoretical combustion temperature.
It expresses the ratio of the heat transferred to a fireplace surface to the maximum heat which can be transferred by radiation. It characterizes the heat transfer
in boiler fireplaces. Their inner space is considered as an absolutely black body.
Info: [A23].
Ludwig Boltzmann (p. 205).
8.1.3
Christensen Number Chr
Chr 5
w2 A
a2
w (m s21) motion velocity; A (m2) surface; L (m) characteristic length,
weld width; a (m2 s21) thermal diffusivity.
It is often called the dimensionless weld width. It expresses the coupling
between the weld dimension and the dynamic thermal process in electric arc welding. It is valid for both 3D and 2D solutions.
Info: [B37].
N. Christensen.
8.1.4
Chvorinov Cast Mould System Number Ch
1
1
τ
Rc Vc λf 2
Ch 5 2
5 FoPV P2λ
L lΔTcf Rf Vf λc
Vc
; Rc, Rf (kg m23) density of
L5
S
casting and mould; l (J kg21) specific latent heat of solidification; ΔT (K) superheating of melting; cf (J kg21 K21) specific heat capacity of a mould;
Vc (m3) casting volume; Vf (m3) mould volume; S (m2) surface area of casting; λf, λc (W m21 K21) thermal conductivity of mould and casting, respectively;
Fo (2) Fourier number (p. 175); PV (2) volume parameter; Pλ (2) thermal conductivity parameter.
τ (s) time; L (m) characteristic length
Technology and Mechanical Engineering
355
In the case of a contact task, it expresses the mutual dynamic coupling between
thermomechanical quantities, during the process of casting solidification in a mould
and the cooling time of the casting. Foundry technology.
Nikolai Chvorinov (15.11.19037.11.1987), Czech engineer of Russian origin.
He was an engineer, but a talented painter too, enabling
him to introduce abstraction into his foundry work, which
was only empirical before. With extensive experiments
researching the solidification and crystallization of steel
and diversely formed castings, he deduced that solidification time is proportional to the second power of the volume-to-surface ratio. His work formed the basis for
subsequent theoretical research on foundry processes.
8.1.5
Chvorinov Shrink Formation Number Chsh
grad T grad T 21
Chsh 5 pffiffiffiffiffiffi
pffiffiffiffiffiffi
wT
wT ref
grad T (K m21) temperature gradient in solidifying casting;WT (K s21) solidification rate of casting; subscript ref limiting value for shrink hole formation.
This criterion is based on Chvorinov’s idea about the influence of the simultaneous action of the geometric and time temperature gradients on a shrink hole in a
solidifying casting. The solution process of the casting solidification dynamics
starts by modeling the temperature field and from it the module and argument
fields of the geometric and time temperature gradients. From these, the distribution
field of the criterion for the shrink hole arising in a casting is determined. In the
casting, the zones having the value of Chsh , 1 point to the place of possible shrink
hole origin. For Chsh $ 1, no shrink hole appears. It is analogous to the Niyam
shrink formation number Φ (p. 358).
Nikolai Chvorinov (see above).
8.1.6
Chvorinov Time Number Ch, Chf
Ch 5
a1 τ
λ1 ΔTl τ
5
ll R1 L21
L21
where l1 5 lls 1 ΔTl c1 5
ð1Þ;
lls
1 c1 ΔTl 5 cekv ΔTl ;
ΔTl
356
Dimensionless Physical Quantities in Science and Engineering
Chf 5
a2 τ
λ2 τ
5
ðcRÞ2 L22
L22
ð2Þ
a1, a2 (m2 s21) thermal diffusivity; τ (s) time; L1, L2 (m) characteristic
length L1 5 VS1 ; L2 5 VS2 ; V1, V2 (m3) volume; S (m2) surface area of casting; λ1, λ2 (W m21 K21) thermal conductivity; ΔTl (K) superheating of melting; l1, lls (J kg21) specific latent heat of solidification; R1, R2 (kg m23) density; c1, c2 (J kg21 K21) specific heat capacity; cekv (J kg21 K21) equivalent
specific heat capacity; subscripts: 1 casting, 2 mould, ls liquidussolidus.
This number expresses the dimensionless solidification time of a casting (1) and
a mould (2). It is a modification of the Fourier number Fo (p. 175).
Nikolai Chvorinov
8.1.7
CrayaCurtet Number Ct
wk
Ct 5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
w2d 2 12w2k
Qm1 1 Qm2
;
where wk 5
RA
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
H1 1 12H2
wd 5
RA
wk (m s21) mean kinematic velocity; wd (m s21) mean dynamic velocity; Qm1,
Qm2 (kg s21) mass flux relative to burner and to induced air stream; R (kg m23) gas density; A (m2) cross section of furnace; H1, H2 (kg m s21) momentum relative to burner and to induced air stream.
It characterizes the radiation flow and the heat transfer in industrial furnaces by
means of a passing mass and the momentum of basic and induced currents. For
Ct-0, the axial flow has great momentum and recirculation. When Ct . 0.74, the
return flow disappears.
Info: [A23].
Antoine Joseph Edouard Craya (19111976), French engineer.
Roger Michel Curtet (born 1923), French engineer.
8.1.8
Heat Diffusivity B
B5
a
;
aref
b5
pffiffiffiffiffiffiffiffi
Rcλ
a (m2 s21) thermal diffusivity; aref (m2 s21) reference value of thermal diffusivity; R (kg m23) material density; c (J kg21 K21) specific heat capacity;
λ (W m21 K21) thermal conductivity.
Technology and Mechanical Engineering
357
Most frequently, it characterizes the diffusivity in a cast-mould system with the
relative value corresponding to the casting diffusivity and the value of b corresponding to the mould. Foundry work technology. Solidification and cooling of
castings.
Info: [B81].
8.1.9
Latent Heat NL
NL 5
lsl
cðTl 2 Tini Þ
NL 5
lls
cðTl 2 Ts Þ
ð1Þ;
ð2Þ
lsl (J kg21) specific latent heat of melting; lls (J kg21) specific latent heat of
solidification; c (J kg21 K21) specific heat capacity; Tl, Ts, Tini (K) temperature of liquidus and solidus, initial mould temperature.
It expresses the latent heat of melting (1) and solidification (2) of a casting in a
mould. Foundry work technology.
Info: [B81].
8.1.10 Loss Dissipation Coefficient η, Ndis
ΔU
η5
;
2πU
where U 5
ð σmax
dε
0
1 σ2
2E
U (J m23) elastic energy in material per volume unit; ΔU (J m23) difference
of dissipation energy between loading and unloading; σmax (Pa) maximal stress
at elastic loading; E (Pa) modulus of elasticity; ε (2) relative elongation
(p. 144).
It expresses the degree which is reached by a material due to vibration energy
dissipation. The cycle can be expressed in various ways, either as fast or slow.
Usually, the coefficient value depends on a timescale or a cycling frequency.
Info: [C65].
8.1.11 Melting Efficiency ηW
ηW 5
wAQV
Q
w (m s21) motion velocity; A (m2) weld surface;QV (J m23) heat per unit
volume; Q (W) heat flux.
358
Dimensionless Physical Quantities in Science and Engineering
It expresses the melting efficiency as a ratio of the power necessary to melt the
welding zone to the net power absorbed in a part. It represents an important aspect
of the welding process. The theoretical maximum value is 0.48 for the 2D heat
propagation process. For the 3D process, it is 0.37.
Info: [B37].
8.1.12 Niyam Shrink Formation Number Φ
Φ 5 rΘ Nη21
rΘ (2) temperature gradient (p. 361); Nη (2) solidification rate constant (p. 360).
It is an approximate criterion of the shrink hole originating in a casting. It is
based on the assumption that geometric and time gradients influence shrink hole
formation in a solidifying casting. It is analogous to the Chvorinov shrink formation
number Chsh (p. 355). Foundry work technology. Solidification of castings.
Info: [B81].
8.1.13 Part of Liquid Phase β
β5
T 2 Ts
Tl 2 Ts
β50
for Ts , T , Tl
for T , Ts
where h 5 href
ðT
cp dT;
ð1Þ;
ð2Þ;
ΔH 5 βL
ð3Þ;
Tref
H 5 h 1 ΔH
where h 5 href
ð4Þ;
ðT
cp dT;
ΔH 5 βL
Tref
T (K) temperature of melting; Ts (K) solidus temperature; Tl (K) liquidus
temperature; H (J) enthalpy of material; h (J) enthalpy of heating up; href (J) reference enthalpy; Tref (K) reference temperature; cp (J kg 2 1 K 2 1) specific heat capacity; ΔH (J) addition of enthalpy; L (J) latent heat of
melting.
In equation (1), it expresses the liquid phase portion of material melting. The
enthalpy increment can be determined from the latent heat of the material and is in
the range of h0; Li for the solidus and liquidus. For the enthalpy of the material,
expression (4) holds.
Technology and Mechanical Engineering
359
8.1.14 Power of Smeltery N
N5
P
cηTd
ð1Þ;
N5
Pz
cηTd
ð2Þ
P (W) furnace power per electrode; c (J kg 2 1 K 2 1) specific heat capacity;
η (Pa s) dynamic viscosity; T (K) temperature; d (m) electrode diameter;
Pz (W) power dissipation per electrode.
It expresses the electrical-to-thermal-power ratio. In the dimensionless state, it
characterizes the electrical power (1) or the power dissipation (2) of an electric
induction furnace.
Info: [A23].
8.1.15 Pressure in Smeltery N
N5
p
g R β ΔT d
p (Pa) pressure in a molten material in smelting furnace; g (m s22) gravitational acceleration; R (kg m22) density; β (K21) volume thermal expansion
coefficient; ΔT (K) temperature difference; d (m) electrode diameter.
It expresses the relationship between the pressure in a molten material and the
pressure caused by the temperature non-uniformity in the molten material. It characterizes the pressure relations in a melting pan of an electric smelting furnace
with free convection of the molten material.
Info: [A23].
8.1.16 Rockwell Scale HR
HR A fHRA; HRB; HRC; ?g
It expresses the hardness of a material based on the depth of a body impression
in a material sample compared to that in a reference material. The HR hardness is a
dimensionless quantity which is characterized by its simplicity and wide application in mechanical engineering and most especially in metallurgy. Several
hardness scales are used, mostly the B and C scales, in which the B scale applies to
soft material (aluminium, bronze and mild steel). In the B scale, hard steel balls
and 100 kg loads are used in the measurement. The measured value is denoted as
HRB. With the C scale, a diamond cone and 150 kg loads are used. For very hard
steel, HRC A h55; 62i. For the steel for spades, axes and chisels, for example,
360
Dimensionless Physical Quantities in Science and Engineering
HRC A h40; 45i. The values below HRC 5 20 are considered unreliable, as are
those over HRC 5 100. Many other scales, including the HRA, have specific
applications.
Info: [C117].
8.1.17 Rosenthal Number Ro
Ro 5
Q
LλΔT
(W) heat flux; L (m) characteristic length, width of weld; λ (W m21 K21) thermal conductivity; ΔT (K) temperature difference.
Together with the Rykalin number Ry (p. 360), it is used for the 3D and 2D thermal process analyses of electric arc welding. It expresses the influence of a characteristic weld dimension in welding. It is analogous to the Pomerantsev heat number
Po (p. 181). Welding.
Info: [B37].
D. Rosenthal.
Q
8.1.18 Rykalin Number Ry
Ry 5
Qw
a2 Q V
(W) heat flux; w (m s21) motion velocity; a (m2 s21) thermal diffusivity;
23
QV (J m ) heat per unit volume.
This number expresses the dynamic thermal process in electric arc welding. It is
used to analyse the heat conduction in melting down. It can be used for 3D and 2D
thermal processes.
Info: [B37].
N. N. Rykalin, Russian engineer.
Q
8.1.19 Solidification Rate Constant Nη
η
Nη 5 pffiffiffi
2 a
η (m s21/2) solidification rate; a (m2 s21) thermal diffusivity.
It expresses the solidification rate of a casting in a mould. Together with the
temperature gradient rΘ (p. 361), it influences the shrink hole formation in a casting. Foundry work technology. Solidification of castings.
Info: [B81].
Technology and Mechanical Engineering
361
8.1.20 Solidification Shrinkage β, Nsh
β5
vl 2 vs
vs
vl, vs (m3 kg21) specific volume of liquidus and solidus.
It expresses the volume changes with different molten materials and casting
shrinkage during solidification. Foundry work technology. Solidification of
castings.
Info: [B81].
8.1.21 Specific Heat of Melting Nmelt
Nmelt 5
ΔTc ðT 2 Tmin Þc
5
lsl
lsl
T (K) temperature; c (J kg21 K21) specific heat capacity; lsl (J kg21) specific latent heat of melting.
It characterizes the specific heat which is necessary to melt the material in a
furnace.
Info: [A23].
8.1.22 Temperature Gradient rΘ
rΘ 5
pffiffiffiffiffi
aτ rTref
Tl 2 Tini
a (m2 s21) thermal diffusivity; τ (s) time; rTref (K m21) reference temperature gradient; Tl (K) liquidus temperature; Tini (K) initial mould temperature.
It expresses the temperature gradient in a solidifying casting during time
τ, related to the temperature difference between the molten material and a
mould. Together with the solidification rate constant Nη (p. 360), it is the main
quantity acting during shrink hole formation. Foundry work technology.
Solidification of castings.
Info: [B81].
8.1.23 Temperature Range of Solidification Θsl
Θsl 5
Tl 2 Ts
Tl 2 Tini
Tl, Ts, Tini (K) temperature of solidus and liquidus, initial mould temperature.
362
Dimensionless Physical Quantities in Science and Engineering
It expresses the temperature range during casting solidification in a mould.
Foundry work technology. Solidification of castings.
Info: [B64].
8.1.24 Tikhonov Number (1.) Ti
rffiffiffiffiffi
τn
Ti 5 w
D
w (m s21) reaction rate; τ n (s) surface saturation time; D (m2 s21) diffusivity.
This number expresses the ratio of the velocity of transferring an element from
the surroundings on the surface of an object to that of the element propagation in a
body surface layer. It characterizes the process of diffusion saturation of a rigid
body surface by various atoms from the gas or vapour phase, for example, in
cementation, oxidation or chrome plating of steel or cast iron parts. It is the criterion of the thermochemical treatment of materials.
Info: [A33].
Andrey Nikolayevich Tikhonov (30.10.19068.11.1993),
Russian mathematician and physicist.
He was engaged in research in topology, functional analysis and differential equations. He focused especially on
mathematical problems in geophysics, the physics of
plasma, gas dynamics and electrodynamics, and solved
many mathematical and physical problems in the area of
thermomechanics, for example, in heating, cooling and
solidification. He formulated and proved the Tikhonov theorem in the field of topology.
8.1.25 Tikhonov Number (2.) Ti
αchar pffiffiffi
τ ð1Þ;
b
αchar pffiffiffiffiffi αmed
Ti 5
τ1
b
αchar
ð τ1
where αmed 5
α dτ
Ti 5
ð2Þ;
0
22
21
α (W m K ) heat transfer coefficient at the meltmould interface; αchar
(W m22 K21) characteristic heat transfer coefficient; αmed (W m22 K21) mean
heat transfer coefficient from the initial time of mould heating-up to the time
3
τ 1; b ðkg s2 2 K 21 Þ mould capability of heat accumulation; τ (s) heating time
of monitored mould section; τ 1 (s) heating time.
Technology and Mechanical Engineering
363
It characterizes the mould heating time by considering the heat transfer on the
boundary between molten material and a mould, and the thermal accumulating
capabilities of the mould, whereas the Chvorinov cast mould system number Ch
(p. 354) solves the heat transfer between a casting and a mould as a contact problem (a fourth-type boundary condition).
Info: [B33].
Andrey Nikolayevich Tikhonov (see above).
8.1.26 Weber Agglomerate Number Weagl
Weagl 5
RDw2
Γ
Weagl 5
RDðw 2 w0 Þ2
Γ
ð1Þ;
ð2Þ
R (kg m23) particle density; D (m) particle diameter; w (m s21) particle
velocity; w0 (m s21) limiting velocity at which no agglomeration of particles
occurs; Γ (J m22) surface energy.
It expresses the ratio of the dynamic agglomeration energy to the surface energy.
Expression (2) is a modification of expression (1), a modification taking into consideration the limiting velocity at which no agglomeration occurs.
Ernst Heinrich Weber (p. 92).
Wilhelm Eduard Weber
8.1.27 Weld Size Parameter PW
PW 5
wA
La
w (m s21) motion velocity; A (m2) weld surface; L (m) characteristic weld
length; a (m2 s21) thermal diffusivity.
It is often called the dimensionless weld width. It characterizes the weld width in
considering 3D and 2D thermal processes.
Info: [B37].
8.1.28 Weld Width Nww
Nww 5
wA
La
364
Dimensionless Physical Quantities in Science and Engineering
w (m s21) motion velocity; A (m2) weld surface; L (m) characteristic length,
weld width; a (m2 s21) thermal diffusivity.
It expresses the weld width in 2D analysis of an electric arc welding process.
Together with the Rykalin number Ry (p. 360) and the Rosenthal number Ro
(p. 360), it is among the fundamental welding criteria. Welding technology.
Info: [B37].
8.2
Mechanical Engineering
In mechanical engineering, many dimensionless quantities are applied in all aspects
of the design and construction of machines or parts thereof. This includes, for
example, the design and construction of water pumps, compressors, water and
steam turbines, fans and mixers [A29].
These quantities relate to rotating parts dynamics in moving blades, airscrews,
propellers and worm conveyers. Other dimensionless quantities relate to the design
of bearings, their loading and lubrication, dimensionless cryogenic cooling of
machines and devices, the observation of water drops in steam, including the flow
through part of the last stages of steam turbines, and the dimensioning of combustion chambers, rocket engines and other machines and devices. Of the applied
dimensionless quantities, the Harrison and Addison numbers are used, for example,
to design a water turbine or a rotary pump. Similar to the Harrison and Ocvirk
numbers for bearings, the Leroux and Thoma numbers can be used for the cavitation phenomena in water machines, and the Tomson number can be used for fuel
consumption depending on environmental resistance in aircraft and water
machines. Many of the numbers are modifications of applied dimensionless
quantities.
8.2.1
Addison Shape Number Ad
Ad 5 103 Kn ;
where Kn is specific velocity
1
Kn 5
nP2
1
5
for water turbines;
R2 ðghÞ4
1
Kn 5
nQ2
3
ðghÞ4
for rotary pumps
n (2) revolutions per time interval; P (W) power; R (kg m23) density;
g (m s22) gravitational acceleration; h (m) hydrostatic head of machine;
Q (m3 s21) volume flow.
Technology and Mechanical Engineering
365
This number is a multiple of the specific velocity. It is used with water
machines. It is an empirically obtained dimensionless number.
Info: [A29].
Herbert Addison (born 1889), English engineer.
8.2.2
J5
Advance Ratio J
w
nD
w (m s21) motion velocity; n (s21) revolutions per second; D (m) outer
diameter of screw or propeller, impeller.
It characterizes the relation between the velocity and the geometric parameters
of an airscrew or a ship propeller. With the ultrafiltration of mixing chambers,
it serves to predict the permeability and clogging degree thereof. It enables
dimensioning of a drive system.
Info: [A29].
8.2.3
Bearing Damping Coefficient Ndam, D0
Ndam 5
DωR
FηðTÞ
D (m kg2 s22) general damping coefficient; ω (s21) angular frequency; R (m) radius of bearing journal; F (N) bearing loading; η (Pa s) dynamic viscosity
T (K) temperature.
This coefficient expresses the plain bearing damping capabilities. Mechanical
engineering. Tribology. Bearings.
8.2.4
Bearing Modulus Nmod
Nmod 5
ηn
p
η (Pa s) dynamic viscosity; n (s21) revolutions per second; p (Pa) bearing
pressure.
It is an important bearing characteristic, according to which three bands of the
bearing friction coefficients can be distinguished (thin film friction, transient
boundary and hydrodynamic friction). Mechanical engineering. Tribology. Plain
bearings.
Info: [C77].
366
Dimensionless Physical Quantities in Science and Engineering
8.2.5
Bearing Number Nbea, S0
Nbea 5
F
ηðTÞRωL
F (N) loading force; η (Pa s) dynamic viscosity; T (K) temperature; R (m) radius of bearing journal; ω (s21) angular frequency; L (m) characteristic length
of bearing.
It expresses the plain bearing load size. It is a modification of the Sommerfeld
number Sm (p. 385).
8.2.6
Bearing Stiffness Nst
Nst 5
kst
EðTÞtp
kst (N m21) stiffness; E(T ) (Pa) modulus of elasticity; tp (m) thickness of
lubricating film.
It expresses the stiffness of various kinds of fluid and gas plain bearings.
Mechanical engineering. Plain bearings.
8.2.7
Betz Number Be
Be 5
ðωrÞ2 L
w2 r
ω (s21) angular frequency; r (m) radius; L (m) characteristic length, chord
of profile; ω (m s21) flow velocity.
It expresses the flow in rotating channels of flow machines.
Info: [A24].
8.2.8
Cavitation Number Ncav , σ
Ncav 5
p 2 pv
Rw2
Ncav Th 5
ð1Þ;
ha 2 hs 2 hv
h
ð2Þ
Technology and Mechanical Engineering
367
p (Pa) total static pressure; pv (Pa) partial vapour pressure; R (kg m23) density; w (m s21) mean flow velocity; h, ha, hs, hv (m) total, atmospheric, sucking and water vapour pressure heads; Th (2) Thoma number (1.) (p. 90).
It characterizes the formation of cavitation in a flowing fluid. Above all, it
depends on the flow rate, the kind and purity of the fluid, and the inlet flow turbulence. In equation (1), it expresses the ratio of the static pressure drop, between the
fluid and vapour, to the fluid dynamic pressure. It characterizes the thermohydrodynamic process of the cavitation origin in flow through parts of water machines. It is
a modification of the Euler number (1.) Eu (p. 61). In equation (2), it is called the
Thoma number (1.) Th (p. 90), and it expresses the cavity relations in water pumps.
See the Leroux number Lx (p. 374). Hydrodynamics. Hydraulics. Water machines.
Info: [A7],[A26].
8.2.9
Critical Number of Brake Disc Ndisc, crit
Ndisc; crit 5
Pe 5
η20 ws u
R2 σ2max
uΔx
.2
a
ð1Þ;
ð2Þ
η0 (Pa s) dynamic viscosity of oil at the temperature at input to contact area;
ws (m s21) shear speed; u (m s21) surface speed; R (m) disc radius; σmax (Pa) maximum Hertz stress under different loading, not leading to seizing; Δx (m) grid
space; a (m2 s21) thermal diffusivity.
