Heat Transfer: Matlab 2D Conduction Question

Question

Charles 2012-3-27

4
链接

此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/33651-heat-transfer-matlab-2d-conduction-question

评论： ARJUN MODIA 2020-6-23

1. The problem statement, all variables and given/known data

Having trouble with code as seen by the gaps left where it asks me to add things like the coefficient matrices. Any Numerical Conduction matlab wizzes out there?

A long square bar with cross-sectional dimensions of 30 mm x 30 mm has a specied temperature on each side, The temperatures are:

Tbottom = 100 C
Ttop = 150 C
Tleft = 250 C
Tright = 300 C

Assuming isothermal surfaces, write a software program to solve the heat equation to determine the two-dimensional steady-state spatial temperature distribution within the bar. Your analysis should use a finite difference discretization of the heat equation in the bar to establish a system of equations:

2. Relevant equations

AT = C

Must use Gauss-Seidel Method to solve the system of equations

3. The attempt at a solution

    clear all
  close all
  %Specify grid size
  Nx = 10;
  Ny = 10;
  %Specify boundary conditions
  Tbottom = 100
  Ttop = 150
  Tleft = 250
  Tright = 300
  % initialize coefficient matrix and constant vector with zeros
  A = zeros(Nx*Ny);
  C = zeros(Nx*Ny,1);
  % initial 'guess' for temperature distribution
  T(1:Nx*Ny,1) = 100;
  % Build coefficient matrix and constant vector
  % inner nodes
  for n = 2:(Ny-1)
  for m = 2:(Nx-1)
  i = (n-1)*Nx + m;
  A(i,i+Nx) = 1;
  A(i,i-Nx) = 1;
  A(i,i+1) = 1;
  A(i,i-1) = 1;
  A(i,i) = -4;
  end
  end
  % Edge nodes
  % bottom
  for m = 2:(Nx-1)
  %n = 1
  i = m;
A(i,i+Nx) = 1;
A(i,i+1) = 1;
A(i,i-1) = 1;
A(i,i) = -4;
C(i) = -Tbottom;
end
%top:
for m = 2:(Nx-1)
% n = Ny
i = (Ny-1)*Nx + m;
A(i,i-Nx) = 1;
A(i,i+1) = .5;
A(i,i-1) = .5;
A(i,i) = -5;
C(i) = -Ttop;
end
%left:
for n=2:(Ny-1)
%m = 1
i = (n-1)*Nx + 1;
A(i,i+Nx) = .5;
A(i,i+1) = 1;
A(i,i-Nx) = .5;
A(i,i) = -2;
end
%right:
for n=2:(Ny-1)
%m = Nx
i = (n-1)*Nx + Nx;
A(i,i+Nx) = 1;
A(i,i-1) = 1;
A(i,i-Nx) = 1;
A(i,i) = -4;
C(i) = -Tright;
DEFINE COEFFICIENT MATRIX AND CONSTANT VECTOR ELEMENTS HERE
end
% Corners
%bottom left (i=1):
i=1
A(i,Nx+i) = 1;
A(i,2) = 1;
A(i,1) = -4;
C(i) = -(Tbottom + Tleft);
%bottom right:
i = Nx
A(i,Nx+i) = 1;
A(i,2) = 1;
A(i,1) = -4;
C(Nx) = -(Tbottom + Tright);
%top left:
i = (Ny-1)*Nx + 1;
A(i,i+1) = .5
A(i,i) = 
%top right:
i = Nx*Ny;
DEFINE COEFFICIENT MATRIX AND CONSTANT VECTOR ELEMENTS HERE
%Solve using Gauss-Seidel
residual = 100;
iterations = 0;
while (residual > 0.0001) % The residual criterion is 0.0001 in this example
7% You can test different values
iterations = iterations+1
%Transfer the previously computed temperatures to an array Told
Told = T;
%Update estimate of the temperature distribution
INSERT GAUSS-SEIDEL ITERATION HERE
%compute residual
deltaT = abs(T - Told);
residual = max(deltaT);
end
iterations % report the number of iterations that were executed
%Now transform T into 2-D network so it can be plotted.
delta_x = 0.03/(Nx+1)
delta_y = 0.03/(Ny+1)
for n=1:Ny
for m=1:Nx
i = (n-1)*Nx + m;
T2d(m,n) = T(i);
x(m) = m*delta_x;
y(n) = n*delta_y;
end
end
T2d
surf(x,y,T2d)
figure
contour(x,y,T2d)

10 个评论
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Jonathan Ayala 2019-11-14

Hello, any luck with modifying the code with values for K and h? I am working on a similar problem, Thanks.

