Automatic valves
It has been stated elsewhere in literature on the subject that a control valve is a variable restriction, and this description is apt. It follows that a study of the flow of fluid through an orifice will assist in understanding the behaviour of a control valve.
The loss of head associated with air or water flow through a duct or pipe is discussed in section 15.1, and the equation quoted is:
This implies that the rate of fluid flow is proportional to the square root of the pressure drop along the pipe and to the crosssectional area of the pipe. We can therefore write a basic equation for the flow of fluid through any resistance, for example a valve:
(13.7) 
Q = Ka(2g(hu — hd))1/2
Where q = volumetric flow rate in m3 s1,
A = crosssectional area of the valve opening, in m2,
Hu, hd = upstream and downstream static heads, in m of the fluid,
K = a constant of proportionality.
If it is assumed that the position of the valve stem, z, is proportional to the area of the valve opening, then
(13.8) 
Q = KlZ(2g(hu — hA))m
Where K is a new constant of proportionality.
Unfortunately, the picture is not as simple as this and the flow rate is not directly proportional to the position of the valve stem; the constants of proportionality are not true constants (they depend on the Reynold’s number, just as does the coefficient/in equation (15.2)) and the area of the port opened by lifting the valve stem is not always proportional to the lift. A more realistic picture of the behaviour is obtained if the flow of fluid is considered through a pipe and a valve in series, under the influence of a constant difference of head across them. Figure 13.9 illustrates the case.
The head loss in the pipe, plus that across the valve, must equal the driving force produced by the difference of head between the reservoir, Hu and the sink, H0.
The Fanning friction factor,/, is sometimes reexpressed as the Moody factor, fm (= 4/). Using the Moody factor the head lost in the pipe, hh is given by (fmlv2)/(2gd). Also
Q2 = a2v2 = (nd2/4)2v2 and hence
Hi Fig. 13.9 A source and sink of water provide a constant head that is used as the driving force for a Simple analysis of valve performance. 
H = (Sfmlq2)/(n2c?) (13.9)
The head lost across the valve is then (H — H0 h). Thus equation (13.8) for the flow through the valve can be rewritten:
Q = Klz{2g[H] H0 (8fjq2/n2gcf))m
Q2 = K*z22g[Hx — H0 (8fmlq2/n2gd5)]
Write (3 = (16 fmlK2/n2d5) and the equation simplifies to
SHAPE \* MERGEFORMAT
(13.10) 
Q = KiZ 
2g(Hi — H0T1/2
So it is seen that even if K is a constant, the flow rate is not directly proportional to the lift, z. In the equation, Pz2 is the ratio of the loss of head along the pipe (equation (13.9)) to that lost across the valve (from equation (13.8)):
8fjq2 j q2 = 16fJK2 n2gd5 / K2z22g ji2d5
For the smaller pipe sizes the influence of d5 increases and pz2 becomes large. Thus, for a constant difference of head the flow rate falls off as the pipe size is reduced, as would be expected, the drop through the valve reducing as the drop through the pipe increases. Hence, the presence of a resistance in series with the valve alters the flow through the valve, under conditions of a constant overall difference of head. Whether this conclusion is valid also for the case of circulation by a centrifugal pump through a piping circuit depends on where the pressurevolumetric flow rate characteristic curve for the piped circuit intersects that of the pump: for intersections on or near the flat part of the pump
Curve and for comparatively small volumetric flow rate changes, elsewhere on the curve, it is probably valid.
If a valve is to exercise good control over the rate of flow of fluid passing through it, the ideal is that qlq0 shall be directly proportional to z/zo, where q0 is the maximum flow and Zo is the maximum valve lift (when the valve is fully closed, q and z are both zero). A direct proportionality between these two ratios is not attainable in practice.
The ratio of the pressure drop across the valve when fully open to the pressure drop through the valve and the controlled circuit, is termed the ‘authority’ of the valve. For example, if a valve has a loss of head of 5 metres when fully open and the rest of the piping circuit has a loss of 15 metres, the authority of the valve is 0.25.
The effect of valve authority on the flowlift characteristic of a valve can be seen by
Means of an example.
