Trying to calculate the temperature from the equilibrium heat equation

Hello everyone,

I am trying to calculate the temperature from the equation :

The state of each parameter:

m = const.

= const.

=const.

= vector of n constants (length = n)

n=const

T = variable/vector (length = n)

How will you find the vector T?

Thanks !!!!

Answer 1

[Sam Chak]

Hi @tom shneor

Edit: This is a little difficult for me to imagine since the values of the parameters are not provided. assuming that m, Cp, Qemit are all ones, create and save the odefcn.m function m-file:

function dTdt = odefcn(t, T, Qtot)

dTdt = (- 1*T^4 + Qtot)/(1*1);

end

Run at the Command Window to create the parameters needed for the ODE solver

Qtot = 1:5;
Tfinal = 10;		        % final simulation time
Ts = Tfinal/length(Qtot);	% uniform time interval
T0 = 0;                     % initial Temperature value

Create a for loop to pass the value of Qtot at certain time intervals for ode45 to solve and plot T vs. t continuously at each time interval.

for k = 1:length(Qtot)
    tspan = [Ts*(k-1) Ts*k];
    [t, T] = ode45(@(t, T) odefcn(t, T, Qtot(k)), tspan, T0); 
    T0    = T(end);         % last value of T assigned as new initial value for next interval
    plot(t, T, 'linewidth', 1.5, 'b-')
    hold on                 % hold current figure to continue plotting for the next interval
end
hold off

Display axes grid lines and add axis labels for informative reasons.

grid on
xlabel('t', 'fontsize', 14)
ylabel('T', 'fontsize', 14)

[tom shneor]

hi, thanks!

The only problem is T(t) needs to be calculated for Q(t), because Q is a vector with the same length as T.

How do you do that?

[tom shneor]

And what does "0:0.1:10" in the ode45 means? t vec needs to be in second:

t = [0,1,2,3,4...]

Thanks again!!

[Sam Chak]

@tom shneor, It creates 0.1s-interval-spaced vector for time t from 0 to 10 seconds.

If you want a 1s-interval-spaced vector, then set 0:1:10, but the simulation may inaccurate due to high nonlinearity of .

I overlooked the vectorized . How does this term affect the temperature dynamics?

Is a discrete parameter? Or a piecewise linear function described by intervals?

[tom shneor]

ohh ok thanks.

And do you have a solution for the problem in my first commet?

[tom shneor]

The system is satellite around the moon.

Qtot contains the heat the satellite absorb from inside (heat being emitted by the inner component) and from outside(solar radiation, IR from the moon and albedo).

The satellite uses radiator to emit the Qtot.

Qtot change between day and night. In the figure you can see the heat that the satellite absorb during 10 days (It takes 7620 sec to orbit the moon for the satellite)

[Sam Chak]

Hi @tom shneor

Thanks for the clarification. Say if has 11 discrete values, which a vector, then it is possible to "force" the time vector t to have the same length of .

t = linspace(tStart, tFinal, length(Qtot));

However, after seeing the image, I think is considered a function of time, as measured continuously from the 10-day data shown in the image. Once you loaded the data into the Workspace, you can solve the ODE with the time-dependent term by following the example shown on this link:

https://www.mathworks.com/help/matlab/ref/ode45.html#bu3l43b

This idea is to use interpolation interp1(). Since is fairly linear, I think the interpolated result is acceptable.

https://www.mathworks.com/help/matlab/ref/interp1.html

[tom shneor]

Hi can you please write the code with the "force" of the time vector t to have the same length of Qtot?

Does the ODE needs to be in for loop?

[Sam Chak]

Hi @tom shneor

Since you are hesitate to provide at least some sample values, I have edited my Answer and tried to make the code as informative as possible. But that is not the way I normally run simulation due to the lack of the time-dependent data information about Qtot. Anyway, hope it helps.

[Torsten]

time = 0:50;
Qtot = 10:10:510;
tfinal = 50;		        
T0 = 0;                     
fun = @(t,T) (- 1*T^4 + interp1(time,Qtot,t))/(1*1);
tspan = [0 tfinal];
[t,T] = ode45(fun,tspan,T0); 
plot(t,T)