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W = w, - W , K 5 = H% - Hcs UA UA
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dTv 1
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aVdt + UA(T, - To)
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TV = WV, - H,,bV + mlC1
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From Eq. (21.15), we see that the steam temperature &&depends on the steam flow rate WV and the water temperature TL. The combination of Eqs. (21.9) and (21.15) give the dynamic response of the water temperature to changes in water flow rate, inlet water temperature, and steam flow rate. These equations are represented by a portion of the block diagram of Fig. 21.4. Before completing the analysis of the control system, we must consider the effect of valve-stem position on the steam flow rate.
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Analysis of Valve
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The flow of steam through the valve depen& on three variables: steam supply pressure, steam pressure in the jacket, and the valve-stem position, which we shall assume to be proportional to the pneumatic value-top pressure p. For simplicity, assume the steam supply pressure to be constant. with the result that the steam flow rate is a function of only the two remaining variables; thus (2l.L) WV = f@1PY) Because of the assumption that the steam in the jacket is always saturated, we know that pv is a function of T,; thus
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THEORETICAL
ANALYSIS
COMPLEX
PROCESSES
This functional relation can be obtained from the saturated steam tables. Equations (21.16) and (21.17) can be combined to give
WV = fb,g(~,)l =
fl(p> T )
The function ft@, T,,) is in general nonlinear, and if an analytic expression* is available, the function can be linearized as described previously:In this example, we shall assume that an analytic expression is not available. The linearized form of fi(P, T,) can be obtained by making some experimental tests on the valve. If the valve-top pressure is fixed at its steady-state (or average) value and wV is measured for several values of TV (or p,), a curve such as the one shown in Fig. 21.3~ can be obtained. If the steam temperature T, (or p,) is held constant and the flow rate is measured at several values of valve-top pressure, a curve such as that shown in Fig. 21.3.b can be obtained. These two curves can now be used to evaluate the partial derivatives in the-linear expansion of fl(p,T,) as we shall now demonstrate. Expanding w,. about the operating point ps ,T,, and retaining only the linear terms give
This equation can be written in the form
Vv - Tv,)
(21.20)
1 -= R,
*The flow of steam through a controJ valve dn &en be represented by the relationship. WY I~:&& &q ** . where ps = supply pressure of steam pv = pressure downstream of valve S & = cross-sectionai area for flow of steam through valve C, = constant of the vaive
(21.18)
For a linear valve, A0 is propokonal to stem position and the stem position is proportional to valve-top pressure p; under these conditions, Eq. (21.18) takes the form
w=c:PvF-E .
(21.19)
326 PROCESS
APPLICATIONS
IP, P0) T,, = T,,
FIGURE 213 Linearization of valve characteristics from experimental tests.
The coefficients K, and -l/R, in Eq. (21.20) am the slopes of the curves of Fig. 21.3 at the operating point ps, TV,. This follows from the definition of a partial derivative. Notice that l/R, has been defined as the negative of the slope so that R, is a positive quantity. The experimental approach described here for obtaining a linear form for the flow characteristics of a valve is always possible in principle. However, it must be emphasized that the linear form is useful only for small deviations from the operating point. If the operating point is changed considerably, the coefficients K, and l/R,, must be reevaluated. Notice that, in writing Eq. (21.20), we have assumed the valve to have no dynamic lag between p and stem position. This assumption is valid for a system having large time constants, such as a steam-jacketed kettle, as was demonstrated in Chap. 10.
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