... instrumentation1.1
Analytical (chemical composition) measurement is undeniably more complex and diverse than flow measurement, but analytical measurement covers a great deal of specific measurement types. As a single process variable, flow measurement is probably the most complex. 
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... pipe1.2
Sometimes referred to as a plug of fluid. 
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...differential1.3
What really matters in Newton's Second Law equation is the resultant force causing the acceleration. This is the vector sum of all forces acting on the mass. Likewise, what really matters in this scenario is the resultant pressure acting on the fluid plug, and this resultant pressure is the difference of pressure between one face of the plug and the other, since those two pressures impart two forces on the fluid mass in direct opposition to each other. 
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...piezometers1.4
Think of a piezometer tube as nothing more than a manometer tube: the greater the fluid pressure at the bottom of the tube, the higher the liquid will rise inside the tube. 
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... constant1.5
This is a very sound assumption for liquids, and a fair assumption for gases when pressure changes through the venturi tube are modest. 
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... system1.6
One of the simplifying assumptions we make in this derivation is that friction plays no significant role in the fluid's behavior as it moves through the venturi tube. In truth, no industrial fluid flow is totally frictionless (especially through more primitive flow elements such as orifice plates), and so our “theoretical” equations must be adjusted a bit to match real life. 
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... tube1.7
To see a graphical relationship between fluid acceleration and fluid pressures in a venturi tube, examine the illustration found in section [*] beginning on page [*]. 
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... improvement1.8
This re-write is solidly grounded in the rules of algebra. We know that $\sqrt{a} \sqrt{b} = \sqrt{ab}$, which is what allows us to do the re-write. 
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... number1.9
For positive numbers only! 
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...somewhere1.10
With so many modern instruments being capable of digitally implementing this square-root function, one must be careful to ensure it is only done once in the loop. I have personally witnessed flow-measurement installations where both the pressure transmitter and the indicating device were configured for square-root characterization. This essentially performed a fourth root characterization on the signal, which is just as bad as no characterization at all! Like anything else technical, the key to successful implementation is a correct understanding of how the system is supposed to work. Simply memorizing that “the instrument must be set up with square-root to measure flow” and blindly applying that mantra is a recipe for failure. 
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... technology1.11
Despite the impressive craftsmanship and engineering that went into the design of pneumatic square root extractors, their obsolescence is mourned by no one. These devices were notoriously difficult to set up and calibrate accurately, especially as they aged. 
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... choice1.12
L.K. Spink, in his book Principles and Practice of Flow Meter Engineering, notes that drain holes intended to pass solid objects may be useless in small pipe sizes, where the hole is so small it will probably become plugged with solid debris and cease to provide benefit. In such installations he recommends re-orienting the pipe vertically instead of horizontally. This allows solids to pass through the main bore of the orifice without “damming” on the upstream side of the orifice plate. I would add the suggestion to consider a different primary element entirely, such as a venturi tube. The small size of the line will limit the cost of such an element, and the performance is likely to be far better than an orifice plate anyway. 
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... number1.13
To read more about the concept of Reynolds number, refer to section [*] beginning on page [*]. 
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... error1.14
One significant source of error for customer-drilled tap holes is the interior finish of the holes. Even a small “burr” of metal left where the hole penetrates the inner surface of the pipe wall will cause substantial flow measurement errors! 
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... readings1.15
What this means is that a “pipe tap” installation is actually measuring permanent pressure loss, which also happens to scale with the square of flow rate because the primary mechanism for energy loss in turbulent flow conditions is the translation of linear velocity to angular (swirling) velocity in the form of eddies. This kinetic energy is eventually dissipated in the form of heat as the eddies eventually succumb to viscosity. 
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... below1.16
One installation error seen in this photograph is a green plastic impulse tube with a bend extending above the upper flange tap. Any elevated portion of the impulse tube system will tend to collect gas bubbles over time, possibly causing measurement errors. A better installation would ensure the impulse tubes never extend above the flange tap they connect to on the liquid-bearing pipe. 
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... reverse1.17
If an orifice plate is a “donut,” the V-cone is a “donut hole.” 
