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X, fraction of maximum lift
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FIGURE 20-3 Inherent valve characteristics (pressure drop across valve is constant) I linear, II incmasing sensitivity (e.g. equal percentage valve), III decreasing sensitivity.
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In general, the flow through a control valve for a specific fluid at a given temperature can be expressed as:
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(20.2)
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where q = volumetric flow rate L = valve stem position (or lift) po = upstream pressure pt = downstream pressure The inherent valve characteristic is determined for fixed values of pa and p 1, for which case, Eq. (20.2) becomes
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(20.3)
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For convenience let: and x = LIL,, m = dqmax where qmax is the maximum flow when the valve stem is at its maximum lift L,, (valve is full-open) x is the fraction of maximum lift m is the fraction of maximum flow. Equation (20.3) may now be written m = 4hmax = f(L~Lnlax) or
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(20.4) m = f(x) The types of valve characteristics can be defined in terms of the sensitivity of the valve, which is simply the fractional change in flow to the fractional change in stem position for fixed upstream and downstream pressures; mathematically, sensitivity may be written
sensitivity = dmldx In terms of valve characteristics, valves can be divided into three types: decreasing sensitivity, linear, and increasing sensitivity. These types are shown in Fig. 20.3 where the fractional flow m is plotted against fractional lift x . For the decreasing sensitivity type, the sensitivity (or slope) decreases with m . For the
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linear type, the sensitivity is constant and the characteristic curve is a straight line. For the increasing sensitivity type, the sensitivity increases with flow. Valve characteristic curves, such as the ones shown in Fig. 20.3, can be obtained experimentally for any valve by measuring the flow through the valve as a function of lift (or valve-top pressure) under conditions of constant upstream and downstream pressures. Ikro types of valves that are widely used are the linear valve and the logarithmic (or equal percentage) valve. The linear valve is one for which the sensitivity is constant and the relation between flow and lift is linear. The equal percentage valve is of the increasing sensitivity type. It is useful to derive mathematical expressions for these types of valves. For the linear valve, dmldx = a (20.5) where Q is a constant. Assuming that the valve is shut tight when the lift is at lowest position, we have that m = 0 at x = 0. For a single-seated valve that is not badly worn, the valve can be shut off for x = 0. Integrating Eq. (20.5) and introducing the limits m = 0 at x = 0 and m = 1 at x = 1 gives
dm =
Integrating this equation and inserting limits gives a=1 Recall that the definitions of x and m require that m = 1 at x = 1. For CY = 1, Eq. (20.5) can now be integrated to give m = x (linear valve) (20.6) (20.7) For the equal percentage valve, the defining equation is dmldx = pm where p is constant. Integration of this equation gives (20.8) or
where mo is the flow at x = 0. Equation (20.9) shows that a plot of m versus x on semi-log paper gives a straight line. A convenient way to determine if a valve is of the equal percentage type is to plot the flow versus lift on semi-log paper. The relation expressed by Eq. (20.9) is the basis for calling the valve characteristic logarithmic. The basis for calling the valve characteristic equal percentage can be seen by rearranging Eq. (20.7) in the form dmlm = /3dx or hmlm = @Ax
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