Notes
Notes - notes.io |
clear all
close all
%k=0.011; % conductivity W/(cm*K)
h = 0.002; % heat transfer coefficient W/(cm^2*K)
qs=0.2; % heat flux W/cm^2
T_amb = 23; % ambient temperature C
Ly = 0.5; % height cm
Lx = 1.5; % width cm
Lz = 3.0; % depth cm
a = 0.25; % cm
N = 39; % number of pieces for both axis
dx = 2*Lx/N; % grid size for x
dy = Ly/N; % grid size for y
dz = Lz/N; % grid size for z
%Number of qs points
%G = floor((Lx+a)/m+1)-ceil((Lx-a)/m+1)+1;
%qs=ones(G,1)*0.2; % heat flux W/cm^2
k = 0.011*ones((N+1)^3,1);
A = zeros((N+1)^3,(N+1)^3); % coefficient matrix
T_dist = zeros((N+1),(N+1),(N+1)); % matrix for plotting temperature distribution
b = zeros((N+1)^2,1);
%%%%% Interior Points %%%%%
for r=2:N
for j=2:N
for i=2:N
p = (r-1)*(N+1)^2 + (j-1)*(N+1) + i;
A(p,p) = -2*k(p)*(k(p+1)/(k(p)+k(p+1))/dx^2 + k(p-1)/(k(p)+k(p-1))/dx^2 + ...
k(p+N+1)/(k(p)+k(p+N+1))/dy^2 + k(p-N-1)/(k(p)+k(p-N-1))/dy^2 + ...
k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2 + k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2);
A(p,p+1) = 2*k(p)*k(p+1)/(k(p)+k(p+1))/dx^2;
A(p,p-1) = 2*k(p)*k(p-1)/(k(p)+k(p-1))/dx^2;
A(p,p+N+1) = 2*k(p)*k(p+N+1)/(k(p)+k(p+N+1))/dy^2;
A(p,p-N-1) = 2*k(p)*k(p-N-1)/(k(p)+k(p-N-1))/dy^2;
A(p,p+(N+1)^2) = 2*k(p)*k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2;
A(p,p-(N+1)^2) = 2*k(p)*k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2;
b(p,1)=0;
end
end
end
%%%%% Boundary Conditions on Edges %%%%%
i=1;
r=1;
for j=2:N % at x=0 z=0 %
p = (r-1)*(N+1)^2 + (j-1)*(N+1) + i;
A(p,p) = -k(p)*(k(p+1)/(k(p)+k(p+1))/dx^2 + k(p+N+1)/(k(p)+k(p+N+1))/dy^2 + ...
k(p-N-1)/(k(p)+k(p-N-1))/dy^2 + k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2);
A(p,p+1)=2*k(p)*k(p+1)/(k(p)+k(p+1))/dx^2;
A(p,p+N+1)=k(p)*k(p+N+1)/(k(p)+k(p+N+1))/dy^2;
A(p,p-N-1)=k(p)*k(p-N-1)/(k(p)+k(p-N-1))/dy^2;
A(p,p+(N+1)^2) = 2*k(p)*k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2;
b(p,1)=0;
end
i=N+1;
r=1;
for j=2:N % at x=2Lx z=0 %
p = (r-1)*(N+1)^2 + (j-1)*(N+1) + i;
A(p,p) = -k(p)*(2*k(p-1)/(k(p)+k(p-1))/dx^2 + k(p+N+1)/(k(p)+k(p+N+1))/dy^2 + ...
k(p-N-1)/(k(p)+k(p-N-1))/dy^2 + 2*k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2);
A(p,p-1)=2*k(p)*k(p-1)/(k(p)+k(p-1))/dx^2;
A(p,p+N+1)=k(p)*k(p+N+1)/(k(p)+k(p+N+1))/dy^2;
A(p,p-N-1)=k(p)*k(p-N-1)/(k(p)+k(p-N-1))/dy^2;
A(p,p+(N+1)^2) = 2*k(p)*k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2;
b(p,1)=0;
end
i=1;
r=N+1;
for j=2:N % at x=0 z=Lz %
p = (r-1)*(N+1)^2 + (j-1)*(N+1) + i;
A(p,p) = -k(p)*(2*k(p+1)/(k(p)+k(p+1))/dx^2 + k(p+N+1)/(k(p)+k(p+N+1))/dy^2 + ...
