Final Control Elements - Control Valves - Page 15
Control valve characterization
When control valves are tested in a laboratory setting, they are connected to a piping system that is able to provide a nearly constant pressure difference between upstream and downstream (P1 − P2). With a fluid of constant density and a constant pressure drop across the valve, flow rate becomes a direct function of flow coefficient (Cv). This is clear from an examination of the basic valve capacity equation:
(If pressures and specific gravity are constant . . .)
Q = kCv
The amount of “resistance” offered by a restriction of any kind to a turbulent fluid depends on the cross-sectional area of that restriction and also the proportion of fluid kinetic energy dissipated in turbulence. If a control valve is designed such that the combined effect of these two parameters vary linearly with stem motion, the Cv of the valve will likewise be proportional to stem position. That is to say, the Cv of the control valve will be approximately half its maximum rating with the stem position at 50%; approximately one-quarter its maximum rating with the stem position at 25%; and so on.
If such a valve is placed in a laboratory flow test piping system with constant differential pressure and constant fluid density, the relationship of flow rate to stem position will be linear:
However, most real installations do not place the control valve under the same conditions. Due to frictional pressure losses in piping and changes in supply/demand pressures that vary with flow rate, a typical control valve “sees” substantial changes in differential pressure as its controlled flow rate changes. Generally speaking, the pressure drop available to the control valve will decrease as flow rate increases.
The result of this pressure drop versus flow relationship is that the actual flow rate of the same valve installed in a real process will not linearly track valve stem position. Instead, it will “droop” as the valve is further opened. This “drooping” graph is called the valve’s installed characteristic, in contrast to the inherent characteristic exhibited in the laboratory with constant pressure drop:
Each time the stem lifts up a bit more to open the valve trim further, flow increases, but not as much as at lower-opening positions. It is a situation of diminishing returns, where we still see increases in flow as the stem lifts up, but to a lesser and lesser degree.
In my years of teaching, I have found this concept of “installed characteristic” to be elusive for many students to grasp. In the interest of clarifying the concept, I wish to present a pair of contrasting scenarios using realistic numbers.
First, let us imagine a control valve installed at the base of a dam, letting water out of the reservoir. Given a constant height of water in the reservoir, the upstream (hydrostatic) pressure at the valve will likewise be constant. Let’s assume this constant upstream pressure will be 20 PSI (corresponding to approximately 46 feet of water column above the valve inlet). With the valve discharging into the air, downstream pressure will essentially be zero. This set of upstream and downstream conditions guarantees a constant pressure drop of 20 PSI across our control valve at all times, for all flow conditions:
Furthermore, let us assume the control valve has a “linear” inherent characteristic and a maximum flow capacity (Cv rating) of 18. This means the valve’s Cv will be 18 at 100% open, 13.5 at 75% open, 9 at 50% open, 4.5 at 25% open, and 0 at fully closed (0% open). We may plot the behavior of this control valve at these four stem positions by graphing the amount of flow through the valve for varying degrees of pressure drop across the valve. The result is a set of characteristic curves28 for our hypothetical control valve:
Each curve on the graph traces the amount of flow through the valve at a constant stem position, for different amounts of applied pressure drop. For example, looking at the curve representing 50% open (Cv = 9), we can see the valve should flow about 42 GPM at 22 PSI, about 35 GPM at 15 PSI, about 20 GPM at 5 PSI, and so on. Of course, we can obtain these same flow figures just by evaluating the formula (which is in fact what I used to plot these curves), but the point here is to learn how to interpret the graph.
We may use this set of characteristic curves to determine how this valve will respond in any installation by superimposing another curve on the graph called a load line29, describing the pressure drop available to the valve at different flow rates. Since we know our hypothetical dam applies a constant 20 PSI across the control valve for all flow conditions, the load line for the dam will be a vertical line at 20 PSI:
By noting the points of intersection30 between the valve’s characteristic curves and the load line, we may determine the flow rates from the dam at those stem positions:
| Opening (%) || Cv ||Flow rate (GPM)|
|75|| 13.5 ||60.4|
If we were to graph this table, plotting flow versus stem position, we would obtain a very linear graph. Note how 50% open gives us twice as much flow as 25% open, and 100% open nearly twice as much flow as 50% open. This tells us our control valve will respond linearly when pressed into service on this dam, with a constant pressure drop.