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...slurries1.18
A “slurry” is a suspension of solid particles within a liquid. Mud is a common example of a slurry. 
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... corner1.19
This phenomenon may be observed when watching the flow of water through a turn in a river, especially if the river is fast-moving. Water level at the far (outside) bank of the turn will be higher than the water level at the near (inside) bank of the turn, due to radial acceleration of the water and the pressure difference that acceleration generates. In fact, that difference in water height may even be used to estimate the river's flow rate! 
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... generated1.20
The fact that a pipe elbow generates small differential pressure is an accuracy concern because other sources of pressure become larger by comparison. Noise generated by fluid turbulence in the elbow, for example, becomes a significant portion of the pressure sensed by the transmitter when the differential pressure is so low (i.e. the signal-to-noise ratio becomes smaller). Errors caused by differences in elbow tap elevation and different impulse line fill fluids, for example, become more significant as well. 
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... process1.21
This is not always the case, as primary elements are often found on throttled process lines. In such cases where a control valve normally throttles the flow rate, any energy dissipated by the orifice plate is simply less energy that the valve would otherwise be required to dissipate. Therefore, the presence or absence of an orifice plate has no net impact on energy dissipation when used on a process flow throttled by a control valve. 
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... turbulence1.22
This is not to be confused with micro-turbulence in the fluid, which cannot be eliminated at high Reynolds number values. In fact, “fully-developed turbulent flow” is desirable for head-based meter elements such as orifice plates because it means the flow profile will be relatively flat (even velocities across the pipe's diameter) and frictional forces (viscosity) will be negligible. The thing we are trying to avoid is large-scale turbulent effects such as eddies, swirl, and asymmetrical flow profiles, which compromise the ability of most flowmeters to accurate measure flow rate. 
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... flowstream1.23
L.K. Spink mentions in his book Principles and Practice of Flow Meter Engineering that certain tests have shown flow measurement errors induced from severe disturbances as far as 60 to 100 pipe diameters upstream of the primary flow element. The April 2000 update of API standard 14.3 (for custody-transfer measurement of natural gas using orifice plates) calls for upward of 145 pipe diameters of straight-length pipe upstream of the orifice plate! 
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... tolerance1.24
Flow elements with low beta ratio values tolerate greater disturbance in the flow pattern because they accelerate the flowstream to a greater degree. This may be best visualized by a thought experiment where we imagine an orifice plate with a very large beta ratio (i.e. one where the bore size is nearly as large as the pipe diameter): such an orifice plate would hardly accelerate the fluid at all, which would mean a mis-shapen flow profile entering the bore would probably remain mis-shapen exiting it. The acceleration imparted to a flowstream by a low-beta element tends to overshadow any asymmetries in the flow profile. However, there are disadvantages to using low-beta elements, one of them being increased permanent pressure loss which may translate to increased operating costs due to energy loss. 
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... unattractive1.25
Beauty is truly in the eye of the beholder. While a piping designer might see straight-run lengths of pipe in awkward locations – necessitating more pipe and/or more bends elsewhere in the system to accommodate – as wasteful and ugly, the instrument engineer sees it as a thing of beauty. 
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... discrepancy1.26
Richard W. Miller, in his outstanding book Flow Measurement Engineering Handbook, states that venturi tubes may come within 1 to 3 percent of ideal, while a square-edged orifice plate may perform as poorly as only 60 percent of theoretical! 
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... conditioner1.27
Specified in Part 2 of the AGA Report #3, section 2.6.5, page 22. A major reason for this is von Kármán vortex shedding caused by the gas having to flow around the width of the thermowell. The “street” of vortices shed by the thermowell will cause serious pressure fluctuations at the orifice plate unless mitigated by a flow conditioner, or by locating the thermowell downstream so that the vortices do not reach the orifice. 
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... measurement1.28
This is especially true in the gas exploration industry, where natural gas coming out of the well is laden with mineral debris. 
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... purposes1.29
Liquids can and do compress, the measurement of their “compressibility” being what is called the bulk modulus. However, this compressibility is too slight to be of any consequence in most flow measurement applications. A notable exception is the metering of diesel fuel through a high-pressure injection pump, where liquid pressures range in the tens of thousands of PSI, and the compressibility of the liquid diesel fuel may affect the precise timing of individual injections into the engine cylinders. 
