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Cal Poly Pomona CHE 426 - Problem Set #3

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_______________________ Last Name, First CHE426: Problem set #31. An isothermal, first-order, irreversible reaction A  B with rate constant k takes place inthe liquid phase in a constant-volume reactor. Due to imperfect mixing, the reactor systemcan be modeled as a two-tank system with back mixing as shown in the sketch below3.FCA0VC1A1VC2A2F + FRFRFCA2Assuming F and FR are constant, write the differential equations describing the concentrationCA1 and CA1 as function of time. Do not solve the equations.SolutionV1 A1dCdt = FCA0 + FRCA2  (F + FR)CA1  V1kCA1V2 A2dCdt = (F + FR)CA1  FCA2  FRCA2  V2kCA22. If a forcing function f(t) has the Laplace transformf(s) = 1s + 22s se es- -- 3ses-Plot f(t) for 0  t  5. Note: The unit step function u(t  2) is represented by Matlab asheaviside(t-2).Solution3. Solve the following equation for y(t) 2 0( )ty dt t� = dydt + 3y y(0) = 1Solutiony(s) = 2e-2t  e-t4. Express the function given in Fig. 3-4 in the t-domain and the s-domainFigure 3-4Solutionf=u(t-1)+(t-2)u(t-2)-(t-3)u(t-3)-u(t-3)-(t-5)u(t-5)f(s) = 2 3 3 52 2s s s s se e e e es s s s s- - - - -+ - - -5. Sketch the following functions:a) f(t) = u(t)  2u(t  1) + u(t  3)b) f(t) = 3tu(t)  3u(t  1)  u(t  2)Solutiona) f(t) = u(t)  2u(t  1) + u(t  3)b) f(t) = 3tu(t)  3u(t  1)  u(t  2)6. Determine f(t) at t = 1.5 and at t = 3 for the following function:f(t) = 0.5 u(t) − 0.5 u(t  1) + (t  3) u(t  2)Solutionf (1.5) = 0 and f (3) = 0.7. Find and sketch the solution to the following differential equations using LaplaceTransforms.a) y′ + y =δ (t) , y(0) = 0b) y′+ y =δ (t −1), y(0) = 0Solutiona) y′ + y =δ (t) , y(0) = 0y(s) = 11s +  y(t) = e-t u(t)b) y′+ y =δ (t −1), y(0) = 0y(s) = 1ses-+ y(t) = e-(t-1) u(t  1)8. For the following transforms, find limt�� f(t).(a) f(s) = 21( 1)s s +(b) f(s) = 21( 1)s s -Solution(a) f(s) = 21( 1)s s +limt�� f(t) = 0lims� sf(s) = 0lims�21( 1)s + = 1(b) f(s) = 21( 1)s s -Final value theorem does not apply, because of pole in the RHP (s=1).f (t)→∞ as t →∞ , because of exp(t) terms in the solution.9. Develop the dynamic model equations for the continuous stirred tank (CST) thermal mixershown in Figure 3.9.Figure 3.9 Schematic of a CST thermal mixing process.The process parameters and variables are defined as:F1: mass flow rate of stream 1 (initially 5 kg/s)F2: mass flow rate of stream 2 (5 kg/s)M: mass of liquid in the mixer (100 kg) = constant (perfect level control)T1: temperature of stream 1 (25oC)T2: temperature of stream 2 (75oC)t: time (s)v: the time constant for the flow controller on stream 1 (2 s)Ts: the time constant for the temperature sensor on the product stream (6 s)At time equal to 10 seconds, a step change in the special flow rate for stream 1 is made from5 kg/s to 4 kg/s. Plot the product stream temperature and the sensor temperature from 0 to100 s. Use Matlab to plot and label the graph with your name using the Title command.SolutionMdTdt= F1T1 + F2T2  (F1 + F2)T Sensor model: sdTdt= 1Tst(T 


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