The number size expresses the brake disc critical temperature and the possibility
of it subsequently seizing up. Usually, a high disc velocity and arising superficial
thermal instabilities cause brake disc oscillations. The Pe´clet heat number Pe
(p. 180), expressing heat transfer by convection and conduction, is the characteristic dimensionless number for formation of motion thermal field. In the case of disc
brakes, the value of this number is Pe 10.5 With the numerical solution using the
finite elements method (FEM) application, expression (2) is valid for the selection
of the grid space.
Info: [B16].
8.2.10 Cryogenic Cooling of Superconductor Nsupra
Nsupra 5
λqV1 O
21
5 Bi 21 Poiso Posupra
αqV1 A
λ (W m21 K21) thermal conductivity of isolation; qV1, qV2 (W m23) volume
density of heat flux in isolation and superconductor; O (m) helium cooled
368
Dimensionless Physical Quantities in Science and Engineering
perimeter; α (W m22 K21) effective heat transfer coefficient; A (m2) cross section area of supraconductor; Bi (2) Biot number (p. 173); Poiso (2) Pomerantsev number for isolation; Posupra (2) Pomerantsev number for superconductor, see Pomerantsev heat number Po (p. 181).
It characterizes the affect of insulation on superconductor cooling. Cryogenic
cooling.
Info: [B118].
8.2.11 Damping Coefficient Ndb
Ndb 5
hωkd
pa bL
h (m) bearing clearance; ω (s21) angular frequency of bearing journal;
kd (kg m21 s21) damping coefficient of bearing; pa (Pa) ambient pressure;
b (m) bearing width; L (m) bearing length.
It characterizes the plain bearing damping. It depends on the geometric properties, revolving speed and damping properties of the bearing. Mechanical engineering. Aerostatic bearings.
8.2.12 Delivery Number ND
ND 5
Q
Au
(m3 s21) volume flow; A (m2) impeller surface A 5 14πd 2 ; u (m s21) peripheral speed.
It characterizes the volume throughflow in rotary machines. It is used in
throughflow blade machines.
Info: [A35].
Q
8.2.13 Diameter Group ND
1
1 1
1 2 2
πD
ND 5
ð2ghÞ4 w2
4
D (m) diameter of screw propeller, impeller; g (m s22) gravitational acceleration; h (m) hydrostatic head; w (m s21) flow velocity.
It serves to dimension water and turbojet machines.
Technology and Mechanical Engineering
369
8.2.14 Discharge Number NΦ
See the flow coefficient φ (p. 370).
8.2.15 Droplets Formation in Jet N
N5
ηl rmin
pffiffiffiffiffiffiffi
Rl L2 rT1
N5
λrmin
pffiffiffiffiffiffiffi
cl Rl L2 rT1
N5
σRl L
η2l
N5
L
rmin
N5 5
cl
r
ð1Þ;
ð2Þ;
ð3Þ;
ð4Þ;
ð5Þ;
rT1 Rl
ð6Þ;
p1
pffiffiffiffiffiffiffi
LR rT1
N5 l
ð7Þ
ηl
N5
ηl (Pa s) dynamic viscosity; rmin (m) radius of minimal jet cross section;
Rl (kg m23) drop density; L (m) characteristic length, initial drop diameter;
r (J kg21 K21) specific gas constant; T1 (K) temperature in input cross section; p1
(Pa) pressure in input cross section; λ (W m21 K21) thermal conductivity; cl
(J kg21 K21) drop specific heat capacity; σ (N m21) surface tension.
The set of criteria (1)(7) describes gas flow influence on the laws of droplets
crumbling in nozzles without considering evaporation and condensation. The fractional composition of droplets is presumed according to the law of normal logarithmic distribution. Temperature dependence of physical properties is neglected.
Info: [B119]
8.2.16 Eccentricity of the Bearing ε
ε5
e
c
e (m) eccentricity (distance between centre of journal and bearing); c (m) interspace between journal and bearing.
370
Dimensionless Physical Quantities in Science and Engineering
It expresses the bearing journal deviation which occurs in the hydrodynamic
mode. Sliding bearings, lubrication.
Info: [B31].
8.2.17 Ekman Number (2.) Ek
Ek 5
w
ωL2
w (m s21) motion velocity; ω (s21) angular frequency; L (m) characteristic
length.
It characterizes, for example, the inertia to centrifugal forces ratio during
rotation of a vessel with the fluid in motion. It appears in cases of heat and mass
transfer solutions in rotary systems.
Info: [A5].
8.2.18 Elastic Component of Indentation Energy ηIT
ηIT 5
Welast
100;
Wcelk
where Wcelk 5 Welast 1 Wplast
Welast, Wplast, Wcelk (J) elastic, plastic and total indentation energy, respectively.
It characterizes the hardness measurement. It expresses the portion of the elastic
energy in measurement of macro-, micro-, and nanohardness by means of diverse
kinds of indenters. The elastic energy component is given with the test conditions,
of which the first is the loading force in newtons (N) and the second is the loading
time in seconds (s), for example, ηIT (0, 5/10) 5 36, 5%.
Info: [B111].
8.2.19 Fineness Coefficient, Waterplane Coefficient Ψ
1
Ψ 5 LW 3
L (m) characteristic length; W (m3) volume draft.
It is applied in ship modelling and design.
8.2.20 Flow Coefficient φ
φ5
Q
nD3
ð1Þ;
Technology and Mechanical Engineering
φ5
φ5
φ5
Q
1
2
2πD u2
ð2Þ;
Q
1
2
2
4πð1 2 Dr ÞD u2
v
u2
371
ð3Þ;
ð4Þ
Q (m3 s21) volume flow; n (s21) rotational frequency; D (m) diameter of
flow part; u2 (m s21) peripheral flow velocity; Dr (2) ratio of inner to outer
diameter; v (m s21) mean flow velocity.
It expresses the throughflow through turbojet rotary machines. In equation (1), it
is often called the outlet number. Together with modifications (2) through (4), it is
used in the hydrodynamics and aerodynamics of rotary machines, fans and blowers.
Info: [B20].
8.2.21 Froude Number (3.) Rotation Fr3
Fr3 5
Dn2
g
D (m) diameter of impeller; n (s21) revolutions; g (m s22) gravitational
acceleration.
It expresses the influence of rotation on the flow through a pump running wheel.
Hydrodynamics. Water machines. Mixing.
Info: [A29].
William Froude (p. 63)
8.2.22 Geometric Parameter of Water Jet A
A5
Rz r d
Nr12
Rz (m) curvature radius of liquid flow in sprayer chamber; rd (m) radius of
jet; N (2) number of canals; r1 (m) radius of input canals.
It characterizes the geometric parameter of a fluid sprayer. It is applied in the
heat transfer with condensation of the sprayed fluid.
Info: [B39].
8.2.23 Gravitational to Centrifugal Acceleration Ratio Ngc
Ngc 5
q
ω2 R
372
Dimensionless Physical Quantities in Science and Engineering
g (m s22) gravitational acceleration; ω (s21) angular frequency; R (m) radius.
It characterizes the dynamic force relations, for example, in rotary systems.
Info: [A5].
8.2.24 Head Coefficient CH, Ψ
CH 5
H
n2 d 2
H (J kg21) specific pressure energy; n (s21) rotational frequency; d (m) impeller diameter of pump, ventilator, etc.
It characterizes the flow in flow machines. It serves to dimension rotary pumps
and ventilators.
Info: [A4],[A29].
8.2.25 Head Coefficient, Loading Coefficient, Energy Transfer
Coefficient Ψ
Ψ5
ΔP
Rn2 D2
ð1Þ;
Ψ5
ΔP
Rω3 D5
ð2Þ;
Ψ5
gh
n2 D 2
Ψ5
gh
u2
ð3Þ;
ð4Þ
ΔP (Pa) pressure difference; R (kg m23) density; n (s21) rotational frequency; D (m) diameter of flow part; ω (s21) angular frequency; g (m s22) gravitational acceleration; h (m) hydrostatic head; u (m s21) peripheral speed.
It is used in rotary machine dynamics. Expressions (1) and (2) are used to design
fans and rotary blowers. Expressions (3) and (4) are applied in the designing of
water pumps and turbines.
Info: [A29].
8.2.26 Homochronicity Number Ho
Ho 5 nτ
n (s21) revolutions; τ (s) mixing time.
Technology and Mechanical Engineering
373
It is used for time comparison of mixing, whisking and other processes in time.
Info: [A29].
8.2.27 Homochronous Number Ho
Ho 5
wτ
L
w (m s21) motion velocity; τ (s) transfer time to the distance L; L (m) distance.
It serves to choose the timescale for similar processes.
Info: [A29].
8.2.28 Kennedy Number Ω
Ω5
τ da
pf Rb
Tda (s) deaerating time; pf (m3 kg21 s21) constant of material permeability
(specific volume per second); Rb (kg m23) material density.
It expresses the ratio of the deaerating time to the delivery time of the unit material amount in pneumatic transport. Mechanical engineering. Pneumatic transport.
8.2.29 Lagrange Number (2.) Lg2
Lg2 5
P
5 Rerot Np ;
ηL3 n2
where
Np 5
P
L5 Rn3
P (W) power input of impeller; η (Pa s) dynamic viscosity; L (m) characteristic length; n (Hz) revolutions per second; R (kg m23) density; Rerot (2) Reynolds number (p. 81); Np (2) power input criterion.
It characterizes the energy loss from mixing viscous fluids in vessels, depending
on the physical properties of the fluid and the dimensions and speed of the mixer.
Info: [A14],[A24].
Joseph-Louis Lagrange (p. 70).
8.2.30 Lautrec Number Lau
Lau 5
L
δs
ð1Þ;
374
Dimensionless Physical Quantities in Science and Engineering
Lau 5
R
δk
ð2Þ
L (m) characteristic length of the wall; R (m) characteristic length (gap thickness) between two walls; δs (m) penetration depth in solid wall; δk (m) penetration depth in liquid.
It characterizes the thermoacoustic phenomenon of the thermal and acoustic
energy transformations in a wall (1) or in a gap between two walls (2) in thermoacoustic devices such as thermal pumps and cooling devices. It appears together
with the Mach number M (p. 73) and the Prandtl number Pr (p. 197), especially in
observing the interaction between porous surroundings and a sound field in pipes
and pipelines.
Info: [C128].
8.2.31 Leroux Number Lx, σc
Lx 5
p 2 pv
1
2
2Rl w
p (Pa) local static pressure; pv (Pa) vapour pressure; Rl (kg m23) liquid density; w (m s21) velocity.
This number expresses the ratio of the difference between the local static pressure head and the vapour head to the velocity head. In water machines it characterizes the cavitation phenomenon with water flow through. See the cavitation
number Ncav (p. 366).
Info: [A35].
8.2.32 Load Parameter of Journal Bearing NB
NB 5
ηur 2
FL2
η (Pa s) dynamic viscosity; u (m s21) peripheral speed; r (m) radius; F (N) force; L (m) characteristic length.
It is another variant of the Sommerfeld number Sm (p. 385). Mechanical engineering. Tribology.
8.2.33 Loading Number of Bearing NM, M
α2
NM 5 q
ηus R2
1
3
Technology and Mechanical Engineering
375
q (N m21) length loading; α (m2 N21) coefficient of piezoviscosity; η (Pa s) input dynamic viscosity; us (m s21) sum of tangential surface velocities: R (m) effective radius curvature.
It expresses the ratio of the plain bearing load to surface friction forces.
Elastohydrodynamic lubrication. Bearings.
Info: [B77].
8.2.34 Loading Number of Sliding bearing NB
NB 5
ηuR2
5 2Gu
Fh2
η (Pa s) dynamic viscosity; u (m s21) peripheral speed; R (m) radius of
bearing journal; F (N m21) length force loading of bearing; h (m) thickness of
round gap; Gu (2) Gűmbel number (1.) (p. 164).
It expresses the ratio of the dynamic friction force to the loading force of a bearing. It is another variant of the Sommerfeld number Sm (p. 385). Tribology.
Lubrication. Bearings.
8.2.35 Lock Number (1.) Lk
Lk 5
RR4 la
I
R (kg m23) liquid density; R (m) rotor radius; l (m) blade chord;
a (m2 kg21) steepness of lifetime curve of rotor; I (m4) moment of inertia of
blade at hangings.
It expresses the dynamics of rotor blades. It is an empirical dimensionless
number.
Info: [A35].
Christopher Noel Hunter Lock (18941949), English mathematician.
8.2.36 Lock Number (2.) Lk
Lk 5
dCL RlR4
dα I
dCL
(2) angular coefficient of rotor lifting curve; CL, (2) lifting coefficient
dα
of rotor; α (rad) angle of pitch of copter rotor; R (kg m23) liquid density;
l (m) blade chord; R (m) rotor radius; I (m2 kg) inertia moment of rotary
rotor blades.
376
Dimensionless Physical Quantities in Science and Engineering
It expresses the buoyancy to inertia forces ratio in helicopters.
Info: [29].
Christopher Noel Hunter Lock (see above).
8.2.37 Mach Blowing Number Mb
Mb 5
wb
a
Wb (m s21) inject velocity at surface of porous boundary; a (m s21) mean stagnation sound speed.
It is the basic criterion of combustion chambers. The effect of fuel viscous dissipation is important with a low Mb value occurring with small fuel injection, high
frequency and kinematic viscosity, which is possible in hot gases. With a very
small Mach number (p. 73) corresponding to slight fuel injection, the solution
approaches the injectionless Stokes solution. Jet engines. Combustion chambers.
Info: [B119].
Ernst Mach (p. 73).
8.2.38 Mach Propeller Blade Tip Number Ma
Ma 5
πnD
a
n (s21) revolutions per second; D (m) diameter; a (m s21) sound speed.
It characterizes the velocity of a propeller blade tip or of a pump propeller running wheel, as a couple of examples.
Ernst Mach (see above).
8.2.39 Mechanical Load Group NM
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi
F
NM 5 ασHZ 5 αE
2πLER
α (m2 N21) pressure coefficient of viscosity; σHZ (Pa) maximal dry (Hertzian)
stress; E (Pa) reduced modulus of elasticity; F (N) force loading of bearing;
L (m) characteristic length (width of bearing); R (m) reduced radius of
curvature.
It expresses the mechanical contact load influence on the oil film thickness and
pressure in a bearing. Electrohydrodynamic lubrication. Tribology. Lubrication.
Plain bearings.
Info: [B117].
Technology and Mechanical Engineering
377
8.2.40 Mishken Number Mi
Mi 5
Def L
q
Def (m2 s21) effective diffusivity; L (m) characteristic length (endless screw
diameter); q (m3 s21) volume flow.
It describes the mixing process in worm conveyors from the kinematic point of
view.
8.2.41 Modified Power Number NPM
D
Δs0:5
NPM 5 Np
L ðnb ns Þ0:67
D (m) effective diameter of blade wheel; L (m) effective wheel length;
Δs (2) wall closeness factor; nb, ns (2) blade number of impeller or stator;
Np (2) power coefficient, power number (p. 379).
It is used to recalculate the power output of rotary mixers and pumps after
changing the rotating wheel parameters.
8.2.42 Muramtsev Number Mo
Mo 5
Ra λb ccp
ηd
R (kg m23) fluid density; λ (W m21 K21) thermal conductivity; cp
(J kg21 K21) specific heat capacity; η (Pa s) dynamic viscosity; a, b, c, d (2) dimensionless constants.
It is the criterion used to choose the heat transferring material with the heat
transfer properties in a limited range of working temperatures, specifically from the
point of view of heat transfer intensity under the equal velocity condition. It can be
used to design energy generating plants. It is an empirical dimensionless number.
Info: [A33].
8.2.43 Nusselt Modified Number Numod
Numod 5
αλcrit
;
λv
where λcrit 5 2π
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
σ
gðRl 2 Rv Þ
α (W m22 K21) heat transfer coefficient; λcrit (m) critical wavelength;
λv (W m21 K21) vapour thermal conductivity; σ (N m21) surface tension;
378
Dimensionless Physical Quantities in Science and Engineering
g (m s22) gravitational acceleration; Rl, Rv (kg m23) density of liquid and
vapour, respectively.
It expresses the ratio of the load due to convection heat transition on a corresponding transfer surface to the nucleon film boiling, to the thermal load by conduction in vapour. Cryogenic cooling of machines and plants.
Ernst Kraft Wilhelm Nusselt (p. 196).
8.2.44 Ocvirk Number Oc
Oc 5
q Δs 2r 2
ηw r L
q (kg s22) length loading; η (Pa s) dynamic viscosity; w (m s21) surface
speed of bearing journal; Δs (m) bearing clearance; r (m) bearing radius;
L (m) bearing length.
This number expresses the ratio of the bearing load to the friction force.
Tribology. See also Hersey number He (p. 165) and Sommerfeld number Sm
(p. 385).
Info: [A29].
Frederick A. Ocvirk (19131967), American engineer.
8.2.45 Orifice Design Parameter Nλ, λ
p
ffiffiffi
3
2cd D2 ηL
Nλ 5 pffiffiffiffiffiffi 30
Rpe D
cd (2) orifice outflow coefficient; D0 (m) nozzle diameter; η (Pa s) dynamic viscosity; L (m) characteristic bearing length; R (kg m23) lubricant
density p (Pa) pressure; e (m) eccentricity clearing; D (m) bearing journal
diameter.
Mechanical engineering. Tribology. Bearings.
8.2.46 Péclet Brake Number Pebrake
Pebrake 5
uL
. 2;
a
where u 5 ωr
u (m s21) sliding speed; L (m) characteristic length; a (m2 s21) thermal diffusivity; ω (s21) angular speed; r (m) radius of disc.
It characterizes the convectively diffusive heat transfer in disc brakes from the
point of view of the finite elements method solution. Incorrect size of the brake
Technology and Mechanical Engineering
379
disc causes often inadmissible oscillations and instability. This is one o- the main
problems of brakes, in which the typical size of the number Pebrake is approximately 105.
Info: [B123].
Jean Claude Eugène Péclet (p. 180).
8.2.47 Penetration Number Npen, Sp
Npen 5
w3z
5 Mb3 Nω22 Re 21
ω2 νL
wz (m s21) inject velocity at surface of porous boundary; ω (s21) angular frequency; v (m2 s21) kinematic viscosity; L (m) characteristic length, radius
Mb (2) Mach blowing number (p. 376); Nω (2) wave number, angular speed
(p. 389); Re (2) Reynolds number based on sound speed (p. 383).
This number governs the depth and structure of the unsteady boundary layer,
e.g. in jet engines, particularly the shape and dimension of the outer casing Jet
engines. Combustion chambers.
Info: [B71].
8.2.48 Petrov Number Pt
Pt 5
Δp1 1 Δp2
Δp3
Δp1 (Pa) pressure difference on water part of boiler; Δp2 (Pa) pressure difference on economizer; Δp3 (Pa) pressure difference on vapourliquid part and
superheating part.
It characterizes the stability of the flow parameters of a heat transferring
medium which moves in a steam boiler heating system. For steam boilers, Pt . 1
is valid. Generally, it is Pt A h0.1; 10i.
Info: [A33].
P. A. Petrov.
8.2.49 Power Coefficient, Power Number NP
NP 5
M
Rn2 D5
ð1Þ;
NP 5
Mω
Rn3 D5
ð2Þ;
380
Dimensionless Physical Quantities in Science and Engineering
NP 5
P
Rn3 D5
ð3Þ;
where P 5 2πnM
M (m2 kg s21) force moment; R (kg m23) fluid density; n (s21) rotational
frequency; D (m) rotor diameter; ω (rad s21) angular speed; P (W) shaft
power.
This coefficient relates to the dynamic criteria for rotary machines. For example,
it represents the hydrodynamic resistance on the mixer blade against the inertia
force. It expresses the energy loss in mixer vessels. In the shapes (1) and (2), these
numbers express the ratio of the shaft moment to the inertia force. In the shape (3),
it is the ratio of the thrust force (resistance) to the inertia force. It characterizes the
power output of some rotary machines, such as pumps, turbines, fans, mixers and
other devices. It is among the widely used dimensionless numbers in mechanical
engineering. Mechanical engineering.
Info: [A29],[A20],[A35],[B20].
8.2.50 Power Pressure Number NP
NP 5
NP 5
Δp
RD2S n2
P
1
3
2RAw
ð1Þ;
ð2Þ;
Δp (Pa) pressure difference; R (kg m23) fluid density; Ds (m) impeller
diameter; n (s21) rotation of mixer, etc.; P (W) impeller power; A (m2) effective surface; w (m s21) wind speed.
This number expresses the ratio of the pressure gradient to the dynamic pressure. It serves to design and dimension rotary mixers (1) and wind energy
plants (2).
Info: [B19].
8.2.51 Pressure Number (2.) NP2
1 2 21
NP2 5 e u
2
e (m2 s22) specific energy; u (m s21) peripheral speed.
It describes the flow through turbines, pumps and other flow machines.
Info: [17].
Technology and Mechanical Engineering
381
8.2.52 Propeller Torque Coefficient Mt
Mt 5
MF
Rn2 D5
MF (N m) torque of propeller or impeller; R (kg m23) density; n (s21) revolutions per second; D (m) diameter of propeller or impeller.
It expresses the torque caused by the force acting on a rotating propeller or a
rotating wheel in rotary machines. It is not derived strictly from similarity theory
but from empirical experience.
Info: [A29].
8.2.53 Pulsation Number Pu
Pu 5
fdR
qm
f (s21) frequency of pulsation; d (m) equivalent pipe or channel diameter;
R (kg m23) fluid density; qm (kg m22 s21) mass flux density.
This number characterizes the pulsation transport of fluids.
8.2.54 Rayleigh Centrifugal Number Rac
Rac 5
ω2 TβΔCR3
5 Re2 ScRo
wD
ω (s21) angular frequency; T (K) temperature; β (K21) volume thermal coefficient of expansion; ΔC (2) difference of ion concentration; R (m) characteristic length, vessel diameter with fluid; w (m s21) travel speed; D (m2 s21) diffusivity; Re (2) Reynolds number (p. 81); Sc (2) Schmidt number (p. 263);
Ro (2) Rossby number (p. 406).
It expresses the influence of temperature, mass diffusion and motion in rotating
systems, such as volume conveyors and containers, as some examples. Mechanical
engineering.
Lord Rayleigh (p. 187).
8.2.55 Rayleigh Modified Number Ramod
Ramod 5
gλ3cr Rl 2 Rv
ν v av R v
ð1Þ;
382
Dimensionless Physical Quantities in Science and Engineering
Ramod 5 PrGrmod
where λcrit 5 2π
ð2Þ;
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
σ
gðRl 2 Rv Þ
g (m s22) gravitational acceleration; λcrit (m) critical wavelength; ν v (m2 s21) vapour kinematic viscosity; av (m2 s21) vapour thermal diffusivity; Rl, Rv (kg m23) liquid and vapour density; σ (N m21) surface tension; Pr (2) Prandtl number
(p. 197); Grmod (2) Grashof modified number (p. 186).
