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We have now completed the analysis of the kettle and valve. A block diagram of the control system, based on Eqs. (21.9), (21.15), and (21.20) is shown in Fig. 21.4. The controller action is not specified but merely denoted by G, in the block diagram. Also, the feedback element is denoted as H. From Fig. 21.4, we see that the steam-jacketed kettle is a multiloop control system. Furthermore, the loops overlap. The block diagram can be used to obtain the overall transfer function between any two variables by applying the methods of Chap. 12. After considerable algebraic manipulation, the following result is obtained: + G(l + WRv) T, _ G3U + WWW (21.21) D(s) i D(s) where D(s) = 1 + GsIR, + G,G2G5KvH - G2G4. The terms Gl,G2,G3,G4, Gs,G,, and Hare defined in Fig. 21.4. For example, if G, = K, and H = 1, one obtains from Eq. (21.21) the transfer function K -= (21.22) R T2S2 + 2lTS + 1
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FlGURE 21-4 Block diagram for control of steam-jacketed kettle.
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D1 = 1 + 2 + K,K,K2K5 - Kz Y It is seen that the response of the control system is second-order when proportional control is used and the measuring element does not have dynamic lag. Notice that the parameters K, TV, and XT in Eq. (21.22) am positive. This follows from the fact that the parameters K,, K,, K2, Kg, Ry, T,, and 7W am all positive and that K2 < 1. When a block diagram of a control system becomes very complicated, such as the one in this example, it is convenient to simulate the control system with a computer. When computer simulation is selected as the means of studying the transient response of the control system, ,the block diagram can be translated directly into a computer program. This computer-simulation technique will be coveted in detail in Chap. 34.
PROCRSS
AF FUCATIONS
DYNAMIC RESPONSE OF A GAS ABSORBER
Another example of a complex system is the plate absorber* shown in Fig. 21 S. In this process, air containing a soluble gas such as ammonia is contacted with fresh water in a two-plate column in order to remove part of the ammonia from the gas. The action of gas bubbling through the liquid causes thorough mixing of the two phases on each plate. During the mixing process, ammonia diffises from the bubbles into the liquid. In an industrial operation, many plates may be used; however, for simplicity, we consider only two plates in this example, since the basic principles am unaffected by the number of plates. Our problem is to analyze the system for its dynamic response. In other words, we want to know how the concentrations of liquid and gas change as a result of change in inlet composition or flow rate. Throughout the analysis, the following symbols am used: L, = flow of liquid leaving nth plate, moles/n-& V, = flow of gas leaving nth plate, moles/mm XII = concentration of liquid leaving nth plate, mole fraction NH3 y, = concentration of gas leaving nth plate, mole fraction NH3 H, = holdup (or storage) of liquid on nth plate, moles
*The reader who has not studied gas absorption may find this subject presented in any textbook on chemical engineering unit operations. For example, see Bennett and Myers (1982).
FIGURE 21-5 Bubble-cap gas absorber.
THEORETICAL
ANALYSIS
COMPLEX
PROCESSES
In order to avoid too many complicating details, the following assumptions will be used: 1. The temperature and total pressure throughout the column are uniform and do not vary with changes in flow rates. 2. The entering gas stream is dilute (say 5 mole percent NHs) with the consequence that we can neglect the decrease in total molar flow rate of gas as ammonia is removed. Likewise, we can assume that the molar flow rate of liquid does not increase as ammonia is added. 3. The plate efficiency is 100 percent, t which means that the vapor and liquid streams leaving a plate are in equilibrium. Such a plate is called an i&l equilibrium stage. 4. The equilibrium relationship is linear and is given by the expression Yn = rnxz + b (21.23)
where m and b are constants that depend on the temperature and total pressure of the system, and xz is the concentration of liquid in equilibrium with gas of concentration yn. For an ideal plate x, = x,* 5. The holdup of liquid H, on each plate is constant and independent of flow rate. Furthermore, the holdup is the same for each plate, that is, H 1 = Hz = H . 6. The holdup of gas between plates is negligible. As a consequence of this assumption and assumption 2, the flow rate of gas from each plate is the same and equal to the entering gas flow rate; that is,
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