_______________________ 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 5106 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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