hw9solution - Problem1. Find all the steady states for the...

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Unformatted text preview: Problem1. Find all the steady states for the reactor at the specified operating conditions assuming CAf and Tj are the two manipulated inputs and CA and T are the two controlled outputs. function f=hw9prob1(x) k0=3e7; dH=5000; E=12000; rhoCp=400; UA=150; R=1.987; V=0.9; Caf=10; Tf=298; Tj=298; q=1; Ca=x(1); T=x(2); f(1)=(q/V)*(Caf-Ca)-k0*exp(-E/R/T)*Ca; f(2)=(q/V)*(Tf-T)+dH/rhoCp*k0*exp(-E/R/T)*Ca-UA/V/rhoCp*(T-Tj); M-file : x01=[15 300]; x1=fsolve('hw9prob1',x01) x02=[0 400]; x2=fsolve('hw9prob1',x02) Optimization terminated: first-order optimality is less than options.TolFun. x1 = 9.4318 303.1659 Optimizer appears to be converging to a minimum that is not a root: Sum of squares of the function values is > sqrt(options.TolFun). Try again with a new starting point. x2 = 3.6399 369.5571 (MATCAD can be used as well OR simple ‘while’ loop in MATLAB can be used to get the lowest convergence steady state values) Linear state space model : Problem2. Prove that steady state gain can be calculated by K = -C A-1 B At steady state MATLAB code A=[-1.178 0.0415; 0.837 -1.0091]; B=[1.1111 0;0 0.417]; C=[1 0;0 1]; D=[0;0]; K=-C*inv(A)*B K= 0.9716 0.0150 0.8059 0.4257 The preferred control loop pairing is output T controlled by input Tj and output Ca controlled by Caf. The Niederlinski index: The closed loop system is unstable if index = 0.9708 => System is unstable. Problem 3 : To find transfer functions Or can do it in MATLAB >> sys=ss(A,B,C,D); >> g=tf(sys) Transfer function from input 1 to output... 1.111 s + 1.121 #1: --------------------s^2 + 2.187 s + 1.154 0.93 #2: --------------------s^2 + 2.187 s + 1.154 Transfer function from input 2 to output... 0.01731 #1: --------------------s^2 + 2.187 s + 1.154 0.417 s + 0.4912 #2: --------------------s^2 + 2.187 s + 1.154 Problem 4: For Ca/Caf case: For MATLAB : P= 1.95 ; I= 1.032; D=0.889 For Ca/Caf case: For MATLAB : P= 4.40; I= 2.328; D=2.0064 Problem 5 10 Caf 1 4 .065 s+ 1 Ca set PID Controller 1 Transfer Fcn Ca model _ hw 9 To Workspace 2 S-Function Tf T 298 Tj 1 0 . 856 s+ 1 T set PID Controller 2 Transfer Fcn 1 Tj To Workspace 1 To Workspace 3 Caf To Workspace PID PID Problem 6 Reactant setpoint +5Kmol/m^3 300 16 295 CAf [kmol/m3] 0 20 40 time [h] 60 14 Tj [K] 290 12 285 10 0 20 40 time [h] 60 304.5 14 304 CA [kmol/m3] 0 20 40 time [h] 60 12 T [K] 303.5 10 303 8 0 20 40 time [h] 60 Reactant setpoint -5kmol/m^3 310 10 CAf [kmol/m3] 0 20 40 time [h] 60 305 8 6 4 2 Tj [K] 300 295 0 20 40 time [h] 60 303.5 10 302.5 302 301.5 CA [kmol/m3] 0 20 40 time [h] 60 303 8 6 4 2 T [K] 0 20 40 time [h] 60 T setpoint +10K 330 320 10 310 300 290 CAf [kmol/m3] 0 20 40 time [h] 60 Tj [K] 9.5 9 0 20 40 time [h] 60 315 9.5 310 CA [kmol/m3] 0 20 40 time [h] 60 9 T [K] 305 8.5 300 8 0 20 40 time [h] 60 T setpoint -10K 300 10 290 CAf [kmol/m3] 0 20 40 time [h] 60 9.5 Tj [K] 280 9 270 8.5 0 20 40 time [h] 60 305 9.5 300 CA [kmol/m3] 0 20 40 time [h] 60 T [K] 9 295 290 8.5 0 20 40 time [h] 60 Tf +10K 300 10 290 CAf [kmol/m3] 0 20 40 time [h] 60 Tj [K] 9.5 280 270 9 0 20 40 time [h] 60 308 9.5 306 CA [kmol/m3] 0 20 40 time [h] 60 T [K] 9 304 302 8.5 0 20 40 time [h] 60 Tf -10K 330 320 10 CAf [kmol/m3] 0 20 40 time [h] 60 9.5 Tj [K] 310 300 290 9 8.5 0 20 40 time [h] 60 304 9.5 302 CA [kmol/m3] 0 20 40 time [h] 60 T [K] 9 300 298 8.5 0 20 40 time [h] 60 330 320 10 CAf [kmol/m3] 0 20 40 time [h] 60 9.5 Tj [K] 310 300 290 9 8.5 0 20 40 time [h] 60 304 9.5 302 CA [kmol/m3] 0 20 40 time [h] 60 T [K] 9 300 298 8.5 0 20 40 time [h] 60 The control loop pairings and controller performance both seem to work adequately . The setpoint is attained quickly after disturbance. Problem 7. Observability: A=[-1.178 0.0415; 0.837 -1.0091]; C=[1 0]; rank(obsv(A,C)) det(obsv(A,C)) ans = 2 ans = 0.0415 The rank= no. of rows/columns and matrix is not singular. So, the system is observable. Prblem 8. Desired characteristic equation: Characteristic equation Matching two equations- Problem 9 Gamma =5 , Tj +5K 306 temperature [K] 305 T T estimate 304 303 0 5 10 15 time [h] 20 25 30 concentration [kmol/m 3] 9.5 9.45 9.4 9.35 9.3 CA CA estimate 0 5 10 15 time [h] 20 25 30 Gamma =5 , Tj -5K 304 temperature [K] 303 T T estimate 302 301 0 5 10 15 time [h] 20 25 30 concentration [kmol/m 3] 9.5 CA CA estimate 9.45 9.4 0 5 10 15 time [h] 20 25 30 For gamma=1, Tj+5K 306 temperature [K] 305 T T estimate 304 303 0 5 10 15 time [h] 20 25 30 concentration [kmol/m 3] 9.6 CA 9.5 CA estimate 9.4 9.3 0 5 10 15 time [h] 20 25 30 For gamma=1 , Tj-5K 304 temperature [K] 303 T T estimate 302 301 0 5 10 15 time [h] 20 25 30 concentration [kmol/m 3] 9.5 9.45 9.4 9.35 9.3 CA CA estimate 0 5 10 15 time [h] 20 25 30 The estimated temperature fairly matches with actual temperature, but estimated concentration and actual concentration do not match. ...
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This document was uploaded on 02/07/2011.

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