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Cal Poly Pomona CHE 426 - Problem Set #2

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_______________________ Last Name, First CHE426: Problem set #21. 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?Answer(a) x(t) = 1  exp(t/2)cos 32t� �� �� �� �  33 exp(t/2)sin32t� �� �� �� �(b) x(t) = 1  te-t  e-t = 1  e-t(t + 1)(c) x(t) = 1  1.17exp( 0.382t) + 0.17exp( 2.618t)As the coefficient of dx/dt increases, the response becomes more sluggish and lessoscillatory.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) = 2Answer(a) x(t) = 0.5t2 + t  1 + 0.5cos(t)  0.5sin(t) + 0.5e-t (b) q(t) = 2 + 33t + 2exp(t)3. 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- - +-Answer(a) y(t) = cos t  cos 2t (b) y(t) = 15 + 110et sin 2t 15 cos 2t(c) y(t) = 2t + 1 + tet +2et4.2 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.AnswerV = 0.0408 m35.2 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. Answert = 1.94 h6. In tank A are 200 gal of brine containing 80 lbs of dissolved salts. Solution from this tankruns 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 BAnswert = 30 min  B = 0.2583 lb/gal7. ( )254 3s s s+ + = As + 1Bs r+ + 2Cs r+In this equation, r1 < r2. Determine B and C.AnswerB =  2.5C = 0.833338. Given f(t) = 3  4(t  1)U(t  1) + 4(t  3)U(t  3), determine f(2) and f(5).Answera) f(2) = 3  4(t  1) = 3  4 =  1b) f(5) = 3  4(t  1) + 4(t  3) = 3 + 4  12 =  59. Find the Laplace transform of e-2tcos 3tAnswerL{e-2tcos 3t} = 22( 2) 9ss++ +10. Find the inverse ofF(s) = 236 18ss s+- + Answerf(t) = e3tcos 3t + 2e3tsin 3t11. Figure 6 shows the schematic of a process for treating residential sewage. In this simplified 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 and recycled back to the aeration tank. The treated sewage leaving the settling tank has no bacteria in it while the recycle sewage contains a bacterial concentration of 1.0 lb/gal. Both the aeration and the settling tanks have the same volume of 5106 gallons. You can assume the liquid (sewage) density remains constant throughout the process and neglect the mass loss 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.AnswerQtreated = 8000 gal/minQrecycle = 2000 gal/minReferences1. 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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Cal Poly Pomona CHE 426 - Problem Set #2

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