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... “swamped1.30
“Swamping” is a term commonly used in electrical engineering, where a bad effect is overshadowed by some other effect much larger in magnitude, to the point where the undesirable effect is negligible in comparison. 
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... element1.31
This includes elaborate oil-bath systems where the laminar flow element is submerged in a temperature-controlled oil bath, the purpose of which is to hold temperature inside the laminar element constant despite sudden changes in the measured fluid's temperature. 
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... constant1.32
If we know that the plummet's weight will remain constant, its drag area will remain constant, and that the force generated by the pressure drop will always be in equilibrium with the plummet's weight for any steady flow rate, then the relationship $F = P A$ dictates a constant pressure. Thus, we may classify the rotameter as a constant-pressure, variable-area flowmeter. This stands in contrast to devices such as orifice plates, which are variable-pressure, constant-area. 
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... weir1.33
Orifice plates are variable-pressure, constant-area flowmeters. Rotameters are constant-pressure, variable-area flowmeters. Weirs are variable-pressure, variable-area flowmeters. As one might expect, the mathematical functions describing each of these flowmeter types is unique! 
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... flumes1.34
It is also possible to operate a Parshall flume in fully submerged mode, where liquid level must be measured at both the upstream and throat sections of the flume. Correction factors must be applied to these equations if the flume is submerged. 
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... sources1.35
These figures are reported in Béla Lipták's excellent reference book Instrument Engineers' Handbook – Process Measurement and Analysis Volume I (Fourth Edition). To be fair to closed-pipe elements such as orifice plates and venturi tubes, much improvement in the classic 3:1 rangeability limitation has been achieved through the use of microprocessor-based differential pressure sensors. Lipták reports rangeabilities for orifice plates as great as 10:1 through the use of such modern differential pressure instruments. However, even this pales in comparison to the rangeability of a typical weir or flume, which Lipták reports to be 75:1 for “most devices” in this category. 
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... transfer1.36
“Custody transfer” refers to measurement applications where a product is exchanging ownership. In other words, someone is selling, and someone else is buying, quantities of fluid as part of a business transaction. It is not difficult to understand why accuracy is important in such applications, as both parties have a vested interest in a fair exchange. Government institutions also have a stake in accurate metering, as taxes are typically levied on the sale of commodity fluids such as natural gas. 
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... values1.37
It is important to note that the vortex-shedding phenomenon ceases altogether if the Reynolds number is too low. Laminar flow produces no vortices, but rather stream-line flow around any object placed in its way. 
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... second)1.38
Note that if flow rate is to be expressed in units of gallons per minute as is customary, the equation must contain a factor for minutes-to-seconds conversion: $f = {kQ \over 60}$ 
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... meter1.39
This $k$ factor is empirically determined for each flowmeter by the manufacturer using water as the test fluid (a factory “wet-calibration”), to ensure optimum accuracy. 
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... liquid1.40
In a practical sense, only liquid flows are measurable using this technique. Gases must be super-heated into a plasma state before they are able to conduct electricity, and so electromagnetic flowmeters cannot be used with most industrial gas flowstreams. 
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... velocity1.41
This is an application of the transitive property in mathematics: if two quantities are both equal to a common third quantity, they must also be equal to each other. This property applies to proportionalities as well as equalities: if two quantities are proportional to a common third quantity, they must also be proportional to each other. 
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... formula1.42
The colloquial term in the United States for this sort of thing is fudge factor. 
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... close1.43
The obvious solution to this problem – relocating the pipes to give more clearance between flowmeters – would be quite expensive given the large pipe sizes involved. A “compromise” solution is to tilt the magnetic flowtubes as far as possible without the electrodes touching the adjacent flowtube. Horizontal electrode installation is ideal for horizontal pipes, but an angled installation will be better than a vertical installation. 
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... typical1.44
As always, check the manufacturer's literature for specific requirements, as variations do exist for different models and sizes of magtube. 
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... solids1.45
Even electrically non-conducting solid matter is tolerated well by magnetic flowmeters, since the conducting liquid surrounding the solids still provides continuity from one electrode to the other. 
