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9.1 Noninteracting: 7, = 247, G, = 0.60. Interacting: 7, = 337, G1 = 0.115. P.P (a) 70 = 80 min with 4R = -45 , (b) R = 12.7 min, (c) G1 = 0.38. 9.3 P = 269%. 2.4 Flow in excess of 60 percent. 2.5 C,/CR = 2. A manual valve installed in series with the control valve can bring about a reduction in C R. As CR decreases, valve size must increase to deliver a specified maximum flow. P.6 The process is essentially dead-time plus single-capacity, in that T,/T~ = 3.91. The dead time varies inversely with flow, which can very likely be compensated for by using an equal-percentage valve.
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P = 640%; R = 1.0 sec. r. = 6.72 sect; P = 875%. The estimate of 70 is close, that of P is conservative. A large number of equal noninteracting capacities approaches dead time. 3.3 P = 148%; G, = 1.6 psi/gpm. 3.4 7. = 1.24 sec. The manometer would increase the natural period of the flow loop considerably.
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3.6 With the pump off, 7 varies from 1.4 to 4.2 min; with the pump on, 7, is 1.33 min. The pump provides some mixing, reducing the dynamii gain of the system. 3.7 P = 6000/,.
4 4.1 $R = -60"; R = 0.28rd; P = 4009& The period of the loop changes less with controller phase lag in the dead-time process. 4.2 For a 5 percent step in Am, e = lo /,; for dm/dt of 5 %/30 min, e = 3.3 %; for a 5 percent sine wave of 2 hr-period, a 4.6 percent sinusoidal error will result. 4.3 P = 30%; R = 0.75; D = 1.5. Masimum D//R = 0 . 2 5 , o c c u r r i n g when D = R. 4 . 4 P, = 4rd; P = 190r,L/r1; PR/lOO = I.~~T~~/T~. D and R ought to be set equal, but a 2: 1 ratio is only slight,ly subol)timum, owing to interaction (D /R = 0.222). 4.5
ru = 27 min; R = 7.5 min; P = 33K,. 4.6 A sampling controller is most effective and reliable for coping with dead time alone. The control interval Al, should be set to a very low value, e.g.,
Answers to Problems
0.1 min; then the sample interval must be 2.1 min. For critical damping, R = 0.25 min. E/Am would lie between 5.12 and 10.4 min.
5 5.1 The loop will limit-cycle with a peak-to-peak amplitude of 25 percent. Reduction of the band to 5 percent will produce no significant change in the limit cycle. 5.2 TV = 40 set; d = 1.6 F, peak to peak. 5.3 T,, = 75 set; 14 = 3.O F, peak to peak. 5.4 2 = 3.8%; 7, = 3.8 sec. 5.5 Preload 9 = 30%; td = 3.6 min; et = 4.45 F. 5 . 6 P = 18%.
6.1 The natural period of the secondary loop is about 8 min. The primary loop oscillates at about a 22-min period, contrasted to 35 min without cascade control. 6.P RP = 1.75 set; 70 = 5.0 sec. 6.3 Two level controllers must be used, one set for high level, the other low. The outputs of the flow controller and the high-level controller go to a high selector; its output is compared to that of the low-level controller in a low selector, whose output drives the valve. 6.4 At ra = 0.71, r, = 2.0, G = 0.32; at rd = 1.18, 7, = 8.0, G = 1.25. 6.5 At rd = 0.71, r0 = 1.8, G = 0.35; at 7d = 1.18, the loop is unstable, oscillating with a very high gain at a very low period. The effective values of reset and derivative are 4 : 1 apart for the interacting controller, rendering it less sensitive to changes in dead time. Control settings must be relaxed when dead time is variable, unless adaptation is possible.
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