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:

v^{2}/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.

Figure 1. Orifice creates a pressure drop that’s a function of flow velocity. |

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 (v^{2}/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 corresponds to 1.55 feet of water (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.

Figure 2. Electromagnetic flowmeters utilize Faraday’s law for conductive fluids. |

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 Gaspard-Gustave de Coriolis, born in 1792, became a student and teacher of mechanics, engineering 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:

F_{c} = -2m(? v)

where

F_{c} = the Coriolis force (perpendicular to the axis of rotation and flow)

m = the mass acted upon

? = the rate of angular rotation (a vector)

v = the velocity of the mass

Figure 3. Induced Coriolis forces twist the tube an amount proportional to the mass flowrate. |

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.

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:

f_{r} = f_{s} {(c/c-v)}

where

f_{r} = the observed frequency

f_{s} = the source frequency

c = the speed of sound in the medium (liquid in our case)

v = the velocity of the liquid

Figure 4. The Dopper frequency difference between source and reflected beams are a measure of flow velocity. |

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

Figure 5. Fluids with Reynolds numbers in the transitional region are neither laminar nor turbulent. |

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 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.)

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. |

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.

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 greg.livelli@us.abb.com or 215 674-6000.

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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. To request a FREE “Pioneers of Instrumentation” 2007 calendar, contact Jesse Yoder, president of Flow Research, at jesse@flowresearch.com or 781 245-3200. Photos courtesy of the American Institute of Physics’ Emilio Segre Visual Archives (www.aip.org). |

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