ARJUN MODIA 2020-6-23

If we work on steady-state problem then the values of h & k will not affect your results. @ Jonathan and @ Bimal.

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Answer 1

Charles 2012-3-28

0
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/33651-heat-transfer-matlab-2d-conduction-question#answer_42305

在 MATLAB Online 中打开

clear all
close all
%Specify grid size
Nx = 10;
Ny = 10;
%Specify boundary conditions
Tbottom = 100
Ttop = 150
Tleft = 250
Tright = 300
% initialize coefficient matrix and constant vector with zeros
A = zeros(Nx*Ny);
C = zeros(Nx*Ny,1);
% initial 'guess' for temperature distribution
T(1:Nx*Ny,1) = 100;
% Build coefficient matrix and constant vector
% inner nodes
for n = 2:(Ny-1)
for m = 2:(Nx-1)
i = (n-1)*Nx + m;
A(i,i+Nx) = 1;
A(i,i-Nx) = 1;
A(i,i+1) = 1;
A(i,i-1) = 1;
A(i,i) = -4;
end
end
% Edge nodes
% bottom
for m = 2:(Nx-1)
%n = 1
i = m;
A(i,i+Nx) = 1;
A(i,i+1) = 1;
A(i,i-1) = 1;
A(i,i) = -4;
C(i) = -Tbottom;
end
%top:
for m = 2:(Nx-1)
% n = Ny
i = (Ny-1)*Nx + m;
A(i,i-Nx) = 1;
A(i,i+1) = 1;
A(i,i-1) = 1;
A(i,i) = -4;
C(i) = -Ttop;
end
%left:
for n=2:(Ny-1)
%m = 1
i = (n-1)*Nx + 1;
A(i,i+Nx) = 1;
A(i,i+1) = 1;
A(i,i-Nx) = 1;
A(i,i) = -4;
end
%right:
for n=2:(Ny-1)
%m = Nx
i = (n-1)*Nx + Nx;
A(i,i+Nx) = 1;
A(i,i-1) = 1;
A(i,i-Nx) = 1;
A(i,i) = -4;
C(i) = -Tright;
end
% Corners
%bottom left (i=1):
i=1;
A(i,Nx+i) = 1;
A(i,2) = 1;
A(i,1) = -4;
C(i) = -(Tbottom + Tleft);
%bottom right:
i = Nx;
A(i,i+Nx) = 1;
A(i,i-1) = 1;
A(i,i) = -4;
C(i) = -(Tbottom + Tright);
%top left:
i = (Ny-1)*Nx + 1;
A(i,i+1) = 1;
A(i,i) = -4;   
A(i,i-Nx) = 1;
C(i) = -(Ttop + Tleft);
%top right:
i = Nx*Ny;
A(i,i-1) = 1;
A(i,i) = -4;
A(i,i-Nx) = 1;
C(i) = -(Tright + Ttop);
%Solve using Gauss-Seidel
residual = 100;
iterations = 0;
while (residual > 0.0001) % The residual criterion is 0.0001 in this example
% You can test different values
iterations = iterations+1;
%Transfer the previously computed temperatures to an array Told
Told = T;
%Update estimate of the temperature distribution
for n=1:Ny
for m=1:Nx
i = (n-1)*Nx + m;
Told(i) = T(i);
end
end
% iterate through all of the equations
for n=1:Ny
for m=1:Nx
i = (n-1)*Nx + m;
%sum the terms based on updated temperatures
sum1 = 0;
for j=1:i-1
sum1 = sum1 + A(i,j)*T(j);
end
%sum the terms based on temperatures not yet updated
sum2 = 0;
for j=i+1:Nx*Ny
sum2 = sum2 + A(i,j)*Told(j);
end
% update the temperature for the current node
T(i) = (1/A(i,i)) * (C(i) - sum1 - sum2);
end
end
residual = max(T(i) - Told(i));
end
%compute residual
deltaT = abs(T - Told);
residual = max(deltaT);
iterations; % report the number of iterations that were executed
%Now transform T into 2-D network so it can be plotted.
delta_x = 0.03/(Nx+1);
delta_y = 0.03/(Ny+1);
for n=1:Ny
for m=1:Nx
i = (n-1)*Nx + m;
T2d(m,n) = T(i);
x(m) = m*delta_x;
y(n) = n*delta_y;
end
end
T2d;
surf(x,y,T2d)
figure
contour(x,y,T2d)

3 个评论
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Frank Mota 2017-11-4

the code wont run for me , is there something im doing wrong?

Max 2017-12-7

Charles: It's been a while since you posted this, and perhaps you've found the error - but you forgot to include the C vector update in your left edge, which is why the distribution looked odd!

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