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# hw6sol - 1 Consider the following parameters for the...

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1. Consider the following parameters for the remainder of this assignment: k1 = 5/6, k2 = 5/3, k3 = 1/6 mol/L/min, and CAf = 10 mol/L. Find the maximum achievable steady-state concentration of B for the case where the dilution rate (F/V ) is the manipulated input (no graphs allowed yet!). k1=5/6; k2=5/3; k3=1/6; Caf=10; u=[0:0.01:10]; Cb=(-k1*(k1+u)+k1*sqrt((k1+u).^2+4*k3*Caf.*u))./(2*k3*(k2+u)); [Cbmax, i]=max(Cb) u(i) Cb(i) This returns, maximum Cb = 1.266, for F/V=1.29. 2. Since we've already de_ned u and CB as arrays above, all we have to do is type plot(u,CB) and we get the plot shown below.

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3. The maximum achievable concentration of B corresponds to a singularity, and the system is uncontrollable at this point. Instead consider both of the steady-states corresponding to CB = 1.117 mol/L (F/V = 0.5714, and F/V = 2.8744 ), and find the linear state-space model for both steady-state operating points. 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4
For F/V =0.5714, ? 1 = 2.4047 0 0.8333 2.2381 ? 1 = 7 1.117 For F/V=2.8744, ? 1 = 5.7367 0 0.8333 4.5411 ? 1 = 3.913 1.117 4. Compute the eigenvalues of each state-space model and determine the stability of each steady state. Matlab: >> eig(A1) ans = -2.2381 -2.4047 >> eig(A2) ans = -4.5411 -5.7367 Since both of these sets of characteristic values are real and negative, both steady states are stable. We would expect a perturbation from the steady state to decay without oscillation. 5. Find the transfer function G(s) = C B (s)/D(s) corresponding to each state-space model (D = F/V ). I can find it easily using MATLAB >> [num1,den1] = ss2tf(A1,B1,C1,D1); G1=tf(num1,den1) Transfer function: -1.117 s + 3.147 --------------------- s^2 + 4.643 s + 5.382 >> [num2,den2] = ss2tf(A2,B2,C2,D2); G2=tf(num2,den2)

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Transfer function: -1.117 s - 3.147 --------------------- s^2 + 10.28 s + 26.05 6. Finding zeroes and poles Matlab:
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hw6sol - 1 Consider the following parameters for the...

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