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Determine the actual gain (psig per inch displacement) and the integral time. 10.2. A unit-step change in error is introduced into a PID controller. If K, = 10, q = 1, and ~0 = 0.5, plot the response of the controller, P(t). 10.3. An ideal PD controller had the transfer function
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P - = KJq,s + 1) E
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An actual PD controller had the transfer function P -= KC TDS + 1 E (TdP)S + 1 where /3 is a large constant in an industrial controller. If a unit-step change in error is introduced into a controller having the second transfer function, show that P(t) = K,(l +Ae-@ D) where A is a function of p which you are to determine. For p = 5 and K, = 0.5, plot P(t) versus t/~. As p + m, show that the unit-step response approaches that for the ideal controller. 10.4. A PID controller is at steady state with an output pressure of 9 psig. The set point and pen point are .initially together. At time t = 0, the set point is moved away from the pen point at a rate of 0.5 in./min. The motion of the set point is in the direction of lower readings. If the knob settings are
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K, = 2 psiglin. of pen travel
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= 1.25min = 0.4min TD plot the output pressure versus time. 10.5. The input (e) to a PI controller is shown in Fig. P10.5. Plot the output of the controller if K, = 2 and ~1 = 0.50 min.
FIGURE PlO-5
CHAPTER
BLOCK DIAGRAM OF A CHEMICAL-REACTOR CONTROL SYSTEM
To tie together the principles developed thus far and to illustrate further the procedure for reduction of a physical control system to a block diagram, we consider in this chapter the two-tank chemical-reactor control system of Fig. 11.1. This entire chapter serves as an example and may be omitted by the reader with no loss in continuity. Description of System A liquid stream enters tank 1 at a volumetric flow rate F cfm and contains reactant A at a concentration of CO moles Alft3. Reactant A decomposes in the tanks according to the irreversible chemical reaction A-B The reaction is first-order and proceeds at a rate r = kc where r = moles A decomposing/(ft3)(time) c = concentration of A, moles Alft3 k = velocity constant, a function of temperature 135
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Controller
FIGURE 11-1
Control of a stirred-tank chemical nxctor.
The reaction is to be carried out in a series of two stirred tanks. The tanks are maintained at different temperatures. The temperature in tank 2 is to be greater than the temperature in tank 1, with the result that kz, the velocity constant in tank 2, is greater than that in tank 1, kl. We shall neglect any changes in physical properties due to chemical reaction. The purpose of the control system is to maintain ~2, the concentration of A leaving tank 2, at some desired value in spite of variation in inlet concentration CO. This will be accomplished by adding a stream of pure A to tank 1 through a control valve.
Reactor Ikansfer Functions
We begin the analysis by making a material balance on A around tank 1; thus V% = Fco - F + ; cl - klVcl + m ( i where m = molar flow rate of pure A through the valve, lb moles/mm PA = density of pure A, lb moles/ft3 V = holdup volume of tank, a constant, ft3 (11.1)
It is assumed that the volumetric flow of A through the valve m/PA is much less than the inlet flow rate F with the result that E!q. (11.1) can be written V% + (F + klV)cl = Fco + m This last equation may be written in the form (11.2)
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