k(p-N-1)/(k(p)+k(p-N-1))/dy^2 + 2*k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2);
A(p,p+1)=2*k(p)*k(p+1)/(k(p)+k(p+1))/dx^2;
A(p,p+N+1)=k(p)*k(p+N+1)/(k(p)+k(p+N+1))/dy^2;
A(p,p-N-1)=k(p)*k(p-N-1)/(k(p)+k(p-N-1))/dy^2;
A(p,p-(N+1)^2) = 2*k(p)*k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2;
b(p,1)=0;
end
i=N+1;
r=N+1;
for j=2:N % at x=2Lx z=Lz %
p = (r-1)*(N+1)^2 + (j-1)*(N+1) + i;
A(p,p) = -k(p)*(2*k(p-1)/(k(p)+k(p-1))/dx^2 + k(p+N+1)/(k(p)+k(p+N+1))/dy^2 + ...
k(p-N-1)/(k(p)+k(p-N-1))/dy^2 + 2*k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2);
A(p,p-1)=2*k(p)*k(p-1)/(k(p)+k(p-1))/dx^2;
A(p,p+N+1)=k(p)*k(p+N+1)/(k(p)+k(p+N+1))/dy^2;
A(p,p-N-1)=k(p)*k(p-N-1)/(k(p)+k(p-N-1))/dy^2;
A(p,p-(N+1)^2) = 2*k(p)*k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2;
b(p,1)=0;
end
i=1;
j=N+1;
for r=2:N % at x=0 y=Ly %
p = (r-1)*(N+1)^2 + (j-1)*(N+1) + i;
A(p,p) = -k(p)*(2*k(p+1)/(k(p)+k(p+1))/dx^2 + 2*k(p-N-1)/(k(p)+k(p-N-1))/dy^2 + ...
k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2+k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2)-h/dy;
A(p,p+1)=2*k(p)*k(p+1)/(k(p)+k(p+1))/dx^2;
A(p,p-N-1)=2*k(p)*k(p-N-1)/(k(p)+k(p-N-1))/dy^2;
A(p,p+(N+1)^2) = k(p)*k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2;
A(p,p-(N+1)^2) = k(p)*k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2;
b(p,1)=0;
end
i=N+1;
j=N+1;
for r=2:N % at x=2Lx y=Ly %
p = (r-1)*(N+1)^2 + (j-1)*(N+1) + i;
A(p,p) = -k(p)*(2*k(p-1)/(k(p)+k(p-1))/dx^2 + 2*k(p-N-1)/(k(p)+k(p-N-1))/dy^2 + ...
k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2+k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2)-h/dy;
A(p,p-1)=2*k(p)*k(p-1)/(k(p)+k(p-1))/dx^2;
A(p,p-N-1)=2*k(p)*k(p-N-1)/(k(p)+k(p-N-1))/dy^2;
A(p,p+(N+1)^2) = k(p)*k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2;
A(p,p-(N+1)^2) = k(p)*k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2;
b(p,1)=0;
end
j=N+1;
r=1;
for i=2:N % at z=0 y=Ly %
p = (r-1)*(N+1)^2 + (j-1)*(N+1) + i;
A(p,p) = -k(p)*(k(p-1)/(k(p)+k(p-1))/dx^2 + 2*k(p-N-1)/(k(p)+k(p-N-1))/dy^2 + ...
k(p+1)/(k(p)+k(p+1))/dz^2+2*k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2)-h/dy;
A(p,p-1)=k(p)*k(p-1)/(k(p)+k(p-1))/dx^2;
A(p,p+1)=k(p)*k(p+1)/(k(p)+k(p+1))/dx^2;
A(p,p-N-1)=2*k(p)*k(p-N-1)/(k(p)+k(p-N-1))/dy^2;
A(p,p+(N+1)^2) = 2*k(p)*k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2;
b(p,1)=0;
end
j=N+1;
r=N+1;
for i=2:N % at z=Lz y=Ly %
p = (r-1)*(N+1)^2 + (j-1)*(N+1) + i;
A(p,p) = -k(p)*(k(p-1)/(k(p)+k(p-1))/dx^2 + 2*k(p-N-1)/(k(p)+k(p-N-1))/dy^2 + ...
