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FIGURE 7-5 Effect of interaction on step response of two-tank system. step response of
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7.1. Determine the transfer function H(s)/Q(s) for the liquid-level system shown in Fig. W. 1. Resistances Rl and R2 ate linear The flow rate from tank 3 is maintained constant at b by means of a pump; i.e., the flow rate from tank 3 is independent of head h. The tanks are noninteracting.
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7.2. The mercury thermometer in Chap. 5 was considered to have all its resistance in the convective film surrounding the bulb and all its capacitance in the mercury. A more detailed analysis would consider both the convective resistance surrounding the bulb and that between the bulb and mercury. In addition, the capacitance of the glass bulb would be included. Let Ai = inside ama of bulb, for heat transfer to mercury A0 = outside ama of bulb, for heat transfer from surrounding fluid m = mass of mercury in bulb rnb = mass of glass bulb C = heat capacity of mercury Cb = heat capacity of glass bulb hi = convective coefficient between bulb and mercury h, = convective coefficient between bulb and surrounding fluid T = temperature of mercury Tb = temperature of glass bulb Tf = temperature of surrounding fluid Determine the transfer function between Tf and T. What is the effect of the bulb resistance and capacitance on the thermometer response Note that the inclusion of the bulb results in a pair of interacting systems, which give an overall transfer function somewhat different from that of Eq. (7.24). 7.3. Them are N storage tanks of volume V arranged so that when water is fed into the first tank, an equal volume of liquid overflows from the first tank into the second tank, and so on. Each tank initially contains component A at some concentration CO and is equipped with a perfect stirrer. At time zero, a stream of zero concentration is fed into the first tank at a volumetric rate q. Find the resulting concentration in each tank as a function of time.
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7.4. (a) Find the transfer functions Hz/Q and H$Q for the three-tank system shown in Fig. P7.4 where Hz, H3 and Q are deviation variables. Tank 1 and Tank 2 are interacting. (b) For a unit-step change in q (i.e., Q = l/s), determine H3(0), H3(w), and sketch H3(t) versus t.
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4 A,= 1 -- -2 --= A,= 1 -z n z - ._ a ._ . . - = - h2 RI= 2 Tank 1 Tank 2 R2=2 A3= 0.5 -z I- = h, R,= 4 Tank 3 *
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FIGURE W-4
7.5. Three identical tanks are operated in series in a noninteracting fashion as shown in Fig. P7.5. For each tank, R = 1, T = 1. If the deviation in flow rate to the first tank is an impulse function of magnitude 2, determine (a) An expression for H(s) where H is the deviation in level in the third tank. (b) Sketch the response H(t). (c) Obtain an expression for H(t).
FIGURE W-5
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7.6. In the two-tank mixing process shown in Fig. W.6, x varies from 0 lb salt/ft3 to 1 lb salt/ft3 according to a step function. At what time does the salt concentration in tank 2 reach 0.6 lb salt/ft3 The holdup volume of each tank is 6 ft3.
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Tank 1
Tank 2
FIGURE W-6
7.7. Starting from first principles, derive the transfer functions Hl(s)lQ(s) and Hz(s)/Q(s) for the liquid level system shown in Fig. P7.7. The resistances are linear and R1 = R2 = 1. Note that two streams are flowing from tank 1, one of which flows into tank 2. You are expected to give numerical values of the parameters in the transfer functions and to show clearly how you derived the transfer functions. q(t) R,=2
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