EXAMPLE 13.1
Using the foregoing theory show how the authority of a valve affects the water flow rate for a given degree of valve opening.
Answer
To simplify the arthmetic divide equation (13.10) throughout by K^2g, choosing new units for q and z. We have then
Q = z[(//i — H0)/(l + Pz2)]1/2 (13.10a)
Denote the maximum flow rate by q0, when the valve stem is in its position of maximum lift, zo — For any given valve position other than fully open the valve lift ratio is then zJzo and the corresponding water flow ratio is q/q0. It follows that zJzq can vary from 0 to 1.0, as also can q/q0. Equation (13.11) shows that Pz2 equals h/hv where hv is the loss of head through the valve. The authority of the valve is then defined by:
A = hv/(hi + hv) = 1/(1 + Pzq) (13.12)
Because a is for full flow conditions when the valve is completely open we must use zo in equation (13.12).
Let us suppose that H — H0 is 100 units of head and that the valve lift, z, varies from 0 units (fully closed) to 1 unit (fully open). We may now consider various valve sizes, changing the size of the connected pipe circuit each time so that the head absorbed by both always equals the available driving head of 100 units.
(i) Suppose a small valve is used, with hv = 50 units. Then, for the special case of z = 1.0, equation (13.12) yields a = 0.5 and the constant (3 is 1.0. We may use this value of P and {H — H0) = 100 in equation (13.10a) and we have q = 10z[l/(l + z2)]1/2, which may be applied for all values of z between 0 and 1.0 to give corresponding flow rates, tabulated as follows for the case of a = 0.5.
Z 
0.1 
0.2 
0.3 
0.5 
0.7 
0.9 
1.0 
Z/z0 
0.1 
0.2 
0.3 
0.5 
0.7 
0.9 
1.0 
Q 
0.995 
1.961 
2.874 
4.472 
5.735 
6.690 
7.071 
0.141 
0.277 
0.406 
0.632 
0.811 
0.946 
1.0 
Any particular value of qlq0 is determined by dividing the value of q by the value of q0, which is 7.071. Thus when z = 0.2, q/q0 = 1.961/7.071 = 0.277.
(ii) Suppose a larger valve is used, with hw = 20. Since H{ — H0 is a constant at 100 we must consider a larger piping system with a loss of h = 80. Hence a = 20/(20 + 80) = 0.2 and, by equation (13.12), the constant P is now 4. Equation (13.10a) then becomes q = 10z[l/ (1 + 4z2)]1/2. We may proceed like this for bigger valves, such that a = 0.4, 0.2, 0.1, 0.05, and so on. Similar tabulations can be done and the results are shown as curves in Figure 13.10. We see that the greater the authority the nearer the characteristic is to a straight line. For the control of fluid flow an authority of 0.5 is often chosen.
Valve lift ratio zlz0 Fig. 13.10 Characteristic curves for a conventional valve (full lines) and a valve with an increasing Sensitivity (broken line). 
In air conditioning a nearlinear relationship is desired between valve lift and heat transfer from a cooler coil or heater battery, rather than with fluid flow rate. This introduces a complication because the sensible heat transfer capacity of either is not directly proportional to fluid flow. This is typified by Figure 13.11 where we see that, for a water temperature change of about 10 degrees, there is a proportional reduction in heat transfer from 100 per cent to about 85 per cent or 90 per cent, as the flow drops to about 50 per cent but thereafter the capacity falls off rapidly. A valve with the behaviour shown in Figure 13.10 would not be suitable for controlling cooling or heating capacity. Instead, the ports and plugs of valves are modified, or ‘characterised’, to give them an increasing valve sensitivity, defined as Aq/Az. An arbitrary example of this is shown as a broken line in Figure 13.10. As with uncharacterised valves, the position of the curve depends on the authority but the shape
Water flow ratio q/q0 Fig. 13.11 The relationship between water flow rate and heat transfer for a typical finned tube cooler or heater coil. The curve is for a water temperature rise/drop of about 10 degrees. If the water temperature change is more than this the curve moves closer to the diagonal position. 
Depends on the design of the port or the plug. The valve seat may also be contoured to influence the control achieved. Different possibilities are illustrated in Figure 13.12.