In expression (1), it is a modification of the Rayleigh number (2.) (heat instability)
(p. 187). It is used in thermal calculations of cryogenic cooling systems such as cryogenic turboalternators. Expression (2) is a general equation expressing the natural
convective heat transfer from a heated or cooled wall.
Info: [B60].
Lord Rayleigh (see above).
8.2.56 Reynolds Acoustic Number Rea
Rea 5
ωL2
5 Nω Re
ν
ω (s21) angular frequency; L (m) characteristic length, radius; ν (m2 s21) kinematic viscosity; Nω (2) wave number, angular speed (p. 389); Re (2) Reynolds number based on sound speed (p. 383).
In comparison to the Reynolds number based on sound speed (p. 383), it
involves the influence of the angular velocity of combustion products. Rocket
machines. Combustion chambers.
Info: [B71].
Osborne Reynolds (p. 82).
8.2.57 Reynolds Mixing Number Remix
Remix 5
D2 nRm
ηm
D (m) propeller diameter; n (s21) propeller rotational frequency; Rm (kg m23) mixture density; ηm (Pa s) mixture dynamic viscosity.
It characterizes the ratio of dynamic to viscous forces which act on an airscrew
with mixing. It is used to design and dimension rotary mixers.
Info: [C3].
Osborne Reynolds (see above).
Technology and Mechanical Engineering
383
8.2.58 Reynolds Number Based on Sound Speed Re
Re 5
aL
ν
a (m s21) mean stagnation sound speed; L (m) characteristic length, radius;
ν (m2 s21) kinematic viscosity.
It expresses the ratio of the mean dynamic force of acting combustion products
to the viscous force. Rocket engines. Combustion chambers.
Info: [B71].
Osborne Reynolds (see above).
8.2.59 Reynolds Oscillatory Number Re
Re 5
2πDfA
ν
D (m) column diameter of oscillatory reactor; f (s 21) oscillatory frequency;
A (m) oscillatory amplitude; ν (m2 s21) kinematic viscosity.
It characterizes the intensity of mixing and flow through a circulation reactor.
The fluid (e.g. denatured protein) is delivered into the mixing chamber where it is
vibrated with an oscillating piston. Mixing of fluids. Physical chemistry. Chemical
mechanical engineering.
Info: [C130].
Osborne Reynolds (see above).
8.2.60 Reynolds Rotation Number Rerot
Rerot 5
L2 nR
η
Rerot 5
ωr 2
ν
ð1Þ;
ð2Þ
L D (m) characteristic length (diameter); n (s21) rotational frequency;
R (kg m23) fluid density; η (Pa s) dynamic viscosity; ω (s21) angular frequency; r (m) characteristic radius; ν (m2 s21) kinematic viscosity.
This number characterizes the rotating viscous fluid flow. It expresses the
dynamic similarity of turbojet rotary machines (e.g. fluid mixers).
Info: [A29],[B17].
Osborne Reynolds (see above).
384
Dimensionless Physical Quantities in Science and Engineering
8.2.61 Richardson Gradient Number of Wind Turbine Rig
g ΔT Δw 22
Rig 5
Tmed Δz Δz
g (m s22) gravitational acceleration; Tmed (K) mean air temperature; w (m s21) wind speed; z (m) height above ground.
This number expresses the ratio of the generated turbulence, with vertical temperature (density) difference or buoyancy, to the wind velocity change according to
height. A negative value of Rig corresponds to an unstable condition of the flow in
a layer between the surface and the turbine rotor height. On the contrary, its positive value corresponds to a stable condition. The zero value of Rig represents a
neutral condition under which the buoyancy does not have any influence and the
turbulence is generated only by wind action. Mechanical engineering. Wind turbines. Aerodynamics.
Lewis Fry Richardson (p. 403).
8.2.62 Size Number NSN
1
NSN 5
DðghÞ4
1
Q2
D (m) diameter of reactor or propeller; g (m s22) gravitational acceleration;
H (m) pressure head difference across turbine; Q (m3 s21) volume flow.
It is also called the specific diameter. It serves to dimension water turbines and
pumps. It is an empirical dimensionless number.
Info: [A29].
8.2.63 Slosh Time Nτ
Nτ 5
rffiffiffiffiffiffiffiffi
σ
τ
RR3
σ (N m21) surface tension; R (kg m23) density; R (m) tank radius; τ (s) time.
It expresses the time of non-linear free oscillations of the interface of immiscible
fluids in containers under zero friction and zero gravity conditions. Tanking of fluid
in a low gravitation field. Astronautics. Space mechanical engineering.
Info: [B8].
Technology and Mechanical Engineering
385
8.2.64 Sommerfeld Number Sm
ηn D 2
ð1Þ;
p h
ηn R 2
ð2Þ;
Sm 5
p h
F h 2
Sm 5
ð3Þ;
ηu R
F h 2
ð4Þ
Sm 5
2ηn R
Sm 5
η (Pa s) dynamic viscosity; n (s21) rotational frequency; p (Pa) mean bearing pressure; D (m) shaft diameter; h (m) gap thickness; R (m) shaft radius;
F (N m21) length loading of bearing; u (m s21) peripheral speed of shaft.
In equations (1) and (2), this number expresses the viscosity to loading forces
ratio. In equations (3) and (4), it represents the inverse value. It is analogous to the
Gűmbel number (1.) Gu (p. 164), and in equation (4), it is identical to it. It characterizes the relation of forces in a bearing with lubrication.
For small Sommerfeld numbers ðSm{1Þ; the lubrication pressure represents a
small part of the increase in the loading capability of a plain bearing. Two surface
sides are contacted by means of the roughness. For larger numbers Smc1; the
lubrication pressure wholly supports the bearing load and the friction strain corresponds to the viscous tension in the wholly developed lubrication mode. For
Sm1, a mixed mode exists, with which the friction is influenced both by the viscous resistance and by the roughness of contacts.
Info: [A29],[B20],[C77].
Arnold Johannes Wilhelm Sommerfeld (5.12.1868
26.4.1951), German physicist.
He was engaged in research in atomic physics, electrodynamics, mechanics and partial differential equations. He
modified the Niels Bohr atomic theory by considering
the elliptical path of electrons. Similarly, he also modified
the special relativity theory by formulating the Sommerfeld
model for energy in which the energetic levels are distributed into numerous parts. He was the first to come up with
an exact solution to the diffraction problem by making use
of the Maxwell equations.
386
Dimensionless Physical Quantities in Science and Engineering
8.2.65 Speed of the Bearing Ns
Ns 5
6ηω
pa R2 e 22
η (Pa s) dynamic viscosity; ω (s21) angular frequency; pa (Pa) ambient pressure; R (m) radius of bearing journal; e (m) bearing gap.
It characterizes the lubrication rate of a plain bearing. Tribology. Bearings.
8.2.66 Speed Specific Number N3
1
N3 5
1
nQ2
3
ðghS Þ4
2
3
5 ND2 NP24
n (s21) rotational frequency; Q (m3 s21) volume flow; g (m s22) gravitational acceleration; hS (m) hydrostatic head; ND (2) delivery number
(p. 368); NP2 (2) pressure number (2.) (p. 380).
It expresses the dimensionless velocity in rotary flow machines, such as pumps,
water turbines and compressors.
Info: [A29],[A35],[B17].
8.2.67 Squeeze Number NS
NS 5
12ηL2 ω
pa h 2
η (Pa s) dynamic viscosity; L (m) characteristic length (length of plate);
ω (s21) oscillation frequency; pa (Pa) ambient pressure; h (m) gap height.
It characterizes the squeezing of a thin isothermic film between two flat plates
of which the bottom one is steady and the top one is oscillating. Among other
things, it is used to damp dynamic systems and microsystems.
Info: [B20].
8.2.68 Stability of Cryogenic Cooling Nstab
qmax
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Nstab 5 pffiffiffiffiffip
L Rv 4 σgðRl 2 Rv Þ
qmax (W m22) maximal surface heat flux; L (J kg21) specific latent heat of
evaporation; Rv, Rl (kg m23) vapour and liquid density, respectively; σ (N m21) surface tension; g (m s22) gravitational acceleration.
Technology and Mechanical Engineering
387
It expresses the ratio of the maximum planar thermal surface load, corresponding to the transfer with nucleon film boiling, to the load caused by the surface
strain force and the force of gravity. It characterizes the stability of the cryogenic
cooling process in machines and plants.
8.2.69 Storage Energy Number Nen
Nen 5
Ef
p
E (J) accessible energy; P (W) nominal actual output; f (s21) synchronization frequency.
It expresses the number of basic cycles which should be chosen for accessible
energy to reach the power corresponding to the nominal energy. When
P 5 100 MW, f 5 50 Hz, E 5 20 MW, the number of cycles is Nen 5 10.
Info: [C47].
8.2.70 Strouhal Boundary Layer Number St
St 5
ωL
5 Nω M21
b
wb
ω (s21) angular frequency; L (m) characteristic length, radius, ratio of volume
to half porous part; wb (m s21) inject velocity at surface of porous boundary;
Nω (2) wave number, angular speed (p. 389); Mb (2) Mach blowing number
(p. 376).
It is a factor in the wavelength control of the unsteady swirl velocity of the
oscillatory flow in a cylindrical combustion space with side injection. It provides
precise observation of the amplitude and phase between non-stationary velocity
and pressure. Rocket engines.
Info: [B71].
Vincenc Strouhal (p. 87).
8.2.71 Structural Utilization Number N
N5
gRL
E
g (m s22) gravitational acceleration; R (kg m23) density; L (m) characteristic length; E (Pa) modulus of elasticity.
It expresses the ratio of the weight of a structural part to its rigidity.
388
Dimensionless Physical Quantities in Science and Engineering
8.2.72 Subcooling Number (1.) Nsubc
Nsubc 5
cp ðTs 2 Tl Þ
l
cp (J kg21 K21) specific heat capacity of liquid; TS (K) temperature of saturation; Tl (K) temperature of liquid; l (J K21) specific latent heat of evaporation.
It expresses the heat transfer in subcooled liquid cryogenic materials such as
helium, hydrogen and nitrogen. Cryogenic cooling of machines and plants.
8.2.73 Thoma Number (2.) Tm
Tm 5
pH 2 pV
p2 2 p1
pH (Pa) local total hydrostatic pressure; pV (Pa) vapour pressure under given
temperature; p2 (Pa) absolute input pressure; p1 (Pa) absolute output pressure.
It expresses the ratio of the pressure gradient in cavitation to the pressure
increase in a pump. It characterizes the cavitation in water pumps.
Info: [A29].
Dieter Thoma (18811942), German hydraulic engineer.
8.2.74 Thrust Coefficient CDP
CDP 5
FT
Rn2 D4
FT (N) drag force of propeller or impeller; R (kg m23) fluid density; n (s21) rotational frequency; D (m) propeller diameter.
It characterizes the ratio of thrust to dynamic forces which act on a propeller.
Info: [A29].
8.2.75 Tomson Number To
To 5
gQ
Rw3 L
g (m s22) gravitational acceleration; Q (kg s21) fuel usage; R (kg m23) environment density; w (m s21) motion velocity; L (m) characteristic length.
It expresses the ratio of fuel time unit consumption to environmental resistance.
Flying or floating machines. Aeronautics. Ships.
Info: [B20].
Technology and Mechanical Engineering
389
8.2.76 Torque Coefficient N
N5
Mk
Rv2 D3
ð1Þ;
N5
Mk
Rn2 D5
ð2Þ
Mk (N m) torsion moment; R (kg m23) density; v (m s21) peripheral speed;
D (m) impeller diameter; n (s21) revolutions per second.
It is used to design propellers and running wheels. It is an empirically determined dimensionless number.
Info: [A29],[A35].
8.2.77 Wave Number, Angular Speed Nω
Nω 5
ωL
am
ω (s21) angular frequency; L (m) characteristic length, radius; am (m s21) mean stagnation sound speed.
It expresses the ratio of the dynamic force of the combustion products flow to
the dynamic force corresponding to the stagnation sound velocity. Rocket engines.
Combustion chambers.
Info: [B71].
8.2.78 Weaver Flame Speed Number Wea
Wea 5
w
100
wH
w (m s21) laminar combustion rate of gas; wH (m s21) laminar combustion
rate of hydrogen.
It expresses the ratio of the laminar combustion rate of the considered gas to that
of hydrogen gas (Wea 5 100). It characterizes the tendency of the gas to react. A
lower value of this number represents a lower flame rate. The value of a high combustion rate of the gas is Wea A h32; 45i; of a medium rate, it is Wea A h25; 32i;
and of a low rate, it is Wea A h13; 25i.
Info: [C37].
8.2.79 Weber Number of Damp Steam Wev
Wev 5
Rv ðwv 2 w2 ÞL
σ
390
Dimensionless Physical Quantities in Science and Engineering
Rv (kg m23) vapour density; wv (m s21) absolute vapour speed; w (m s21) absolute speed of water drop; L (m) characteristic length, maximal length of stabilized water drop dimension; σ (N m21) surface tension.
It characterizes the process of the erosive influence of water droplets in damp
steam on the throughflow parts, for example, of the last steam turbine stages.
Wev A h21; 24i.
Info: [C37].
Ernst Heinrich Weber (p. 92).
Wilhelm Eduard Weber.
8.2.80 Wobbe Number, Wobbe Index Wob
pffiffiffiffiffi
λV1 R2
Wob 5
pffiffiffiffiffi
λV2 R1
Wob 5
λV
p
ð1Þ;
ð2Þ
λV1, λV2 (J m23) volume density of energy in gas 1 and 2; R1, R2 (kg m23) density of gas 1 and 2; λV (J m23) gas total volume energy density; p (Pa) gas
pressure.
In equation (1), this number enables a comparison of the energies of changing
gases, provided the burner is the same (having an equal cross section and pressure
drop). In equation (2), it expresses the ratio of the energy density and pressure in
the same gas. Wob A h17.8; 35.8i is valid for a coke oven gas, Wob A h35.8; 71.5i
holds for natural gas, and Wob A h71.5; 87.2i is for liquefied gases (LPG).
Info: [C63].
9 Geophysics and Ecology
To withdraw from Nature would mean to languish physically and a total divorce
would be death.
Jan Neruda (18341891)
9.1
Geophysics, Meteorology and Astrophysics
In recent years, in geophysics, meteorology and astrophysics, the number of dimensionless quantities give evidence of the increasing importance of these scientific
disciplines. Geophysics is about geodynamic, geothermic, geomagnetic, gravimetric
and seismologic processes, to which natural catastrophes are closely related, for
example, eruptions of volcanoes, tsunamis and earthquakes. Meteorology, as atmospheric physics, deals especially with climatological conditions and accompanying
phenomena. In geophysics, the Ekman, Hatta, Rossby, Argand and Love numbers,
as well as the tsunami velocities and the Shida number, are very important. In
meteorology, the Richardson number and the Burger, Davies and Rossby atmospheric numbers are among the best known. In astrophysics, the redshift number
and the Eddington, Golitsyn, Hubble, and Wesson numbers are among the most
significant.
9.1.1
Albedo Number A
A512
4σ0 T 4
εC
σ0 (W m22 K24) Stefan-Boltzmann constant; T (K) absolute temperature;
ε (2) emissivity (p. 206) of the atmosphere, around the height of about 5.5 km
where long-wave radiation is emitted; C (W m22) solar constant.
It expresses the reflectance of surfaces and bodies. It characterizes the climatic
conditions on the Earth which are produced by the solar radiation reflected from
the Earth. The global value of the Albedo number can be determined by measuring
the amount of solar radiation, reflected from the Earth, and back to the Earth during
rotation from the dark surface of the Moon. Meteorology. Climatology. Astronomy.
Info: [B73],[C42].
Dimensionless Physical Quantities in Science and Engineering. DOI: 10.1016/B978-0-12-416013-2.00009-9
© 2012 Elsevier Inc. All rights reserved.
392
Dimensionless Physical Quantities in Science and Engineering
9.1.2
Argand Number Arg
Arg 5
gLRc ðRm 2 Rc Þ
BRm
rffiffiffiffiffi
n L
wc
g (m s22) gravitational acceleration; L (m) characteristic length, thickness of
lithosphere; Rc (kg m23) crustal density; Rm (kg m23) mantle density;
B (N s m22) constant characterizing vertical average of compositional and thermal influences on the lithosphere rheology; wc (m s21) imposed collision velocity; n (2) stressstrain exponent.
It expresses the deformation in the terrestrial crust as the ratio of the buoyancy
force to the boundary force acting on the terrestrial plate. Geophysics.
Info: [B73],[C42].
9.1.3
Argand Number (1.) Arg
Arg 5
ΔRgh
η_ε
ΔR (kg m23) density difference of the oceanic and continental lithospheres;
g (m s22) gravitational acceleration; h (m) thickness; η (Pa s) lithosphere
dynamic viscosity; ε_ (s21) strain rate.
This number expresses the ratio of the tension, induced by the density difference
on the bottom part of the lithosphere, to the ductile deformation of the lithosphere.
With a certain simplification (the Newtonian fluid is considered instead of the
material creep law), it describes the lithosphere rheology. The criterion Arg characterizes the interface area between the ocean and the mainland. For Arg . 1, the lithosphere deformation force cannot lead to lower energetic potential so that the
ocean lithosphere will act on the mainland lithosphere and the interface angle
decreases. Geophysics. Terrestrial crust geology. Rheology.
Info: [B73].
9.1.4
Argand Number (2.) Arg
Arg 5
Rc ghc
C 1 μc Rc hc
Rc (kg m23) density of continental lithosphere; g (m s22) gravitational acceleration; hc (m) thickness of the continental brittle lithosphere, which is unsupported by a juxtaposed oceanic lithosphere; C (kg m21 s22) cohesion;
μc (kg m21 s22) constant dependent on friction coefficient (usually 0.5).
It expresses the ratio of the tension, on the bottom part of the non-hardened
mainland brittle lithosphere, generated with various increases of the mainland and
Geophysics and Ecology
393
oceanic lithospheres, to the brittle tension of the lithosphere. The Arg reproduces
the development of the destruction in the upper lithosphere along the interface of
the mainland plates. Geophysics. Terrestrial crust geology. Rheology.
Info: [B73].
9.1.5
Argand Number (3.) Arg
Arg 5
τ
μσn
τ (Pa) shear stress; μ (2) coefficient of friction between the lithospheres;
σn (Pa) normal stress.
It expresses the shear strain-to-resistance ratio along the interface between the
ocean and the mainland. Therefore, with Arg . 1, the slip of the lithosphere and
its shift down into the asthenosphere can be expected. The shear resistance can be
used to represent the main impediment force in the subduction process (subshifting).
Geophysics. Terrestrial crust geology. Rheology.
Info: [B73].
9.1.6
Beaufort Number B
B5
w 2
3
0:836
B5
w 2
3
1:87
for ½wBm s 21
ð1Þ;
for ½wBmile hour 21
ð2Þ
w (m s21) wind speed.
It characterizes the wind velocity on the sea. The value is in the range
BAh0; 15i.
Info: [A43].
Francis Beaufort (7.5.177417.12.1857), Irish oceanographer.
He formulated the Beaufort scale indicating wind force.
His digital scale extended from zero (calm) to 12 (hurricane)
and determined the wind force and possible thunderstorm
activity. Later, the scale was extended to 17 (tropical
cyclones) and supplemented by a formula to determine the
Beaufort number conversion to wind velocity. He was a
significant contributor to the development of the fields of
geography, astronomy, oceanography and meteorology.
394
Dimensionless Physical Quantities in Science and Engineering
9.1.7
Buoyancy Wave Number NB
rffiffiffiffi
f3
NB 5
ε
f (s21) buoyancy frequency; ε (s21) dissipation of turbulent kinetic energy.
This number characterizes the buoyancy vertical oscillations originating in a
stable environment as considered in meteorology. This is different to gravitational
waves arising in a gravitation field. On the turbulent kinetic energy spectrum, the
wave number separates the buoyancy band from the inertia band. The measurements show a relatively small portion of this number on the spectrum.
Info: [C13].
9.1.8
BusingerDyer Relationship NBD
wmed
1
z2d
z2d
NBD 5
5
1Ψ
ln
wf
k
z0
L
wmed (m s21) mean wind velocity; wf (m s21) friction velocity; k (2) Kármán constant; z (m) height above the ground; d (m) distance of translocation; z0 (m) roughness length; Ψ (2) empirical correction; L (m) MoninObukhov length.
It characterizes the similarity relation between the surface flow of a variable
quantity and the mean profile value of this variable. Micrometeorology of surface
layer. Boundary conditions.
Info: [C15].
9.1.9
Centripetal Acceleration Ac
Ac 5
ac
;
g
where ac 5 ω2 R
ac (m s22) centrifugal acceleration; g (m s22) gravitational acceleration;
ω (s21) angular frequency of body (planet) rotation; R (m) body radius.
It expresses the ratio of the centrifugal acceleration, with uniform circular
motion of a body, to the gravitational acceleration. It is used for bodies in approximate hydrostatic equilibrium. Geophysics. Astrophysics.
Info: [C18].
9.1.10 Dalton Number Dal
Dal 5
wv
2
5 CD;ev Sc23
wðks 2 kv Þ
Geophysics and Ecology
395
wv (m s21) resultant evaporation velocity; w (m s21) wind speed above water
level; ks (2) saturation vapour concentration; kv (2) vapour concentration;
CD,ev (2) drag evaporation coefficient (p. 228); Sc (2) Schmidt number
(p. 263).
It expresses the local evaporation rate of a substance from the free surface into
the atmosphere. It characterizes the atmospheric diffusion in a thin subsurface layer
between the surface and the surrounding atmosphere. Meteorology. Geophysics.
Info: [B48].
9.1.11 Davies Number Dav
Dav 5 CD Re2
CD (2) drag coefficient (p. 60); Re (2) Reynolds number (p. 81).
It is used in meteorology to calculate the final velocity of water droplets falling
from the atmosphere to the earth. It is also called the Best number. Meteorology.
Info: [C30].
9.1.12 Dissipation Number Dn
Dn 5
αgL
cp
α (K21) linear thermal expansion coefficient; g (m s22) gravitational acceleration; L (m) characteristic length (depth); cp (J kg21 K21) specific heat
capacity.
It expresses viscous dissipation with rapidly rotating thermal convection. It follows the dynamics of the terrestrial crust, but without the influence of the magnetic
field. Geology. Geophysics.
9.1.13 Ekman Number (1.) Ek
rffiffiffiffiffiffiffiffi rffiffiffiffiffiffi
v
Ro
5
Ek 5
ð1Þ;
2
ωL
Re
rffiffiffiffiffiffiffiffiffiffiffi
ν
Ek 5
ð2Þ;
2ωL2
Ek 5
ν
2L2 ω sin ϕ
ð3Þ
ν (m2 s21) kinematic viscosity; ω (s21) angular frequency of planetary rotation; L (m) characteristic length (usually vertical); ϕ (2) latitude;
396
Dimensionless Physical Quantities in Science and Engineering
2w sin ϕ (s21) Coriolis frequency; Ro (2) Rossby number (p. 406); Re (2) Reynolds number (p. 81).
