Download In the late 1700s, Swiss physicist Daniel Bernoulli and his father

Innovations in Fluid Flow
6 Men that Shaped the World of Flowmetering
By Greg Livelli
In the late 1700s, Swiss physicist Daniel Bernoulli and his father, Johann, separately
entered a scientific contest sponsored by the University of Paris. They tied for first place.
Johann, shamed by the comparable abilities of his son, banned Daniel from his house.
His father also tried to steal Daniel’s book Hydrodynamica and rename it Hydraulica.
Daniel’s father carried this grudge until his death.
Nonetheless, Daniel Bernoulli’s equation of fluid flow, first published in Hydrodynamica,
stands as the principle behind the operation of many flowmeters today. This article
covers Bernoulli’s equation, as well as five other theories that are fundamental to the
world of flowmetering.
1. Daniel Bernoulli
In 1738, the publication of Hydrodynamica introduced the concept of the conservation of
energy for fluid flows. Bernoulli and his assistant Eular determined that an increase in
the velocity of a fluid increases its kinetic energy while decreasing its static energy. It is
for this reason that any flow restriction causes an increase in the flowing velocity and
also causes a drop in the static pressure of the flowing fluid. For noncompressible fluids,
such as liquids, the equation is:
v2/2 + gh + p/ρ = constant
where
v = fluid velocity along the streamline
g = acceleration due to gravity on earth (32 ft/s/s at 60º latitude)
h = height from an arbitrary point in the direction of gravity
p = pressure along the streamline
ρ = fluid density
As the fluid velocity v increases when it passes through a restriction, the pressure p
must decrease accordingly for the sum in the equation to remain constant. A similar,
more general equation, applies for compressible fluids such as gases.
A flowmeter element, such as an
orifice, provides the restriction
(Figure 1). The pressure loss, or
differential pressure, is a function
of the fluid flowrate. Instrument
engineers often express
permanent pressure loss through a
flowmeter either as a percentage
of the total pressure drop or in
units of velocity heads (v2/2g). For
example, if the velocity of a
flowing fluid is 10 ft/s, the velocity
head is 100/64.4 = 1.55 ft. If the
fluid is water, the velocity head
Figure 1. Orifice creates a pressure drop that’s a function of flow
corresponds to 1.55 feet of water
velocity.
(or 0.67 PSI). If the fluid is air,
then the velocity head corresponds to the weight of a 1.55-ft column of air.
The permanent pressure loss through an orifice depends on its beta ratio (diameter of
the orifice to that of the pipe). The permanent pressure losses for orifices with beta
ratios of 0.6, 0.5, and 0.3 are about 63 percent, 74 percent, and 88 percent of the total
pressure drop. Flowmeters, such as vortex-shedding sensors and PD meters, and
Venturis can reduce these losses by factors of two, four, and eight, respectively.
2. Michael Faraday
Many consider Michael Faraday, born in 1791, to be one of the finest experimental
scientists in history. From a poor family in London, he apprenticed to a bookbinder
during his adolescence. As an apprentice, he developed an interest in science,
performing chemical experiments. In 1810, he joined the City Philosophical Society,
where he broadened his scientific knowledge during weekly lectures.
Faraday eventually became a member of the Royal Institution, where he made
significant contributions in both chemistry and electricity. For example, he discovered
benzene and found a way to liquefy chlorine. But during the 1830s, he turned his
attention to electricity. Working in the field of electro-chemistry, he coined many familiar
words, such as electrode, electrolyte, anode, cathode, and ion.
In 1831, he began a notable series of experiments leading to the discovery of
electromagnetic induction. He wrapped two insulated coils of wire around a massive iron
ring. Passing a transient current through one coil produced a changing magnetic field
that induced a momentary current in the other. This discovery stands as the basis of
operation of electric generators and, of interest here, electromagnetic flowmeters.
Faraday's Law states that the induced voltage is proportional to the rate of change of
the magnetic field.
E = -N dB/dt
where
E = the induced voltage
N = the number of turns of wire in the coil
dB/dt = the time rate of change of the magnetic field
As applied to the magnetic flowmeter, the equation becomes: E = BvDC. This says that
when a conductive liquid moves in a pipe having an inside diameter D with an average
velocity v through a magnetic field of intensity B, it will induce a voltage E. (D represents
the width of the conductor, which in this case is the distance between the sensing
electrodes on either side of the pipe, and C is a constant depending on the dimensional
units). The voltage is proportional to the fluid velocity through the pipe.
Over the past several years, the
performance of magnetic
flowmeters, diagramed in Figure 2,
has improved significantly. Among
the advances are digital signal
processing, insertion designs and
improvement in noise-reducing
capabilities. But the magnetic
flowmeter's basic operating
principle, Faraday’s law of electric
induction, remains the same.