But what if we alter the scenario so that the pressure drop across the valve does not remain constant as flow through the valve changes? Suppose the valve is not closely coupled to the dam, but rather receives water through a narrow (restrictive) pipe:
In this installation, the narrow pipe drops pressure of its own due to friction between the rushing water and the interior walls, leaving less upstream pressure at the valve with greater amounts of flow. The control valve still drains to atmosphere, so its downstream pressure is still a constant 0 PSIG, but now its upstream pressure will diminish as flow increases. How will this affect the valve’s performance?
We may turn to the same set of characteristic curves for an answer to this question. All we must do is plot a new load line describing the pressure available to the valve at different flow rates, and once again look for the points of intersection between this load line and the valve’s characteristic curves. For the sake of our hypothetical example, I have sketched an arbitrary “load line” (actually a load curve) showing how the valve’s pressure falls off as flow rises:
Now we see a definite nonlinearity in the control valve’s behavior. No longer does a doubling of stem position (from 25% to 50%, or from 50% to 100%) result in a doubling of flow rate31:
|Opening (%)|| Cv ||Flow rate (GPM)|
If we plot the valve’s performance in both scenarios (close-coupled to the dam, versus at the end of a restrictive pipe), we see the difference very clearly:
The “drooping” graph shows how the valve responds when it does not receive a constant pressure drop throughout the flow range. This is how the valve responds when installed in a non-ideal process, compared to the straight-line response it exhibits under ideal conditions of constant pressure. This what we mean by “installed” characteristic versus “ideal” or “inherent” characteristic.
Pressure losses due to fluid friction as it travels down pipe is just one cause of valve pressure changing with flow. Other causes exist as well, including pump curves and frictional losses in other system components such as filters and heat exchangers. Whatever the cause, any piping system that fails to provide constant pressure across a control valve will “distort” the valve’s inherent characteristic in the same “drooping” manner, and this must be compensated in some way if we desire linear response from the valve.
Not only does the diminishing pressure drop across the valve mean we cannot achieve the same full-open flow rate as in the laboratory (with a constant pressure drop), but it also means the control valve responds differently at various points along its range. Note how the installed characteristic graph is relatively steep at the beginning where the valve is nearly closed, and how the graph is almost flat at the end where the valve is nearly full-open. The rate of response (rate-of-change of flow Q compared to stem position x, which may be expressed as the derivative dQ dx ) is much greater at low flow rates than it is at high flow rates, all due to diminished pressure drop at higher flow rates. This means the valve will respond more “sensitively” at the low end of its travel and more “sluggishly” at the high end of its travel.
From the perspective of a flow control system, this varying valve responsiveness means the system will be unstable at low flow rates and slow-responding at high flow rates. At low flow rates, there the valve is nearly closed, any small movement of the valve stem will have a relatively large effect on flow. However, at high flow rates, a much greater stem motion will be required to effect the same change in flow. Thus, the control system will tend to over-react at low flow rates and under-react at high flow rates. Oscillations may occur at low flow rates, and large deviations from setpoint at high flow rates as a result of this “distorted” valve behavior.
The root cause of the problem – a varying pressure drop caused by frictional losses in the piping and other factors – generally cannot be eliminated. This means there is no way to regain maximum flow capacity short of replacing the control valve with one having a greater Cv rating32. However, there is a clever way to flatten the valve’s responsiveness to achieve a more linear characteristic, and that is to purposely design the valve such that its inherent characteristic complements the process “distortion” caused by changing pressure drop. In other words, we design the control valve trim so it opens up gradually during the initial stem travel (near the closed position), then opens up more rapidly during the final stages of stem travel (near the full-open position). With the valve made to open up in a nonlinear fashion inverse to the “droop” caused by the installed pressure changes, the two non-linearities should cancel each other and yield a more linear response.
This re-design will give the valve a nonlinear characteristic when tested in the laboratory with constant pressure drop, but the installed behavior should be more linear:
Now, control system response will be consistent at all points within the controlled flow range, which is a significant improvement over the original state of affairs.