k(p+1)/(k(p)+k(p+1))/dz^2+2*k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2)-h/dy;
A(p,p-1)=k(p)*k(p-1)/(k(p)+k(p-1))/dx^2;
A(p,p+1)=k(p)*k(p+1)/(k(p)+k(p+1))/dx^2;
A(p,p-N-1)=2*k(p)*k(p-N-1)/(k(p)+k(p-N-1))/dy^2;
A(p,p-(N+1)^2) = 2*k(p)*k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2;
b(p,1)=0;
end
i=1;
j=1;
for r=2:N % at x=0 y=0 %
p = (r-1)*(N+1)^2 + (j-1)*(N+1) + i;
A(p,p) = -k(p)*(2*k(p+1)/(k(p)+k(p+1))/dx^2 + 2*k(p+N+1)/(k(p)+k(p+N+1))/dy^2 + ...
k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2 + k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2);
A(p,p+1)=2*k(p)*k(p+1)/(k(p)+k(p+1))/dx^2;
A(p,p+N+1)=2*k(p)*k(p+N+1)/(k(p)+k(p+N+1))/dy^2;
A(p,p+(N+1)^2) = k(p)*k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2;
A(p,p-(N+1)^2) = k(p)*k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2;
b(p,1)=-qs/dy;
end
i=N+1;
j=1;
for r=2:N % at x=2Lx y=0 %
p = (r-1)*(N+1)^2 + (j-1)*(N+1) + i;
A(p,p) = -k(p)*(2*k(p-1)/(k(p)+k(p-1))/dx^2 + 2*k(p+N+1)/(k(p)+k(p+N+1))/dy^2 + ...
k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2 + k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2);
A(p,p-1)=2*k(p)*k(p-1)/(k(p)+k(p-1))/dx^2;
A(p,p+N+1)=2*k(p)*k(p+N+1)/(k(p)+k(p+N+1))/dy^2;
A(p,p+(N+1)^2) = k(p)*k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2;
A(p,p-(N+1)^2) = k(p)*k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2;
b(p,1)=-qs/dy;
end
r=1;
j=1;
for i=2:N % at z=0 y=0 %
p = (r-1)*(N+1)^2 + (j-1)*(N+1) + i;
A(p,p) = -k(p)*(k(p+1)/(k(p)+k(p+1))/dx^2 + 2*k(p+N+1)/(k(p)+k(p+N+1))/dy^2 + ...
2*k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2 + k(p-1)/(k(p)+k(p-1))/dz^2);
A(p,p+1)=k(p)*k(p+1)/(k(p)+k(p+1))/dx^2;
A(p,p+N+1)=2*k(p)*k(p+N+1)/(k(p)+k(p+N+1))/dy^2;
A(p,p-1) = k(p)*k(p-1)/(k(p)+k(p-1))/dx^2;
A(p,p+(N+1)^2) = 2*k(p)*k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2;
b(p,1)=-qs/dy;
end
r=N+1;
j=1;
for i=2:N % at z=Lz y=0 %
p = (r-1)*(N+1)^2 + (j-1)*(N+1) + i;
A(p,p) = -k(p)*(k(p+1)/(k(p)+k(p+1))/dx^2 + 2*k(p+N+1)/(k(p)+k(p+N+1))/dy^2 + ...
2*k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2 + k(p-1)/(k(p)+k(p-1))/dz^2);
A(p,p+1)=k(p)*k(p+1)/(k(p)+k(p+1))/dx^2;
A(p,p+N+1)=2*k(p)*k(p+N+1)/(k(p)+k(p+N+1))/dy^2;
A(p,p-1) = k(p)*k(p-1)/(k(p)+k(p-1))/dx^2;
A(p,p-(N+1)^2) = 2*k(p)*k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2;
b(p,1)=-qs/dy;
end
%%%%% Boundary Conditions on Faces %%%%%
r=1;
for i=2:N % at z=0 %
for j=2:N
p = (r-1)*(N+1)^2 + (j-1)*(N+1) + i;
A(p,p) = -k(p)*(k(p+1)/(k(p)+k(p+1))/dx^2 + k(p+N+1)/(k(p)+k(p+N+1))/dy^2 + ...