3. Gaspard-Gustave de Coriolis
The French engineer GaspardGustave de Coriolis, born in 1792,
became a student and teacher of
mechanics, engineering
Figure 2. Electromagnetic flowmeters utilize Faraday’s law for
conductive fluids.
mathematics, friction, hydraulics,
machine performance, and
ergonomics. Few know that he first introduced the scientific term “work” as a function of
force and distance, permitting a comparative measure of the effort expended by a
person, horse, or steam engine. He also coined the term “kinetic energy” with its present
scientific meaning.
But he is best known for his explanation of the "Coriolis effect" that bears his name —
the apparent deflection of an object in a rotating frame of reference. For example,
suppose you stand near the center of a rotating carousel and you throw a ball to a
stationary friend on the ground. Your friend sees the ball come straight to him. But
because of the carousel's rotation, your perception is that the ball curves away from you
to reach your friend. From your point of view, a mysterious force, now named the
Coriolis force, has curved the ball's flight. Similarly, an airborne artillery shell will seem to
drift sideways because of the earth's rotation. The same is true for winds and ocean
currents.
In mathematical terms, the vector equation that describes the action of Coriolis force is:
Fc = -2m(ώ v)
where
Fc = the Coriolis force (perpendicular to the axis of rotation and flow)
m = the mass acted upon
ώ = the rate of angular rotation (a vector)
v = the velocity of the mass
The Coriolis mass flowmeter
represents a practical application
of this effect. Most flowmeters
measure volumetric flowrate,
which varies directly with the mass
flowrate, only when the fluid has a
constant density and contains no
bubbles. The Coriolis flowmeter
substitutes oscillation for full-circle
rotation. Essentially, this flowmeter
monitors the inertial effects
resulting from induced oscillation
and mass flow through the tube.
Figure 3. Induced Coriolis forces twist the tube an amount
proportional to the mass flowrate.
As indicated in Figure 3, an energy source vibrates the flow tube up and down. A Coriolis
force induced on the tube with fluid moving from left to right bends the tube down. The
Coriolis force on the tube with fluid moving from right to left bends the tube up. As a
result, the tube twists. Optical pickups measure the magnitude of the tube twist, which
is a function of the mass flowrate through the tubes.
The slow evolution of sensors and electronics delayed creation of the first commercial
Coriolis mass flowmeter until the 1970s.
4. Christian Doppler
Christian Doppler, born in 1803 the son of a prosperous stonemason in Austria, would
normally have taken over his father's business. But, being too frail, he turned instead to
the study of mathematics, mechanics, and astronomy. Later, while competing via exams
for professorships, he had to make a living. He unhappily spent more than a year and a
half as a bookkeeper at a cotton-spinning factory. His difficulties moved him to make
arrangements to emigrate to America, but he finally received an appointment for a
position in Prague.
While not a talented mathematician compared to his peers, he did possess a strong
sense of originality and innovation. He presented his most famous idea to the Royal
Bohemian Society in 1842. Seeking an explanation of light colors emanating from double
stars, he hypothesized that the frequency (color) of light waves for a star moving away
from the earth would be different from that of a star moving toward the earth. He was
right about this, but actually the effect was not measurable with the instruments of the
time.
Sound waves, however, were a different matter. In 1845 scientists conducted
experiments with musicians on railway trains playing musical notes. Trained musicians
on the platform noted the apparent pitch of the note as the train approached and as it
receded. In 1847, Doppler published refinements to his new principle, accounting for
both the motion of the source and the observer.
The Doppler equation for the apparent pitch (frequency) of sound as the source moves
away from the observer is:
fr = fs {(c/c-v)}
where
fr = the observed frequency
fs = the source frequency
c = the speed of sound in the medium (liquid in our case)
v = the velocity of the liquid
More than a century passed before
the first ultrasonic Doppler
flowmeter came on the market. To
use the Doppler effect to measure
flowrate in a pipe, the transducer
emits an ultrasonic beam into the
flow stream (Figure 4). The liquid
in the pipe must contain entrained
particles or air bubbles to reflect
the sound beam back to a receiver. The difference between the source and received
frequencies, called the Doppler shift, varies directly with the velocity of the liquid in the
pipe.
5. Osborne Reynolds
Born in Belfast, Ireland in 1842, Osborne Reynolds became interested early on in his
father's hobby of math and mechanics. As a young man he apprenticed to an
engineering firm, studied mathematics at Cambridge, and spent a year as a practicing
civil engineer. In 1868, he became a professor of engineering (one of only two in
England at the time).
Reynolds contributed much to our knowledge of fluid flow — not only the theory, but
also its practical applications. A few of these applications include ship propulsion, pumps,
turbines, estuaries of rivers, cavitation, condensation of steam, thermodynamics of gas
flow, rolling friction, and lubrication. For instance, he explained why a rotating shaft
rides on the film of oil between the shaft and sleeve.
For his work in 1883, leading to the now famous Reynolds Number, he injected ink into
water flowing along a glass pipe. He varied the pipe diameter and the water velocity to
develop the relationships that determine whether the resulting flow was smooth or
turbulent.