If a characterised valve is used to regulate the sensible capacity of a coil we can see the sort of result obtained by using the broken line in Figure 13.10 with the curve in Figure 13.11 that relates qlq0 with hlh0, the heat output ratio. Reading the data from the two curves and tabulating we have:
DZo 
0.1 
0.2 
0.3 
0.4 
0.5 
0.6 
0.7 
0.8 
0.9 
<?/<?<, 
0.045 
0.090 
0.145 
0.190 
0.250 
0.315 
0.395 
0.500 
0.650 
H/h0 
0.09 
0.21 
0.33 
0.45 
0.58 
0.70 
0.81 
0.885 
0.95 
This is a simplification because the curves from Figures 13.10andl3.11 were not for an actual characterised valve or an actual cooler coil. Nevertheless, the results plotted in Figure 13.13 show that an approximation to the ideal, linear relationship between valve lift ratio (zJz0) and heat output ratio (h/h0) can be achieved. Different forms of characterization are possible and each would have an equation different from the simplification of equation (13.10a). One popular form of characterisation is termed ‘equal percentage’, where, for each linear increment of the valve lift, the flow rate is increased by a percentage of the original flow rate. In practice it turns out that an authority of between 0.2 and 0.4 gives the best resemblance to the ideal performance. Figure 13.14 shows the curves usually obtained. It also illustrates the minimum possible controllable flow.
RQ 
(c) Skirted plug to give required flow characteristic — used with larger valve sizes 
(a) Conventional seat (b) Contoured seat to with shaped plug give quick action
Opening
Fig. 13.12 Various ways of characterising valve performance. 
•c 
3 Q. 3 O 
<1) X 
Valve lift ratio z! z0 Fig. 13.13 Performance curve for a notional characterised valve. 
Twoport valves may be single — or doubleseated (see Figure 13.15), arranged to be normally open (upon failure of power or air pressure or upon plant shutdown) or normally closed, depending on system operational requirements or safety. Valves with single seats can give a tight shutoff but there is always an outofbalance pressure across the plug which is a restricting factor in valve selection: a valve must be able to withstand the maximum likely pressure difference. Double seated valves (Figure 13.15) do not have the same outofbalance forces across the pair of plugs, although there is still a pressure drop across the valve as a whole. Such valves will not give tight shutoff.
Controllable Flow Fig. 13.14 Relationship between heat transfer from a finned tube heater battery or cooler coil and valve lift, for a typical equal percentage characterised valve with authorities of 0.2 and 0.4. 
Threeport valves are intended to give constant flow rate but do not always achieve this. In Figure 13.16(a) we see a diagram of a mixing valve, defined as a valve in which two fluid streams, A and B, mix to give a common stream, AB. In Figure 13.6(b) it is seen that the constancy of the combined flow depends on how the individual plugs, A and B, are characterised. A diverting valve is much less used than a mixer and is defined as one which splits a single entering fluid stream into two divergent streams. A mixing valve, as shown in Figure 13.16(a), must not, under any circumstances, be piped up as a diverter. Figure 13.16(c) shows that this is an unstable arrangement: as the plug moves from its central position towards either seat the pressure drop across the plug increases and forces it on to the seat. The plug then bounces off the seat on to the opposite seat and control is impossible. Figure 13.16(d) shows a stable diverting valve arrangement with two plugs: as a plug moves towards a seat the pressure drop increases and tends to push the pair of plugs back to the neutral, central position. The actuator then has something to do and stable control is obtained.
Threeport valves can be used in mixing or diverting applications, as Figure 13.17 shows. The most common and cheapest arrangement is subfigure (a) but there may be
Fig. 13.15 Single and double seat twoport valves.
(c) Threeport, unstable, uncontrollable, diverting valve 
(d) Threeport, stable, controllable, diverting valve 
B: Open B: Closed Valve port position (b) Threeport mixing valve flow charcteristics 
Fig. 13.16 Threeport valves, (a) Threeport mixing valve, (b) Threeport mixing valve flow characteristic. (c) Threeport, unstable, uncontrollable, diverting valve, (d) Threeport, stable,
Controllable, diverting valve.
