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FIGURE P6-9
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6.10. A tank having a cross-sectional area of 2 ft2 and a linear resistance of R = 1 ft/cfm is operating at steady state with a flow rate of 1 cfm. At time zero, the flow varies as shown in Fig. P6.10. (a) Determine Q(t) and Q(s) by combining simple functions. Note that Q is the deviation in flow rate. (b) Obtain an expression for H(t) where H is the deviation in level. (c) Determine H(r) at r = 2 and t = 00.
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6.11. Determine Y(5) if Y(s) = ee3$/[s(7s + l)].
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6.12. Derive the transfer function H/Q for the liquid level system shown in Fig. P6.12. The resistances are linear. H and Q are deviation variables. Show clearly how you derived the transfer function. You are expected to give numerical values in the transfer function.
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PHYSICAL EXAMPLES OF FIRST-ORDER SYSTEMS
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R2=5
FIGURE P6-12
6.13. The liquid-level system shown in Fig. P6.13 is initially at steady state with the inlet flow rate at 1 cfm. At time zero, one ft3 of water is suddenly added to the tank; at t = 1, one ft3 is added, etc. In other words, a train of unit impulses is applied to the tank at intervals of one minute. Ultimately the output wave train becomes periodic as shown in the sketch. Determine the maximum and minimum values of this output.
Train of imwlses H H max-------~-~L$-J--J Hmin -
I I I I ,n n+l n+2
I I I I n+3
FIGURE
P6-13
6.14. The two-tank mixing process shown in Fig. P6.14 contains a recirculation loop that transfers solution from tank 2 to tank 1 at a flow rate of (;Y q o. (a) Develop a transfer function that relates the concentration in tank 2, ~2, to the concentration in the feed, x; i.e. Cz(s)/X(s) where C2 and X are deviation variables. For convenience, assume that the initial concentrations are x = Cl = c2 = 0. (b) If a unit-step change in x occurs, determine the time needed for c2 to reach 60 percent of its ultimate value for the cases where (Y = 0, 1, and 00. (c) Sketch the response for (Y = 00. Assume that each tank has a constant holdup volume of 1 ft3. Neglect transportation lag in the line connecting the tanks and the recirculation line. Try to answer parts (b) and (c) by intuition.
CHAPTER
RESPONSE OF FIRST-ORDER SYSTEMS IN SERIES
Introductory Remarks
Very often a physical system can be represented by several first-order processes connected in series. To illustrate this type of system, consider the liquid-level systems shown in Fig. 7.1 in which two tanks are arranged so that the outlet flow from the first tank is the inlet flow to the second tank. Two possible piping arrangements are shown in Fig. 7.1. In Fig. 7. la the outlet flow from tank 1 discharges directly into the atmosphere before spilling into tank 2 and the flow through R t depends only on h 1. The variation in h 2 in tank 2 does not affect the transient response occurring in tank 1. This type of system is referred to as a noninteracting system. In contrast to this, the system shown in Fig. 7. lb is said to be interacting because the flow through RI now depends on the difference between h 1 and h2. We shall consider first the noninteracting system of Fig. 7.1~.
Noninteracting System
As in the previous liquid-level example, we shall assume the liquid to be of constant density, the tanks to have uniform cross-sectional area, and the flow resistances to be linear. Our problem is to find a transfer function that relates h2 to 4, that is, Hz(s)lQ(s). The approach will be to obtain a transfer function
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