Square-root characterization (Pressure based flowmeter)

It should be apparent by now that the relationship between flow rate (whether it be volumetric or mass) and differential pressure for any fluid-accelerating flow element are non-linear: a doubling of flow rate will not result in a doubling of differential pressure. Rather, a doubling of flow rate will result in a quadrupling of differential pressure. Likewise, a tripling of flow rate results in nine times as much differential pressure developed by the fluid-accelerating flow element.

When plotted on a graph, the relationship between flow rate (Q) and differential pressure (∆P) is quadratic, like one-half of a parabola. Differential pressure developed by a venturi, orifice plate, pitot tube, or any other acceleration-based flow element is proportional to the square of the flow rate:

An unfortunate consequence of this quadratic relationship is that a pressure-sensing instrument connected to such a flow element will not directly sense flow rate. Instead, the pressure instrument will be sensing what is essentially the square of the flow rate. The instrument may register correctly at the 0% and 100% range points if correctly calibrated for the flow element it connects to, but it will fail to register linearly in between. Any indicator, recorder, or controller connected to the pressure-sensing instrument will likewise register incorrectly at any point between 0% and 100% of a range because the pressure signal is not a direct representation of flow rate.

In order that we may have indicators, recorders, and controllers that do register linearly with flow rate, we must mathematically “condition” or “characterize” the pressure signal sensed by the differential pressure instrument. Since the mathematical function inherent to the flow element is quadratic (square), the proper conditioning for the signal must be the inverse of that: square root. Just as taking the square root of the square of a number yields the original number9, taking the square root of the differential pressure signal – which is itself a function of flow squared – yields a signal directly representing flow.

The traditional means of implementing the necessary signal characterization was to install a “square root” function relay between the transmitter and the flow indicator, as shown in the following diagram:

The modern solution to this problem is to incorporate square-root signal characterization either inside the transmitter or inside the receiving instrument (e.g. indicator, recorder, or controller). Either way, the square-root function must be implemented somewhere in the loop in order that flow may be accurately measured throughout the operating range.

In the days of pneumatic instrumentation, this square-root function was performed in a separate device called a square root extractor. The Foxboro model 557 (left) and Moore Products model 65 (right) pneumatic square root extractors are classic examples of this technology:

Pneumatic square root extraction relays approximated the square root function by means of triangulated force or motion. In essence, they were trigonometric function relays, not square-root relays per se. However, for small angular motions, certain trigonometric functions were close enough to a square-root function that the relays were able to serve their purpose in characterizing the output signal of a pressure sensor to yield a signal representing flow rate.

The following table shows the ideal response of a pneumatic square root relay:

Input Signal (PSI)	Input (%)	Output (%)	Output Signal (PSI)
3	0.0	0.0	3.00
4	8.33	28.87	6.464
5	16.67	40.82	7.899
6	25.0	50.0	9.00
7	33.33	57.74	9.928
8	41.67	64.55	10.75
9	50.0	70.71	11.49
10	58.33	76.38	12.17
11	66.67	81.65	12.80
12	75.0	86.60	13.39
13	83.33	91.29	13.95
14	91.67	95.74	14.49
15	100.0	100.0	15.00

As you can see from the table, the square-root relationship is most evident in comparing the input and output percentage values. For example, at an input signal pressure of 6 PSI (25%), the output signal percentage will be the square root of 25%, which is 50% (0.5 = √0.25) or 9 PSI as a pneumatic signal. At an input signal pressure of 10 PSI (58.33%), the output signal percentage will be 76.38%, because 0.7638 = √0.5833, yielding an output signal pressure of 12.17 PSI.

When graphed, the function of a square-root extractor is precisely opposite (inverted) of the quadratic function of a flow-sensing element such as an orifice plate, venturi, or pitot tube:

When cascaded – the square-root function placed immediately after the flow element’s “square” function – the result is an output signal that tracks linearly with flow rate (Q). An instrument connected to the square root relay’s signal will therefore register flow rate as it should.

