Vortex flowmeter (Velocity-based)

When a fluid moves with a high Reynolds number past a stationary object (a “bluff body”), there is a tendency for the fluid to form vortices on either side of the object. Each vortex will form, then detach from the object and continue to move with the flowing gas or liquid, one side at a time in an alternating fashion. This phenomenon is known as vortex shedding, and the pattern of moving vortices carried downstream of the stationary object is known as a vortex street.

It is commonplace to see the effects of vortex shedding on a windy day by observing the motion of flagpoles, light poles, and tall smokestacks. Each of these objects has a tendency to oscillate perpendicular to the direction of the wind, owing to the pressure variations caused by the vortices as they alternately form and break away from the object:

This alternating series of vortices was studied by Vincenc Strouhal in the late nineteenth century and later by Theodore von K´arm´an in the early twentieth century. It was determined that the distance between successive vortices downstream of the stationary object is relatively constant, and directly proportional to the width of the object, for a wide range of Reynolds number values35. If we view these vortices as crests of a continuous wave, the distance between vortices may be represented by the symbol customarily reserved for wavelength: the Greek letter “lambda” (λ).

The proportionality between object width (d) and vortex street wavelength (λ) is called the Strouhal number (S), approximately equal to 0.17:

λS = d; λ ≈ d/0.17

If a differential pressure sensor is installed immediately downstream of the stationary object in such an orientation that it detects the passing vortices as pressure variations, an alternating signal will be detected:

The frequency of this alternating pressure signal is directly proportional to fluid velocity past the object since the wavelength is constant. This follows the classic frequency-velocity-wavelength formula common to all traveling waves (λf = v). Since we know the wavelength will be equal to the bluff body’s width divided by the Strouhal number (approximately 0.17), we may substitute this into the frequency-velocity-wavelength formula to solve for fluid velocity (v) in terms of signal frequency (f) and bluff body width (d).

Thus, a stationary object and pressure sensor installed in the middle of a pipe section constitute a form of flowmeter called a vortex flowmeter. Like a turbine flowmeter with an electronic “pickup” sensor to detect the passage of rotating turbine blades, the output frequency of a vortex flowmeter is linearly proportional to the volumetric flow rate. The pressure sensors used in vortex flowmeters are not standard differential pressure transmitters, since the vortex frequency is too high to be successfully detected by such bulky instruments. Instead, the sensors are typically piezoelectric crystals. These pressure sensors need not be calibrated, since the amplitude of the pressure waves detected is irrelevant. Only the frequency of the waves matter for measuring flow rate, and so nearly any pressure sensor with a fast enough response time will suffice.

Like turbine meters, the relationship between sensor frequency (f) and volumetric flow rate (Q) may be expressed as a proportionality, with the letter k used to represent the constant of proportionality for any particular flowmeter:

Where,

f = Frequency of output signal (Hz)

Q = Volumetric flow rate (e.g. gallons per second)36

k = “K” factor of the vortex shedding flowtube (e.g. pulses per gallon)

This means vortex flowmeters, like electronic turbine meters, each has a particular “k factor” relating to the number of pulses generated per unit volume passed through the meter37. Counting the total number of pulses over a certain time span yields total fluid volume passed through the meter over that same time span, making the vortex flowmeter readily adaptable for “totalizing” fluid volume just like turbine meters. The direct proportion between vortex frequency and volumetric flow rate also means vortex flowmeters are linear-responding instruments just like turbine flowmeters. Unlike orifice plates which exhibit a quadratic response, turbine and vortex flowmeters alike enjoy a wider range (turndown) of flow measurement and do not require special signal characterization to function properly.

Since vortex flowmeters have no moving parts, they do not suffer the problems of wear and lubrication facing turbine meters. There is no moving element to “coast” as in a turbine flowmeter if fluid flow suddenly stops, which means vortex flowmeters are better suited to measuring erratic flows.

A significant disadvantage of vortex meters is a behavior known as low flow cutoff, where the flowmeter simply stops working below a certain flow rate. The reason for this is the laminar flow: at low flow rates (i.e. low Reynolds number values) fluid viscosity becomes sufficient to prevent vortices from forming, causing the vortex flowmeter to register zero flow even when there may be some (laminar) flow through the pipe. At high flow rates (i.e. high Reynolds number values), fluid momentum is enough to overcome viscosity and produce vortices, and the vortex flowmeter works just fine.