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... braided1.46
Braided conductors do a better job of shunting radio-frequency currents, because at very high frequencies the skin effect makes the surface area of a conductor a greater factor in its conductivity than its cross-sectional area. 
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... varies1.47
For example, in a condition of no liquid flow through the tube, the electrodes will intercept no voltage at all when the magnetic excitation is 60 Hz AC. When liquid moves slowly in the forward direction through the tube, a low-amplitude 60 Hz millivoltage signal will be detected at the electrodes. When liquid moves rapidly in the forward direction through the tube, the induced 60 Hz AC millivoltage will be greater in amplitude. Any liquid motion in the reverse direction induces a proportional 60 Hz AC voltage signal whose phase is 180$^{o}$ shifted from the excitation signal driving the magnetic coils of the flowtube. 
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... well1.48
We know this because the largest electrical noise sources in industry are electric motors, transformers, and other power devices operating on the exact same frequency (60 Hz in the United States, 50 Hz in Europe) as the flowtube coils. 
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... transducer1.49
In the industrial instrumentation world, the word “transducer” usually has a very specific meaning: a device used to process or convert standardized instrumentation signals, such as 4-20 mA converted into 3-15 PSI, etc. In the general scientific world, however, the word “transducer” describes any device converting one form of energy into another. It is this latter definition of the word that I am using when I describe an ultrasonic “transducer” – a device used to convert electrical energy into ultrasonic sound waves, and vice-versa. 
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... upstream1.50
This phenomenon is analogous to paddling a canoe across the width of a river, with the canoe bow angled upstream versus angled downstream. Angled upstream, the canoeist must overcome the velocity of the river and therefore takes longer to reach the other side. Angled downstream, the river's velocity aids the canoeist's efforts and therefore the trip takes less time. 
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... fluid1.51
If you would like to prove this to yourself, you may do so by substituting path length ($L$), fluid velocity ($v$), and sound velocity ($c$) for the times in the flow formula. Use $t_{up} = {L \over {c-v}}$ and $t_{down} = {L \over {c+v}}$ as your substitutions, then algebraically reduce the flow formula until you find that all the $c$ terms cancel. Your final result should be $Q = {2kv \over L}$. 
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... analysis1.52
An instrument called a gas chromatograph is able to provide live measurement of gas composition, with a computer calculating the average speed of sound for the gas given the known types and percentages of each molecular compound comprising the gas mixture. It just so happens that gas composition analysis by chromatograph is something typically done for custody transfer flow measurement of natural gas anyway, for the primary purpose of calculating the gas's heating value as a fuel, and therefore no additional investment of instrumentation is necessary to calculate the gas's speed of sound in this application. 
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... transition1.53
See page 10 of Friedrich Hofmann's Fundamentals of Ultrasonic Flow Measurement for industrial applications paper. 
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... problems1.54
Most notably, the problem of achieving good acoustic coupling with the pipe wall so signal transmission to the fluid and signal reception back to the sensor may be optimized. Also, there is the potential for sound waves to “ring around the pipe” instead of travel through the fluid with clamp-on ultrasonic flowmeters because the sound waves must travel through the full thickness of the pipe walls in order to enter and exit the fluid stream. 
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... other1.55
Recall from algebra that we may perform any arithmetic operation we wish to any equation, so long as we apply that operation equally to both sides of the equation. Dividing one equation by another equation obeys this principle, because both sides of the second equation are equal. In other words, we could divide both sides of the first equation by $P_A V_A$ (although that would not give us the solution we are looking for), but dividing the left side by $P_A V_A$ and the right side by $nR T_A$ is really doing the same thing, since $nR T_A$ is identical in value to $P_A V_A$. 
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... ratio1.56
Division by $t$ does not alter the equation at all, since we are essentially multiplying the left-hand side by $t \over t$ which is multiplication by 1. This is why we did not have to apply $t$ to the right-hand side of the equation. 
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... multiple1.57
The wonderful thing about standards is that there are so many to choose from! 
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... occupy1.58
In some applications, such as the custody transfer of natural gas, we are interested in something even more abstract: heating value. However, in order to calculate the gross heating value of a fuel gas stream, we must begin with an accurate mass flow measurement – volumetric flow is not really helpful. 