This number expresses the ratio of the viscosity force to the Coriolis force. It
characterizes the flow in rotating channels. In expression (2) Ek, as defined by relation (1), it is called the velocity pressure criterion. Sometimes, equation (2) is
applied. In magnetohydrodynamic processes, it expresses the viscous fluid friction
influence and the effect of a secondary carrying motion, for example, the rotation of
the Earth. In equation (3), it describes geophysical phenomena in oceans and in the
atmosphere. With a small Ek, the arising disturbances can propagate due to a weakening friction effect. Magnetohydrodynamics. Geophysics of viscous friction forces.
Info: [C45].
Vagn Walfrid Ekman (3.5.18749.3.1954), Swedish
oceanographer and physicist.
He is among the most famous oceanographers of his
time. He observed the movement of icebergs and found that
they did not move in the wind direction, but had a deviation
of 2040 . He published his theory of the Ekman spiral, in
which this phenomenon was explained as the equilibrium
between friction effects and the fictive forces due to the
Earth’s rotation. He examined how the steady-state winds
formed a thin layer on the ocean surface, the eponymous
Ekman layer, and defined its velocity and thickness.
9.1.14 Froude Biomechanical Number Fr
Fr 5
w2
gLl
w (m s21) forward speed of animal; g (m s22) gravitational acceleration;
Ll (m) leg length, usually measured from coxa.
It expresses the classic Froude number (1.) Fr (p. 62), but with parameters which
correspond to the motion of a person or animal. It is an important similarity criterion
in modelling the motion of a person under conditions of reduced gravitation, for
example, of the astronauts on other planets. Biomechanics. Astrophysics.
Astronautics.
Info: [B29].
William Froude (p. 63).
9.1.15 Geometry Erosion Parameter Pg
rffiffiffi
h
Pg 5
L
h (m) depth of surface water; L (m) slope surface size.
Geophysics and Ecology
397
It characterizes the geometric parameters of the hydrodynamic and kinematic
processes of soil erosion. The parameter value is very small, among other things,
with respect to physical conditions connected to a channel acting with surface
water flow.
Info: [B65].
9.1.16 Golitsyn Energy Balance Number Goenergy, E
Goenergy 5
qabs 1 qint
Tef 4
5
qabs
Tekv
qabs (W m22) absolute planet radiation; qint (W m22) internal planet radiation;
Tef (K) effective temperature; Tekv (K) reference temperature.
This number expresses the thermal equilibrium between an internal heat source
and planet radiation. When Goenergy . 1.27, the internal heat source can be
expected to play an important role in controlling atmospheric dynamics.
Geophysics and astrophysics. Dynamics of the atmosphere.
Info: [B111].
Georgy Sergeyevich Golitsyn (see below).
9.1.17 Golitsyn Number Go
Go 5
τ dyn
;
τ rad
where τ dyn 5
Rp
;
a
τ rad 5
cp pef
σ0 gTef3
τ dyn (h) dynamic time constant; τ rad (day) radiative time constant; Rp (m) planetary radius; a (m s21) sound speed; cp (J kg21 K21) specific heat capacity;
pef (Pa) effective pressure characterizing the condition when hydrogen
optical depth is 1; σ0 (W m22 K24) StefanBoltzmann constant; g (m s22) gravitational acceleration; Tef (K) effective temperature.
It characterizes the thermal inertia of the atmosphere or its structure, which is
important from an atmospheric radiative heating or cooling point of view, as compared to the dynamics thereof. With Go{1, the atmosphere would be in equilibrium from a radiative point of view. In this case, the thermal mode plays a
controlling role over atmospheric dynamics and leads to low temperature gradients.
For Go $ 1, the global circulation requires longer time periods for the transference
of heat. The atmosphere is closed for local radiation equilibrium originating and
the dynamics is set on this thermal mode. The Golitsyn number Go is the time measuring number Nτ (p. 30) for the atmosphere. Geophysics and astrophysics.
Dynamics of the atmosphere.
Info: [B111].
Georgy Sergeyevich Golitsyn, Russian astrophysicist.
398
Dimensionless Physical Quantities in Science and Engineering
9.1.18 Grouch Number Gro
Gro 5
wv ω
5 Fr Sh 21
g
wv (m s21) vertical landing velocity; ω (s21) natural frequency of mass
dynamical system during impact (incidence); g (m s21) gravitational acceleration; Fr (2) Froude number (1.) (p. 62); Sh (2) Strouhal number (p. 87).
It expresses the ratio of the vertical Froude number (1.) Fr (p. 62) and the vertical Strouhal number Sh (p. 87). In dimensionless form, it represents the landing
velocity of a person or animal under various gravitational conditions. Astrophysics.
Astronautics.
Info: [B29].
9.1.19 Hubble Number Hu, z
Hu 5
δλ H
5 r
λ
c
δλ (m) wavelength shift of spectral lines; λ (m) wavelength of spectral line;
H (s21) Hubble constant; c (m s21) speed of light; r (m) distance of each
nebula from the Earth.
Essentially, it expresses Hubble’s law. According to it, the electromagnetic radiation propagating from the Universe to the Earth has shifted towards longer
wavelengths (the so-called redshift in the visible light band) and is proportional to
the distance the radiation travelled. Together with the Doppler law, it confirms the
expansion of the Universe.
Info: [C44],[C71].
Edwin Powell Hubble (20.11.188928.9.1953), American
astronomer.
He was a prestigious astronomer because of his discovery
of galaxies behind the Milky Way and his cosmological substantiation of the wavelength redshift. He was the first to
argue that this shift was due to the Doppler effect, induced
by the expansion of the Universe. He was one of the leading
astronomers of his time who laid the foundation for physical
cosmology. Shortly before his death, the Hale telescope was
finished and Hubble was the first to use it.
9.1.20 Jeffreys Phase Φ
0
Φ5
2π
1
ðFo=2Þ3
Ai
1
X
2
Fo
@
A
1
ðFo=2Þ3
;
rffiffiffi
g
where Fo 5 τ
;
h
X5
x
h
Geophysics and Ecology
399
τ (s) time; g (m s22) gravitational acceleration; h (m) depth of ocean; x (m) distance; Fo (2) Fourier number characterizing tsunami propagation; X (2) geometrical coordinates (p. 15).
It characterizes the phase shift of tsunamis. Geophysics. Oceanography.
9.1.21 Kibel Number Ki
See the Rossby number Ro (p. 406).
9.1.22 Kirpichev Number of Solar Radiation KiS
KiS 5
εSL
λT
ε (2) relative absorption of solar radiation; S (W m22) solar constant, density
of solar radiation; L (m) characteristic length; λ (W m21 K21) thermal conductivity of absorption environment; T (K) temperature.
It expresses the ratio of the specific solar heat flow, transferred by the solar
radiation to the system, to the specific heat flow transferred by conduction in the
system. It characterizes the solar radiation heat flow onto the surface of a body. It
is the modified Kirpichev heat number Ki. (p. 176)
Info: [A23].
Mikhail Viktorovich Kirpichev (p. 177).
9.1.23 Kolmogorov Parameter Kol
Rgwff c @w 21
Kol 5
w2m
@z
where
R5
Rw 2 R
;
R
wm 5
rffiffiffi
τ
;
R
0 # Kol , 1
R (2) relative density; g (m s22) gravitational acceleration; wff (m s21) free
fall velocity of droplets in air; c (2) droplet concentration in thin layer;
21
wm (m s21) shear stress velocity; @w
@z ðs Þ gradient of flow velocity in the
21
z axis; w (m s ) mean flow velocity at water level; Rw (kg m23) water density; R (kg m23) air density; τ (Pa) shear stress.
For the suspension of droplets, it expresses the ratio of the energy dissipation
rate to the turbulent energy generation rate. The acceleration rate and the creation
of water spray connected with it on the surface of oceans is an example. The
Coriolis effect and the effect of cooling act with this, causing water droplets to be
400
Dimensionless Physical Quantities in Science and Engineering
formed and to evaporate. It is a specific analogy of the Kolmogorov number Kol
(p. 104). Geophysics. The physics of water spray generation in ocean surf.
Info: [C75].
Andrey Nikolayevich Kolmogorov (p. 104).
9.1.24 Langmuir Turbulent Number Lat
Lat 5
rffiffiffiffiffi
uf
uS
uf (m s21) water friction velocity; uS (m s21) Stoke’s drift velocity.
It expresses the ratio of the friction force to the Stokes drift velocity. It characterizes the turbulent process in the ocean surface layer with this process causing
mixing in a layer which becomes deeper due to Langmuir circulation. The
Langmuir turbulence has various characteristics, depending on the wind acting at
the surface. Geophysics.
Info: [C35].
Irving Langmuir (p. 345).
9.1.25 Lochtin Number Lo
Lo 5
Q
pffiffiffiffiffiffiffiffi
d2 gdL
Q (m3 s21) 2 volume flow; d (m) size of particles; g (m s22) gravitational
acceleration; L (m) characteristic length.
It characterizes the steady-state natural river flow which depends on the size of
particles of sediments, forming the river bed, and on the bottom inclination and
flow. It determines the width and depth. It is analogous to the Froude river inflow
number Frin (p. 417).
Info: [B52].
V.M. Lochtin.
9.1.26 Love Number Lov, k
Lov 5
Vred r 3
Vtide R
Lov 5
21
3
19w2s
11
2gr
2
ð1Þ;
Lov 5
ΔVred;2 r 3
Vtide R
ð3Þ;
Lov 3 gr
19 w2s
ð2Þ;
ð4Þ;
Geophysics and Ecology
401
Vred (m2 s22) deformation potential due to influence of tidal mass redistribution;
Vtide (m2 s22) 2 tidal potential; ΔVred,2 (m2 s22) increase of deformation potential at excitation8; r (m) local radius of planet; R (m) outer radius of planet;
ws (m s21) shear speed dependent up to tidal excitation; g (m s22) gravitational
acceleration.
The Love numbers characterize the influence of the Moon’s rotation around the
Earth and its periodic action on oceans (ebb and flow) and on the atmosphere and
rigid surface of the Earth. Due to the tide, this surface is deformed alternatively
and changes the gravitational field. As a result, the tide potential is created, which
differs on opposite sides of the Earth and causes two different shifts. The Love
number expresses the ratio of the vertical shift, leading to surface heightening and
to the tide potential. The expressions (1)(4) are valid for homogeneous and
incompressible planets. For the Earth, it is Lov , 10 and for the Mars Lov . 10.
The expression (4) gives an approximate value. Geophysics. Mineralogy.
Info: [C81].
Augustus Edward Hough Love (17.4.18635.6.1940),
English mathematician.
He became famous by his work on the mathematical
elasticity theory. He was also engaged in the wave propagation and elaborated the mathematical model of terrestrial
surface waves which are known as the Love waves. The
book Some Problems in Geodynamics is among his most
popular works. Also important is his two-volume work, A
Treatise of the Mathematical Theory of Elasticity.
9.1.27 Moment Magnitude Scale Mw, Me , Mn
2
M0
29:1
Mw 5
log10
ðN mÞ
3
2
Es
Me 5 log10
22:9
ðN mÞ
3
where
ð1Þ;
ð2Þ;
Es 5 1024:8 M0 5 1:6 3 1025 M0 ;
Mn 5
2
mtnt
log10
ðkgÞ
3
ð3Þ pro ½mtnt 5 kg;
Mn 5
2
mtnt
log
3 10 ðktÞ
ð4Þ pro ½mtnt 5 kt;
Mn 5
2
mtnt
log10
ðMtÞ
3
ð5Þ
pro ½mtnt 5 Mt;
402
Dimensionless Physical Quantities in Science and Engineering
M0 (N m) seismic moment of the total amount of energy that is transformed during an earthquake; Me (N m) small fraction of the moment M0 converted into
radiation seismic energy; Mn (N m) comparative nuclear detonation moment;
Es (J) radiation seismic energy; mtnt (kg, kt, Mt) mass of explosive TNT serving for comparison.
The expression (1) represents the successor of the well-known Richter’s scale. It
serves to compare the energy loosened during the earthquake. At this logarithmic
scale, the rise by 1 degree corresponds to the loosened energy increase multiplied
by 101.5 5 31.65. The degree 2 corresponds to 103 5 1000 times greater energy.
The expression (2) shows a part of the energy, released during the earthquake, in
the form of the radiation seismic energy. Expressions (3) and (4) present the comparison with nuclear detonations of the trinitrotoluene in kg, kilotons and
megatons.
Info: [C92].
9.1.28 Pratt Number Pra
Pra 5
Ls
τw
5
Ll
Ll
Ls (m) step length; Ll (m) leg length, usually measured from the coxa; τ (s) time duration of step; w (m s21) forward speed of animal motion.
Together with the Froude biomechanical number Fr (p. 396), it expresses the
fundamental similarity criteria in modelling human motion under reduced gravity
conditions, such as in the motion of astronauts on other planets. Biomechanics.
Astrophysics. Astronautics.
9.1.29 Redshift Number Z
See the Hubble Number Hu (p. 398).
9.1.30 Richardson Bulk Number Rib
Rib 5
gTV21 ΔTV Δz
ðΔw1 Þ2 1 ðΔw2 Þ2
ð1Þ;
Rib 5
gΔRh
R0 ðΔwÞ2
ð2Þ
g (m s22) gravitational acceleration; TV (K) absolute virtual temperature;
ΔTV (K) virtual potential temperature difference across a layer of thickness Δz;
h Δz (m) layer thickness; Δw1, Δw2 (m s21) changes in horizontal wind
components across the same layer; ΔR (kg m23) change of density across a
layer; R0 (kg m23) initial density; Δw (m s21) change of resulting horizontal
wind component across a layer.
Geophysics and Ecology
403
In meteorology, it represents the relation between the vertical flow stability and
the vertical slip. A high value indicates the instability and a low one shows stability. The value RibAh50; 100i represents advantageous development. It is the difference analogy of the Richardson gradient number Rig (p. 384) and expresses the
approximation of local gradients by finite differences across a layer. Both numbers
characterize a slip instability model which expresses mixing. It is used to simulate
daily fluctuations in the depth of an instable layer and to determine the flow profiles, for example from subtropical regions. A complete mathematical model also
involves the influence of thermal flows and wind tension on layer instability. The
number Rib itself expresses the approximation of local gradients by finite differences across a layer. It is used in expression (1) or (2). The mathematical difference
model works, for example with the so-called surface box, and with the slip being
greater than the stratification density necessary to support it (Rib . 0.55), the properties of both adhering boxes are averaged because both have been stirred. This
process drops continuously up to Rib . 0.65. Meteorology.
Info: [C12].
Lewis Fry Richardson (11.10.188130.9.1953), English
mathematician, physicist, meteorologist, psychologist and
pacifist.
His interest was concentrated on meteorology and, above
all, on the design of a mathematical model, using differential equations (1922), to predict the weather, which is still
used today. He was also engaged in atmospheric turbulence
and executed many terrestrial experiments. The dimensionless parameters of turbulence theory are named after him.
9.1.31 Richardson Flux Number Rif
Rif 5
21
gTVP
wTV
uwðrUÞ 1 vwðrVÞ
g (m s22) gravitational acceleration; TVP (K) virtual potential
temperature (theoretic temperature of dry air having the same density as wet air);
TV (K) virtual temperature (coming from state equation of dry air for wet air
except substitution of T for TV); U, V (m s21) horizontal Cartesian wind components from the west and from the east; rU, rV (s21) vertical wind velocity gradients @[email protected]; @[email protected]; u, v, w (m s21) axial components of wind velocity.
It expresses the ratio of the buoyancy force limit of turbulent kinetic energy to
the negative slip limit. It is the measure of the dynamic instability describing the
flow capability to stay turbulent in case of a slip wind and certain static stability.
In the expression, the denominator is usually negative. Therefore, with a positive
numerator, the Rif is negative for statically instable flow. For Rif , 1, the flow is
dynamically instable and turbulent. For Rif . 1, the flow is dynamically stable, the
404
Dimensionless Physical Quantities in Science and Engineering
turbulence tends to decrease and, according to the number Rif, the flow cannot be
defined. Meteorology.
Info: [C12].
Lewis Fry Richardson (see above).
9.1.32 Richardson Gradient Number Rig
Rig 5
gTA21 rTP
ðrw1 Þ2 1 ðrw2 Þ2
ð1Þ;
Rig 5 g
@R 21 @w 22
R0
@z
@z
ð2Þ
g (m s22) gravitational acceleration; TP (K) virtual potential temperature;
TA (K) absolute virtual temperature; w1, w2 (m s21) wind components towards
the east and north; R (kg m23) air density; R0 (kg m23) initial density; z (m) height of the layer.
This number is used in expression (1) or (2). It expresses the capability of generation or dissipation of the turbulence, divided according to direction, in the slip
development thereof. It is used to determine the dynamic stability and formation of
turbulence. For Rig less than the Richardson critical number (approximately
Rig # 0.25), the air becomes dynamically instable and turbulent. The origin of hysteresis can be considered too, with which the laminar flow must sink under the
above-mentioned critical value in order to pass to a turbulent flow. However, turbulent flow can exist up to Rig 5 1.0, provided it was primarily laminar before.
Meteorology.
Info: [C12].
Lewis Fry Richardson (see above).
9.1.33 Richardson Meteorology Number (1.) Rim
Rim 5
gαrT
ðrwÞ2
g (m s22) gravitational acceleration; α (K21) linear thermal expansion coefficient; rT 5 @[email protected] (K m21) temperature gradient expressing vertical stability;
rw 5 @[email protected] (m s22) velocity gradient expressing characteristic vertical wind
shear.
It expresses the buoyancy-to-inertia forces ratio or, alternatively, the capability
to suppress turbulence in the formation of the turbulence slip. It is used as a stability measure to determine whether turbulence arises. Meteorology.
Info: [C12].
Lewis Fry Richardson (see above).
Geophysics and Ecology
405
9.1.34 Richardson Meteorology Number (2.) Ri
Ri 5
gLΔR
5 Gr Re 22
Rw2
ð1Þ;
Ri 5
gαΔTL
w2
ð2Þ
g (m s22) gravitational acceleration; L (m) characteristic length; ΔR (kg m23)
density difference; R (kg m23) density; w (m s21) flow velocity; α (K21) linear thermal expansion coefficient; ΔT (K) temperature difference; Gr () Grashof heat number (p. 185); Re () Reynolds number (p. 81).
It expresses the ratio of buoyancy effects to vertical slip effects. With Ri{1, the
buoyancy influences the turbulence with forced convection. For natural convection,
it is Ric1. At Ri1, it expresses mixed convection. The value of the number
Ri 5 1 is known as the MoninObukhov length L (p. 21). Mechanics of fluids.
Meteorology.
Info: [C12].
Lewis Fry Richardson (see above).
9.1.35 Richardson Overall Number Ri
Ri 5
g0 L
w2
g0 (m s22) reduced gravitational acceleration; L (m) characteristic length;
w (m s21) velocity imposed by the boundary condition of the problem.
In the above-mentioned expression, this number presents a basic view of its
physical meaning only. A more detailed description is in the following Richardson
numbers.
Lewis Fry Richardson (see above).
9.1.36 Rossby Atmospheric Number (1.) Roatm
Roatm 5
τ rot
a
B
τ dyn Rp ω
τ rot (h) time period of rotation; τ dyn (h) dynamic time constant; a (m s21) sound speed; Rp (m) planet radius; ω (s21) angular frequency of planet
rotation.
It expresses the importance of planet rotation velocity (the Coriolis force) to create atmospheric motion. Alternatively, it expresses the ratio of the rotation period
to the dynamic time scale. With Roatm , 1, the Coriolis forces play an important
role in atmospheric motion control.
Carl Gustaf Arvid Rossby (see below).
406
Dimensionless Physical Quantities in Science and Engineering
9.1.37 Rossby Atmospheric Number (2.) Ro
Ro 5
τ rot
w
;
5
τ conv
ωH
where τ conv 5
H
;
w
τ rot 5
1
ω
τ rot (h) time period of rotation; τ conv (h) time period of convection; w (m s21) vertical velocity in the convective layer; ω (s21) angular frequency of planet
rotation; H (m) atmospheric height at static equilibrium.
This number characterizes the rotation acting on the vertical motion which
causes buoyancy. Therefore, it is also called the Rossby buoyancy number. With
Ro , 1, the planetary rotation causes the deviation of moving objects.
Carl Gustaf Arvid Rossby (see below).
9.1.38 Rossby Concentration Number RoC
RoC 5 β C ΔC
β C (M21) volume ion concentration coefficient11; ΔC (M) ion concentration
difference between two electrodes12.
It expresses the change of the ion concentration in the atmosphere or other systems. Geophysics. Meteorology.
Carl Gustaf Arvid Rossby (see below).
9.1.39 Rossby Heat Number RoT
RoT 5 βΔT
β (K21) volume thermal expansion coefficient; ΔT (K) temperature difference
between surfaces of warm and cold systems.
It expresses the thermal dilatability of solid, liquid or gaseous materials.
Mechanics. Geophysics. Meteorology.
Carl Gustaf Arvid Rossby (see below).
9.1.40 Rossby Number Ro
Ro 5
w
2ωL sin α
ð1Þ;
Ro 5
w
ωL
ð2Þ
w (m s21) fluid flow velocity; ω (s21) angular frequency; L (m) characteristic length; α () angle between the axis of rotation and the direction of fluid
motion.
Geophysics and Ecology
407
It expresses the ratio of the fluid inertia force to the Coriolis force in the atmosphere or in an ocean, based on the planetary rotation. It characterizes the influence
of the rotation of the Earth on the flow in channels and pipelines. It describes the
geophysical phenomena in oceans and in the atmosphere. With a large number Ro,
usually Ro . 1, for example in the tropics or lower geographical latitudes, the planetary rotational influence is insubstantial and can be neglected. If the Ro number is
small (Ro , 1), this influence cannot be neglected. It is also used in expression (2).
It is also known as the Kibel number (p. 399).
Info: [A21],[B20].
Carl Gustaf Arvid Rossby (28.12.189819.8.1957),
Swedish-American meteorologist and oceanographer.
He was the first to clarify the wide movements of the terrestrial atmosphere by using fluid mechanics. He introduced
a mathematical description (model) for these movements.
He was engaged in atmospheric thermodynamics, in mixing
and turbulence, and in the oceanatmosphere interaction.
He identified and characterized nozzle flowing and Rossby
waves in the atmosphere. After the Second World War, he
modified his previous mathematical model of atmospheric
dynamics for the computer and weather predictions.
