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7
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7.1 Throughput is the load; distillate and bottoms composition are product quality; levels are inventory variables. The extra manipulated variable can be adjusted to the most economic value. 7.2 m2 ml
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l/PI - l/P, l/P, - l/pz l/PI - l/P l/P - l/m p l/P1 - l/P, l/p, - l/p2 7.3 The manipulated variable that affects density most should be the product of the outputs of both controllers. The other flow can be manipulated directly by the flow controller. 7.4
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1 Appendix
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7.5 The relative-gain terms are -t m, indicating that pressure and temperature are dependent on one another. Control of either one results in control of the other. Coolant teml)erature should be manipulated for control of either, while flow sets the throughput.
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8.1
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dTz = dllW,/Kll~,; dT, = 4 F at Il., = 50%, and 8 F at TT-, = 25%. 8.2 L = kFz/y. L is manipula.ted directly proportional to P but to the first power of kx rather than the half-power. 8.3 The peak will occur about 2 min following a step-load change because this is the point of greatest departure between the curves. Lead time should be about 1.6 min, and lag time about 3 min. 8.4 (dc/dg)dt = (r/q)(~~ - T,~) = $2 min. [(dm - m)/m] dt = 71 - 72 = +I.5 min. Area of compensated curve = 0.5 min. 8.5 Integrated area reduced by (a) 2: 1, (b) 4: 1, (c) 10: 1. Variable dead time would prevent the three-mode controllers from being optimally tuned, particularly the noninteracting controller. So the amount of improvement possible may only be realized with t,he forward loop.
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9 9.1 Flow of cold water could be manipulated directly by the flow controller, while hot water is the product of the outputs of the flow and temperature controllers. 9.2 At 120 lb/min feed rate, oil flow must be infinite; at 80 lb/min, oil flow is reduced to 133 lb/min. At 160 F inlet, oil flow must be 964 lb/min; at 240 F inlet, it is reduced to.150 lb/min. 9.3 Process gain is O.O41 F/(lb/ min) at 100 lb/min feed, and 0.28a F/ (lb/min) at 80 Ib/min, a ratio of almost 7. Process and valve gain products are 16.8k and 40.3k for the same conditions, a ratio of only 2.4. 9.4 & = [llyc(T, - T&]/(l/Ak + 1/2C,). H e a t t r a n s f e r a p p e a r s t o b e linear with flow, but actually [I contains constant t,erms including metal conductivity and condensing film coefficient; in addition, the liquid film coefficient actually varies with the 0.8 power of flow. 9.5 Steam temperature in a drum boiler is controlled principally with spray. The once-through boiler has one less controlled variable (liquid level), which frees a manipulated variable (feedwater flow) for temperature control. 9.6 Coefficient k, = 7.716 X IOeG ft/rpm2, and kz = 0.01 ft/gpm2; N = 3,120 rpm. HHP at 3,120 rpm is 0.632 HP; at 3,600 rpm it is 0.948 HP.
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Tc = T - RT=/E = 206.2 F. TT = -3 min; 70 = 4.8 min; P = 71%. The loop is marginally stable. 10.3 719 = -7.8 min; 70 = 4.3 min; P = 72%. For the second calculation, TT = +lO min; 70 = 3.9 min; P = 65%.
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10.1 lo.2
Answers to Problems 10.4 A feedback optimizing controller must be used to hold minimum conductivity, but the process must be made self-regulating first. This can be accomplished by feeding solvent from one storage tank while flowing into another, switching when the feed t,ank is empty. 10.5 i2t neutrality, the pH is 9.366. The slope of the curve at that point is 9,330. 10.6 Equation (10.22) and the answers to Prob. 10.2 and 10.3 indicate that the dynamic gain is independent of temperature and concentration.
For V/F = 5, D/Ii = 0.471, x = 0.0822, dy/d(D/P) = -0.5. For V/F = 2.5, D/F = 0.393, x = 0.196, dy/d(D/F) = -0.2. F o r c o n s t a n t V/P, B = F(1 - z)/(l - z). F o r c o n s t a n t V, B = F[l - ,(a - bF)]. 11.4 Let z1 and z2 be the mole fractions of propane and isobutane, respectively, in the feed; then D = 1f(0.929z1 + 0.912z2 + 0.068). 1 1 . 5 Optimum V/F for z = 0.50 is about 8. 11.6 Optimum V/I; for 2 = 0.60 is about 7; V/F can be programmed with z or with D/b as shown in Fig. 11.20.
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