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...mole1.59
A “mole” is equal to a value of $6.022 \times 10^{23}$ entities. Therefore, one mole of carbon atoms is 602,200,000,000,000,000,000,000 carbon atoms. For a more detailed examination of this subject, refer to section [*] beginning on page [*]. 
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... grams1.60
I am purposely ignoring the fact that naturally occurring carbon has an average atomic mass of 12.011, and naturally occurring oxygen has an atomic mass of 15.9994. 
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... lbm1.61
The British unit of the “pound” is technically a measure of force or weight and not mass. The proper unit of mass measurement in the British system is the “slug.” However, for better or worse, the “slug” is rarely used, and so engineers have gotten into the habit of using “pound” as a mass measurement. In order to distinguish the use of “pound” to represent mass (an intrinsic property of matter) as opposed to the use of “pound” to represent weight (an incidental property of matter), the former is abbreviated lbm (literally, “pounds mass”). In Earth gravity, “lbm” and “lb” are synonymous. However, the standard Newtonian equation relating force, mass, and acceleration ($F = ma$) does not work when “lbm” is the unit used for mass and “lb” is used for force (it does when “slug” is used for mass and “lb” is used for force, though!). A weird unit of force invented to legitimize “pound” as an expression of mass is the poundal (“pdl”): one “poundal” of force is the reaction of one “pound” of mass (lbm) accelerated one foot per second squared. By this definition, a one-pound mass (1 lbm) in Earth gravity weighs 32 poundals! 
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... nature1.62
One could argue that orifice plates and other pressure-based flowmeters respond primarily to mass flow rather than volumetric flow, since their operation is based on the pressure created by accelerating a mass. However, fluid density does affect the relationship between mass flow rate and differential pressure (note how the density term $\rho$ appears in the mass flow equation $W = k\sqrt{\rho (P_1 - P_2)}$, where it would not if differential pressure were a strict function of mass flow rate and nothing else), and so the raw output of these instruments must still be “compensated” by pressure and temperature measurements. 
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... technologies1.63
The impeller-turbine and twin-turbine mass flowmeter types are examples of mechanical true-mass flow technologies. Both work on the principle of fluid inertia. In the case of the impeller-turbine flowmeter, an impeller driven by a constant-speed electric motor imparts a “spin” to a moving fluid, which then impinges on a stationary turbine wheel to generate a measurable torque. The greater the mass flow rate, the greater the impulse force imparted to the turbine wheel. In the twin-turbine mass flowmeter, two rotating turbine wheels with different blade pitches are coupled together by a flexible coupling. As each turbine wheel attempts to spin at its own speed, the inertia of the fluid causes a differential torque to develop between the two wheels. The more mass flow rate, the greater the angular displacement (offset) between the two wheels. 
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...density1.64
In fact, this density-measuring function of Coriolis flowmeters is so precise that they often find use primarily as density meters, and only secondarily as flowmeters! 
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...undulate1.65
An interesting experiment to perform consists of holding a water hose in a U-shape and gently swinging the hose back and forth like a pendulum, then flowing water through that same hose while you continue to swing it. The hose will begin to undulate, its twisting motion becoming visually apparent. 
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... equation1.66
This is an example of a vector cross-product where all three vectors are perpendicular to each other, and the directions follow the right-hand rule. 
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... hose1.67
The Coriolis force generated by a flowing fire hose as firefighters work to point it in a different direction can be quite significant, owing to the high mass flow rate of the water as it flows through the hose and out the nozzle! 
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... directions)1.68
For those readers with an automotive bent, this is the same principle applied in opposed-cylinder engines (e.g. Porsche “boxer” air-cooled 6-cylinder engine, Volkswagen air-cooled 4-cylinder engine, BMW air-cooled motorcycle twin engine, Citroen 2CV 2-cylinder engine, Subaru 4- and 6-cylinder opposed engines, etc.). Opposite piston pairs are always 180$^{o}$ out of phase for the purpose of maintaining mechanical balance: both moving away from the crankshaft or both moving toward the crankshaft, at any given time. 