The equation for the dimensionless Reynolds Number, Re, for a circular pipe is:
Re = ρvD/µ
where
ρ = the fluid density
v = the mean fluid velocity
D = the pipe diameter
µ = the absolute fluid viscosity
As shown in Figure 5,
Laminar flow generally
occurs at low Reynolds
numbers (Re < 2000),
where viscous forces
dominate, and is
characterized by smooth,
constant fluid motion.
Turbulent flow, on the
other hand, occurs at high Figure 5. Fluids with Reynolds numbers in the transitional region are neither
laminar nor turbulent.
Reynolds numbers (Re >
4000), dominated by
inertial forces. Turbulent flow produces random eddies, vortices, and other flow
fluctuations. Engineers tend to avoid piping situations where Re falls between 2000 to
4000 to ensure the flow is either laminar or turbulent.
6. Theodore von Kármán
Born in 1881 a native of Budapest, Hungary, Theodore Von Kármán eventually became
an American citizen who contributed many key advances in aerodynamics. Like
Reynolds, von Kármán used mathematical tools to study fluid flow and apply the results
to practical designs.
His work helped to optimize the shape and lift of airplanes. For example, he was the first
to discover the importance of swept-back wings in jet aircraft. Later he became an
expert in supersonic and hypersonic airflow. He was named the first director of the Jet
Propulsion Laboratory at CalTech in 1944.
Early in his career, working with a newly constructed wind tunnel at the University of
Göttingen in Germany, von Kármán analyzed the flow of fluid past a cylindrical obstacle
at right angles. He found the wake separated into two rows that created wavelets, which
came to be called von Kármán vortices. This work led to the streamlining of many
structures, ships, and air ships.
He proposed vortex flowmeters based on the von Kármán vortices in 1954. These
flowmeters contain a bluff body with a broad flat front that faces the flow (Figure 6).
Distances between traveling vortices in the wake are constant, regardless of fluid
velocity. Sensors and electronics count the vortices in the wake. Flow velocity varies
directly with the frequency of vortices generated, according to the equation:
v = fL/St
where
v = the fluid velocity
f = the frequency of von Kármán vortices
L = the length across the bluff body
St = called the Strouhal number (a function of the Reynolds number, but generally
ranges from 0.18 to 0.2.)
Introduced to industrial markets in
the early 1970s, vortex flowmeters
have became relatively popular,
especially for steam flow
measurement. Compared to
differential-pressure flowmeters,
they offer reduced pressure drop
at a reasonable price.
Most of the equations discussed
above provide a value for the fluid
flow velocity within a pipe.
Knowledge of the flow profile (to
be covered in subsequent articles)
and the pipe diameter permits
calculation of the volumetric flow
rate.
Figure 6. Bluff obstacle in Vortex flowmeters creates eddies in the
wake separated by a constant distance. So the frequency of eddies
generated is a function of flow velocity.
This is the first article in a five-part series on the history and operation of flowmeter
technology. Part II will appear in the May issue.
Greg Livelli is the senior product manager for ABB Inc., based in Warminster, Pa. He
has more than 15 years experience in the design and marketing of flow measurement
equipment. Mr. Livelli earned an MBA from Regis University and a bachelor’s degree in
Mechanical Engineering from New Jersey Institute of Technology. Mr. Livelli can be
reached at [email protected] or 215 674-6000.
www.abb.com
Pioneers of Instrumentation
Daniel Bernoulli
Daniel Bernoulli (1700 – 1782) formulated what is known today
as Bernoulli’s theorem. This principle states that, in a flowing
stream, the sum of a fluid’s static energy, kinetic energy, and
potential energy is conserved across a constriction in the pipe.
Michael Faraday
Michael Faraday (1791 – 1867) is best
known in the history of flow for formulating
Faraday’s Law of Electromagnetic Induction.
According to this principle, a voltage is
developed when a conductor is passed
through a magnetic field, and the resulting
voltage is proportional to the velocity of the
conductor moving through the field.
Gaspard-Gustave de Coriolis
Gaspard-Gustave Coriolis (1792 – 1843) is well known for his
work on the Coriolis effect. The Coriolis effect forms the basis for
the operation of Coriolis flowmeters today. He was also the first
person to apply the term “work” to the product of force and
distance.
Christian Doppler
Christian Doppler (1803 – 1853) is known
for proposing what is today called the
Doppler effect. This has to do with the
apparent change in wavelength and
frequency of a wave perceived by the
observer to be moving relative to the wave
source.
Osborne Reynolds
Osborne Reynolds (1842 – 1912) is known in the history of flow
for his formulation of the Reynolds number. This number states
the relation between the inertial forces and
the viscous forces in a flowing stream.
Theodore von Kármán
Von Kármán (1881 – 1963) made early
studies in 1912 on the use of bluff bodies
and vortex swirls. The series of vortex swirls
are now called the von Kármán vortex
street.