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Cal Poly Pomona CHE 426 - Problem Set #6

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_______________________ Last Name, First CHE426: Problem set #611. A step change of magnitude 4 is introduced into a system having the transfer function ( )( )Y sX s = 2101.6 4s s+ +A. Determine (a) Percent overshoot, (b) Rise time, (c) Maximum value of Y(t), (d)Ultimate value of Y(t), and (e) Period of oscillation. Use the following formulas:Step response for  < 1.Y(t)/10 = 1  211 z- exptzt� �-� �� � 22 11sin 1 tantzzt z-� �� �-� �� �� �- +� �� �� �� �� �� �Overshoot = exp 21pzz� �� �-� �-� �f = 1T = 12p21 zt-B. Determine graphically from Simulink the questions from (A).2. The two-tank system shown in FigureE-2 is operating at steady state. At time t= 0, 10 ft3 of water is quickly added tothe first tank. Determine the maximumdeviation in level (feet) in both tanksfrom the ultimate steady-state value andthe time at which each maximum occurs.Data: A1 = A2 = 10 ft2, R1 = 0.1 ft/cfm, R2= 0.35 ft/cfm.Note: Q1 = h1/R1, if y(t) = (0) (unitimpulse) then Y(s) = 1. Figure E-23.1 Determine y(t = 0), y(t = 0.6), and y(t = ∞) if Y(s) = 1s 225( 1)2 25ss s++ +R1R2A1A2h1h2Q1Q22 0 f t / m i n31 0 f t3R1R2A1A2h1h2Q1Q2Q04. The two-tank liquid-level system shown in Figure E-3 is operating at steady state when a step change is made in the flow rate to tank 1. The transient response is critically damped, and it takes 1.0 min for the change in level of the second tank to reach 50 percent of the total change. If the ratio of the cross-sectional areas of the tanks is A1/A2 = 2, calculate the ratio R1/R2.Calculate the time constant for each tank. How long does it take for the change in level of thefirst tank to reach 90 percent of the total change? Note: 1 = 2 for critical damping.Figure E-35.1 Sketch the response y(t) if Y(s) = exp(2s)/[s2 + 1.2s + 1]. Determine y(t) for t = 0, 1, 5, ∞.6.2 The two tanks shown in Fig. E-6 are connected in an interacting fashion. The system isinitially at steady state with q = 10 cfm. The following data apply to the tanks: A1 = 1 ft2, A2= 1.25 ft2, R1 = 1 ft/cfm, and R2 = 0.8 ft/cfm.(a) If the flow changes from 10 to 11 cfm according to a step change, determineH2(s),i.e., the Laplace transform of H2 where is the deviation in h2.(b) Determine H2(1), H2(4), and H2(∞).R1R2A1A2h1h2Q2QFigure E-67. The two-tank heating process shown in Fig. E-7 consists of two identical, well-stirredtanks in series. A flow of heat can enter tank 2. At time t = 0, the flow rate of heat to tank 2suddenly increases according to a step function to 1000 Btu/min, and the temperature of theinlet water Ti drops from 60oF to 52oF according to a step function. These changes in heatflow and inlet water temperature occur simultaneously.(a) Develop a block diagram that relates the outlet temperature of tank 2 to the inlettemperature to tank 1 and the flow of heat to tank 2.(b) Obtain an expression for T2’(s) where T2’ is the deviation in the temperature of tank2. This expression should contain numerical values of the parameters.(c) Determine T2 (2) and T2 (∞).(d) Sketch the response T2’(t)versus t.Initially, Ti = T1 = T2 = 60 oF and q = 0. The following data apply:w = 250 lb/minHoldup volume of each tank = 5 ft3Density of fluid = 50 lb/ft3Heat capacity of fluid = 1 Btu/(lboF)wwqTiT1T2T a n k 1 T a n k 2Figure E-78.1 The overhead vapor from a depropanizer distillation column is totally condensed in awater-cooled condenser at 120oF and 230 psig. The vapor is 98 mol % propane and 2 mol %isobutene. The vapor design flow rate is 40,000 lb/h and average latent heat of vaporization is128 Btu/lb. Cooling water inlet and outlet temperatures are 75 and 100oF, respectively. Thecondenser heat transfer area is 1000 ft2. The cooling water pressure drop through thecondenser at design rate is 50 psi. A linear-trim control valve (air-to-closed, when CO = 20mA, PV = 15 psig) is installed in the cooling water line. The pressure drop over the valve is25 psi at design with the valve half open. The process pressure is measured by anelectronic (4-20 mA) pressure transmitter whose range is 150-400 psig. An analog electronicproportional controller with a gain of 2 is used to control process pressure by manipulatingcooling water flow. The electronic signal from the controller (CO) is converted into apneumatic signal in the I/P transducer.R e f l u x d r u mP TP CI / PP MS PC OP VC o n t r o l v a l v eC o o l i n gw a t e rC o n d e n s e rV a p o ra) Calculate the cooling water flow rate (gpm) at design conditions. Water density is 62.3lb/ft3 and 1 ft3 = 7.48 gal.b) If the cooling water flow rate is 250 gpm at design conditions, calculate the sizecoefficient (Cv) of the control valve.c) Calculate the value of the signal PM at design condition. d) Calculate the value of the signal PV at design conditions _________e) Suppose the process pressure jumps 20 psi, determine value for CO.9. Express the function given the graph in the t-domain10. A thermometer having first-order dynamics with a time constant of 1 min is at 100oF. Thethermometer is suddenly placed in a bath at 110oF at t = 0 and left there for 0.167 min, after which it is immediately returned to a bath at 100oF. Calculate the thermometer reading at t = 0.5


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