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Cal Poly Pomona CHE 426 - Problem Set #2

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_______________________ Last Name, First CHE426: Problem set #2 (Matlab solutions are not acceptable)1. Solve the following equations by using Laplace transforms1.(a)22d xdt + dxdt + x = 1 x(0) = x’(0) = 0 (Note: x’ = dxdt)(b)22d xdt + 2dxdt + x = 1 x(0) = x’(0) = 0(c)22d xdt + 3dxdt + x = 1 x(0) = x’(0) = 0Use Matlab to plot the behavior of these solutions on a single graph for 0  t  10. Use thetitle command to label the graph with your name. What is the effect of the coefficient ofdx/dt?2. Solve the following differential equations by Laplace transforms1.(a)44d xdt + 33d xdt = cos t x(0) = x’(0) = x’’’(0) = 0, x’’(0) = 1.(b)22d qdt + dqdt = t2 + 2t q(0) = 4, q’(0) = 23. Invert the following transforms1.(a)2 23( 1)( 4)ss s+ +, (b) 21( 2 5)s s s- +, (c) 3 22 23 3 2( 1)s s ss s- - +-4.2 Two consecutive, first order reactions take place in a perfectly mixed, isothermalcontinuous reactor (CSTR).A B Ck12kVolumetric flow rates (F) and density are constant. The reactor operates at steady state. Theinlet stream to the reactor contains only A with CA,in = 10 kmol/m3. If k1 = 2 min-1, k2 = 3min-1, and F = 0.1 m3/min, find the tank volume that maximized the concentration ofcomponent B in the product stream. Show all your work.5.2 A tank containing 3.8 m3 of 20% (by volume) NaOH solution is to be purged by adding pure water at a rate of 4.5 m3/h. If the solution leaves the tank at a rate of 4.5 m3/h, determinethe time necessary to purge 90% of the NaOH by mass from the tank. Assume perfect mixing. Specific gravity of pure NaOH is 1.22. 6. In tank A are 200 gal of brine containing 80 lbs of dissolved salts. Solution from this tank runs at a rate of 4 GPM into a second tank, B, which contains initially 100 gal of brine with aconcentration of 0.2 lb/gal of solution. Similarly, solution runs from tank B at the same rate. Determine the concentration of salt in tank B after 30 minutes. pure water4GPM4GPM 4GPMA B7. ( )254 3s s s+ + = As + 1Bs r+ + 2Cs r+In this equation, r1 < r2. Determine B and C.8. Given f(t) = 3  4(t  1)U(t  1) + 4(t  3)U(t  3), determine f(2) and f(5).9. Find the Laplace transform of e-2tcos 3t10. Find the inverse ofF(s) = 236 18ss s+- +11. Figure 6 shows the schematic of a process for treating residential sewage. In thissimplified process, sewage (without bacteria) at a rate of 6000 gal/min is pumped into a well-mixed aeration tank where the concentration of bacteria CB,aration is maintained at 0.25 lb/gal.The treated sewage is then pumped to a settling tank where the bacterial is separated andrecycled back to the aeration tank. The treated sewage leaving the settling tank has nobacteria in it while the recycle sewage contains a bacterial concentration of 1.0 lb/gal. Boththe aeration and the settling tanks have the same volume of 5106 gallons. You can assumethe liquid (sewage) density remains constant throughout the process and neglect the massloss due to the generation of CO2 leaving the aeration tank.S e w a g eQi nA i rC O2A i rA e r a t i o n t a n kP r o c e s s p u m pR e c y c l e p u m p S e t t l i n gt a n kQt r e a t e dQo u tQr e c y c l eFigure 6 A process for treating residential sewage.If 6000 gal/min of sewage enters and leaves the treatment facility, determine the twovolumetric flow rates Qtreated and Qrecycle.References1. D.R. Coughanowr and S. LeBlanc, Process Systems Analysis and Control, McGraw-Hill,3nd edition, 2008.2. Mass Transfer by Hines and Maddox.3. Process Modeling, Simulation, and Control for Chemical Engineers by


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