Download Static of fluids

Static of fluids
A fluid has constant volume and the shape of its container
Law n 1: a fluid at rest cannot produce a force tangent to the surface
(no shear force), the normal force does not depend on the orientation
of the surface
Pressure
F
F
N
p=
, [ 2 ≡ P ascal = P a]
A
m
p → mean pressure; F → normal force acting on the surface
A.
hydraulic
A
press
Law n. 2 (Pascal): the pressure in a fluid
(neglecting the gravitational force) is the same
everywhere in the fluid.
m
Kg
ρ=
, [ 3]
Density
F1
F2
V
m
≡
A1
A2
ρ → (local) density of the body; m mass of the body contained in a section of volume V of the body.
F0
Stevino’s law
Fg = ρAhg
p = p0 + ρgh
F
p → pressure acting at dept h; p0 → pressure acting on the
free surface of the liquid; ρ → density of the liquid.
Communicating vessels
p A ≡ pB → h A ≡ h B
if p0 is the same in both the vassels
P0
Manometer
h
Pa
Pa
F A ≡ FB
p A ≡ pB
Relative (gauge) pressure
∆p = pa − p0 = ρg(ha − h0 ) = ρgh
1 atm = 760 mm Hg = 1.0133 × 105 Pa
Archimedes’ law
F2 − F1 = (p2 − p1 )A = Aρf g(h2 − h1 ) = V ρf g = mf g
FA = mf g
Fg = mc g
p1
p2
FA = ρf · g · Vc
FA → Archimedes’ force (acting upward); ρf → density of
the fluid; Vc → volume of the body immersed in the fluid
ρf > ρc Archimedes’ force wins on the gravity force
ρf = ρc equilibriumù
ρf < ρc gravity force wins on Archimedes’ force
Vi
ρc
=
Vc
ρf
ρc = density of the body ( Vmc ); Vi volume of the immersed
part of the body; Vc total volume of the body
Dynamics of ideal fluids
Continuity equation
ρf · A · v = cost = mass flow rate
ρf = fluid density; A = area of the conductor section in
which the fluid flows; v fluid velocity through the section
Q = Vt volume flow rate
For a incompressible fluid (ρ = cost) we have: A · v =
m3
Q = cost , [ s ] Q = Vt = Ad
d = A · v = volume flow rate
v
Bernoulli’s equation
(stationary and incompressible fluids)
p + ρgh + 12 ρv 2 = cost
p pressure acting on the considered section, ρ fuid density; h
mean level of the section w.r.t. a reference; v fluid velocity
through the section
Dynamics of real fluids
real fluid moves laminar flow
(es: blood in small vassells)
Fη =
ηSvm
R
=
η(2πRL)v
R
= 2πηLvm
Poiseulle’s law (flow on a cylindric conductor at laminar regime)
πR4 (p1 − p2 )
Q=
8η
L
R = radius of the pipeline; L = length of the pipeline;
p1 − p2 = difference of pressure at the ends of the pipeline;
·s
η[ N
m2 ] = friction coefficient due to the fluid viscosity.
Poiseuille’s law: the flow of fluid through a section of the
pipeline is proportional to the difference of pressure in the
pipeline and to the fourth power of the pipeline radius
vm
(p1 − p2 )R2
=
4ηL
Surface tension
On the surfaces of separation between liquid and other materials act
forces of molecular attraction
In small vessels (capillary < 1 mm) the surface forces
raise or lower the liquid according to the law:
2τ cosα
h=
R·ρ·g
Where: τ [N/m] = is the surface force: force per unit of
length tangent to the surface.
R vessel radius; ρ fluid density; α angle between the surface
of the liquid and the walls of the vessel; h is the level ho de
meniscus ( > 0 concave meniscus, < 0 convex meniscus) of
the fluid.
α
α
Heat and temperature
Thermometric scales
Triple point of water 273.16 K (0.01°C)
Boiling point of
water
100.00°C
373.15 K
212.00°F
Freezing point of
water
0.00°C
273.15 K
32.00°F
Absolute zero
-273.15°C
0K
-459.67°F
Thermal expansion in solids and fluids
∆l = li · λ · ∆T
∆A = Ai · σ · ∆T
∆V = Vi · γ · ∆T
∆l, ∆A, ∆V variation of length, area or volume; li , Ai , Vi
initial values, ∆T variation of temperature; λ, σ, γ[o C −1 ]
coefficients of expansion.
Heat absorbed or lost by a body
Q = m · c · ∆T = C · ∆T dove C = m · c
Q heat ABSORBED (Q > 0) or LOST (Q < 0) by a body
(in which there are not phase transitions); ∆T variation of
temperature; m mass of the body; c [cal / Kg o C] specific
heat; C [cal / o C] heat capacity of the body
In purely thermal transformations the heat is conserved
Heat (latent) of transition A = Lm, where L is the
specific latent heat
cH2 0 = 1.00 gcal
0C
Thermal
equilibrium
Heat is a form of energy
1Cal = 4.186J
Heat is a form of energy, thus it is expressed in
Joule in the IS. Another unit of common use is the
calorie (cal)
As we will see, in general energy + heat is conserved
in an isolated system (first principle of thermodynamics)
Heat transfer mechanisms
Conduction:
The exchanged heat Q in a time t through a section of
area A of a body of length l with thermal conductivity k
[cal/ m s o C] whose ends are at temperatures T1 e T2 is:
T2 − T1
Q=k·A
·t
l
Convection:
Irradiation:
Power P [Watt] irradiated or absorbed (Tc > Ta or Ta >
Tc ) by a surface of area A with emissivity � ∈ [0, 1] at
temperature Tc in an environment at temperature Ta is:
σ = 5.67 · 10−8 W / m2 K4 costante di Stefan-Boltzmann.