9.1.41 Rossby Temporal Number Roτ
Roτ 5
2π
τω
τ (s) time, time of the one rotation period; ω (s21) angular frequency of the
Earth’s rotation.
It compares the local velocity changes in the atmosphere to the Coriolis force.
Geophysics. Meteorology.
Info: [C48].
Carl Gustaf Arvid Rossby (see above).
9.1.42 Seismic Efficiency η
η5
Erad
Δσ
;
5
Etot
2σ
where σ 5 σ1 1
Δσ
2
Erad (J) radiated seismic energy; Etot (J) total loading energy; Δσ (Pa) difference of stresses before and after dislocation; σ (Pa) average stress during
dislocation; σ1 (Pa) stress after dislocation.
It expresses the ratio of the radiated seismic wave energy during earthquakes to
the total energy. Alternatively, it also expresses the ratio of the tension increment
to the mean tension value. Geophysics. Earthquakes.
Info: [C41].
408
Dimensionless Physical Quantities in Science and Engineering
9.1.43 Shida Number Shi, l
Shi 5
uλg sin λ
R gradλ V
ð1Þ;
Shi 5
uϑ g
R gradϑ V
ð2Þ
uλ, uϑ (m) components of horizontal shift; g (m s22) gravitational acceleration; λ, ϑ () shift angles; R (m) outer Earth radius; V (m2 s22) tide
potential.
Like the Love number, it characterizes the influence of the moon’s rotation
around the Earth, which causes the action of the tide on the surface deformation
thereof. The number expresses the ratio of the angles of horizontal shifts with two
components of the tide potential gradient.
9.1.44 Smith Erosion Number Sm
Sm 5
εw
L
ε () scale factor, usually ε{1; w (m s 21) mean effective velocity of rain,
incident to Earth surface; τ (s) time; L (m) characteristic length of a surface,
vertical or horizontal.
It characterizes the erosion time during which measurable changes occur on the
soil surface. Especially, it is used with physical modelling and analysis of landscape terrain changes. Geophysics. Morphology. Ecology.
Info: [B65].
9.1.45 Strouhal Biomechanical Number St
St 5
wv Ll21
wn
ð1Þ;
St 5
wh Ll21
ws
ð2Þ
wv (m s21) vertical velocity of landing; Ll (m) length of leg, usually measured
from coxa; ωn (s21) natural frequency; wh (m s21) horizontal speed at jumping
motion; ωs (s21) jumping frequency at horizontal motion.
It is a modification of the basic Strouhal number Sh (p. 87) for the dynamics of
vertical motion with landing (1) or with horizontal spring motion (2). Astrophysics.
Astronautics.
Info: [B29].
Vincenc Strouhal (p. 87)
Geophysics and Ecology
409
9.1.46 Tidal Horizontal Displacement Number Nhor
uϑ
Shi @V
5
ð1Þ;
R
g @ϑ
uλ
Shi @V
5
ð2Þ;
Nhor 5
R
g sin λ @λ
Nhor 5
uϑ, uλ () components of horizontal displacement; R (m) outer Earth radius;
g (m s22) gravitational acceleration; V (m2 s22) tide potential; ϑ, λ () angles of deviation; Shi () Shida number (p. 408).
It expresses the horizontal displacement components caused by the tidal redistribution of the terrestrial mass.
Info: [C81].
9.1.47 Tidal Radial Displacement Number Nrad
Nrad 5
ur
Lov2 Vtide
5
R
g
ur () radial displacement; R (m) outer Earth radius; Vtide (m2 s22) tide potential; g (m s22) gravitational acceleration; Lov2 () Love number.
It expresses the radial displacement caused by the tidal redistribution of the
mass of the Earth.
Info: [C81].
9.1.48 Tide Height Ratio H, h
H 5 Lov 5 Shi 5 0
Hn 5 1 1 Lovn
ð1Þ;
H 5 Shi 5 1
ð2Þ;
ð3Þ
Lov () Love number (p. 400); Shi () Shida number (p. 408); Lovn () Love number (p. 400) at excitation potential of nth degree (here n 5 2).
It expresses the ratio of the main tide height to the static height of the sea tide.
The expression (1) holds for a fully rigid body, the expression (2) for a liquid body
and the expression (3) for real body. If they have equal density, then Lov2 5 3=2
and H2 5 5=4 are valid. Geophysics.
Info: [C81].
9.1.49 Tsunami Frequency Number Ω
sffiffiffi
h
Ω5ω
ð1Þ;
g
Ω2 5 K tanh K
ð2Þ;
where K 5 kh
410
Dimensionless Physical Quantities in Science and Engineering
ω (s21) angular frequency; g (m s22) gravitational acceleration; h (m) characteristic length, depth of flat ocean; k (m21) dimension wave number; K () wave number (p. 33).
In expression (1), it represents the tsunami propagation frequency on the ocean
surface. In expression (2), it represents the dispersion relation for created gravitational waves. Geophysics. Oceanography.
Info: [B18].
9.1.50 Tsunami Velocities C, W
c
K2
C 5 pffiffiffiffiffi 5 1 2
6
gh
ð1Þ;
w
K2
W 5 pffiffiffiffiffi 5 1 2
2
gh
ð2Þ
c, w (m s21) velocity propagation of water wave; g (m s22) gravitational acceleration; h (m) sea depth; K () wave number (p. 33).
It expresses the approximate velocity, provided the sea depth is constant and the
velocity and phase approximation is quadratic. In addition, it is valid for long-wave
tsunamis (K{1). Geophysics. Oceanography.
9.1.51 Wesson Coupling Constant β
β5
G
B3 3 1023 Bα
pc
G (m3 kg21 s22) Newtonian constant of gravitation; p (kg21 m2 s21) Wesson’s
empirical parameter expressing the ability of planets, stars and galaxies to super-clustering; c (m s 2 1) speed of light; α () electromagnetic coupling constant (p. 306).
It is an analogy to the gyromagnetic ratio, but is more suitable from the general
relativity point of view. Astronomy. Astrophysics.
9.2
Ecology and Biology
Above all, ecology, as a multidisciplinary field, is based on biology and other
branches such as physics, chemistry, meteorology, geology and other technical as
well as non-technical fields. The increasing significance of ecology is demonstrated
in a lot of dimensionless quantities. They involve general ecology as well as
Geophysics and Ecology
411
industrial ecology, ecology of seas, landscapes and forests, and global ecology.
Among the dimensionless quantities, the Brown number for atmospheric pollution,
the Burger number for the influence of the earth’s rotation on the water movement
in water systems, and the Imberger number to express the tendency of a surface
water layer to be mixed are presented. Several dimensionless quantities relate to
exergy and industrial ecology. As for biology, the following numbers are presented:
the sperm number for viscous and elastic tensions in microbiologic fibres, the radiation weighting factor and the breakup wavenumber expressing the influence of
biological heterogeneity on intercellular relations in wave propagation in cardiac
tissue. A survey of fundamental dimensionless quantities for biological systems,
with more detailed comments on their formulation is in [B95]. In addition to the
dimensionless quantities in biology, microbiology, and agrobiology, these quantities are presented for pulmonology and botany as well.
9.2.1
Absorbed Agrobiological Dose Ndose
Ndose 5
where ed B
Red
E2 γτ
E2 γτ
R
for ω{ωcr ;
ed B
E2 ε
R
for ωcωcr
R (kg m23) density of soil; ed (J kg21) specific energy; E (V m21) electric
field intensity; γ (S m21) specific electrical conductance; τ (s) time; ω (s21) frequency of supply; ωcr (s21) critical frequency; ε (F m21) permittivity.
Together with other dimensionless quantities, it characterizes both the capability
of soil to absorb electric field energy and the efficiency of the delivered energy. It
is used in agricultural soil refining technology, based on the electric field method,
enabling inactivation of various harmful microorganisms in the soil. Agrobiology.
Info: [B113].
9.2.2
Association Constant kA
kA 5
kon C0
koff
kon (m3 mol21 s21) kinetic speed constant of direct reaction; C0 (mol m23) volume concentration of the target DNA in solution; koff (s21) kinetic speed
constant of reversed reaction.
It expresses the ratio of the maximum direct biochemical reaction rate to the
maximum reversed reaction rate. It characterizes the biochemical processes, for
example, in DNA microorganisms in the hybridizing process and diffusion and
412
Dimensionless Physical Quantities in Science and Engineering
convective motions. In kA {1; it corresponds to the equilibrium relative molar surface concentration of the formation of hybrid pairs. Microbiology. Microorganisms.
Info: [B86].
9.2.3
Bond Natural Number Bo
Bo 5
RghL
2σ
R (kg m23) sap density; g (m s22) gravitational acceleration; h (m) height in
capillary; L (m) characteristic length, capillary radius; σ (N m21) surface
tension.
It expresses the ratio of the gravitational force to that of the surface tension
force in the sap movement in plants. It characterizes this movement originating
from water loss as a result of evaporation from leaves. In the trunk or stem of a
plant, the radius of capillary tubes equals several tens of micrometres, and the
radius of leaves equals several tenths of micrometres. Biomechanics. Botany.
Info: [A49].
Wilfrid Noel Bond (18971937), English physicist.
9.2.4
Breakup Wave Number NS
rffiffiffiffiffiffiffiffi
SR D
NS 5 τ r
τ
τ r (s) enhanced time of output phase; SR (m22) density downtrend per unit
area; D (m2 s21) diffusion coefficient; τ (s) transfer time rising between
inhomogeneities.
This number expresses the heterogeneity of a biological system and the properties of a medium capable of being excited. The isolated heterogeneity causes transfer wave decomposition. Microbiology. The influence of heterogeneity and
intercellular coupling on wave propagation in the heart tissue.
Info: [B104],[B108].
9.2.5
Brown Number Bro
sffiffiffiffiffiffiffiffi wp 2 wp 0 ð1Þ;
5
Bro 5
wf 2 wx 0 where wf 0 5
rffiffiffiffiffiffiffiffiffiffi
8kT
;
πmf
Geophysics and Ecology
w5
1
cf λ
2
413
ð2Þ;
1
kT
where λ 5 pffiffiffi
5 pffiffiffi
2
2p
2πndm
2πdm
wp ; wf (m s21) mean velocity of particles (p) and fluid (f); wp 0 ; wf 0 (m s21) mean
temperature fluctuation of the velocity of particles (p) and fluid (f); k (J K21) Boltzmann constant; T (K) absolute temperature; mf (kg) mass of fluid;
w (m s21) velocity of fluid; cf (m s21) sound velocity in fluid; λ (m) mean free
path of molecules in gas; n (cm23) numerical density of molecules in gas; dm (m) diameter of gas molecule; p (Pa) pressure in fluid.
Most frequently, it characterizes aerosols as suspensions of solid or liquid particles in a fluid and in the air. Dust, smoke, smog and mist are common aerosols. It
describes a colloid system in which the dispersed phase is represented by sprayed
solid or liquid particles. Usually, the radius of aerosol particles is in the range of
0.0110 μm. The Brown number represents an important ecological criterion
expressing atmospheric pollution conditions, among other things. Ecology.
Meteorology.
Info: [A0].
Robert Brown (21.12.177310.6.1858), English botanist.
Above all, he dedicated himself to botany. In Australia,
his research over many years led him to discover roughly
2000 hitherto unknown plants. He edited a whole range of
works about the results of his Australian flora research. He
became most famous for his microscopic observations of
the random movement of dust pollen flecks and for formulating the law of this movement. To honour him, this movement has been named Brown movement. He also
discovered and described vegetal cells’ nuclei.
9.2.6
Burger Number, Si Number Bur
2 2
fB LV
Ro
Rr
Bur 5
5
5
ωLh
L
Fr
fB (s21) buoyancy frequency; LV (m) vertical depth of flow; ω (s21) inertial
angular frequency of the Earth; Lh (m) characteristic horizontal length of flow;
Rr (m) Rossby deformation radius; L (m) characteristic length; Ro (2) Rossby number (p. 406); Fr (2) Froude number (1.) (p. 62).
It is also called the Si number. It expresses the influence of the earth’s rotation
on the motion in a lake or other large water basin. With its value less than the critical one (Bur 5 1), the waves have the character of Kelvin or topographic waves.
414
Dimensionless Physical Quantities in Science and Engineering
For Bur , 1, the earth’s rotation influences the dynamics and the waves have an
oscillating character with most of their energy in the form of kinetic energy. The
Burger number characterizes the atmospheric or oceanographic flow expressed by
the ratio of the vertical stratification density to the horizontal terrestrial rotation.
For Bur-0, the rotational flow predominates, whereas the vertical stratification
does for Burc0: Geophysics. Ecology. Meteorology.
Info: [B52],[C14].
Alewyn Burger, mathematician and meteorologist.
9.2.7
Crop Coefficient KC
KC 5
ETC
ET0
ETC (mm day21) crop evapotranspiration; ET0 (mm day21) reference crop
evapotranspiration.
It expresses the ratio of the real evaporation rate from plants and the soil under
them to the relative rate, generally determined by calculation according to the
PenmanMonteith equation for grass. The coefficient value is used to determine
the amount of water in soil irrigation. The evaporation coefficient value is usually
within the range KCAh0.1; 1.5i. Ecology. Botany. Agriculture.
Info: [C26].
9.2.8
Damkőhler Biological Number Dabiol
Dabiol 5
kon Hmax h
Dmol
kon (m3 mol21 s21) constant of kinetic speed of direct reaction; Hmax (mol m22) molar surface concentration freely connecting the position of specific places on the
probe; h (m) height of liquid layer; Dmol (m2 s21) molecular diffusion
coefficient.
It expresses the ratio of the maximum direct reaction rate to the maximum normal diffusion rate. It characterizes biochemical processes, for example, those in
DNA microorganisms related to the hybridizing process and diffusion and convective motions. In the case of convective hybridization ðDabiol {1Þ; the hybridizing
time dependencies (in the probe covering place) are almost identical to the average
value of the relative molar superficial concentration of created hybrid pairs on the
probe surface. For Dabiol $ 1, pure diffusive hybridization occurs. Microbiology.
Microorganisms.
Info: [B86].
Gerhard Friedrich Damkő hler (19081944), German physical chemist.
Geophysics and Ecology
9.2.9
415
Deep Parameter Pd
rffiffiffi
h
Pd 5
L
h (m) depth of surface water; L (m) characteristic length of surface gradient.
It expresses the relative depth of the surface water on an observed surface section. Geophysics and geomorphology. Ecology.
Info: [B65].
9.2.10 Depletion Number Dp
Dp 5
εd
5 Dp ðΨ; Φ; ΩÞ
εl
εd (kW, MJ year21) exergy depletion rate; εl (kW, MJ year21) exergy loss rate,
exergy destruction; Ψ (2) exergy cycling fraction number (p. 415); Φ (2) exergy
efficiency number (p. 416); Ω (2) renewed exergy fraction number (p. 422).
It expresses a dimensionless indicator of exergy consumption for the loss exergy
unit. The depletion number Dp is the function of three dimensionless indicators:
the exergy cycling fraction number Ψ (p. 415), the exergy efficiency number Φ
(p. 416) and the renewed exergy fraction number Ω (p. 422).
Info: [B25].
Catherine Preston Koshland (p. 416)
9.2.11 Exergy Cycling Fraction Number Ψ
Ψ 5 rm rq 5
mR Δε1
;
mC Δε2
where rm 5
mR
;
mC
rq 5
Δε1
Δε2
rm (2) mass recovery factor; rq (2) quality recovery factor; mR (kg s21) mass
flux of resources recovered from output of consumption process; mC (kg s21) mass flux of consumption process; Δε1 (kJ kg21) specific exergy difference of
recovered resources; Δε2 (kJ kg21) specific exergy difference of consumption
process.
It expresses the ratio of the velocity of the exergy return to sources to that transferred from the sources. Ecology.
Info: [B25].
Catherine Preston Koshland (see page 416).
416
Dimensionless Physical Quantities in Science and Engineering
9.2.12 Exergy Efficiency Number Φ
Φ5
ε1 2 ε2
ε1
ε1 (kW, MJ year21) exergy removal rate; ε2 (kW, MJ year21) exergy loss rate.
It expresses the ratio of the difference between transferred and lost exergies and
converted exergy or, alternatively, the ratio of the exergy, distributed into other
sources, to the transferred one. Ecology.
Info: [B25].
Catherine Preston Koshland (born 11.5.1950), American
engineer and ecologist.
She concentrates mainly on the problems of combustion
and arising emissions. Among other things, she is engaged
in research related to reducing the influence of temperature,
oxygen and other substances on the environment, especially
in relation to older industrial facilities, such as incinerators
and incineration furnaces, among others. She is engaged in
work related to increasing plant efficiency and to reducing
atmospheric contamination. She investigates the areas that
are the origins of toxic industrial by-products. Above all,
she tries to optimize industrial systems to achieve significant reductions of emissions.
9.2.13 Föpplvon Kármán Number NFK, γ
NFK 5
EL2
k
E (Pa) modulus of elasticity; L (m) characteristic length (radius of roundel,
virus etc.); k (Pa m2) flexural rigidity.
In the case of Newtonian fluids, it characterizes, for example, the behaviour of
the cytoplasm of erythrocytes flowing through a circular cross section of a capillary
tube. It also characterizes changes in spherical swirl walls, among other things. Its
value is NFKAh103; 108i for elastic walls. Medicine. Biology.
Info: [B82],[B68].
Theodore von Kármán (p. 67).
9.2.14 Froude Dam-Breach Number Frdb
qd
Frdb 5 pffiffiffiffiffiffi
gV
Geophysics and Ecology
417
qd (m2 s21) maximum outflow rate for unit width of breach; g (m s22) gravitational acceleration; V (m3) volume outflow from hydrograph.
It characterizes the outflow wave propagation from a breached dam.
Geophysics. Ecology. Floods.
Info: [B89].
William Froude (p. 63).
9.2.15 Froude River Inflow Number Frin
wi
Qi
5 pffiffiffiffiffiffiffi
Frin 5 pffiffiffiffiffiffiffi
;
g0 R
g0 R Ai
where g0 5
ΔRi
g
R0
wi (m s21) inflow velocity; g0 (m s22) corrected gravitational acceleration;
R (m) hydraulic radius of the underflow; Qi (m3 s21) inflow volume rate;
Ai (m2) inflow area; ΔRi (kg m23) density difference between inflowing river
water and surface lake water; R0 (kg m23) average density of water in the lake;
g (m s22) gravitational acceleration.
It characterizes the river water inflow into a lake or similar water system. It is
an important hydrodynamic criterion in describing solid particle sedimentation. The
critical value is Frin 5 1. For Frin , 1, inflow strokes of discharge occur.
Geophysics. Hydrodynamics. Ecology.
Info: [B52].
William Froude (see above).
9.2.16 Froude River Outflow Number Frout
Qout
Frout 5 pffiffiffiffiffiffiffiffiffi
gH 5
Qout (m3 s21) outflow volume rate; g (m s22) gravitational acceleration; H (m) total depth of lake.
It characterizes the lake water outflow into a river or other water system. It is
often applicable as an important hydrodynamic criterion in describing solid particle
sedimentation and dispersion. The critical value is Frout 5 1. Geophysics.
Hydrodynamics. Ecology.
Info: [B52].
William Froude (see above).
9.2.17 Heating Agrobiological Temperature Qheat
Qheat 5
cp Tτ
a
418
Dimensionless Physical Quantities in Science and Engineering
cp (J kg21 K21) specific heat capacity; T (K) temperature; τ (s) time;
a (m2 s21) thermal diffusivity of ground.
It characterizes temperature penetration into soil which depends on specific thermal capacity and diffusivity, and on action time. It is used in the cultivation of
agricultural soil using electrical field technology.
Info: [B113].
9.2.18 Human Inhalation Number Ninh
Ninh 5
DQ
D
5
m
Dexp
D (kg m23) personal dose or exposition to concentration; Q (m3 s21) volume
air flux in input; m (kg s21) mass flux intensity of polluted gas; Dexp (kg m23) fully mixed gas concentration.
It expresses the ratio of the inhaled concentration dose to the fully mixed gas
concentration. Pulmonology. Ecology.
Info: [B51].
9.2.19 Imberger Number Im
Im 5
h
1
h
1 2 ð1 2 e 2 Rk Þ 1 ;
L0
Rk
L1
where Rk 5 kh
h (m) reference depth (e.g. daily thermocline); L0, L1 (2) Monin-Obukhov
length for solar wave radiation and for remaining heat fluxes at the lake
surface (p. 21); Rk (2) function of surface water mixing; k (s21) damping
coefficient.
On a water surface, it expresses the surface water tendency to mix. The critical
value is Im 5 0.2. With Im 5 0.3, the tendency appears to cause stratification and
mixing at the surface. Geophysics. Ecology.
Info: [B52].
Jőrg Imberger (born 10.9.1942), Australian engineer and
limnologist.
He is engaged in research on fluid dynamics, especially
in mixing and streaming in lakes, at river mouths and at the
sea coast. His work consists of designing computer models
to investigate the movement of water with biochemical
activity. It concerns interactions between physical, biological and chemical processes. With the range of his work and
the standard of its results, he has contributed markedly
to the improvement of the environment in many countries
of the world.
Geophysics and Ecology
419
9.2.20 Internal Mixing in the Metalimnion RS
RS 5
τs
h2
5
Lak; where h1 , h2
τi
h2 2 h1
h1 (m) depth of the upper layer; h2 (m) depth of the lower layer; τ s (s) time
of internal wave period for reserved water system (e.g. lake or pond); τ i (s) time
of free wave period; Ts (K) temperature of surface water layer; Ti (K) temperature of lower water layer; Lak (2) lake number (p. 420).
It expresses the time of one strong wave period. With h2 . (h2 h1) and
Ts/Ti , 1, shear instability may be expected with increased mixing in the top part
of the fluid surface. With h2/(h1 h2) . 1 and Ts/Ti , 1,26 vehement oscillation
occurs (Lak , 1), which is connected to bottom water rising to the surface. The
critical value is RS 5 1. Geophysics. Ecology.
Info: [B52].
9.2.21 Joule Agrobiological Number Joargo
Joagro 5
L2 E2 γ
λT
L (m) characteristic length; E (V m21) electric field intensity; γ (S m21) specific electric conductance; λ (W m21 K21) thermal conductivity; T (K) temperature; ω, ωcr (s21) supply and critical frequency of electric field.
It expresses the ratio of the total energy of a low frequency ðω{ωcr Þ electric
field to the generated heat. It describes the amount of energy dispersed during the
process of soil cultivation using the electric field method, which deactivates various
detrimental microorganisms in the soil. Agrobiology.
Info: [B113].
James Prescott Joule (p. 298).
9.2.22 Knudsen Dispersion Number Kn
Kn 5
2λ 4Ma
5
d
Re
λ (m) mean free path; d (m) particle diameter; Ma (2) Mach dispersion
number (p. 421); Re (2) Reynolds dispersion number (p. 423).