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... tubes1.69
An alternative to splitting the flow is to plumb the tubes in series so they must share the exact same flow rate, like series-connected resistors sharing the exact same amount of electrical current. 
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... frequency1.70
The force coil is powered by an electronic amplifier circuit, which receives feedback from the sensor coils. Like any amplifier circuit given positive (regenerative) feedback, it will begin to oscillate at a frequency determined by the feedback network. In this case, the feedback “network” consists of the force coil, tubes, and sensor coils. The tubes, having both resilience and mass, naturally possess their own resonant frequency. This mechanical resonance dominates the feedback characteristic of the amplifier loop, causing the amplifier circuit to oscillate at that same frequency. 
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... loop1.71
This usually takes the form of a simple analog oscillator circuit, using the tubes and sensors as feedback elements. It is not unlike a crystal oscillator circuit where the mechanical resonance of a quartz crystal stabilizes the circuit's frequency at one value. The feedback system naturally finds and maintains resonance, just as a crystal oscillator circuit naturally finds and maintains the resonant frequency of the quartz crystal when provided with ample regenerative (positive) feedback. As fluid density inside the tubes changes, the tubes' mass changes accordingly, thus altering the resonant frequency of the system. The analog nature of this mechanical oscillator explains why some early versions of Coriolis flowmeters sometimes required a minor shake or tap to the flowtubes to start their oscillation! 
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... density1.72
If you consider each tube as a container with a fixed volume capacity, a change in fluid density (e.g. pounds per cubic foot) must result in a change in mass for each tube. 
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... temperature1.73
An important caveat is that the RTD sensing tube temperature in a Coriolis flowmeter really senses the tubes' outside skin temperature, which may not be exactly the same as the temperature of the fluid inside the tube. If the ambient temperature near the flowmeter differs substantially from the fluid's temperature, the tube skin temperature reading may not be accurate enough for the flowmeter to double as a fluid temperature transmitter. 
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... mixed-phase1.74
Significant technological progress has been made on mixed-phase Coriolis flow measurement, to the point where this may no longer be a serious consideration in the future. 
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... amount1.75
For example, the specific heat of water is 1.00 kcal / kg $\cdot$ $^{o}$C, meaning that the addition of 1000 calories of heat energy is required to raise the temperature of 1 kilogram of water by 1 degree Celsius, or that we must remove 1000 calories of heat energy to cool that same quantity of water by 1 degree Celsius. Ethyl alcohol, by contrast, has a specific heat value of only 0.58 kcal / kg $\cdot$ $^{o}$C, meaning it is almost twice as easy to warm up or cool down as water (little more than half the energy required to heat or cool water needs to be transferred to heat or cool the same mass quantity of ethyl alcohol by the same amount of temperature). 
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... width1.76
In a laminar flowstream, individual molecules do not cross paths, but rather travel in parallel lines. This means only those molecules traveling near the wall of a tube will be exposed to the temperature of the wall. The lack of “mixing” in a laminar flowstream means molecules traveling in the inner portions of the stream never contact the tube wall, and therefore never serve to transfer heat directly to or from the wall. At best, those inner-path molecules transfer heat by conduction with adjacent molecules which is a less efficient transfer mechanism than convection. 
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... equate1.77
The proper mass flow rate value corresponding to these two measurements would be 45.0 lb/h. 
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... equation1.78
While this may seem like a very informal definition of differential, it is actually rooted in a field of mathematics called nonstandard analysis, and closely compares with the conceptual notions envisioned by calculus' founders. 
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... time1.79
To be precise, the equation describing the function of this analog differentiator circuit is: $V_{out} = -RC {dV_{in} \over dt}$. The negative sign is an artifact of the circuit design – being essentially an inverting amplifier with negative gain – and not an essential element of the math. 
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... plate1.80
This is not always the case, as primary elements are often found on throttled process lines. In such cases where a control valve normally throttles the flow rate, any energy dissipated by the orifice plate is simply less energy that the valve would otherwise be required to dissipate. Therefore, the presence or absence of an orifice plate has no net impact on energy dissipation when used on a process flow throttled by a control valve, and therefore does not affect cost over time due to energy loss. 
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