Having trouble modelling 2-dimensional transient heat transfer

Question

Mark McNallly 2013-2-12

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I'm having trouble modelling 2-dimensional transient heat transfer in MATLAB. I've reduced the problem down to a very simple model, simply an insulated box split into a 2x2 matrix with a heat generation term inside, but the values I'm getting out are way off what should be correct. The code makes use of the ODE45 command in MATLAB. This is the code I'm using, any help is much appreciated! The version of MATLAB that I'm using is 7.10 (R2010a)

This is the code I'm using:

Main file

%%Basic 2 dimensional heat transfer code
clear all
close
clc
global l h k mass spec_heat T_amb m n z
% Box parameters
l = 1; % length
h = 1; % height
k = 0.2; % W/m.K
mass = 100; % kg
spec_heat = 4000; % J/kg
% Initial conditions
T_amb = 288; % Kelvin
% Number of elements
m = 2; % x direction
n = 2; % y direction
z = m*n; % total number of elements
T0 = zeros(m*n, 1);
for i = 1:(m*n)
  T0(i,1) = T_amb; % T0(i,1) is initial temp of the material in K
end
tdur = 3456000;                         % Duration of simulation in seconds (s).
tstep = 3600;                           % Time step (of 1 hour) in seconds.
tspan = [0:tstep:tdur];  
[t, T] = ode45(@twod_function, tspan, T0);
plot (t,T(:,1),'-r',t,T(:,2),'b',t,T(:,3),'-y',t,T(:,4),'--')

And for the function file

function dTdt = twod_function(t,T)
global l h k mass spec_heat T_amb m n z
R = (l/m)/(k*l*h); % resistance of material in box
Q_gen = 50;
 Q_cond = zeros(1,(m*n));
 for i = 1:(m*n)
    if i == 1
        Q_cond(i) = (T(i) - 288)/R;
    else
    Q_cond(i) = (T(i)-T(i-1))/R; % heat transfer by conduction
    end
end
 dTdt = zeros(4,1);
 for i = 1:(m*n)
    if i == 1
        dTdt(i,1) = (Q_gen - Q_cond(i+1) - Q_cond(i+2))/((1/m)*mass*spec_heat);
    elseif i == 2
        dTdt(i,1) = (Q_cond(i-1) - Q_cond(i+2))/((1/m)*mass*spec_heat);
    elseif i == 3
        dTdt(i,1) = (Q_cond(i-2) - Q_cond(i+1))/((1/m)*mass*spec_heat);
    else
        dTdt(i,1) = (Q_cond(i-2) + Q_cond(i-1))/((1/m)*mass*spec_heat);
    end
 end
 end

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Youssef Khmou 2013-2-12

Hi , i tried your code, but it gives the same oscillatory Temp in the end, i think you should work on a 2d matrix , can you ?

Youssef Khmou 2013-2-12

在 MATLAB Online 中打开

try this as preliminary test :

 surf(10*log10(abs(T)))
 >> shading interp

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Answer 1

Youssef Khmou 2013-2-12

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/63234-having-trouble-modelling-2-dimensional-transient-heat-transfer#answer_74842

编辑：Youssef Khmou 2013-2-12

在 MATLAB Online 中打开

hi, please, change the duration as :

tspan=[0:4:7200];

and after running the file, type in the command prmpt :

surf(T);

Does that mesh make sens as Decreasing Gradient of Temperature ??

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Mark McNallly 2013-2-13

Oh wait, I read your reply as 2nd, not 2-d. I will try it with a 2-d matrix.

Youssef Khmou 2013-2-13

hi, Mark,

yes you try per example initializing a 2D T matrix , and put the equation you want to solve as Laplace(T) = constant ( just as example, you put those convection and conduction equations in general form ) , next it will be easy to write a function that differentiate the 2D T .

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