This number expresses the ratio of the mean free path of molecules to the characteristic length (i.e. a particle diameter). It is a modification of the Knudsen number (1.) Kn (p. 69) for colloidal systems with dispersed solid or liquid particles.
Ecology. Meteorology.
Info: [A0].
420
Dimensionless Physical Quantities in Science and Engineering
Martin Hans Christian Knudsen (15.2.187127.5.1949),
Danish physicist and oceanographer.
He was engaged especially in research related to statistical mechanics, continuum mechanics and fluid mechanics.
Above all, he is known because of his studies in molecular
gas flow and for the development of the Knudsen chambers, which is the basic component of molecular epitaxial
systems. The problems connected with the Knudsen number
are related, for example, to dust particle movement in the
atmosphere, to satellite movement and to the aerodynamics
of aircraft wings.
9.2.23 Lake Number Lak
Lak 5
SSch ðH 2 ht Þ
3
w2 A2L ðH 2 hV Þ
SSch (m5 s22) Schmidt stability; H (m) total depth of the lake; ht (m) depth
of diurnal thermocline; w (m s21) water shear velocity due to wind; AL (m2) surface area of the lake; hV (m) height of water centre.
It expresses the ratio of the acting forces prohibiting mixing and the forces
delivering the energy for system mixing. In the expression, the first number (SSch)
expresses the stability, which depends on the water system size. With Lak . 1,
there is no flow of a lower water layer to the surface. With Lak , 1, the cold and
the depth cause a flow towards the surface. The critical value is reached with
Lak 5 1. Geophysics. Ecology.
Info: [B52]
9.2.24 Littoral Water Exchange Nlit, CV
1
τf
Vh3
Nlit 5
5
; where w 5
7
τc
w tan ϕL3 τ c
sffiffiffiffiffiffiffiffiffiffiffiffiffi
3 αqA gh
cR
τ f (s) flushing time; τ c (s) time over which night cooling acts; V (m3) volume of the lake; h (m) total depth of the lake; w (m s21) convective flow
velocity; ϕ (2) bottom slope angle in littoral area; L (m) characteristic
length (equivalent radius of the lake bottom); α (K21) thermal expansion coefficient of water; qA (W m22) surface heat flux; g (m s22) gravitational acceleration; c (J kg21 K21) specific heat capacity of water; R (kg m23) water density.
It characterizes the influence of the night cooling of shallow littoral waters on the
exchange of this colder water with the warmer water from superficial layers of the
water system in its deeper parts by free convection. The critical value is Nlit 5 1.
Geophysics and Ecology
421
When Nlit , 1, in the shallow littoral zone the cooled water will flow into deeper
waters of the lake because of natural (gravitational) flow, and the warmer water takes
its place from the superficial layer at the deeper areas of the lake. When Nlit . 1 in
the littoral zone, the night cooling destroys the thermal equilibrium of the water, but
this water exchange is not influenced significantly. Geophysics. Ecology.
Info: [B52]
9.2.25 Mach Dispersion Number Ma
Ma 5
wp 2 wf cf
wp, wf (m s21) velocity of particles (p) and fluid (f); cf (m s21) sound speed.
It expresses the ratio of the difference between particle and fluid velocities to
the sound velocity under equal conditions of temperature and pressure. Ecology.
Meteorology.
Info: [A0].
Ernst Mach (p. 73).
9.2.26 Manning Roughness Coefficient Nn, n
5 1
Nn 5
A3 S2 wini
2
QO3
A (m2) flow cross section; S (m m21) inclination of water level; wini (m s21) inlet velocity; Q (m3 m21 s21) water volume flux on 1 m ground width; O (m) wetted perimeter.
It characterizes the irrigation of furrowed clayey soil at diverse inflow velocities
and surface inclinations with various kinds of grain crops. Geophysics. Biosystems.
Info: [B104]
9.2.27 Péclet Solar Number Pesol
Pesol 5
H2
aτ
H (m) water height in solar reservoir; a (m2 s21) thermal diffusivity; τ (s) daytime duration of solar radiation.
It expresses the convective-to-conductive heat transfer ratio as water flows into
a solar reservoir. Ecology.
Info: [B48].
Jean Claude Eugène Péclet (p. 180).
422
Dimensionless Physical Quantities in Science and Engineering
9.2.28 Radiation Weighting Factor wrad
wrad 5
H
D
H (Sv 5 keV μm21) equivalent dose in sievert (Sv); D (Sv) absorbed dose by
the biological tissue.
The radiation weighting factor is used because some types of radiation are more
harmful biologically than others, provided the absorbed dose is equal. It characterizes the influence of various kinds of radiation on biological damage to body tissue. By means of the wrad, the dose absorbed by biological tissue can be
recalculated to an equivalent radiation dose by distinguishing the harmful effects of
diverse kinds of absorbed radiation. wrad 5 1 holds for photons and electrons,
wradAh5; 20i for neutrons, wrad 5 5 for protons, and wrad 5 20 for alpha particles.
Info: [C50].
9.2.29 Ratio of Spot Radius R
R5
r
h
r (m) spot radius; h (m) height of liquid layer.
It expresses the ratio of the spot radius to the fluid layer height. It represents a
geometric parameter, of which a large value corresponds to a large target, and vice
versa. Microbiology. Microorganisms.28
Info: [B39],[B86].
9.2.30 Renewed Exergy Fraction Number Ω
Ω5
εren
εren 1 εnon
εren (kW, MJ year21) exergy from renewed resources; εnon (kW, MJ year21) exergy from nonrenewable resources.
This number expresses the exergy from renewable sources which is multiplied
by the inverse value of the sum of the exergies from both renewable and nonrenewable sources. It influences the depletion number Dp (p. 415) by expressing supplies
to the system which have been derived from renewable sources. The renewable
resources in question are, for example, solar radiation, wind or tide energy, and the
supply sources include biomass, among other sources.
Info: [B25].
Catherine Preston Koshland (p. 416).
Geophysics and Ecology
423
9.2.31 Resuspension Number Nres, NRS
d 2 g0p LH
ws
pffiffiffiffiffiffi
5
wb
18ν CD hl wi
pffiffiffiffiffi
d 2 g0p
c D w i hl
; wb 5
;
where ws 5
18ν
LH
Nres 5
g0p 5
ΔRp
g
R
ws (m s21) sedimentation rate at the bottom; wb (m s21) shear velocity at the
bottom; d (m) effective sedimentation size; g0p (m s22) corrected gravitational
acceleration; L (m) characteristic length; H (m) total depth of the lake;
ν (m2 s21) kinematic viscosity; hl (m) depth of upper water layer; wi (m s21) phase speed of the internal wave; ΔRp (kg m23) density difference between the
particles and water; R (kg m23) water density; g (m s22) gravitational acceleration; CD (2) drag coefficient (p. 60).
This number is an important ecological criterion determining whether a sedimentation will settle on the bottom or be dispersed during the activity of waves in a lake
or other similar water systems. In compliance with the data in [C36], the value
Nres , 1 holds for the particle diameter d 5 0.1 mm, as does the value Nres , 1024
for the diameter d 5 1 μm. The critical value is Nres 5 1. Geophysics. Ecology.
Info: [A30],[B25].
9.2.32 Reynolds Dispersion Number Re
wp 2 wf d
4Ma
5
Re 5
ν
Kn
wp, wf (m s21) velocity of particles (p) and fluid (f); d (m) particle diameter;
ν (m2 s21) kinematic viscosity; Ma (2) Mach dispersion number (p. 421);
Kn (2) Knudsen number (1.) (p. 69).
It is a modification of the Reynolds number Re (p. 81) for colloidal systems with
dispersed solid or liquid particles. Ecology. Meteorology.
Info: [A0].
Osborne Reynolds (p. 82).
9.2.33 Schmidt Dispersion Number Sc
ν
nf λd4
5
4
D
2 21
ν (m s ) kinematic viscosity; D (m2 s21) diffusivity; nf (m23) numerical
gas density; λ (m) mean free path; d (m) particle diameter.
It is a modification of the Schmidt number Sc (p. 263) for colloidal systems with
dispersed solid or liquid microparticles. Ecology. Meteorology.
Info: [A0].
Ernst Schmidt (p. 264).
Sc 5
424
Dimensionless Physical Quantities in Science and Engineering
9.2.34 Sediment Number Eis
qb
Eib 5 pffiffiffiffiffiffiffiffiffi
RgDD
ð1Þ;
qt
Eit 5 pffiffiffiffiffiffiffiffiffi
RgDD
ð2Þ;
where qt 5 qb 1 qs ;
R5
Rs
21
R
qb (m2 s21) volume bedload transport rate per unit width of cross-section diameter; qt (m2 s21) total volume bed material transport rate per unit width;
qs (m2 s21) volume suspended load transport rate per unit width; R (2) sediment submerged specific density; g (m s22) gravitational acceleration; D (m) characteristic sediment size; Rs (kg m23) sediment density; R (kg m23) water
density.
It expresses the sedimentation process with water flow in rivers, channels and
other water courses. It is also called the Einstein sedimentation number.
Geophysics. Ecology. Morphodynamics.
9.2.35 Smith Number Smi
Smi 5
εwτ
L
ε (2) scale factor, its value ε{1; w (m s21) effective rain velocity; τ (s) time connected with measurable surface changes; L (m) characteristic length.
It characterizes the influence of rain in changing the character of a land surface.
Geophysics and geomorphology. Ecology.
Info: [B65].
9.2.36 Sperm Number Sp
Sp 5 L
ηω1
4
k
L (m) characteristic fibre length; η (Pa s) dynamic viscosity; ω (s21) angular
frequency of motion; k (N m21) linear stiffness.
This number expresses the viscous-to-elastic tensions ratio on microbiological
fibres, such as bacteria and many eukaryotic cells moving by means of a hair
structure (flagellum). For the so-called simple swimmers, internal elasticity dominance holds with a low value of Sp. With high values of Sp, the influence of viscous friction predominates. Microbiology. Microphysics.
Info: [B32].
Geophysics and Ecology
425
9.2.37 Tank Solar Number Ntank
Ntank 5
Rcp V
Sατ
R (kg m23) water density; cp (J kg21 K21) specific heat capacity of water;
V (m23) water volume in reservoir; S (m2) surface area of reservoir;
α (W m22 K21) heat transfer coefficient; τ (s) daytime duration of solar
radiation.
It expresses the ratio of the amount of water in an exchanger to the absorbing
area. It is an important quality in designing of solar collector equipment utilizing
natural water convection. Ecology.
Info: [B48].
9.2.38 Timescale Parameter Pτ
Pτ 5
L
wΔτ
L (m) declivity length of surface; w (m s21) mean declivity velocity of surface
water; Δτ (s) time interval of surface change.
It characterizes the development changes caused by hydrodynamic action of surface water on a landscape. Geophysics and geomorphology. Ecology.
Info: [B65].
9.2.39 Volumetric Concentration of Target DNA C 0 0
C00 5
h
C0
Hmax
h (m) height of liquid layer; Hmax (mol m22) molar additional value of volume
concentration of free bonded parts in spot place on the probe; C0 (mol m23) initial volume concentration on target DNA in the solution.
It characterizes the ratio of the number of DNA targets, represented over the
spots, to the number of tied up sides in a spot in the instant τ 5 0. Explicitly, it
expresses the parameter determining the need to demarcate a target from outside
of the spot surface. Its value acts on the relative molar surface concentration of
formed hybrid pairs. Microbiology. Microorganisms.
Info: [B86].
References
A
Books and Proceedings
[A0] Ahmadi G. Particle transport, deposition and removal. In: Proc. ME 437/ME 537,
Dept. of Mechanical and Aeronautical Engineering. Clarkson University, Potsdam,
New York. 1997. 8 p.
[A1] Bauer Chr. Low Reynolds number k and empirical transition models for oscillatory
pipe flow and heat transfer. NASA TM 109785. Ph.D. Thesis. Cleveland State
University, 1993. 99 p.
[A2] Becker H.A. Dimensionless parameters: Theory and methodology. New York: Wiley,
1976. 130 p.
[A3] Berthier J., Silberzan P. Microfluidics for biotechnology. Chapter 1: Dimensionless
numbers in microfluidics. Second edition. Boston: Artech House, 2010. 483 p.
[A4] Bolz R.E., Tuve G.L. (editors). CRC Handbook of chemistry and physics. Second edition. Cleveland: CRC Press, 1973, pp. 1028 1040.
[A5] Boonpongmanee T. Numerical and experimental investigation of heat and mass transfer in rotating systems. Ph.D. Thesis. Case Western Reserve University, Department
of Mechanical and Aerospace Engineering, 2005. 191 p.
[A6] Bosnjakovitz F. Engineering thermodynamics (in German). Dresden, Leipzig: Verlag
Th. Steinkopff, 1960.
[A7] Brennen Ch.E. Cavitation and bubble dynamics. Oxford: Oxford University Press,
1995. 410 p.
[A8] Brennen Ch.E. Fundamentals of multiphase flows. Second edition. Cambridge:
Cambridge University Press, 2005. 410 p.
[A9] Collyer A.A., Clegg D.W. Rheological measurement. Second edition. London:
Chapman and Hall, 1998. 779 p.
[A10] Croft D.R., Lilley D.G. Heat transfer calculations using finite difference equations.
London: Applied Science Publishers, 1977. 283 p.
[A11] Courant R., Hilbert D. Methods of mathematical physics. Vol. II Partial differential
equations. New York: Wiley, 1962. 830 p.
[A12] Delattre N., Moretto P. In: Estivalet M., Brisson P. (editors). The engineering of sport 7.
Chapter: A new dimensionless number for dynamic similarity during running (P250).
Vol. 2. Paris: Springer-Verlag, 2008, pp. 535 542.
[A13] Djakonov G.K. Questions in similarity theory of the physico-chemical processes (in
Russian). Moscow: AN SSSR, 1956. 209 p.
[A14] Douglas J.F. An introduction to dimensional analysis for engineers. Appendix 2:
Dimensionless parameters. Toronto: Pitman Publishing Corp., 1969. 134 p.
[A15] Forsnes V.C., Ulrich R.D. A literature review and discussion of liquid particle
breakup in gas streams. China Lake (California): Naval Weapons Center, 1968. 53 p.
[A16] Green D.W., Perry R.H. (editors). Perry’s chemical engineers’ handbook. Eighth
edition. New York: McGraw-Hill, 2007.
[A17] Honner M. Microthermomechanics in new technologies (in Czech). Ph.D. Thesis.
Pilsen: University of West Bohemia, Faculty of Applied Sciences, 1999. 124 p.
428
References
[A18] Ho sek J. Application of similarity theory on flying animals (in Czech). Prague:
CSAV, 1976. Study of CSAV No. 6., 97 p.
[A19] Ingard K.U. Fundamentals of waves and oscillations. Cambridge: Cambridge
University Press, 1988. 612 p.
[A20] Ipsen D.C. Units, dimensions and dimensionless numbers. New York: McGraw-Hill,
1960. 236 p.
[A21] Jacob J.D. Encyclopedia of fluid mechanics. Kentucky: University of Kentucky,
Lexington, 2003. 59 p.
[A22] Keulegan G.H. Tidal flow in entrances. Vicksburg: US Army Corps of Engineers,
Committee on Tidal Hydraulics, 1967. Tech. Bull. 14.
[A23] Kuneš J. Modeling of thermal processes (in Czech). Prague: SNTL, 1989. 425 p.
[A24] Kuneš J., Vavroch O., Franta V. Fundamentals of modeling (in Czech). Prague:
SNTL, 1989. 264 p.
[A25] Kutateladze S.S., Ljachovskij D.N., Permjakov V.A. Modeling of thermal apparatus
(in Russian). Moscow: Energija, 1968. 347 p.
[A26] Land N.S. Compilation of nondimensional numbers. NASA SP-274, 1972. 123 p.
[A27] Landsberg H.E., Van Mieghem J. Advances in geophysics. Vol. 4. New York:
Academic Press, 1958. 456 p.
[A28] Lindeman M.A. Microcalorimetry and the transition-edge sensor. Ph.D. Thesis.
University of California at Davis, Lawrence Livermore National Laboratory, 2000.
216 p.
[A29] Massey B.S. Units, dimensional analysis and physical similarity. London: Van
Nostrand Reinhold Company, 1971. 140 p.
[A30] McCave I.N. Erosion, transport and deposition of fine grained marine sediments.
In: Stow D.A.V., Piper D.J.W. (editors). Fine grained sediments: Deep sea processes
and facies. Oxford: Geological Society of London, Blackwell Scientific Publication,
1984, pp. 15:35 15:69. Special publication.
[A31] McGraw-Hill dictionary of scientific and technical terms. Sixth edition. New York:
McGraw-Hill, 2003. 2380 p.
[A32] McGraw-Hill encyclopedia of science and technology. Tenth edition. New York:
McGraw-Hill, 2007. 15600 p.
[A33] Minakovskij V.M. Generalized variables in transfer theory (in Russian). Kiev: Vyscaja
skola, 1978. 181 p.
[A34] Moore D.F. Principles and applications of tribology. First edition. Oxford: Pergamon
Press, 1975. 388 p.
[A35] Parker S.P. (editor). Fluid mechanics source book. New York: McGraw-Hill, 1987.
274 p.
[A36] Pope S.B. Turbulent flows. Cambridge: Cambridge University Press, 2000.
[A37] Roshko A. On the development of turbulent wakes from vortex streets. Ph.D. Thesis.
Pasadena: Caltech, 1952. 96 p.
[A38] Roza-Butler N. The interplay between seasonal variations and latitude: Implications
for lake management in South America. Bc. Thesis. University of Western Australia,
Department of Environmental Engineering, 2001. 94 p.
[A39] Schuring D.J. Scale models in engineering. Fundamentals and application. Oxford:
Pergamon Press, 1977. 299 p.
[A40] Shive J.N., Weber R.L. Similarities in physics. Bristol: Adam Hilger Ltd., 1982. 275 p.
[A41] Soloviev A., Lukas R. The near-surface layer of the ocean. Structure, dynamics and
application. Dordrecht: Springer, 2005. Atmospheric and Oceanographic Science
Library, 556 p.
References
429
[A42] Sundén B., Faghri M. (editors). Modelling of engineering heat transfer phenomena.
Part 2. Boston, Southampton: Computational Mechanics Publications, 2003. 368 p.
[A43] Szirtes T. Applied dimensional analysis and modeling. New York: McGraw-Hill,
1997. 820 p.
[A44] Tretjăcenko G.N. Modeling at strength determination of constructions (in Russian).
Kiev: Naukova dumka, 1979. 230 p.
[A45] Tropea C., Yarin A.L., Foss J.F. (editors). Springer handbook of experimental fluid
mechanics. New York: Springer Verlag, 2007.
[A46] Tzou D.Y. Thermal shock phenomena under high rate response in solids. In: Tien C.L.
(editor). Annual review of heat transfer. Chapter 3, Vol. IV. Washington DC: Hemisphere,
1991, pp. 111 185.
[A47] Warrick A.W. (editor). Soil physics companion. Cleveland: CRC Press, 2002. 389 p.
[A48] Yalin M.S. Theory of hydraulic model. London: Macmillan, 1971. 266 p.
[A49] Zlokarnik M. Dimensional analysis and scale-up in chemical engineering. Berlin:
Springer Verlag, 1991. 176 p.
[A50] Zlokarnik M. Scale-up (in German). New York: Wiley VCH, 2000. 224 p.
B
Articles
[B1] Ainshtein V.G. Generalization of penetration theory. Chemistry and Technology of
Fuels and Oils. 1996, Vol. 32, No. 1, pp. 18 23.
[B2] Alexandrou A.N. et al. Compression of materials with yield stress. In: Fifth GRACM
International Congress on Comp. Mechanics, Limassol, 2005.
[B3] Andŕe S., Serra J. et al. Parameter estimation of thermo-optical glass properties.
In: Internat. Conf. on Inverse Problems in Engn. Rio de Janeiro, Brazil, 2002.
[B4] Bakunin O.G. Percolation regime of turbulent transport in weak compressible flows.
Physica A: Statistical Mechanics and its Applications. 2005, Vol. 351, No. 2 4,
pp. 241 250.
[B5] Balmor R.T. Steady and unsteady isochoric simple extensional flows. Journal of
Non-Newtonian Fluid Mechanics. 1977, Vol. 2, No. 4, pp. 307 322.
[B6] Barenblatt G.I., Chorin A.J., Prostokishin V.M. A note concerning the Lighthill
‘sandwich model’ of tropical cyclones. Proceedings of the National Academy of
Sciences of the United States of America. 2005, Vol. 102, No. 32, pp. 11148 11150.
[B7] Bash C.E., Patel Ch.D., Sharma R.K. Inkjet assisted spray cooling of electronics.
In: Proc. of IPACK03 Conference, July 6 11, 2003, Maul, Hawaii, USA. pp. 1 2.
[B8] Bauer H.F., Eidel W. Nonlinear liquid oscillations in spherical systems under zerogravity. Acta Mechanica. 1987, Vol. 65, No. 1 4, pp. 107 126.
[B9] Beismann H., Wilhelmi H., Baillères H., Spatz H.-Ch, Bogenrieder A., Spec T.
Brittleness of twig bases in the genus Salix: Fracture mechanics and ecological relevance. Journal of Experimental Botany. 2000, Vol. 51, No. 344, pp. 617 633.
[B10] Berger S.A., Talbot L., Yao L.S. Flow in curved pipes. Annual Review of Fluid
Mechanics. 1983, Vol. 15, pp. 461 512.
[B11] Boucher D.F., Alves G.E. Dimensionless numbers 1. Chemical Engineering Progress.
1959, Vol. 55, No. 9, pp. 55 64.
[B12] Boucher D.F., Alves G.E. Dimensionless numbers 2. Chemical Engineering Progress.
1963, Vol. 59, No. 8, pp. 75 83.
[B13] Cagnoli B., Manga M. Granular mass flows and Coulomb’s friction in shear cell
experiments: Implications for geophysical flows. Journal of Geophysical Research.
2004, Vol. 109, No. F04005, p. 12.
430
References
[B14] Calderón O.G., Melle S. Dynamics of simple magnetorheological suspensions under
rotating magnetic fields with modulated Mason number. Journal of Physics D:
Applied Physics. 2002, Vol. 35, No. 20, pp. 2492 2498.
[B15] Cao X., Tobita Y. Drag correlations for a hot particle/droplet with vapor film.
Journal of Nuclear Science and Technology. 2001, Vol. 38, No. 9, pp. 721 728.
[B16] Carper H.J., Ku P.M. Thermal and scuffing behavior of disks in sliding-rolling contact. ASLE Transactions. 1975, Vol. 18, No. 1, pp. 39 47.
