Using Symbolic Variable in ODE45 without Separate Function and other advice

Hi,

So, I'm trying to evaluate a system of ODEs

I can replace dC with the reaction rate and arrive upon:

At this point, we're unable to solve the problem analytically.

First, I'd like to know how to solve this using ODE45. I have attempted this using the following code:

syms T t
% constants
Tm = 300;     %K
Cao = 2;      %mol/L
Cpa = 20;     %cal/mol-K
Hr = -10000;  %cal/mol
EaR = 2660;   %K
km = 0.01725; %L/mol-min
Ta = 300;     %K
UAVr = 0.1;   %cal/mol-L-K
% system equations
k = km*exp(-EaR*(1/T-1/Tm))
Ca = Cao/(Cao*k*t + 1)
% ODE
dT =  Hr*k*Ca^2/(2*Cao*Cpa) + UAVr/(2*Cao*Cpa)*(Ta-T)
tspan=[0 30];
% ODE solver
[t,T] = ode45(@(t,T)dT,tspan,300)

The problem is that dT is a sym variable, I understand that. I don't understand how I can turn it into something that ODE45 will recognize.

I believe I could just paste the equation output of dT, but that doesn't help the next step. It needs to be something that can take the characters from the output of dT and turn it into a function handle. I've tried multiple methods, but it seems I fundamentally don't understand the code.

The ultimate goal of this code is to determine a value of such that

The behavior of T is that is trends to a max and then trails off, but the location of the max changes dramatically depending on the value of . I was planning to run a loop until this was true. I'm sure there are better ways. Any ideas?

Answer 1

[Walter Roberson]

dT =  matlabFunction(Hr*k*Ca^2/(2*Cao*Cpa) + UAVr/(2*Cao*Cpa)*(Ta-T), 'vars', {t, T});

and

[t,T] = ode45(dT,tspan,300);