The phenomenon of low-flow cutoff for a vortex flowmeter at first seems analogous to the minimum linear flow limitation of a turbine flowmeter. However, vortex flowmeter low-flow cutoff is actually a far more severe problem. If the volumetric flow rate through a turbine flowmeter falls below the minimum linear value, the turbine continues to spin, albeit slower than it should. If the volumetric flow rate through a vortex flowmeter falls below the low-flow cutoff value, however, the flowmeter’s signal goes completely to zero, indicating no flow at all.

The following photograph shows a Rosemount model 8800C vortex flow transmitter:

The next two photographs show close-up views of the flowtube assembly, front (left) and rear (right):

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Turbine flowmeter (Velocity-based)

Turbine flowmeters use a free-spinning turbine wheel to measure fluid velocity, much like a miniature windmill installed in the flow stream. The fundamental design goal of a turbine flowmeter is to make the turbine element as free-spinning as possible, so no torque will be required to sustain the turbine’s rotation. If this goal is achieved, the turbine blades will achieve a rotating (tip) velocity directly proportional to the linear velocity of the fluid:

The mathematical relationship between fluid velocity and turbine tip velocity – assuming frictionless conditions – is a ratio defined by the tangent of the turbine blade angle:

For a 45 blade angle, the relationship is 1:1, with tip velocity equaling fluid velocity. Smaller blade angles (each blade closer to parallel with the fluid velocity vector) result in the tip velocity being a fractional proportion of fluid velocity.

Turbine tip velocity is quite easy to sense using a magnetic sensor, generating a voltage pulse each time one of the ferromagnetic turbine blades passes by. Traditionally, this sensor is nothing more than a coil of wire in proximity to a stationary magnet, called a pickup coil or pickoff coil because it “picks” (senses) the passing of the turbine blades. Magnetic flux through the coil’s center increases and decreases as the passing of the steel turbine blades presents a varying reluctance (“resistance” to magnetic flux), causing voltage pulses to equal in frequency to the number of blades passing by each second. It is the frequency of this signal that represents fluid velocity, and therefore volumetric flow rate.

A cut-away demonstration model of a turbine flowmeter is shown in the following photograph. The blade sensor may be seen protruding from the top of the flowtube, just above the turbine wheel:

Note the sets of “flow conditioner” vanes immediately before and after the turbine wheel in the photograph. As one might expect, turbine flowmeters are very sensitive to swirl in the process fluid flowstream. In order to achieve high accuracy, the flow profile must not be swirling in the vicinity of the turbine, lest the turbine wheel spin faster or slower than it should represent the velocity of a straight-flowing fluid.

Mechanical gears and rotating cables have also been historically used to link a turbine flowmeter’s turbine wheel to indicators. These designs suffer from greater friction than electronic (“pickup coil”) designs, potentially resulting in more measurement error (less flow indicated than there actually is, because the turbine wheel is slowed by friction). One advantage of mechanical turbine flowmeters, though, is the ability to maintain a running total of gas usage by turning a simple odometer-style totalizer. This design is often used when the purpose of the flowmeter is to track total fuel gas consumption (e.g. natural gas used by a commercial or industrial facility) for billing.

In an electronic turbine flowmeter, volumetric flow is directly and linearly proportional to pickup coil output frequency. We may express this relationship in the form of an equation:

Where,

f = Frequency of output signal (Hz, equivalent to pulses per second)

Q = Volumetric flow rate (e.g. gallons per second)

k = “K” factor of the turbine element (e.g. pulses per gallon)

Dimensional analysis confirms the validity of this equation. Using units of GPS (gallons per second) and pulses per gallon, we see that the product of these two quantities is indeed pulses per second (equivalent to cycles per second, or Hz):

Using algebra to solve for flow (Q), we see that it is the quotient of frequency and k factor that yields a volumetric flow rate for a turbine flowmeter:

The inherent linearity of a turbine flowmeter is a tremendous advantage over nonlinear flow elements such as venturi tubes and orifice plates because this linearity results in a much greater turndown ratio for accurate flow measurement. Contrasted against common orifice-type meters which are usually limited to turndown ratios of 4:1 at best, turbine meters commonly exceed turndown ratios of 10:1.