P = � · σ · A · (Tc4 − Ta4 )
Ideal gas
Dimensionless, massless moleculas interacting only through perfect elastic
collisions
VT = V0 (1 + γT ) with γ =
1
273.15o C
Ideal gas law
p · V = nRT
p e V pressure and volume of the ideal gas; T temperature
in KELVIN; R = 8.3144 [J / K mole] Universal constant of
gasses; n number of moles.
A mole of a substance contains NA = 6.02·1023 molecules;
NA Avogadro number.
k=
R
Na
= 1.38 · 10−23 J/ K is the Boltzmann constant.
Ideal gas transformations
p · V = nRT
p e V pressione e volume
del gas;pressure
T temperatura
in KELVIN;
Transformation
at constant
(ISOBARE):
8.3144 [J / K mole] COSTANTE UNIVERSALE DEI
pR==cost
GAS; n numero moli.
Transformation
at constant volume (ISOCORE): 23
Una mole di qualsiasi sostanza contiene NA = 6.02·10
Vmolecole;
= cost N numero di Avogadro.
A
Transformation at constant temperature (ISOTERME):
T = cost → p · V = cost
Transformation without heat exchange (ADIABATIC):
Q = cost
Thermal bath
Q = Cp ∆T : Cp = specific heat at constant pressure.
Q = Cv ∆T : Cv = specific heat at constant pressure.
Equipartition of energy
In the motion of molecules in a gas at a given temperature, to every independent component of the motion
(degree of freedom) corresponds the same kinetic energy
Km
1
= kT
2
where k = NRA = 1.38 · 10−23 J / mole K is the Blotzmann
constant
The total kinetic energy of a molecule is
Km =
λ
kT
2
λ = degrees of freedom:
monoatomic molecules λ = 3;
diatomic molecules λ = 5;
pluriatomic molecules λ = 6.
Thermodynamics
First principle of thermodynamics
The variation of internal energy of a system (∆U ) is
equal to the energy exchanged with the external with the
external environment (∆U = −∆EE ).
∆U = Q − W
∆U = variation of internal energy of the system; Q heat
absorbed (> 0) by the system; W work done (> 0) by the
system
Thermodynamics
∆U = Q − W
∆U = variation of internal energy of the system; Q heat
absorbed (> 0) by the system; W work done (> 0) by the
system
Reversible transformations
Constant volume: ∆V = 0 ⇒ W = p∆V = 0
∆U = Q − W = Q = nCv ∆T , Q > 0 ⇒ ∆U > 0
Constant pressure: ∆p = 0, W = nR∆T , Q = nCp ∆T
∆U = Q − W = ncp ∆T − nR∆T = ncv ∆T → cp − cv = R
Constant temperature:
∆T = 0 → ∆U = 0 → Q = W
Thermal bath
Adiabatic:
Q = 0 → ∆U = −W = nCv ∆T
Carnot cycle
Reversible transformations
Rendimento del ciclo di Carnot
Wciclo = Q2 − Q1 =,
Q1 = Heat absorbed by the refrigerator,
Q2 = Heat absorbed by the source.
Q1
T2 −T1
T1
ciclo
=
1
−
=
=
1
−
η = WQ
Q2
T2
T2
2
⇒0≤η<1
Second principle of thermodynamics
It is impossible to realize a cyclic machine whose
only result is to transform in work all the heat
absorbed by a homogeneous source
(Kelvin-Planck formulation).
The efficiency ηirr of a real cyclic machine (Irreversible)
working among two temperatures T1 and T2 is less than the
efficiency of a corresponding Carnot cycle ηirr < ηrev .
Chaos in the universe tends to increase
The chaotic configurations are statistically more probable
Entropy
is a function of state which can be defined for reversible
processes by mens of the heat ∆Qrev exchanged by the
system and the corresponding instantaneous temperature
T:
∆Qrev
∆S =
, [J/K]
T
Variation of entropy in a reversible cycle: ∆Srev = 0.
∆Q
Otherwise: ∆QTirr < Tref
For every real isolated system (adibatic): ∆S ≥ 0
Entropy
is a function of state which can be defined for reversible
processes by mens of the heat ∆Qrev exchanged by the
system and the corresponding instantaneous temperature
T:
∆Qrev
∆S =
, [J/K]
T
The entropy describes the disorder of a system: if the
macroscopic system is characterised by D possible equivalent micro-states (disorder), then
S ∝ log D
Second principle ⇒ the chaos in the universe increases.
Third principle of thermodynamics
The absolute entropy of a crystallin solid at the temperature of 0 K is zero.
Real gases
Phase transitions
Q = m · cl
In the isotherm process of phase transition a body absorbs
or give a quantity Q of heat where m is the mass and cL is
the latent heat.
Newton’s law of cooling
T (t) = Ti e−kt , k = gradient of the cooling.