[B17] Catchpole J.P., Fulford G. Dimensionless groups. Industrial and Engineering
Chemistry. 1966, Vol. 58, No. 3, pp. 46 60.
[B18] Chapman Ch. The Asian tsunami in Sri Lanka: A personal experience. Eos,
Transactions American Geophysical Union. 2005, Vol. 86, No. 2, p. 13.
[B19] Chapple D., Kresta S.M., Afacan A. The effect of impeller and tank geometry on
power number for a pitched blade turbine. Chemical Engineering Research and
Design. 2002, Vol. 80, No. A4, pp. 364 372.
[B20] Chart of dimensionless numbers (poster). Omega Engineering, 1995.
[B21] Chen W.J. Bearing dynamic coefficients of flexible-pad journal bearings. Tribology
Transactions. 1995, Vol. 38, No. 2, pp. 253 260.
[B22] Chen X.W., Li Q.M. Shear plugging and perforation of ductile circular plates struck
by a blunt projectile. International Journal of Impact Engineering. 2003, Vol. 288,
No. 5, pp. 513 536.
[B23] Choueiri E. On the thrust of self-field MPD thrusters. In: 25th International Electric
Propulsion Conference, Cleveland, Ohio, August 24 28, 1997.
[B24] Chu H.-S., Lin S., Lin Ch.-H. Non-Fourier heat conduction with radiation in an
absorbing, emitting and isotropically scattering medium. Journal of Quantitative
Spectroscopy and Radiative Transfer. 2002, Vol. 73, No. 6, pp. 571 582.
[B25] Connelly L., Koshland C.P. Exergy and industrial ecology. Part 1: An exergy-based
definition of consumption and a thermodynamic interpretation of ecosystem evolution. Exergy, An International Journal. 2001, Vol. 1, No. 3, pp. 146 165.
[B26] Connelly L., Koshland C.P. Exergy and industrial ecology. Part 2: A non-dimensional
analysis of means to reduce resource depletion. Exergy, An International Journal.
2001, Vol. 1, No. 4, pp. 234 255.
[B27] Cunha F.R., Sobral Y.D. Characterization of the physical parameters in a process of
magnetic separation and pressure-driven flow of a magnetic fluid. Physica A.
Statistical Mechanics and its Applications. 2004, Vol. 343, pp. 36 64.
[B28] Delaplace G., Guérin R., Leuliet J.-C. Dimensional analysis for planetary mixer:
Modified power and Reynolds numbers. AIChE Journal. 2005, Vol. 51,
pp. 3094 3100.
[B29] Donelan J.M., Kram R. Exploring dynamic similarity in human running using simulated reduced gravity. The Journal of Experimental Biology. 2000, Vol. 203, No. 16,
pp. 2405 2415.
[B30] Dorfman A. Transient heat transfer between a semi-infinite hot plate and flowing
cooling liquid film. Journal of Heat Transfer
Transactions of the ASME. 2004,
Vol. 126, No. 2, pp. 149 154.
[B31] Dowson D., Song E.S., Tailor C.M. Non-dimensional groups in elastohydrodynamic
lubrication. In: Thin Films in Tribology. Proc. 19th Leeds-Lyon Symposium, Leeds,
UK, 8 11, September 1992. Elsevier, 1993. pp. 237 242.
[B32] Dreyfus R., Baudry J., Roper M.L., Fermigier M., Stone H.A., Bibette J. Microscopic
artificial swimmers. Nature. 2005, Vol. 437, pp. 862 865.
References
431
[B33] Džagparov U.S. Teplootvod v formu v period zaplněnija. Izvěstija. VUZ
Mašinostrojenije. 1974, Vol. 3, pp. 137 141.
[B34] Eggleton C.D., Ferziger J.H., Pulliam T.H. Moderate Reynolds number entry flow and
boundary-layer approximations for a viscoelastic fluid. Physics Fluids. 1994, Vol. 6,
No. 2, pp. 700 709.
[B35] Escobar R.A., Amon C.H. Influence of phonon dispersion on transient thermal
response of silicon-on-insulator transistors under self-heating conditions. Journal of
Heat Transfer Transactions of the ASME. 2007, Vol. 129, pp. 790 797.
[B36] Franta V. Hybrid simulation of multidimensional temperature fields using Monte
Carlo method (in Czech). Mechanical Engineering. 1979, Vol. 30, No. 1, pp. 62 74.
[B37] Fuersbach P.W., Eisler G.R. Determination of material properties for welding models
by means of arc weld experiments. In: Proc. Sixth Int. Trends in Welding Research,
Pine Mountain, Georgia, April, 15 19, 2002. pp. 1 5.
[B38] Gao H.J., Ji B.H. Modeling fracture in nanomaterials via a virtual internal bond
method. Engineering Fracture Mechanics. 2003, Vol. 70, No. 14, pp. 1777 1791.
[B39] Ghassemieh E., Versteeg H.K, Acar M. Effect of nozzle geometry on the flow characteristics of hydroentangling jets. Textile Research Journal. 2003, Vol. 73, No. 5,
pp. 444 450.
[B40] Ghosh M.K., Nagraj A. Rotordynamic characteristics of a multilobe hybrid journal
bearing in turbulent lubrication. Proceedings of the Institution of Mechanical
Engineers. Part J: Journal of Engineering Tribology. 2004, Vol. 218, No. 1,
pp. 61 68.
[B41] Gur O., Rosen A. Propeller performance at low advance ratio. Journal of Aircraft.
2005, Vol. 42, No. 2, pp. 435 441.
[B42] Gyenge E.L. Dimensionless numbers and correlating equations for analysis of the
membrane-gas diffusion electrode assembly in polymer electrolyte fuel cells. Journal
of Power Sources. 2005, Vol. 152, pp. 105 121.
[B43] Herrera L., Pavón D. Hyperbolic theories of dissipation: Why and when do we need
them? Physica A. Statistical Mechanics and its Applications. 2002, Vol. 307,
No. 1 2, pp. 121 130.
[B44] Holzbecher E. On the relevance of oscillatory convection regimes in porous media
review and numerical experiment. Computers and Fluids. 2001, Vol. 30, No. 2,
pp. 189 209.
[B45] Honner M., Kuneš J. On the wave diffusion and parallel nonequilibrium heat conduction. Journal of Heat Transfer
Transaction of the ASME. 1999, Vol. 121, No. 3,
pp. 702 707.
[B46] Hošek J. Strouhal number, its physical sense and importance (in Czech). Mechanical
Engineering. 1968, Vol. 19, No. 5, pp. 25 55.
[B47] Hu Y.Q. Application of response number for dynamic plastic response of plates subjected to impulsive loading. International Journal of Pressure Vessels and Piping.
2000, Vol. 77, No. 12, pp. 711 714.
[B48] Huang B.-J. Similarity theory of solar water heater with natural circulation. Solar
Energy. 1980, Vol. 25, No. 2, pp. 105 116.
[B49] Huang C.H. Pasquill’s influence: On the evaporation from various liquids into the
atmosphere. Journal of Applied Meteorology. 1997, Vol. 36, No. 8, pp. 1021 1026.
[B50] Huang R., Prévost J.H., Huang Z.Y., Suo Z. Channel-cracking of thin films with
extended finite element method. Engineering Fracture Mechanics. 2003, Vol. 70,
No. 18, pp. 2513 2526.
432
References
[B51] Hyun S., Kleinstreuer C. Numerical simulation of mixed convection heat and mass
transfer in a human inhalation test chamber. International Journal of Heat and Mass
Transfer. 2001, Vol. 44, No. 12, pp. 2247 2260.
[B52] Imberger J. Characterizing the dynamical regimes of a lake. In: Physical Mathematics
Seminar. Massachusetts Institute of Technology, Cambridge, December 4, 2001.
[B53] Iverson R.M. The physics of debris flows. Reviews of Geophysics. 1997, Vol. 35, No. 3,
pp. 245 296.
[B54] Iverson R.M., Denlinger R.P. Flow of variably fluidized granular masses across threedimensional terrain. 1. Coulomb mixture theory. Journal of Geophysical Research.
2001, Vol. 106, No. B1, pp. 537 552.
[B55] Kadaksham J., Batton J., Singh P., Aubry N. Microfluidic platform for manipulation
of micro- and nanoscale particles. In: Proc. IMECE 03. 2003 ASME Int. Mechanical
Engineering Congress, Washington DC.
[B56] Kandlikar S.G. Flow boiling in microchannels. Non-dimensional groups and heat
transfer mechanisms. In: Congres de la Socie´te´ Francois de Thermique 2003,
Grenoble, June 4, 2003. pp. 2 6.
[B57] Kandlikar S.G. Heat transfer mechanisms during flow boiling in microchannels. Journal
of Heat Transfer Transactions of the ASME. 2004, Vol. 126, No. 1, pp. 8 16.
[B58] Käpylä P.J., Korpi M.J., Ossendrijver M., Tuominen I. Estimates of the Strouhal number from numerical models of convection. Astronomische Nachrichten. 2005,
Vol. 326, pp. 186 189.
[B59] Käpylä P.J., Korpi M.J., Ossendrijver M., Tuominen I. Local models of stellar convection III. The Strouhal number. Astronomy and Astrophysics. 2006, Vol. 448,
No. 2, pp. 433 438.
[B60] Kobus C.J. Investigation of natural convection heat transfer from uniformly heated
(isoflux), thin, stationary circular disks at arbitrary angles of inclination. Experimental
Thermal and Fluid Science. 2007, Vol. 31, No. 3, pp. 191 195.
[B61] Kong M., Partoens B., Peeters F.M. Structural, dynamical and melting properties of
two-dimensional clusters of complex plasmas. New Journal of Physics. 2003, Vol. 5,
pp. 23.1 23.17.
[B62] Kubat R. Moving on Mars. An overview (compiled unpublished sources from Ph.D.
thesis draft of J. Pratt (2000) and from personal observations of test subjects aboard
NASA KC-135).
[B63] Kuneš J., Krejčı́ L., Honner M. Wave heat conduction in technologies. Mechanical
Engineering. 1996, Vol. 47, No. 6, pp. 345 404.
[B64] Kuzhir P., López M.T., Bossis G. Magnetorheology of fiber suspensions. II. Theory.
Journal of Rheology Transactions of the Society of Rheology. 2009, Vol. 53, No. 1,
pp. 127 151.
[B65] Lawrence D.S.L. Physically based modeling and the analysis of landscape developments. In: The Scientific Nature of Geomorphology. Proc. 27th Binghamton
Symposium, September, 1996. Wiley, 1996. pp. 273 288.
[B66] Li Q.M., Chen X.W. Dimensionless formulae for penetration depth of concrete target
impacted by a non-deformable projectile. International Journal of Impact
Engineering. 2003, Vol. 28, No. 1, pp. 93 116.
[B67] Li Q.M., Jones N. On dimensionless numbers for dynamic plastic response of structural members. Archive of Applied Mechanics. 2000, Vol. 70, No. 4, pp. 245 254.
[B68] Lidmar J., Mirny L., Nelson D.R. Virus shapes and buckling transitions in spherical
shells. Physical Review. E: Statistical, Nonlinear, and Soft Matter Physics. 2003,
Vol. 68, No. 5, p. 10.
References
433
[B69] Loudon C.A., Tordesillas A. The use of dimensionless Womersley number to characterize the unsteady nature of internal flow. Journal of Theoretical Biology. 1998,
Vol. 191, No. 1, pp. 63 78.
[B70] Mahmud S., Fraser R.A. Mixed convection-radiation interaction in a vertical porous
channel: Entropy generation. Energy. 2003, Vol. 28, No. 15, pp. 1557 1577.
[B71] Majdalani J., van Moorhem W.K. Multiple-scales solution to the acoustic boundary
layer in solid rocket motors. Journal of Propulsion and Power. 1997, Vol. 13, No. 2,
pp. 186 193.
[B72] Makinde O.D., Gbolagade A.W. Second law analysis of incompressible viscous flow
through an inclined channel with isothermal walls. Romanian Journal of Physics.
2005, Vol. 50, No. 9 10, pp. 923 930.
[B73] Mart Y., Aharonov E., Mulugeta G., Ryan W., Tentler T., Goren L. Analogue modelling of the initiation of subduction. Geophysical Journal International. 2005, Vol.
160, No. 3, pp. 1081 1091.
[B74] Melle S., Calderon O.G., Rubio M.A., Fuller G.G. Microstructure evolution in magnetorheological suspensions governed by Mason number. Physical Review E. 2003,
Vol. 68, No. 4, p. 11.
[B75] Metselaar H.S.C., Kerkwijk B., Mulder E.J., Verweij H., Schipper D.J. Wear of
ceramic due to thermal stress: A thermal severity parameter. Wear. 2001, Vol. 249,
No. 10 11, pp. 962 970.
[B76] Minkowycz W.J., Haji-Sheikh A., Vafai K. On departure from local thermal equilibrium in porous media due to a rapidly changing heat source: The Sparrow number.
International Journal of Heat and Mass Transfer. 1999, Vol. 42, No. 18,
pp. 3373 3385.
[B77] Moes H. Optimum similarity analysis with applications to elastohydrodynamic lubrication. Wear. 1992, Vol. 159, No. 1, pp. 57 66.
[B78] Molski A. On the use of the Knudsen number and aerosol Knudsen number in the
kinetics of Brownian coagulation of aerosols. Colloid and Polymer Science. 1984,
Vol. 262, No. 5, pp. 403 405.
[B79] Moreno-Atanasio R., Ghadiri M. Mechanistic analysis and computer simulation of
impact breakage of agglomerates: Effect of surface energy. Chemical Engineering
Science. 2006, Vol. 61, No. 8, pp. 2476 2481.
[B80] Ng S.L., Brennen C. Experiments on the dynamic behavior of cavitating pumps.
Journal of Fluids Engineering. 1978, Vol. 100, No. 2, pp. 166 176.
[B81] Niyama E., Anzai K. Solidification velocity and temperature-gradient in infinitely
thick alloy castings. Materials Transactions, JIM. 1995, Vol. 36, No. 1, pp. 61 64.
[B82] Noguchi H., Gompper G. Shape transitions of fluid vesicles and red blood cells in
capillary flows. Proceedings of the National Academy of Sciences of United States of
America. 2005, Vol. 102, No. 40, pp. 14159 14164.
[B83] Ortiz C., Rodriguez J., Vazquez C. Dimensional analysis of biological systems. In:
Congress of Biofluid Dynamics of Human Systems at University of Puerto Rico,
Mayagũez, July 10, 2004.
[B84] Ozawa S., Sasajima Y., Heermann D.W. Monte Carlo simulations of film growth.
Thin Solid Films. 1996, Vol. 272, No. 2, pp. 172 183.
[B85] Pantokratoras A. Effect of viscous dissipation in natural convection along a heated
vertical plate. Applied Mathematical Modelling. 2005, Vol. 29, No. 6, pp. 553 564.
[B86] Pappaert K., Desmet G. A dimensionless number analysis of the hybridization process
in diffusion- and convection-driven DNA microarray systems. Journal of
Biotechnology. 2006, Vol. 123, No. 4, pp. 381 396.
434
References
[B87] Passoth E., Kudrna P., Csambal C., Behnke J.F., Tichý M., Helbig V. An experimental
study of plasma density determination by a cylindrical Langmuir probe at different
pressures and magnetic fields in a cylindrical magnetron discharge in heavy rare gases.
Journal of Physics D: Applied Physics. 1997, Vol. 30, pp. 1763 1777.
[B88] Peratt A.L. Advances in numerical modeling of astrophysical and space plasmas.
Astrophysics and Space Science. 1997, Vol. 242, No. 1/2, pp. 93 163.
[B89] Ponce V.M., Rao Y.R.S., Mansury N.M. Time of opening of irrigation canal gates.
Journal of Hydraulic Engineering. 1999, Vol. 125, No. 9, pp. 979 980.
[B90] Ponce V.M., Taher-shamsi A., Shetty A.V. Dam-breach flood wave propagation
using dimensionless parameters. Journal of Hydraulic Engineering. 2003, Vol. 129,
No. 10, pp. 777 782.
[B91] Popov V.L., Psakhie S.G., Shilko E.V. et al. Friction coefficient in ‘rail wheel’
contacts as a function of material and loading parameters. Physical Mesomechanics.
2002, Vol. 5, No. 3, pp. 17 24.
[B92] Povilaitienė I., Kamaitis I.Z., Podagėlis I. Influence of gauge width on rail side wear
on track curves. Journal of Civil Engineering and Management. 2006, Vol. 12,
No. 3, pp. 255 260.
[B93] Puri P., Kythe P.K. Discontinuities in velocity gradients and temperature in the
Stokes’ first problem with nonclassical heat conduction. Quarterly of Applied
Mathematics. 1997, Vol. 55, No. 1, pp. 167 176.
[B94] Regier A.A. The use of scaled dynamic models in several aerospace vehicle studies.
In: ASME Colloquium on the use of models and scaling in simulation of shock and
vibration. Philadelphia, 1963.
[B95] Riehl R.R., Ochterbeck J.M., Seleghim P. Effect of condensation in microchannels
with a porous boundary. Journal of the Brazilian Society of Mechanical Sciences.
2002, Vol. 24, No. 3, pp. 1 31.
[B96] Rodd L.E. et al. The inertio-elastic planar entry flow of low-viscosity elastic fluids
in micro-fabricated geometries. Journal of Non-Newtonian Fluid Mechanics. 2005,
Vol. 129, No. 1, pp. 1 22.
[B97] Rodd L.E., Scott T.P., Boger D.V, et al. Planar entry flow of low viscosity elastic
fluids in micro-fabricated geometries. In: Proc. 14th Int. Congr. on Rheology,
August 22 27, 2004, Seoul, Korea.
[B98] Rothstein J.P., McKinley G.H. Non-isothermal modification of purely elastic flow
instabilities in torsional flows of polymeric fluids. Physics of Fluids. 2001, Vol. 13,
No. 2, pp. 382 396.
[B99] Roy S., Raju R., Chuang H.F., Cruden B.A., Meyyppan M. Modeling gas flow
through microchannels and nanopores. Journal of Applied Physics. 2003, Vol. 93,
No. 8, pp. 4870 4879.
[B100] Sahoo P., Roy Chowdhury S.K.R. Normal impact of rough surfaces in presence of
adhesion. Tribology International. 2004, Vol. 37, No. 8, pp. 667 675.
[B101] Sanchez-Lavega A. A similarity approach to the atmospheric dynamics of giant
extrasolar planets and brown dwarfs. Astronomy and Astrophysics. 2001, Vol. 377,
No. 1, pp. 354 360.
[B102] Sarfatti J. A fast coherence and a secret bond. US Navy surface warfare research on
experimental unified field phenomena. Advanced intelligence Agency, 2000. 10 p.
[B103] Sánchez A.L., Lépinette A., Bollig M., Liňán A., Lázaro B. The reduced kinetic
description of lean premixed combustion. Combustion and Flame. 2000, Vol. 123,
No. 4, pp. 436 464.
References
435
[B104] Sepaskhah A.R., Bondar H. Estimation of manning roughness coefficient for bare
and vegetated furrow irrigation. Biosystems Engineering. 2002, Vol. 82, No. 3,
pp. 351 357.
[B105] Shen H.H. Granular shear flows
constitutive relations and internal structures.
In: 15th ASCE Engineering Mechanics Conference, June 2 5, 2002, Columbia
University, New York.
[B106] Sheu L.-J., Lin J.-D., Chen J.-R. Numerical analysis on the hot spot in reactive
chemical storage. Journal of Loss Prevention in the Process Industries. 1999,
Vol. 12, No. 2, pp. 125 136.
[B107] Smith D.S., Williams D.T. Selection of sediment transport relations: Part II. Ranges
of dimensionless numbers. In: Proc. North American Water and Environment
Congress ’96, June 22 28, 1996, Anaheim, California, USA. ASCE
American
Society of Civil Engineers, 1996. pp. 2837 2842.
[B108] Steinberg B.E., Glass L., Shrier A., Bub G. The role of heterogeneities and intercellular coupling in wave propagation in cardiac tissue. Philosophical Transactions of
the Royal Society A: Mathematical, Physical and Engineering Sciences. 2006,
Vol. 364, No. 1842, pp. 1299 1311.
[B109] Stevenson P. Dimensional analysis of foam drainage. Chemical Engineering
Science. 2006, Vol. 61, No. 14, pp. 4503 4510.
[B110] Suo Z., Vlassak J., Wagner S. Micromechanics of macroelectronics. China
Particuology. 2005, Vol. 3, No. 6, pp. 321 328.
[B111] Tabor D. A simple theory of static and dynamic hardness. Proceedings of the Royal
Society of London. Series A: Mathematical, Physical and Engineering Sciences.
1948, Vol. 192, No. 1029, pp. 247 274.
[B112] Taha T., Cui Z.F. CFD modelling of slug flow in vertical tubes. Chemical
Engineering Science. 2006, Vol. 61, No. 2, pp. 676 687.
[B113] Takhistov P. Dimensionless analysis of the electric field-based food processes for
scaleup and validation. Journal of Food Engineering. 2007, Vol. 78, No. 3,
pp. 746 754.
[B114] Trangenstein J.A. Numerical analysis of reservoir fluid flow. Multiphase flow in
porous media. Mechanics, Mathematics and Numerics
Lectures Notes in
Engineering. 1988, Vol. 34.
[B115] Tufenkji N., Elimelech M. Correlation equation for predicting single-collector efficiency in physicochemical filtration in saturated porous media. Environmental
Science and Technology. 2004, Vol. 38, No. 2, pp. 529 536.
[B116] Tzou D.Y., Chiu K.S. Temperature-dependent thermal lagging in ultrafast laser heating. International Journal of Heat and Mass Transfer. 2001, Vol. 44, No. 9,
pp. 1725 1734.
[B117] van Leeuwen H.J., Haagh G.A.A.V., van Bavel P.G.M. Transient EHD lubrication of
concentrated line contacts Part 1: Steady-state and pure squeeze cases. In: Lubricants
and Lubrication, Proc. 21st Leeds-Lyon Symposium on Tribology, Leeds University,
Leeds, UK, September 6 9, 1994. Tribology series 28, 1995, pp. 525 543.
[B118] Vdovin V.F. Problems of cryogenic cooling of superconductor and semiconductor
receivers in the range 0.1 1 THz. Radiophysics and Quantum Electronics. 2005,
Vol. 48, No. 10 11, pp. 779 791.
[B119] Vukasinovic B., Glezer A., Smith M.K. Droplet formation during the atomization of
a viscous liquid. In: 19th Annual Gallery of Fluid Motion exhibit, held at the 54th
Annual Meeting of the American Physical Society, Division of Fluid Dynamics
436
References
[B120]
[B121]
[B122]
[B123]
[B124]
[B125]
[B126]
[B127]
[B128]
[B129]
[B130]
C
(San Diego, California, November 2001). Abstracts and selected images appeared in
a special section of the Physics of Fluids, 2002, Vol. 14, No. 9.