If pickup signal frequency directly represents volumetric flow rate, then the total number of pulses accumulated in any given time span will represent the amount of fluid volume passed through the turbine meter over that same time span. We may express this algebraically as the product of average flow rate (Q), average frequency (f), k factor, and time:

A more sophisticated way of calculating total volume passed through a turbine meter requires calculus, representing a change in volume as the time-integral of instantaneous signal frequency and k factor over a period of time from t = 0 to t = T:

We may achieve approximately the same result simply by using a digital counter circuit to totalize pulses output by the pickup coil and a microprocessor to calculate volume in whatever unit of measurement we deem appropriate.

As with the orifice plate flow element, standards have been drafted for the use of turbine flowmeters as precision measuring instruments in gas flow applications, particularly the custody transfer of natural gas. The American Gas Association has published a standard called Report #7 specifying the installation of turbine flowmeters for high-accuracy gas flow measurement, along with the associated mathematics for precisely calculating flow rate based on turbine speed, gas pressure, and gas temperature.

Pressure and temperature compensation is relevant to turbine flowmeters in gas flow applications because the density of the gas is a strong function of both pressure and temperature. The turbine wheel itself only senses gas velocity, and so these other factors must be taken into consideration to accurately calculate mass flow (or standard volumetric flow; e.g. SCFM).

In high-accuracy applications, it is important to individually determine the k factor for a turbine flowmeter’s calibration. Manufacturing variations from flowmeter to flowmeter make precise duplication of k factor challenging, and so a flowmeter destined for high-accuracy measurement should be tested against a “flow prover” in a calibration laboratory to empirically determine its k factor. If possible, the best way to test the flowmeter’s k factor is to connect the prover to the meter on site where it will be used. This way, any effects due to the piping before and after the flowmeter will be incorporated in the measured k factor.

The following photograph shows three AGA7-compliant installations of turbine flowmeters for measuring the flow rate of natural gas:

Note the pressure-sensing and temperature-sensing instrumentation installed in the pipe, reporting gas pressure and gas temperature to a flow-calculating computer (along with turbine pulse frequency) for the calculation of natural gas flow rate.

Less-critical gas flow measurement applications may use a “compensated” turbine flowmeter that mechanically performs the same pressure- and temperature-compensation functions on turbine speed to achieve true gas flow measurement, as shown in the following photograph:

The particular flowmeter shown in the above photograph uses a filled-bulb temperature sensor (note the coiled, armored capillary tube connecting the flowmeter to the bulb) and shows total gas flow by a series of pointers, rather than gas flow rate.

A variation on the theme of turbine flow measurement is the paddlewheel flowmeter, a very inexpensive technology usually implemented in the form of an insertion-type sensor. In this instrument, a small wheel equipped with “paddles” parallel to the shaft is inserted in the flowstream, with half the wheel shrouded from the flow. A photograph of a plastic paddlewheel flowmeter appears here:

A surprisingly sophisticated method of “pickup” for the plastic paddlewheel shown in the photograph uses fiber-optic cables to send and receive light. One cable sends a beam of light to the edge of the paddlewheel, and the other cable receives light on the other side of the paddlewheel. As the paddlewheel turns, the paddles alternately block and pass the light beam, resulting in a pulsed light beam at the receiving cable. The frequency of this pulsing is, of course, directly proportional to the volumetric flow rate.

The external ends of the two fiber optic cables appear in this next photograph, ready to connect to a light source and light pulse sensor to convert the paddlewheel’s motion into an electronic signal:

A problem common to all turbine flowmeters is that of the turbine “coasting” when the fluid flow suddenly stops. This is more often a problem in batch processes than continuous processes, where the fluid flow is regularly turned on and shut off. This problem may be minimized by configuring the measurement system to ignore turbine flowmeter signals any time the automatic shutoff valve reaches the “shut” position. This way, when the shutoff valve closes and fluid flow immediately halts, any coasting of the turbine wheel will be irrelevant. In processes where the fluid flow happens to pulse for reasons other than the control system opening and shutting automatic valves, this problem is more severe.

Another problem common to all turbine flowmeters is the lubrication of the turbine bearings. The frictionless motion of the turbine wheel is essential for accurate flow measurement, which is a daunting design goal for flowmeter manufacturing engineers. The problem is not as severe in applications where the process fluid is naturally lubricating (e.g. diesel fuel), but in applications such as natural gas flow where the fluid provides no lubrication to the turbine bearings, external lubrication must be supplied. This is often a regular maintenance task for instrument technicians: using a hand pump to inject light-weight “turbine oil” into the bearing assemblies of turbine flowmeters used in gas service.