k(p-1)/(k(p)+k(p-1))/dx^2 + k(p-N-1)/(k(p)+k(p-N-1))/dy^2 + 2*k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2);
A(p,p+1)=k(p)*k(p+1)/(k(p)+k(p+1))/dx^2;
A(p,p-1)=k(p)*k(p-1)/(k(p)+k(p-1))/dx^2;
A(p,p+N+1)=k(p)*k(p+N+1)/(k(p)+k(p+N+1))/dy^2;
A(p,p-N-1)=k(p)*k(p-N-1)/(k(p)+k(p-N-1))/dy^2;
A(p,p+(N+1)^2) = 2*k(p)*k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2;
b(p,1)=0;
end
end
r=N+1;
for i=2:N % at z=Lz %
for j=2:N
p = (r-1)*(N+1)^2 + (j-1)*(N+1) + i;
A(p,p) = -k(p)*(k(p+1)/(k(p)+k(p+1))/dx^2 + k(p+N+1)/(k(p)+k(p+N+1))/dy^2 + ...
k(p-1)/(k(p)+k(p-1))/dx^2 + k(p-N-1)/(k(p)+k(p-N-1))/dy^2 + 2*k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2);
A(p,p+1)=k(p)*k(p+1)/(k(p)+k(p+1))/dx^2;
A(p,p-1)=k(p)*k(p-1)/(k(p)+k(p-1))/dx^2;
A(p,p+N+1)=k(p)*k(p+N+1)/(k(p)+k(p+N+1))/dy^2;
A(p,p-N-1)=k(p)*k(p-N-1)/(k(p)+k(p-N-1))/dy^2;
A(p,p-(N+1)^2) = 2*k(p)*k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2;
b(p,1)=0;
end
end
i=1;
for r=2:N % at x=0 %
for j=2:N
p = (r-1)*(N+1)^2 + (j-1)*(N+1) + i;
A(p,p) = -k(p)*(2*k(p+1)/(k(p)+k(p+1))/dx^2 + k(p+N+1)/(k(p)+k(p+N+1))/dy^2 + ...
k(p-(N+1)^2)/(k(p)-k(p+(N+1)^2))/dz^2 + k(p-N-1)/(k(p)+k(p-N-1))/dy^2 + ...
k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2);
A(p,p+1)=2*k(p)*k(p+1)/(k(p)+k(p+1))/dx^2;
A(p,p+N+1)=k(p)*k(p+N+1)/(k(p)+k(p+N+1))/dy^2;
A(p,p-N-1)=k(p)*k(p-N-1)/(k(p)+k(p-N-1))/dy^2;
A(p,p+(N+1)^2) = k(p)*k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2;
A(p,p-(N+1)^2) = k(p)*k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2;
b(p,1)=0;
end
end
i=N+1;
for r=2:N % at x=2Lx %
for j=2:N
p = (r-1)*(N+1)^2 + (j-1)*(N+1) + i;
A(p,p) = -k(p)*(2*k(p-1)/(k(p)+k(p-1))/dx^2 + k(p+N+1)/(k(p)+k(p+N+1))/dy^2 + ...
k(p-(N+1)^2)/(k(p)-k(p+(N+1)^2))/dz^2 + k(p-N-1)/(k(p)+k(p-N-1))/dy^2 + ...
k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2);
A(p,p-1)=2*k(p)*k(p-1)/(k(p)+k(p-1))/dx^2;
A(p,p+N+1)=k(p)*k(p+N+1)/(k(p)+k(p+N+1))/dy^2;
A(p,p-N-1)=k(p)*k(p-N-1)/(k(p)+k(p-N-1))/dy^2;
A(p,p+(N+1)^2) = k(p)*k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2;
A(p,p-(N+1)^2) = k(p)*k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2;
b(p,1)=0;
end
end
j=1;
for r=2:N % at y=0 %
for i=2:N
p = (r-1)*(N+1)^2 + (j-1)*(N+1) + i;
A(p,p) = -k(p)*(k(p-1)/(k(p)+k(p-1))/dx^2 + k(p+1)/(k(p)+k(p+1))/dx^2 + 2*k(p+N+1)/(k(p)+k(p+N+1))/dy^2 + ...