Although analog electronic square-root relays have been built and used in industry for characterizing the output of 4-20 mA electronic transmitters, a far more common application of electronic square-root characterization is found in DP transmitters with the square-root function built-in. This way, an external relay device is not necessary to characterize the DP transmitter’s signal into a flow rate signal:

Using a characterized DP transmitter, any 4-20 mA sensing instrument connected to the transmitter’s output wires will directly interpret the signal as flow rate rather than as pressure. A calibration table for such a DP transmitter (with an input range of 0 to 150 inches water column) is shown here:

∆P (“WC)	Input %	Output % = √Input %	Output signal (mA)
0	0	0	4
37.5	25	50	12
75	50	70.71	15.31
112.5	75	86.60	17.86
150	100	100	20

Once again, we see how the square-root relationship is most evident in comparing the input and output percentages. Note how the four sets of percentages in this table precisely match the same four percentage sets in the pneumatic relay table: 0% input gives 0% output; 25% input gives 50% output, 50% input gives 70.71% output, etc.

An ingenious solution to the problem of square-root characterization, more commonly applied before the advent of DP transmitters with built-in characterization, is to use an indicating device with a square-root indicating scale. For example, the following photograph shows a 3-15 PSI “receiver gauge” designed to directly sense the output of a pneumatic DP transmitter:

(Pic Courtesy: Ashcroft)

Note how the gauge mechanism responds directly and linearly to a 3-15 PSI input signal range (note the “3 PSI” and “15 PSI” labels in small print at the extremes of the scale and the linearly spaced marks around the outside of the scale arc representing 1 PSI each), but how the flow markings (0 through 10 on the inside of the scale arc) are spaced in a non-linear fashion.

An electronic variation on this theme is to draw a square-root scale on the face of a meter movement driven by the 4-20 mA output signal of an electronic DP transmitter:

As with the square-root receiver gauge, the meter movement’s response to the transmitter signal is linear. Note the linear scale (drawn in black text labeled “LINEAR”) on the bottom and the corresponding square-root scale (in green text labeled “FLOW”) on the top. This makes it possible for a human operator to read the scale in terms of (characterized) flow units. Instead of using complicated mechanisms or circuitry to characterize the transmitter’s signal, a non-linear scale “performs the math” necessary to interpret flow.

A major disadvantage to the use of these non-linear indicator scales is that the transmitter signal itself remains uncharacterized. Any other instrument receiving this uncharacterized signal will either require its own square-root characterization or simply not interpret the signal in terms of flow at all. An uncharacterized flow signal input to a process controller can cause loop instability at high flow rates, where small changes in actual flow rate result in huge changes in differential pressure sensed by the transmitter. A fair number of flow control loops operating without characterization have been installed in industrial applications (usually with square-root scales drawn on the face of the indicators, and square-root paper installed in chart recorders), but these loops are notorious for achieving good flow control at only one setpoint value. If the operator raises or lowers the setpoint value, the “gain” of the control loop changes thanks to the nonlinearities of the flow element, resulting in either under-responsive or over-responsive action from the controller.

Despite the limited practicality of non-linear indicating scales, they hold significant value as teaching tools. Closely examine the scales of both the receiver gauge and the 4-20 mA indicating meter, comparing the linear and square-root values at common points on each scale. A couple of examples are highlighted on the electric meter’s scale:

A few correlations between the linear and square-root scales of either the pneumatic receiver gauge or the electric indicating meter verify the fact that the square-root function is encoded in the spacing of the numbers on each instrument’s non-linear scale.

Another valuable lesson we may take from the faces of these indicating instruments is how uncertain the flow measurement becomes at the low end of the scale. Note how for each indicating instrument (both the receiver gauge and the meter movement), the square-root scale is “compressed” at the low end, to the point where it becomes impossible to interpret fine increments of flow at that end of the scale. At the high end of each scale, it’s a different situation entirely: the numbers are spaced so far apart that it’s easy to read fine distinctions inflow values (e.g. 94% flow versus 95% flow). However, the scale is so crowded at the low end that it’s impossible to clearly distinguish two different flow values such as 4% from 5%.

This “crowding” is not just an artifact of a visual scale; it reflects a fundamental limitation in measurement certainty with this type of flow measurement. The amount of differential pressure separating different low-range values of flow for a flow element is so little, even small amounts of pressure-measurement error equate to large amounts of flow-measurement error. In other words, it becomes more and more difficult to precisely interpret flow rate as the flow rate decreases toward the low end of the scale. The “crowding” that we see on an indicator’s square-root scale is a visual reflection of this fundamental problem: even a small error in interpreting the pointer’s position at the low end of the scale can yield major errors in flow interpretation.