Process fluid viscosity is another source of friction for the turbine wheel. Fluids with high viscosity (e.g. heavy oils) will tend to slow down the turbine’s rotation even if the turbine rotates on frictionless bearings. This effect is especially pronounced at low flow rates, which leads to a minimum linear flow rating for the flowmeter: a flowrate below which it refuses to register proportionately to fluid flow rate.

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Variable-area flowmeter

A variable-area flowmeter is one where the fluid must pass through a restriction whose area increases with flow rate. This stands in contrast to flowmeters such as orifice plates and venturi tubes where the cross-sectional area of the flow element remains fixed.

Rotameters

The simplest example of a variable-area flowmeter is the rotameter, which uses a solid object (called a plummet or float) as a flow indicator, suspended in the midst of a tapered tube:

As fluid flows upward through the tube, a pressure differential develops across the plummet. This pressure differential, acting on the effective area of the plummeting body, develops an upward force (F = P/A). If this force exceeds the weight of the plummet, the plummet moves up. As the plummet moves farther up in the tapered tube, the area between the plummet and the tube walls (through which the fluid must travel) grows larger. This increased flowing area allows the fluid to make it past the plummet without having to accelerate as much, thereby developing less pressure drop across the plummet’s body. At some point, the flowing area reaches a point where the pressure-induced force on the plummeting body exactly matches the weight of the plummet. This is the point in the tube where the plummet stops moving, indicating flow rate by it position relative to a scale mounted (or etched) on the outside of the tube.

The following rotameter uses a spherical plummet, suspended in a flow tube machined from a solid block of clear plastic. An adjustable valve at the bottom of the rotameter provides a means for adjusting gas flow:

The same basic flow equation used for pressure-based flow elements holds true for rotameters as well:

However, the difference in this application is that the value inside the radicand is constant since the pressure difference will remain constant and the fluid density will likely remain constant as well. Thus, k will change in proportion to Q. The only variable within k relevant to plummet position is the flowing area between the plummet and the tube walls.

Most rotameters are indicating devices only. They may be equipped to transmit flow information electronically by adding sensors to detect the plummet’s position in the tube, but this is not common practice.

Rotameters are very commonly used as purge flow indicators for pressure and level measurement systems requiring a constant flow of purge fluid. Such rotameters are usually equipped with hand-adjustable needle valves for manual regulation of purge fluid flow rate.

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Weirs and flumes

A very different style of the variable-area flowmeter is used extensively to measure flow rate through open channels, such as irrigation ditches. If an obstruction is placed within a channel, any liquid flowing through the channel must rise on the upstream side of the obstruction. By measuring this liquid level rise, it is possible to infer the rate of liquid flow past the obstruction.

The first form of the open-channel flowmeter is the weir, which is nothing more than a dam obstructing the passage of liquid through the channel. Three styles of the weir are shown in the following illustration; rectangular, Cippoletti, and V-notch:

A rectangular weir has a notch of simple rectangular shape, as the name implies. A Cippoletti weir is much like a rectangular weir, except that the vertical sides of the notch have a 4:1 slope (rise of 4, run of 1; approximately a 14-degree angle from vertical). A V-notch weir has a triangular notch, customarily measuring either 60 or 90 degrees.

The following photograph shows water flowing through a Cippoletti weir made of 1/4 inch steel plate:

At a condition of zero flow through the channel, the liquid level will be at or below the crest (lowest point on the opening) of the weir. As the liquid begins to flow through the channel, it must spill over the crest of the weir in order to get past the weir and continue downstream in the channel. In order for this to happen, the level of the liquid upstream of the weir must rise above the weir’s crest height. This height of liquid upstream of the weir represents a hydrostatic pressure, much the same as liquid heights in piezometer tubes represent pressures in a liquid flow stream through an enclosed pipe. The height of liquid above the crest of a weir is analogous to the pressure differential generated by an orifice plate. As liquid flow is increased, even more, a greater pressure (head) will be generated upstream of the weir, forcing the liquid level to rise. This effectively increases the cross-sectional area of the weir’s “throat” as a taller stream of liquid exits the notch of the weir.