k(p-N-1)/(k(p)+k(p-N-1))/dy^2 + k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2);
A(p,p-1)=k(p)*k(p-1)/(k(p)+k(p-1))/dx^2;
A(p,p+1)=k(p)*k(p+1)/(k(p)+k(p+1))/dx^2;
A(p,p+N+1)=2*k(p)*k(p+N+1)/(k(p)+k(p+N+1))/dy^2;
A(p,p+(N+1)^2) = k(p)*k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2;
A(p,p-(N+1)^2) = k(p)*k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2;
b(p,1)=-qs/dy;
end
end
j=N+1;
for r=2:N % at y=Ly %
for i=2:N
p = (r-1)*(N+1)^2 + (j-1)*(N+1) + i;
A(p,p) = -k(p)*(k(p-1)/(k(p)+k(p-1))/dx^2 + k(p+1)/(k(p)+k(p+1))/dx^2 + 2*k(p-N-1)/(k(p)+k(p-N-1))/dy^2 + ...
k(p-N-1)/(k(p)+k(p-N-1))/dy^2 + k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2)-h/dy;
A(p,p-1)=k(p)*k(p-1)/(k(p)+k(p-1))/dx^2;
A(p,p+1)=k(p)*k(p+1)/(k(p)+k(p+1))/dx^2;
A(p,p-N-1)=2*k(p)*k(p-N-1)/(k(p)+k(p-N-1))/dy^2;
A(p,p+(N+1)^2) = k(p)*k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2;
A(p,p-(N+1)^2) = k(p)*k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2;
b(p,1)=0;
end
end
%%%%% Boundary Conditions on Corners %%%%%
% at x=0 y=0 z=0 %
i=1;
j=1;
r=1;
p = (r-1)*(N+1)^2 + (j-1)*(N+1) + i;
A(p,p) = -k(p)*(2*k(p+1)/(k(p)+k(p+1))/dx^2 + 2*k(p+N+1)/(k(p)+k(p+N+1))/dy^2 + ...
2*k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2);
A(p,p+1)=2*k(p)*k(p+1)/(k(p)+k(p+1))/dx^2;
A(p,p+N+1)=2*k(p)*k(p+N+1)/(k(p)+k(p+N+1))/dy^2;
A(p,p+(N+1)^2) = 2*k(p)*k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2;
b(p,1)=-qs/dy;
% at x=2Lx y=0 z=0 %
i=N+1;
j=1;
r=1;
p = (r-1)*(N+1)^2 + (j-1)*(N+1) + i;
A(p,p) = -k(p)*(2*k(p-1)/(k(p)+k(p-1))/dx^2 + 2*k(p+N+1)/(k(p)+k(p+N+1))/dy^2 + ...
2*k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2);
A(p,p-1)=2*k(p)*k(p-1)/(k(p)+k(p-1))/dx^2;
A(p,p+N+1)=2*k(p)*k(p+N+1)/(k(p)+k(p+N+1))/dy^2;
A(p,p+(N+1)^2) = 2*k(p)*k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2;
b(p,1)=-qs/dy;
% at x=0 y=0 z=Lz %
i=1;
j=1;
r=N+1;
p = (r-1)*(N+1)^2 + (j-1)*(N+1) + i;
A(p,p) = -k(p)*(2*k(p+1)/(k(p)+k(p+1))/dx^2 + 2*k(p+N+1)/(k(p)+k(p+N+1))/dy^2 + ...
2*k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2);
A(p,p+1)=2*k(p)*k(p+1)/(k(p)+k(p+1))/dx^2;
A(p,p+N+1)=2*k(p)*k(p+N+1)/(k(p)+k(p+N+1))/dy^2;
A(p,p-(N+1)^2) = 2*k(p)*k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2;
b(p,1)=-qs/dy;
% at x=2Lx y=0 z=Lz %
i=N+1;
j=1;
r=N+1;
p = (r-1)*(N+1)^2 + (j-1)*(N+1) + i;
A(p,p) = -k(p)*(2*k(p-1)/(k(p)+k(p-1))/dx^2 + 2*k(p+N+1)/(k(p)+k(p+N+1))/dy^2 + ...