A technical term used to quantify this problem is turndown. “Turndown” refers to the ratio of high-range measurement to low-range measurement possible for an instrument while maintaining reasonable accuracy. For pressure-based flowmeters, which must deal with the non-linearities of Bernoulli’s Equation, the practical turndown is often no more than 3 to 1 (3:1). This means a flowmeter ranging from 0 to 300 GPM might only read with reasonable accurately down to a flow of 100 GPM. Below that, the accuracy becomes so poor that the measurement is almost useless.

Advances in DP transmitter technology have pushed this ratio further, perhaps as far as 10:1 for certain installations. However, the fundamental problem is not transmitter resolution, but rather the nonlinearity of the flow element itself. This means any source of pressure-measurement error – whether originating in the transmitter’s pressure sensor or not – compromises our ability to accurately measure flow at low rates. Even with a perfectly calibrated transmitter, errors resulting from wear of the flow element (e.g., a dulled edge on an orifice plate) or from uneven liquid columns in the impulse tubes connecting the transmitter to the element, will cause large flow-measurement errors at the low end of the instrument’s range where the flow element produces only small differential pressures. Everyone involved with the technical details of flow measurement needs to understand this fact: the fundamental problem of limited turndown is grounded in the physics of turbulent flow and potential/kinetic energy exchange for these flow elements. Technological improvements will help, but they cannot overcome the limitations imposed by physics. If better turndown is required for a particular flow-measurement application, an entirely different flowmeter technology should be considered.

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Mass flow calculations (Pressure based flowmeter)

Measurements of mass flow are preferred over measurements of volumetric flow in process applications where mass balance (monitoring the rates of mass entry and exit for a process) is important. Whereas volumetric flow measurements express the fluid flow rate in such terms as gallons per minute or cubic meters per second, mass flow measurements always express fluid flow rate in terms of actual mass units over time, such as pounds (mass) per second or kilograms per minute.

Applications for mass flow measurement include custody transfer (where a fluid product is bought or sold by its mass), chemical reaction processes (where the mass flow rates of reactants must be maintained in precise proportion in order for the desired chemical reactions to occur), and steam boiler control systems (where the out-flow of vaporous steam must be balanced by an equivalent in-flow of liquid water to the boiler – here, volumetric comparisons of steam and water flow would be useless because one cubic foot of steam is certainly not the same number of H2O molecules as one cubic foot of water).

If we wish to calculate mass flow instead of volumetric flow, the equation does not change much. The relationship between volume (V) and mass (m) for a sample of fluid is its mass density (ρ):

Similarly, the relationship between a volumetric flow rate (Q) and a mass flow rate (W) is also the fluid’s mass density (ρ):

Solving for W in this equation leads us to a product of volumetric flow rate and mass density:

A quick dimensional analysis check using common metric units confirms this fact. A mass flow rate in kilograms per second will be obtained by multiplying a mass density in kilograms per cubic meter by a volumetric flow rate in cubic meters per second:

Therefore, all we have to do to turn our general volumetric flow equation into a mass flow equation is multiply both sides by fluid density (ρ):

It is generally considered “inelegant” to show the same variable more than once in an equation if it is not necessary, so let’s try to consolidate the two densities (ρ) using algebra. First, we may write ρ as the product of two square-roots:

Next, we will break up the last radical into a quotient of two separate square roots:

Now we see how one of the square-rooted ρ terms cancels out the one in the denominator of the fraction:

It is also considered “inelegant” to have multiple radicands in an equation where one will suffice, so we will re-write our equation for aesthetic improvement:

As with the volumetric flow equation, all we need to arrive at a suitable k value for any particular flow element is a set of values taken from that real element in service, expressed in whatever units of measurement we desire.

For example, if we had a venturi tube generating a differential pressure of 2.30 kilopascals (kPa) at a mass flow rate of 500 kilograms per minute of naphtha (a petroleum product having a density of 0.665 kilograms per liter), we could solve for the k value of this venturi tube as such:

Now that we know a value of 404.3 for k will yield kilograms per minute of liquid flow through this venturi tube given pressure in kPa and density in kilograms per liter, we may readily predict the mass flow rate through this tube for any other pressure drop and fluid density we might happen to encounter. The value of 404.3 for k relates the disparate units of measurement for us:

As with volumetric flow calculations, the calculated value for k neatly accounts for any set of measurement units we may arbitrarily choose. The key is first knowing the proportional relationship between flow rate, pressure drop, and density. Once we combine that proportionality with a specific set of data experimentally gathered from a particular flow element, we have a true equation properly relating all the variables together in our chosen units of measurement.