Control valve trim is manufactured in a variety of different “characteristics” to provide the desired installed behavior. The two most common inherent characteristics are linear and equal percentage. “Linear” valve trim exhibits a fairly proportional relationship between valve stem travel and flow capacity (Cv), while “equal percentage” trim is decidedly nonlinear. A control valve with “linear” trim will exhibit consistent responsiveness only with a constant pressure drop, while “equal percentage” trim is designed to counter-act the “droop” caused by changing pressure drop when installed in a process system.
Another common inherent valve characteristic available from manufacturers is quick-opening, where the valve’s Cv increases dramatically during the initial stages of opening, but then increases at a much slower rate for the rest of the travel. Quick-opening valves are often used in pressure-relief applications, where it is important to rapidly establish flow rate during the initial portions of valve stem travel.
The standard “textbook” comparison of quick-opening, linear, and equal-percentage valve characteristics usually looks something like the following graph:
A graph showing valve characteristics taken from actual manufacturers’ data on valve performance33 shows a more moderate picture:
Different valve characterizations may be achieved by re-shaping the valve trim. For instance, the plug profiles of a single-ported, stem-guided globe valve may be modified to achieve the common quick-opening, linear, and equal-percentage characteristics:
Photographs of linear (left) and equal-percentage (right) globe valve plugs are shown side-by-side for comparison:
Cage-guided globe valve trim characteristic is a function of port shape. As the plug rises up, the amount of port area uncovered determines the shape of the characteristic graph:
Ball valve trim characteristic is a function of notch shape. As the ball rotates, the amount of notch area opened to the fluid determines the shape of the characteristic graph. All valve trim in the following illustration is shown approximately half-open (50% stem rotation):
A different approach to valve characterization is to use a non-linear positioner function instead of a non-linear trim. That is, by “programming” a valve positioner to respond in a characterized fashion to command signals, it is possible to make an inherently linear valve behave as though it were quick-opening, equal-percentage, or anywhere in between. All the positioner does is modify the valve stem position as per the desired characteristic function instead of proportionally follow the signal as it normally would.
This approach has the distinct advantage of convenience (especially if the valve is already equipped with a positioner) over changing the actual valve trim. However, if valve stem friction ever becomes a problem, its effects will be disproportionate along the valve travel range, as the positioner must position the valve more precisely in some areas of travel than others when pressed into service as a characterizer.
28For those readers with an electronics background, the concept of “characteristic curves” for a control valve is exactly the same as that of characteristic curves for transistors. Instead of plotting the amount of current a transistor will pass given varying amounts of supply voltage, we are plotting how much water a valve will flow given varying amounts of supply pressure.
29Once again, the exact same concept applied in transistor circuit analysis finds application here in control valve behavior!
30Load line plots are a graphical method of solving nonlinear, simultaneous equations. Since each curve represents a set of solutions to a particular equation, the intersection of two curves represents values uniquely satisfying both equations at the same time.
31Not only is the response of the valve altered by this degradation of upstream pressure, but we can also see from the load line that a certain maximum flow rate has been asserted by the narrow pipe which did not previously exist: 75 GPM. Even if we unbolted the control valve from the pipe and let water gush freely into the atmosphere, the flow rate would saturate at only 75 GPM because that is the amount of flow where all 20 PSI of hydrostatic “head” is lost to friction in the pipe. Contrast this against the close-coupled scenario, where the load line was vertical on the graph, implying no theoretical limit to flow at all! With an absolutely constant upstream pressure, the only limit on flow rate was the maximum Cv of the valve (analogous to a perfect electrical voltage source with zero internal resistance, capable of sourcing any amount of current to a load).
32Even then, achieving the ideal maximum flow rate may be impossible. Our previous 100% flow rate for the valve was 80.5 GPM, but this goal has been rendered impossible by the narrow pipe, which according to the load line limits flow to an absolute maximum of 75 GPM (even with an infinitely large control valve).
33Data for the three graphs were derived from actual Cv factors published in Fisher’s ED, EAD, and EDR sliding- stem control valve product bulletin (51.1:ED). I did not copy the exact data, however; I “normalized” the data so all three valves would have the exact same full-open Cv rating of 50.
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