2*k(p-(N+1)^2)/(k(p)-k(p+(N+1)^2))/dz^2);
A(p,p-1)=2*k(p)*k(p-1)/(k(p)+k(p-1))/dx^2;
A(p,p+N+1)=2*k(p)*k(p+N+1)/(k(p)+k(p+N+1))/dy^2;
A(p,p-(N+1)^2) = 2*k(p)*k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2;
b(p,1)=-qs/dy;
% at x=0 y=Ly z=0 %
i=1;
j=N+1;
r=1;
p = (r-1)*(N+1)^2 + (j-1)*(N+1) + i;
A(p,p) = -k(p)*(2*k(p+1)/(k(p)+k(p+1))/dx^2 + 2*k(p-N-1)/(k(p)+k(p-N-1))/dy^2 + ...
2*k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2)-h/dy;
A(p,p+1)=2*k(p)*k(p+1)/(k(p)+k(p+1))/dx^2;
A(p,p-N-1)=2*k(p)*k(p-N-1)/(k(p)+k(p-N-1))/dy^2;
A(p,p+(N+1)^2) = 2*k(p)*k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2;
b(p,1)=0;
% at x=2Lx y=Ly z=0 %
i=N+1;
j=N+1;
r=1;
p = (r-1)*(N+1)^2 + (j-1)*(N+1) + i;
A(p,p) = -k(p)*(2*k(p-1)/(k(p)+k(p-1))/dx^2 + 2*k(p-N-1)/(k(p)+k(p-N-1))/dy^2 + ...
2*k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2)-h/dy;
A(p,p-1)=2*k(p)*k(p-1)/(k(p)+k(p-1))/dx^2;
A(p,p-N-1)=2*k(p)*k(p-N-1)/(k(p)+k(p-N-1))/dy^2;
A(p,p+(N+1)^2) = 2*k(p)*k(p+(N+1)^2)/(k(p)+k(p+(N+1)^2))/dz^2;
b(p,1)=0;
% at x=2Lx y=Ly z=Lz %
i=N+1;
j=N+1;
r=N+1;
p = (r-1)*(N+1)^2 + (j-1)*(N+1) + i;
A(p,p) = -k(p)*(2*k(p-1)/(k(p)+k(p-1))/dx^2 + 2*k(p-N-1)/(k(p)+k(p-N-1))/dy^2 + ...
2*k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2)-h/dy;
A(p,p-1)=2*k(p)*k(p-1)/(k(p)+k(p-1))/dx^2;
A(p,p-N-1)=2*k(p)*k(p-N-1)/(k(p)+k(p-N-1))/dy^2;
A(p,p-(N+1)^2) = 2*k(p)*k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2;
b(p,1)=0;
% at x=0 y=Ly z=Lz %
i=1;
j=N+1;
r=N+1;
p = (r-1)*(N+1)^2 + (j-1)*(N+1) + i;
A(p,p) = -k(p)*(2*k(p+1)/(k(p)+k(p+1))/dx^2 + 2*k(p-N-1)/(k(p)+k(p-N-1))/dy^2 + ...
2*k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2)-h/dy;
A(p,p+1)=2*k(p)*k(p+1)/(k(p)+k(p+1))/dx^2;
A(p,p-N-1)=2*k(p)*k(p-N-1)/(k(p)+k(p-N-1))/dy^2;
A(p,p-(N+1)^2) = 2*k(p)*k(p-(N+1)^2)/(k(p)+k(p-(N+1)^2))/dz^2;
b(p,1)=0;
%%% Solve the matrix to find temperature distribution %%%
T=Ab;
for r=1:N+1
for j=1:N+1
for i=1:N+1
T_dist(r,j,i)=T((r-1)*(N+1)^2+(j-1)*(N+1)+i)+T_amb;
end
end
end
figure
[X,Y,Z]=meshgrid(0:dx:2*Lx,0:dy:Ly,0:dz:Lz);
% levels=1:1:100;
contour3(X,Y,Z,T_dist,'ShowText','on')
title('Finite Difference Solution')
xlabel('x [cm]')
ylabel('y [cm]')
zlabel('z [cm]')
grid on
c=colorbar;
c.Label.String='Temperature (°C)';
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