This dependence of notch area on flow rate creates a very different relationship between flow rate and liquid height (measured above the crest) than the relationship between flow rate and differential pressure in an orifice plate:

Where,

Q = Volumetric flow rate (cubic feet per second – CFS)

L = Width of crest (feet)

θ = V-notch angle (degrees)

H = Head (feet)

As you can see from a comparison of characteristic flow equations between these three types of weirs, the shape of the weir’s notch has a dramatic effect on the mathematical relationship between flow rate and head (liquid level upstream of the weir, measured above the crest height). This implies that it is possible to create almost any characteristic equation we might like just by carefully shaping the weir’s notch in some custom form. A good example of this is the so-called proportional or Sutro weir:

Sutro weirs are not used very often, due to their inherently weak structure and tendency to clog with debris.

A rare example of a Sutro weir appears in the following photograph, discharging flow from a lake into a stream:

The metal plates forming the weir’s shape are quite thick (about 1/2 inch) to give the weir sufficient strength. A good construction practice seen on this Sutro weir, but recommended on all weir designs, is to bevel the downstream edge of the weir plate much like a standard orifice plate profile. The beveled edge provides a minimum-friction passageway for the liquid as it spills through the weir’s opening.

A variation on the theme of a weir is another open-channel device called a flume. If weirs may be thought of as open-channel orifice plates, then flumes may be thought of as open-channel venturi tubes:

Like weirs, flumes generate upstream liquid level height changes indicative of flow rate. One of the most common flume designs is the Parshall flume, named after its inventor R.L. Parshall when it was developed in the year 1920.

The following formulae relate head (upstream liquid height) to flow rate for free-flowing Parshall flumes:

Where,

Q = Volumetric flow rate (cubic feet per second – CFS)

L = Width of flume throat (feet)

H = Head (feet)

Flumes are generally less accurate than weirs, but they do enjoy the advantage of being inherently self-cleaning. If the liquid stream being measured is drainage- or waste-water, a substantial amount of solid debris may be present in the flow that could cause repeated clogging problems for weirs. In such applications, flumes are often the more practical flow element for the task (and more accurate over the long term as well, since even the finest weir will not register accurately once fouled by debris).

Once a weir or flume has been installed in an open channel to measure the flow of liquid, some method must be employed to sense upstream liquid level and translate this level measurement into a flow measurement. Perhaps the most common technology for weir/flume level sensing is ultrasonic. Ultrasonic level sensors are completely non-contact, which means they cannot become fouled by the process liquid (or debris in the process liquid). However, they may be “fooled” by foam or debris floating on top of the liquid, as well as waves on the liquid surface.

The following photograph shows a Parshall flume measuring effluent flow from a municipal sewage treatment plant, with an ultrasonic transducer mounted above the middle of the flume to detect water level flowing through:

Once the liquid level is successfully measured, a computing device is used to translate that level measurement into a suitable flow measurement (and in some cases even integrate that flow measurement with respect to time to arrive at a value for total liquid volume passed through the element, in accordance with the calculus relationship, 

A technique for providing a clean and “quiet” (still) liquid surface to measure the level of is called a stilling well. This is an open-top chamber connected to the weir/flume channel by a pipe, so the liquid level in the stilling well matches the liquid level in the channel. The following illustration shows a stilling well connected to a weir/flume channel, with the direction of liquid flow in the channel being perpendicular to the page (i.e. either coming toward your eyes or going away from your eyes):

To discourage plugging of the passageway connecting the stilling well to the channel, a small flow rate of clean water may be introduced into the well. This forms a constant purge flow into the channel, flushing out debris that might otherwise find its way into the connecting passageway to plug it up. Note how the purge water enters the stilling well through a submerged tube, so it does not cause splashing on the water’s surface inside the well which could cause measurement problems for the ultrasonic sensor:

A significant advantage that weirs and flumes have over other forms of flow measurement is exceptionally high rangeability: the ability to measure very wide ranges of flow with a modest pressure (height) span. Another way to state this is to say that the accuracy of a weir or flume is quite high even at low flow rates.