Occasions when subfigure (c) is preferred. If the arrangement in Figure 13.17(c) is used then the considerations for valve characterisation are different from those considered hitherto. The water flow rate through the cooler coil (or heater battery) is constant and the cooling or heating capacity is proportional to the logarithmic mean temperature difference, airto — water, as equation (10.9) shows, the {/value for the coil being unchanged. This implies that the control of water flow rate should be related to valve lift in a linear fashion, requiring a valve authority nearer to 0.5 (see Figure 13.10).
The desirable properties of characterisation for the plugs of a threeport mixing valve are asymmetrical: the plug regulating flow through the cooler coil should have an increasing sensitivity while that controlling flow through the port open to the bypass should be characterised to give a constant, combined flow out of the valve. Such asymmetrical characterisation is sometimes available from manufacturers.
The resistance to flow through the bypass should be the same as that through the cooler coil and its connections. This can be done by sizing down the bypass pipe and/or providing a regulating valve in it (as shown in Figure 13.17).
In a throttling application the pressure drop across the twoport valve increases as the valve closes and eventually the force exerted by the valve actuator (through a pneumatic diaphragm or an electric motor) is insufficient to continue closing the valve smoothly. Furthermore, because of the necessary clearance between the plug or skirt and the seating,
Cooler coil 
Cooler coil 

2 
Reg valve 
Reg valve 
(a) Mixing valve in a (b) Diverting valve in a
^ Pump 
Diverting application diverting application
^ Pump 
(c) Mixing valve in a (d) Diverting valve in a
Mixing application mixing application
Fig. 13.17 Mixing and diverting valve installation possibilities.
The uncontrolled leakage becomes an increasing proportion of the flow as the valve closes. There is thus always a minimum flow rate for proportional control and beyond this the control degenerates to twoposition. The curves shown in Figures 13.10,13.13 and 13.16(b) are therefore not quite correct: flow does not modulate smoothly down to zero as the valve closes and there is a minimum controllable flow, as Figure 13.14 illustrates. This gives rise to the concept of rangeability, defined as the ratio of maximum to minimum controllable flow, and turndown ratio, defined as the ratio of maximum to minimum usable flow. The difference arises from the possibility that the valve may be oversized and never required to be fully open in a particular application, the turndown ratio then being less than the rangeability quoted by the valve manufacturer. Commercial valves can have rangeabilities of up to 30:1 but bettermade industrial valves may have as much as 50:1. However, because of the nonlinearity of heat exchanger capacity, with respect to flow rate, and because the valve may often be slightly oversized, proportional control is only possible down to 15 or 20 per cent of heat exchanger capacity. Oversizing the heat exchanger will make this worse.
It must be remembered, when assessing valve authority, that it is the pressure drop in the part of the piping circuit where variable flow occurs that is relevant. Thus for a twoport valve the authority is a fraction of the pump head for the index circuit. This is also the case for twoport valves in parallel subcircuits because a regulating valve in each branch is adjusted to make the branch circuit resistance the same as that for the remainder of the index circuit. With threeport valves the authority is also related to the circuit in which the
flow is variable. Figure 13.17(a) shows an example: the threeport valve at the point 4 should have a loss when fully open that is a suitable fraction (0.2 to 0.4) of the resistance (1 to 2) + (cooler coil) + (3 to 4). The regulating valve in the bypass would be adjusted during commissioning so that the pressure drop from 1 to 4 through the bypass equalled that from 1 to 4 through the cooler coil and its connections.
If valve friction, forces associated with the fluid flow through the valve, or fluctuating external hydraulic pressures prevent a pneumaticallyactuated valve from taking up a position that is proportional to the control pressure, a valve stem positioner is needed. Such a positioner ensures there is only one valve position for any given actuating air pressure. In critical applications it may also be necessary to provide positioners for motorised modulating dampers.
The flow coefficient (or capacity index), Av, is used when sizing valves, in the formula q = Av /Ap, a version of equation (13.7) in which q is the flow rate in m3 s1 and Ap is the pressure drop in Pa. For a given valve, Av represents the flow rate in m3 s1 for a pressure drop of 1 Pa.
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