Wei Y., Hutchinson J.W. Interface strength, work of adhesion and plasticity in the
peel test. International Journal of Fracture. 1998, Vol. 93, pp. 315 333.
Widmann J.F., Davis E.J. Pulsed electromagnetic heating of microparticles.
International Journal of Heat and Mass Transfer. 1998, Vol. 41, No. 24,
pp. 4195 4204.
Wünnemann K., Ivanov B.A. Numerical modelling of the impact crater depth-diameter
dependence in an acoustically fluidized target. Planetary and Space Science. 2003,
Vol. 51, No. 13, pp. 831 845.
Yi Y.-B., Barber J.R., Hartsock D.L. Thermoelastic instabilities in automotive disk
brakes. Finite element analysis and experimental verification. Proc. Third Contact
Mechanics International Symposium, Peniche, Portugal, June 2001.
Yoshimoto S., Tamura J., Nakamura T. Dynamic tilt characteristics of aerostatic
rectangular double-pad thrust bearings with compound restrictors. Tribology
International. 1999, Vol. 32, No. 12, pp. 731 738.
Zhang K., Jones C.A. On small Roberts number magnetoconvection in rapidly rotating systems. Proceedings of the Royal Society of London. Series A: Mathematical,
Physical and Engineering Sciences. 1996, Vol. 452, No. 1947, pp. 981 995.
Zhang L.X., Zhao Y.P. Adhesion of rough surfaces with plastic deformation.
Journal of Adhesion Science and Technology. 2004, Vol. 18, No. 6, pp. 715 729.
Zhao Y.P. Suggestion of a new dimensionless number for dynamic plastic response of
beams and plates. Archive of Applied Mechanics. 1998, Vol. 68, No. 7 8, pp. 524 538.
Zhao Y.P., Wang L.S., Yu T.X. Mechanics of adhesion in MEMS
a review.
Journal of Adhesion Science and Technology. 2003, Vol. 17, No. 4, pp. 519 546.
Zhu Q.S., Li H.Z. Study on magnetic fluidization of group C powders. Powder
Technology. 1996, Vol. 86, No. 2, pp. 179 185.
Zimbardo G., Pommois P., Veltri P. The Kubo number as a parameter governing the
level of chaos in magnetic turbulence. Physica A: Statistical Mechanics and its
Applications. 2000, Vol. 280, No. 1 2, pp. 99 105.
Internet Pages
[C1] 1D sediment transport morphodynamics with applications to rivers and turbidity currents.
http://vtchl.uiuc.edu/people/parkerg/morphodynamics_e-book.htm [cit. 2010-09-02].
[C2] Abbe number. http://en.wikipedia.org/wiki/Abbe_number [cit. 2007-09-21].
Mixing.
[C3] AJ Propeller turbine mixer design calculator. Wastewater treatment
http://www.ajdesigner.com/phpmixing/reynoldsnumbermixing.php [cit. 2007-09-21].
[C4] An emissivity primer. http://www.electro-optical.com/bbrad/emissivity/emisivty.htm
[cit. 2007-09-21].
[C5] Avogadro constant. http://en.wikipedia.org/wiki/Avogadro_constant [cit. 2007-09-21].
[C6] Bejan number. http://en.wikipedia.org/wiki/Bejan_number [cit. 2007-09-21].
[C7] Benard Marangoni instability. http://hmf.enseeiht.fr/travaux/CD0001/travaux/optmfn/
hi/01pa/hyb72/bm/bm.htm [cit. 2010-05-30].
[C8] Boltzmann distribution. http://en.wikipedia.org/wiki/Boltzmann_distribution [cit. 200709-21].
[C9] Books about Brownian motion. http://www.ericweisstein.com/encyclopedias/books/
BrownianMotion.html [cit. 2010-09-02].
References
437
[C10] Boussinesq approximation (buoyancy). http://en.wikipedia.org/wiki/Boussinesq_
approximation_(buoyancy) [cit. 2010-05-30].
[C11] Boyle’s law. http://en.wikipedia.org/wiki/Boyle’s_law [cit. 2010-05-30].
[C12] Bulk Richardson number. http://amsglossary.allenpress.com/glossary/search?id5
bulk-richardson-number1 [cit. 2007-09-21].
[C13] Buoyancy wavenumber. http://amsglossary.allenpress.com/glossary/search?p 5 1&query5
buoyancy 1 wavenumber&submit 5 Search&def 5 on [cit. 2007-09-21].
[C14] Burger number. http://amsglossary.allenpress.com/glossary/browse?s 5 b&p 5 51 [cit.
2007-09-21].
[C15] Businger Dyer relationship. http://amsglossary.allenpress.com/glossary/browse?s 5 b&p 5 52
[cit. 2007-09-21].
[C16] Carnot heat engine. http://en.wikipedia.org/wiki/Carnot_heat_engine [cit. 200709-21].
[C17] Cavitation and bubble dynamics. Chapter 1. Phase change, nucleation and cavitation.
http://authors.library.caltech.edu/25017/1/cavbubdynam.pdf [cit. 2007-09-21].
[C18] Centripetal acceleration. http://theory.uwinnipeg.ca/physics/circ/node6.html [cit. 200709-21].
[C19] Chandrasekhar number. http://en.wikipedia.org/wiki/Chandrasekhar_number [cit. 201010-19].
[C20] Clausius Mossotti relation. http://en.wikipedia.org/wiki/Clausius-Mossotti [cit. 200709-21].
[C21] Coefficient of variation. http://en.wikipedia.org/wiki/Coefficient_of_variation [cit. 200709-21].
[C22] Correlation coefficient. http://mathworld.wolfram.com/CorrelationCoefficient.html
[cit. 2010-05-30].
[C23] Coupling constants for the fundamental forces. http://hyperphysics.phy-astr.gsu.edu/
hbase/forces/couple.html [cit. 2007-09-21].
[C24] Crank-Nicolson method. http://en.wikipedia.org/wiki/Crank%E2%80%93Nicolson_method
[cit. 2007-09-21].
[C25] Crispation number. http://www.answers.com/topic/crispation-number [cit. 2007-09-21].
[C26] Crop evapotranspiration Guidelines for computing crop water requirements. http://
www.fao.org/docrep/X0490E/x0490e00.htm [cit. 2007-09-21].
[C27] Damping. http://en.wikipedia.org/wiki/Damping [cit. 2010-09-02].
[C28] Damping. Part I. Background and theory. http://physics.mercer.edu/hpage/damping.
htm [cit. 2007-09-21].
[C29] Darcy friction factor. http://www.answers.com/Darcy%20friction%20factor [cit. 200709-21].
[C30] Davies number. http://amsglossary.allenpress.com/glossary/search?p 5 1&query5Davies1
number [cit. 2007-09-21].
[C31] de Broglie wavelength. http://scienceworld.wolfram.com/physics/deBroglieWavelength.
html [cit. 2007-09-21].
[C32] de Broglie wavenumber. http://scienceworld.wolfram.com/physics/deBroglieWavenumber.
html [cit. 2007-09-21].
[C33] Debye length. http://en.wikipedia.org/wiki/Debye_length [cit. 2010-05-30].
[C34] Decibel. http://www.diracdelta.co.uk/science/source/d/e/decibel/source.html [cit. 200709-21].
[C35] Deepening of the ocean mixed layer by Langmuir and shear turbulence.
http://ams.confex.com/ams/pdfpapers/78287.pdf [cit. 2007-09-21].
438
References
[C36] Dependency of bulk concentration. http://www.student.dtu.dk/Btsth/reporthoved/
node42.html [cit. 2010-05-30].
[C37] Dimensionless quantity. http://en.wikipedia.org/wiki/Dimensionless_quantity [cit. 200709-21].
[C38] Dimensionless number. http://en.wikipedia.org/wiki/Dimensionless_number [cit. 201005-30].
[C39] Dimensionless numbers. http://seismo.berkeley.edu/Blhsu/dimnum.html [cit. 2010-1019].
[C40] Dimensionless numbers. http://www.rpi.edu/dept/chem-eng/Biotech-Environ/AERATION/
dimnum.htm [cit. 2010-06-30].
[C41] Earthquake source parameters. http://www.earth.northwestern.edu/people/seth/324/
324l8.sp.ppt [cit. 2007-09-21].
[C42] Earthshine measurements of global atmospheric properties. http://www.bbso.njit.edu/
Research/EarthShine/espaper/earthshine_proposal.html [cit. 2007-09-21].
[C43] Eddington number. http://mathworld.wolfram.com/EddingtonNumber.html [cit. 200709-21].
[C44] Edwin Powell Hubble. http://en.wikipedia.org/wiki/Edwin_Hubble [cit. 2007-09-21].
[C45] Ekman number. http://en.wikipedia.org/wiki/Ekman_number [cit. 2007-09-21].
[C46] Energy efficiency. http://en.wikipedia.org/wiki/Energy_efficiency [cit. 2007-09-21].
[C47] Energy storage. http://en.wikipedia.org/wiki/Energy_storage [cit. 2007-09-21].
[C48] Equations governing geophysical flows. http://engineering.dartmouth.edu/BBenoit R
Roisin/books/GFD/chap4.pdf [cit. 2007-09-21].
[C49] Equations of motion and energy in Cartesian coordinates. http://www.owlnet.rice.
edu/Bceng501/Chap6.pdf [cit. 2007-09-21].
[C50] Equivalent dose and radiation weighting factor. http://trshare.triumf.ca/Bsafety/EHS/
rpt/rpt3/node7.html [cit. 2007-09-21].
[C51] Ericksen number. http://en.wikipedia.org/wiki/Ericksen_number [cit. 2010-05-30].
[C52] Erlang unit. http://en.wikipedia.org/wiki/Erlang_unit [cit. 2007-09-21].
[C53] f-number. http://en.wikipedia.org/wiki/F-stop [cit. 2007-09-21].
[C54] Fermi paradox. http://www.en.wikipedia.org/wiki/Fermi_paradox [cit. 2007-09-21].
Nomenclature. http://my.fit.edu/itresources/manuals/
[C55] Fluent 6.3 User’s Guide
fluent6.3/help/html/ug/node1511.htm [cit. 2010-05-30].
[C56] Fluid pressure drop along pipe length of uniform diameter. http://www.engineersedge.com/fluid_flow/pressure_drop/pressure_drop.htm [cit. 2007-09-21].
[C57] Force of friction. http://faculty.wwu.edu/vawter/PhysicsNet/Topics/Dynamics/Forces/
FrictionalForce.html [cit. 2007-09-21].
[C58] Fraction mechanics. http://encyclopedia.thefreedictionary.com/crack 1 propagation
[cit. 2010-09-02].
[C59] Fresnel number. http://en.wikipedia.org/wiki/Fresnel_number [cit. 2007-09-21].
[C60] Friction. http://en.wikipedia.org/wiki/Friction [cit. 2007-09-21].
[C61] Friction loss fluid flow. http://www.engineersedge.com/fluid_flow/friction_loss.htm
[cit. 2007-09-21].
[C62] Gain. http://www.reference.com/browse/wiki/Gain [cit. 2007-09-21].
[C63] Gaseous fuels. http://eyrie.shef.ac.uk/eee/cpe630/comfun5.html [cit. 2007-09-21].
[C64] Glossary of chemical terms
M. http://www.ktf-split.hr/periodni/en/abc/m.html
[cit. 2010-05-30].
[C65] Glossary of materials attributes. http://www.grantadesign.com/resources/materials/
glossary.htm [cit. 2007-09-21].
References
439
[C66] Golitsyn number. http://scienceworld.wolfram.com/physics/GolitsynNumber.html
[cit. 2007-09-21].
[C67] Granulation: Endpoint determination and scale-up. http://www.mcc-online.com/
granulation.htm [cit. 2007-09-21].
[C68] Gravitational constant. http://encyclopedia.thefreedictionary.com/constant 1 of 1
gravitation [cit. 2007-09-21].
[C69] Gravitational coupling constant. http://en.wikipedia.org/wiki/Gravitational_coupling_
constant [cit. 2007-09-21].
[C70] Hagen number. http://en.wikipedia.org/wiki/Hagen_number [cit. 2007-09-21].
[C71] Hubble’s law. http://en.wikipedia.org/wiki/Hubble’s_law [cit. 2007-09-21].
[C72] Inertial parameter. http://unicon.netian.com/dimlesse.html [cit. 2005-02-10].
[C73] Internal Froude number. http://scienceworld.wolfram.com/physics/InternalFroude
Number.html [cit. 2007-09-21].
[C74] Karlovitz number. http://www.cfd-online.com/Wiki/Karlovitz_number [cit. 2010-05-30].
[C75] Kolmogorov microscale. http://amsglossary.allenpress.com/glossary/search?id 5 kolmogorovmicroscale1[cit.2007-09-21].
[C76] Laplace number. http://en.wikipedia.org/wiki/Laplace_number [cit. 2007-09-21].
[C77] Liquid
lubrication.
http://www.roymech.co.uk/UsefulTables/Tribology/Liquid_
Lubrication.htm [cit. 2007-09-21].
[C78] Lockhart-Martinelli parameter. http://en.wikipedia.org/wiki/Lockhart-Martinelli parameter
[cit. 2007-09-21].
[C79] Lorentz factor. http://en.wikipedia.org/wiki/Lorentz_factor [cit. 2007-09-21].
[C80] Lorentz Lorenz equation. http://en.wikipedia.org/wiki/Lorentz-Lorenz [cit. 200709-21].
[C81] Love number. http://scienceworld.wolfram.com/physics/LoveNumber.html [cit. 200709-21].
[C82] Ludolph number π. http://en.wikipedia.org/wiki/Pi [cit. 2010-05-30].
[C83] Mach number. Role in compressible flows. http://www.grc.nasa.gov/WWW/K-12/
airplane/machrole.html [cit. 2007-09-21].
[C84] Magnetic number. http://www.answers.com/topic/magnetic-number [cit. 2007-09-21].
[C85] Mass fraction (chemistry). http://en.wikipedia.org/wiki/Mass_fraction_(chemistry)
[cit. 2010-05-30].
[C86] Mass-to-charge ratio. http://en.wikipedia.org/wiki/Mass-to-charge_ratio [cit. 2007-09-21].
[C87] Maxwell’s equations. http://en.wikipedia.org/wiki/Maxwell’s_equations [cit. 2007-09-21].
[C88] Mechanical efficiency. http://en.wikipedia.org/wiki/Mechanical_efficiency [cit. 200709-21].
[C89] Modeling of oscillating flow past a vertical plate. http://cavity.ce.utexas.edu/kinnas/
fpso/ISOPE2003/ISOPE2003.ppt [cit. 2007-09-21].
[C90] Mole fraction. http://en.wikipedia.org/wiki/Mole_fraction [cit. 2007-09-21].
[C91] Molecular mass. http://en.wikipedia.org/wiki/Molecular_mass [cit. 2007-09-21].
[C92] Moment magnitude scale. http://en.wikipedia.org/wiki/Moment_magnitude_scale
[cit. 2007-09-21].
[C93] Moment of inertia ratio. http://scienceworld.wolfram.com/physics/MomentofInertia
Ratio.html [cit. 2007-09-21].
[C94] Monin Obukhov length. http://scienceworld.wolfram.com/physics/Monin-Obukhov
Length.html [cit. 2007-09-21].
[C95] Monin Obukhov similarity theory. http://amsglossary.allenpress.com/glossary/search?
id 5 monin-obukhov-similaritytheory1 [cit. 2007-09-21].
440
References
[C96] Morozov Hall parameter scaling in magnetized coaxial accelerator channel flows.
http://flux.aps.org/meetings/YR97/BAPSDPP97/abs/G300025.html [cit. 2007-09-21].
[C97] Normalized frequency (fiber optics). http://en.wikipedia.org/wiki/Normalized_
frequency_(fiber_optics) [cit. 2010-05-30].
[C98] Numerical methods applied to chemical engineering. http://www.core.org.cn/NR/
rdonlyres/Chemical-Engineering/10-34Numerical-Methods-Applied-to-ChemicalEngineeringFall2001/D7F8595C-E40C-4978-9CEFBA2EF1704DB9/0/assign03.
pdf [cit. 2007-09-21].
[C99] Optical depth. http://en.wikipedia.org/wiki/Optical_thickness [cit. 2010-05-30].
[C100] Overall Richardson number. http://stommel.tamu.edu/Bbaum/paleo/ocean/node29.
html [cit. 2010-09-02].
[C101] Parasitic drag. Skin friction. http://en.wikipedia.org/wiki/Parasiticdrag#Skinfriction
[cit. 2010-09-02].
[C102] Period doubling ad infinitum. http://www.drchaos.net/drchaos/Book/node44.html
[cit. 2010-05-30].
[C103] Plasma physics. http://www.physics.usyd.edu.au/Bmmmb/plasma/Chapter1.pdf [cit.
2007-09-21].
[C104] Plasma scaling. http://en.wikipedia.org/wiki/Plasma_scaling#Dimensionless parameters in tokamaks [cit. 2007-09-21].
[C105] Porosity. http://en.wikipedia.org/wiki/Porosity [cit. 2007-09-21].
[C106] Propeller. http://en.wikipedia.org/wiki/Propeller [cit. 2007-09-21].
[C107] Propeller performance factors. http://www.epi-eng.com/propeller_technology/
selecting_a_propeller.htm [cit. 2010-06-30].
[C108] Ratio of specific heat. http://www.engineeringtoolbox.com/specific-heat-ratio-d_608.
html [cit. 2007-09-21].
[C109] Rayleigh Benard convection. The Boussinesq approximation. http://home.iitk.ac.in/
Bsghorai/NOTES/benard/node3.html [cit. 2007-09-21].
[C110] Reduced pressure. http://en.wikipedia.org/wiki/Reduced_pressure [cit. 2007-09-21].
[C111] Reference tables
Coefficient of friction. http://www.engineershandbook.com/
Tables/frictioncoefficients.htm [cit. 2007-09-21].
[C112] Rejection coefficient. http://www.msu.edu/course/fsc/229/Notes/Lecture%208.htm
[cit. 2007-09-21].
[C113] Relative permittivity. http://en.wikipedia.org/wiki/Dielectric_constant [cit. 2010-05-30].
[C114] Reynolds number. http://en.wikipedia.org/wiki/Reynolds_number [cit. 2010-09-02].
[C115] Rheological parameters for viscoelastic materials. http://www.vilastic.com/tech3.htm
[cit. 2010-05-30].
[C116] Rheology. http://en.wikipedia.org/wiki/Rheology [cit. 2010-05-30].
[C117] Rockwell scale. http://en.wikipedia.org/wiki/Rockwell_scale [cit. 2007-09-21].
[C118] Roshko number. http://en.wikipedia.org/wiki/Roshko_number [cit. 2007-09-21].
number.
http://scienceworld.wolfram.com/physics/RossbyNumber.html
[C119] Rossby
[cit. 2007-09-21].
[C120] Sedimentation. http://en.wikipedia.org/wiki/Sedimentation [cit. 2010-10-19].
[C121] Size factor. http://scienceworld.wolfram.com/physics/SizeFactor.html [cit. 2007-09-21].
[C122] Sound pressure. http://en.wikipedia.org/wiki/Sound_pressure_level [cit. 2007-09-21].
[C123] Specific heat capacity. http://en.wikipedia.org/wiki/Heat_capacity [cit. 2007-09-21].
[C124] Stratification parameter. http://scienceworld.wolfram.com/physics/Stratification
Parameter.html [cit. 2007-09-21].
[C125] Strehl ratio. http://scienceworld.wolfram.com/physics/StrehlRatio.html [cit. 2007-09-21].
[C126] Surface roughness. http://en.wikipedia.org/wiki/Surface_roughness [cit. 2010-09-02].
References
441
[C127] Swirl flows and flames database. http://www.ae.su.oz.au/thermofluids/swirl.htm [cit.
2007-09-21].
[C128] Systematic derivation of the weakly non-linear theory of thermoacoustic devices.
http://www.win.tue.nl/Bppanhuis/reports/CasaReportErrata.pdf [cit. 2007-09-21].
[C129] The mysterious Eddington Dirac number. http://www.jgiesen.de/astro/stars/diracnumber.htm [cit. 2007-09-21].
[C130] The novel oscillatory flow reactor designs. https://repositorium.sdum.uminho.pt/
bitstream/1822/5624/2/NReis_PhD_file2.pdf [cit. 2007-09-21].
[C131] The plasma parameter. http://farside.ph.utexas.edu/teaching/plasma/lectures/node8.
html [cit. 2007-09-21].
[C132] Thermal conductivity and the Wiedemann Franz law. http://hyperphysics.phy-astr.
gsu.edu/Hbase/thermo/thercond.html [cit. 2010-10-19].
[C133] Thermal de Broglie wavelength. http://en.wikipedia.org/wiki/Thermal_de_Broglie_
wavelength [cit. 2007-09-21].
[C134] Timescale number. http://scienceworld.wolfram.com/physics/TimescaleNumber.html
[cit. 2007-09-21].
[C135] Tortuosity. http://encyclopedia.thefreedictionary.com/tortuosity [cit. 2007-09-21].
[C136] Using filament stretching rheometry to predict strand formation and processability
in adhesives and other non-Newtonian fluids. http://web.mit.edu/nnf/publications/
GHM44.pdf [cit. 2007-09-21].
[C137] Van’t Hoff factor. http://en.wikipedia.org/wiki/Van’t_Hoff_factor [cit. 2007-09-21].
[C138] Volume fraction. http://en.wikipedia.org/wiki/Volume_fraction [cit. 2010-05-30].
[C139] Wavenumber. http://scienceworld.wolfram.com/physics/Wavenumber.html [cit. 200709-21].
[C140] Wiedemann Franz law. http://en.wikipedia.org/wiki/Wiedemann-Franz_law [cit.
2010-10-19].
D
Some Calculators and Graphical Solutions of Dimensionless Numbers
[D1] Calchemy live. http://www.calchemy.com/ [cit. 2007-09-21].
[D2] Fitch B. Graphical solutions on dimensionless-number plots. Journal of the American
Institute of Chemical Engineers. 1955, Vol. 1, No. 4, pp. 475 479.
[D3] Fluid mechanics equations formulas calculator. http://www.ajdesigner.com/index_
fluid_mechanics.php [cit. 2007-09-21].
[D4] Pipe pressure loss calculator. http://www.efunda.com/formulae/fluids/calc_pipe_
friction.cfm [cit. 2007-09-21].
[D5] The fluid flow calculations website. http://www.lmnoeng.com/index.shtml [cit. 2007-09-21].
[D6] The on-line computational and graphics website for mechanical engineers
covering: mechanics, dynamics, design, vibrations, structures, heat transfer, thermodynamics, fluid flow, metalworking, economics and related disciplines.
http://www.mechengcalculations.com/index.html [cit. 2007-09-21].