Here is the scenario, a stirred tank reactor that has constant input and output rates that will change at some time t to different input and output rates that will mimic the following curve:
clear;clc;
A=24;
B=6;
alpha=1.5;
beta=0.25;
t=0:.5:30;
Cp=A*exp(-alpha*t)+B*exp(-beta*t);
semilogy(t,Cp,'o')
using these equation that can be optimized using least squares regression:
Cplin1=Ao*((ri-ro)*t+Vo).^(-ri./(ri-ro))*Vo.^(ro./(ri-ro));
Cplin2=Ao1*((ri1-ro1)*t+Vo1).^(-ri1./(ri1-ro1))*Vo1.^(ro1./(ri1-ro1));
where: Ao and Vo in this example are 300 and 10, respectively; Ao1 and Vo1 are determined at some time t from Cplin1; the t in Cplin2 most likely will require a phase shift; and using a gridded search (because that is what I'm most comfortable doing, but I'd be fine with another method if you wrote the code) to find ro, ri, ro1, ri1 and t.
here is what I have, but it doesn't run:
clear;clc;
Ao=300;
Vo=10;
t12=3;
k=log(2)/t12;
A=24;
B=6
alpha=1.5
beta=0.25
ri=1;
ro=.5;
t=0:.5:30;
phaseTime=t;
Cp=A*exp(-alpha*t)+B*exp(-beta*t);
semilogy(t,Cp,'o')
SSBest=10000
Vo1=Vo
Ao1=Ao
for ri=10:1:20;
for ro=10:1:20;
for ri1=10:1:20;
for ro1=10:1:20;
if (ri*t-ro*t+Vo)>0
if (ri1*t-ro1*t+Vo1)>0
Cplin1=Ao*((ri-ro)*phaseTime+Vo).^(-ri./(ri-ro))*Vo.^(ro./(ri-ro));
Ao1=Ao*((ri-ro)*phaseTime+Vo).^(-ri./(ri-ro))*Vo.^(ro./(ri-ro))*((ri-ro)*phaseTime+Vo);
Vo1=((ri-ro)*phaseTime+Vo);
Cplin2=Ao1*((ri1-ro1)*phaseTime+Vo1).^(-ri1./(ri1-ro1))*Vo1.^(ro1./(ri1-ro1));
Cplin=[Cplin1(1:4) Cplin2(5:end)];
SS=sum((Cp-Cplin).^2);
if (SS<SSBest)
RinBest=ri;
RoutBest=ro;
Rin1Best=ri1;
Rout1Best=ro1;
Ao1Best=Ao1;
Vo1Best=Vo1;
CplinBest=Cplin;
disp([ri,ro,ri1,ro1,SS])
SSBest=SS;
end
end
end
end
end
end
end
Cplin=Ao*(RinBest*t-RoutBest*t+Vo).^(-RinBest./(RinBest-RoutBest))*Vo.^(RoutBest./(RinBest-RoutBest));
hold on
plot(t,CplinBest)
hold off
