How can I integrate a set of ODEs given that my initial conditions also change (linearly) within the concerned time span?

[Torsten]

Please show us the set of ODEs you refer to in the title.

Best wishes

Torsten.

[Maneet Goyal]

Please have a look at the following images:

Answer 1

[Torsten]

function driver

tstart = 0.0;
tend = 1.0;
flag = 1;
[T1 Y1] =ode45(@(t,y)myfun(t,y,flag),[tstart tend],[0.01;600]);

tstart = 1.0;
tend = 2.0;
flag = 2 ;
[T2 Y2] = ode45(@(t,y)myfun(t,y,flag),[tstart tend],[Y1(end,1);Y1(end,2)]);

tstart = 2.0;
tend = 10.0;
flag = 3 ;
[T3 Y3] = ode45(@(t,y)myfun(t,y,flag),[tstart tend],[Y2(end,1);Y2(end,2)]);

function myfun(t,y,flag) 

if flag==1 
  yin = 0.01;
  Tin = 600;
elseif flag==2
  yin = 0.01+(t-1)*0.02;
  Tin = 600+(t-1)*100;
elseif flag==3
  yin = 0.03;
  Tin = 700;
end 

...

Best wishes

Torsten.

[Maneet Goyal]

Thanks a lot Torsten. Function Driver:

tstart = 0;
tend = 1.0;
flag = 1;
[T1, Y1] =ode15s(@(t,y) trancstr(t,y,flag),[tstart tend],[0.01;600]);
tstart = 1.0;
tend = 2.0;
flag = 2 ;
[T2, Y2] = ode15s(@(t,y) trancstr(t,y,flag),[tstart tend],[Y1(end,1);Y1(end,2)]);
tstart = 2.0;
tend = 3.0;
flag = 3 ;
[T3, Y3] = ode15s(@(t,y) trancstr(t,y,flag),[tstart tend],[Y2(end,1);Y2(end,2)]);
t = [T1;T2;T3];
yo = [Y1;Y2;Y3];
plot(t,yo)
grid on

The function used is:

function out = trancstr(t,x,flag)
if flag == 1 
yin = 0.01;
Tin = 600;
elseif flag == 2
yin = 0.01+(t-1)*0.02;
Tin = 600+(t-1)*100;
elseif flag == 3
yin = 0.03;
Tin = 700;
end 
y = x(1);
T = x(2);
% Constants
Cpg = 1070; % J/kgK
Cps = 1000; % J/kgK
epsi = 0.68;
Q = 0.06555; % m3/s
Ctot = 0.0203; % kmol/m3
alpha = 26900; % m2/m3
V = 6*1e-4; % m3
Mcap = 30; % kg/kmol
enthalpy = 2.84*1e8; % J/kmol
rhog = Ctot*Mcap; % kg/m3
rhos = 2500; % kg/m3
% Other Parameters
k1 = 6.70*1e10*exp(-12556/T);
K1 = 65.5*exp(961/T);
r = (0.05*k1*y)/(T*(1+(K1*y))^2);
% Output
out(1,1) = ((Q*Ctot*(yin-y)) - (alpha*V*r))/(epsi*V*Ctot);
out(2,1) = ((Q*Ctot*Mcap*Cpg*(Tin-T)) + (alpha*V*enthalpy*r))/(V*((epsi*rhog*Cpg)+((1-epsi)*rhos*Cps)));
end

The solver was taking too much time, so I moved to ode15s. The system is, in fact, stiff (Very fast dynamics coupled with very slow dynamics). As the reaction proceeds, the concentration keeps on changing before reaching a steady state and subsequently, heat is generated (it's an exothermic reaction). But this heat is low and is not able to bring a big change to the temperature of the mixture. Hence,the rate of change is concentration is very fast compared to the rate of change in temperature due to the large heat capacity of the reaction mixture.