Earlier in this section, you saw a three-image representation of liquid flow through a rectangular weir. As fluid flow rate increased, so did the height (head) of the liquid upstream of the weir:

The height of liquid upstream of the weir depends on the flow rate (volumetric Q or mass W) as well as the effective area of the notch through which the fluid must pass. Unlike an orifice plate, this area changes with flow rate in both weirs and flumes. One way to envision this, by comparison, is to imagine a weir as acting like an elastic orifice plate, whose bore area increases with flow rate. This flow-dependent notch area exhibited by both weirs and flumes means that these devices become more sensitive to changes in flow as the flow rate becomes smaller.

A comparison of transfer function graphs for closed-pipe head elements such as orifice plates and venturi tubes versus weirs and flumes shows this striking difference in characteristics:

Looking at the orifice plate/venturi tube graph near the lower-left corner, you can see how small changes in flow result in extremely small changes in the head (differential pressure), because the function has a very low slope (small dH/dQ ) at that end. By comparison, a weir or flume produces relatively large changes in head (liquid elevation) for small changes in flow near the bottom end of the range, because the function has a very steep slope (large dH/dQ ) at that end.

The practical advantage this gives weirs and flumes is the ability to maintain high accuracy of flow measurement at very low flow rates – something a fixed-orifice element simply cannot do. It is commonly understood in industry that traditional orifice plate flowmeters do not maintain good measurement accuracy much below a third of their full-range flow (a rangeability or turndown of 3:1), whereas weirs (especially the V-notch design) can achieve far greater turndown (up to 500:1 according to some sources).

 

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Laminar flowmeter

A unique form of differential pressure-based flow measurement deserves its own section in this flow measurement chapter, and that is the laminar flowmeter.

Laminar flow is a condition of fluid motion where viscous (internal fluid friction) forces greatly overshadow inertial (kinetic) forces. A flow stream in a state of laminar flow exhibits no turbulence, with each fluid molecule traveling in its own path, with limited mixing and collisions with adjacent molecules. The dominant mechanism for resistance to fluid motion in a laminar flow regime is friction with the pipe or tube walls. Laminar flow is qualitatively predicted by low values of Reynolds number.

This pressure drop created by fluid friction in a laminar flow stream is quantifiable, and is expressed in the Hagen-Poiseuille equation:

Where,

Q = Flow rate

∆P = Pressure dropped across a length of pipe

D = Pipe diameter

μ = Fluid viscosity

L = Pipe length

k = Coefficient accounting for units of measurement

Laminar flowmeter elements generally consist of one or more tubes whose length greatly exceeds the inside diameter, arranged in such a way as to produce a slow-moving flow velocity. An example is shown here:

The expanded diameter of the flow element ensures a lower fluid velocity than in the pipes entering and exiting the element. This decreases the Reynolds number to the point where the flow regime exhibits laminar behavior. The large number of small-diameter tubes packed in the wide area of the element provides adequate wall surface area for the fluid’s viscosity to act upon, creating an overall pressure drop from inlet to the outlet which is measured by the differential pressure transmitter. This pressure drop is permanent (no recovery of pressure downstream) because the mechanism of pressure drop is friction: total dissipation (loss) of energy in the form of heat.

Another common form of laminar flow element is simply a coiled capillary tube: a long tube with a very small inside diameter. The small inside diameter of such a tube makes wall-boundary effects dominant, such that the flow regime will remain laminar over a wide range of flow rates. The extremely restrictive nature of a capillary tube, of course, limits the use of such flow elements to very low flow rates such as those encountered in the sampling networks of certain analytical instruments.

A unique advantage of the laminar flowmeter is its linear relationship between flow rate and developed pressure drop. It is the only pressure-based flow measurement device for filled pipes that exhibits a linear pressure/flow relationship. This means no “square-root” characterization is necessary to obtain linear flow measurements with a laminar flowmeter. The big disadvantage of this meter type is its dependence on fluid viscosity, which in turn is strongly influenced by fluid temperature. Thus, all laminar flowmeters require temperature compensation to derive accurate measurements, and some even use temperature control systems to force the fluid’s temperature to be constant as it moves through the element.

Laminar flow elements find their widest application inside pneumatic instruments, where a linear pressure/flow relationship is highly advantageous (behaving like a “resistor” for instrument airflow) and the viscosity of the fluid (instrument air) is relatively constant. Pneumatic controllers, for instance, use laminar restrictors as part of the derivative and integral calculation modules, the combination of “resistance” from the restrictor and “capacitance” from volume chambers forming a sort of pneumatic time-constant (τ) network.

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