CHE415 : MATLAB assignment (: B)
Consider the second-order exothermic reaction a????b is taking place in the CSTR below. The dynamic model for this process is given as follows:
Nominal process data:
V = 7.08 m3
ρ = 19.2 kgmole/m3
Cp = 1.815E+5 J/kgmole-C
A = 5.4 m2
ko = 0.0744 m3/kgmole-s
ΔHr = −9.86E+7 J/kgmole
U = 3550 J/s-m2-C
E = 1.182E+7 J/kgmole
Fc = 7.392E−3 m3/s
Tc = 27 oC
Ti = 66 oC
CAi = 2.88 kgmole/m3
F = 7.5E-3 m3/s
R = 8314.39 J/kgmole-K
Find the steady state values, i.e., CAss and Tss, using FSOLVE for given operating condition.
Linearize the model and develop the transfer function
Find the dynamic simulation of the states for step change in the nominal value of each of the process inputs, Tc, CAi. Take CA(0) = CAss and T(0) = Tss.
Based on step test in part 3, find the transfer function between CA and CAi and Tc.
Considering Tc is manipulated variable and CAi is disturbance, drive the closed-loop transfer function for CA using proportional controller and the following data:
The concentration transmitter has an input range of 0~5 kmole/m3 and output range of 4~20 mA. Its time constant is 1s.
The control valve has input range of 4~20 mA and output range of 10~100 oC and time constant of 0.5s.
Find the ultimate controller gain. Calculate the PID settings using the Ziegler-Nicholas tuning method.
Simulate the resulting closed-loop transfer function for a setpoint change of 0.5 kmole/m3 in CAi.
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Question # 01
The steady state CSTR modelled with cooling can be written as:
Here, the unknowns are the operating Temperature (T) and the concentration inside the reactor Using FSOLVE in Matlab, the solution is found to be
The Matlab Code used for this is:
%% ques1
fun = @reactor;
x0 = [2, 100] %first value is Conc and 2nd is Temp
x = fsolve(fun, x0)
function F = reactor(x)
V = 7.08; rho = 19.2; Cp = 1.815*10^5; A = 5.4; ko = 0.0744; dHr = -9.86*10^7; U = 3550;
E = 1.182*10^7; Fc = 7.392*10^(-3); Tc = 27; Ti = 66; CAi = 2.88; F = 7.5*10^(-3); R = 8314.39;
F(1) = F/V*(CAi – x(1))-ko*exp(-E/(R*(x(2)+273)))*x(1)^2
F(2) = F/V*(Ti-x(2))-dHr/(Cp*rho)*ko*exp(-E/(R*(x(2)+273)))*x(1)^2-U*A/(V*rho*Cp)*(x(2)-Tc)
end
Question # 02
Now, we are asked to find out the transfer functions. First we will linearize the ODEs and then take its Laplace transform. Linearization is done as follows
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