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GA(S) =
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(Ts + l)sU - e
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-T.V
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(26.36)
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This expression may be simplified to give GI\(S) = K(l - eeTS)eFTS S(TS + 1)
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T-az c(r) delayed by (T-a 7)
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(26.37)
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FIGURE 26-7 Response of first-order, sunpled-data system with transport lag.
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SAMPLED-DATA CONTROL OF A FIRST-ORDER PROCESS WITH TRANSPORT LAG
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Note that the right side of Eq. (26.37) does not include a term involving a nonintegral power of T. Obtaining the Z-transform of GA(S) gives an expression, which is G(z, m). The details are shown in the following steps. GA(Z) = G(z,m) = K(l - z- )z-~~ G(zm)
( s(7s1+ l,}
e-Tfr
K(z - 1)
z _ e-Th
Simplifying this expression gives G(z,m) = K(l - b) z(z - b)
(26.38)
The other terms needed to evaluate C(z,m) in Eq. (26.23) are G(z) and R(z). G(z) is given in Eq. (26.11). For a unit-step change in set point
Substituting Eqs. (26.11), (26.38), and (26.39) into Eq. (26.23) gives, after considerable algebraic manipulation K(1 - b)z (26.40) C(z,m) = (z - l){z2 + [K(l - d) - blz + K(d - b)} Inversion of this expression will give the value of the delayed response c[t - (T UT )]. These values am, of course, the peak values of c(t) as illustrated in Fig.
Mosler (1966) has inverted Eq. (26.40) by partial fraction expansion; the result is a rather complex expression. He used this result to obtain some design rules for determining the values of K and T that will produce a transient with quarter-decay ratio. The development of the rules is quite involved and beyond the scope of this book. SUMMARY In this chapter, the principles of sampled-data theory have been applied to the proportional control of a process, which represents a large class of systems in chemical processing, namely, a process that consists of a first-order process with transport lag [e -aTs/(~~ + l)]. Since the transport lag parameter (UT ) may not be an integral number of sampling periods, the modified Z-transform was used to obtain the pulse transfer function of the system. As the order of the characteristic equation for the closed-loop system increases, the stability criteria become more and mom complex and require that several inequalities be satisfied simultaneously for stability. For the case of proportional control of a first-order system without transport lag, some simple design rules were developed for tuning the proportional controller to obtain a desired decay ratio.
SAMPLED-DATA CONTROL SYSTEMS
PROBLEM
26.1. The stirred-tank control system shown in Fig. P26.1 blends a stream of concentrated solution with a process stream to maintain a desired concentration of solute in the outlet stream. The flow rates and concentrations are indicated in the diagram. The chemical analysis, which must be done manually by withdrawing a small sample from the tank, takes 1.0 min. At the end of each analysis, the chemist sets a dial immediately to a value corresponding to the concentration just determined. The dial, in turn, feeds a concentration signal to the controller. As soon as one sample is analyzed, a new one is withdrawn from the tank and analyzed. The flow rate through the valve varies linearly from 0 to 0.02 liter/min as the valve-top pressure varies from 3 to 15 psig. Under normal conditions, the process stream is free of solute. However, from time to time, a load change may occur in the form of a change in concentration of solute in the process stream entering the tank. (a) Show that the system is equivalent to a sampled-data control system and draw its block diagram. (b) From the design rules developed by Mosler (1966, Eq. 65), one can show that the value of K, required for quarter-decay ratio and fast sampling (T = ur) is 10.3 psi&/l). Using this value of Kc, sketch the transient response for c and q for a step change in ci of magnitude 0.5 g/l. Determine the extreme values of c, p, and q during the transient. Determine the value of c(m). (c) If the chemist uses a continuous analyzer having no lag, but still sets the dial manually as just described, every 60 set, show how the block diagram changes and determine K, to obtain quarter-decay ratio. Use the design rule given by Eq. (26.32) to determine this gain.
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