If we happened to measure 6.1 kPa of differential pressure across this same venturi tube as it flowed seawater (density = 1.03 kilograms per liter), we could calculate the mass flow rate quite easily using the same equation (with the k factor of 404.3):

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Volumetric flow calculations (Pressure based flowmeter)

As we saw in the previous subsection, we may derive a relatively simple equation for predicting flow through a fluid-accelerating element given the pressure drop generated by that element and the density of the fluid flowing through it:

This equation is a simplified version of the one derived from the physical construction of a venturi tube:

As you can see, the constant of proportionality (k) shown in the simpler equation is nothing more than a condensation of the first half of the longer equation: k represents the geometry of the venturi tube. If we define k by the mouth and throat areas (A1, A2) of any venturi tube, we must be very careful to express the pressures and densities incompatible units of measurement.

For example, with k strictly defined by flow element geometry (tube areas measured in square feet), the calculated flow rate (Q) must be in units of cubic feet per second, the pressure values P1 and P2 must be in units of pounds per square foot, and mass density must be in units of slugs per cubic foot. We cannot arbitrarily choose different units of measurement for these variables because the units must “agree” with one another. If we wish to use more convenient units of measurement such as inches of water column for pressure and specific gravity (unitless) for density, the original (longer) equation simply will not work.

However, if we happen to know the differential pressure produced by any flow element tube with any fluid density at a specified flow rate (real-life conditions), we may calculate a value for k in the short equation that makes all those measurements “agree” with one another. In other words, we may use the constant of proportionality (k) as a unit-of-measurement correction factor as well as a definition of element geometry. This is a useful property of all proportionalities: simply insert values (expressed in any unit of measurement) determined by physical experiment and solve for the proportionality constant’s value to satisfy the expression as an equation. If we do this, the value we arrive at for k will automatically compensate for whatever units of measurement we arbitrarily choose for pressure and density.

For example, if we know a particular orifice plate develops 45 inches of water column differential pressure at a flow rate of 180 gallons per minute of water (specific gravity = 1), we may insert these values into the equation and solve for k:

Now we possess a value for k (26.83) that yields a flow rate in units of “gallons per minute” given differential pressure in units of “inches of water column” and density expressed as a specific gravity for this orifice plate. From the known fact of all accelerating flow elements’ behavior (flow rate proportional to the square root of pressure divided by density) and from a set of values experimentally determined for this orifice plate, we now have an equation useful for calculating flow rate given any set of pressure and density values we may happen to encounter with this orifice plate:

Applying our new equation to this orifice plate, we see those 60 inches of water column differential pressure generated by a flow of water (specific gravity = 1) equates to 207.8 gallons per minute of flow:

If we were to measure 110 inches of water column differential pressure across this orifice plate as gasoline (specific gravity = 0.657) flowed through it, we could calculate the flow rate to be 347 gallons per minute:

Suppose, though, we wished to have an equation for calculating the flow rate through this same orifice plate given pressure and density data in different units (say, kPa instead of inches water column, and kilograms per cubic meter instead of specific gravity). To do this, we would need to re-calculate the constant of proportionality (k) to accommodate those new units of measurement. To do this, all we would need is a single set of experimental data for the orifice plate relating flow in GPM, pressure in kPa, and density in kg/m³.

Applying this to our original data where a water flow rate of 180 GPM resulted in a pressure drop of 45 inches water column, we could convert the pressure drop of 45 ”W.C. into 11.21 kPa and express the density as 1000 kg/m3 to solve for a new value of k:

Nothing about the orifice plate’s geometry has changed from before, only the units of measurement we have chosen to work with. Now we possess a value for k (1700) for the same orifice plate yielding a flow rate in units of “gallons per minute” given differential pressure in units of “kilopascals” and density in units of “kilograms per cubic meter.”

If we were to be given a pressure drop in kPa and a fluid density in kg/m³ for this orifice plate, we could calculate the corresponding flow rate (in GPM) with our new value of k (1700) just as easily as we could with the old value of k (26.83) given pressure in ”W.C. and specific gravity.

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