Finite difference temperature distribution with TDMA

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Cubii4 2022-11-29

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1866658-finite-difference-temperature-distribution-with-tdma

编辑： Torsten 2023-3-7

A stainless steel (thermal conductivity, k = 20 W/mK) fin has a circular cross-sectional area (diameter, D=4 cm) and length L=16 cm. The wing is attached to a wall with a temperature of 200C. The ambient temperature of the fluid surrounding the fin is 20C and the heat transfer coefficient is h=10 W/m2K. Fin tip is insulated. Determine the following by applying finite differences with TDMA.

1. Temperatures inside the fin and the amount of heat radiated from the fin. Plot the temperature distribution.

2. Discuss the effect of the number of element dimensions for N = 5, 10, 50, and 100. Plot the temperature distribution.

The temperature distribution from the analitic solution is like this: T1=150.54, T2=111.107, T3=78.671, T4=50.741, T5=25.172, my code does not give these values.

clc
clear all
L=0.16;             %lenght 
N=6;                %number of nodes
dx=L/(N-1);         %lenght between nodes
T=zeros(N+1,1);
Tb=200;             %the wall temperature (boundary condition)
k=6;                %number of iterations
for j=1:1:k
    T(1,1)=Tb;
    T(5,1)=T(7,1);  %the second boundary condition due to the isulation on the tip of the fin
  for i=2:1:N
    T(i,1)=(T(i+1,1)+T(i-1,1)+1.536)/2.0768;    %FDE narrowed down
  end
  T(N,1)=T(N-1,1);
  plot(T);
  hold on
end
hold off

If I can get some help with my code.

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Cubii4 2022-11-30

在 MATLAB Online 中打开

clc
clear all
L=0.16;
N=6;
dx=L/(N-1);
T=zeros(N+1,1);
Tb=200;
k=1000;
for i=1:1:k
    T(1,1)=Tb;
    for j=2:1:N
    T(j,1)=(T(j+1,1)+T(j-1,1)+1.024)/2.0512;
    end
    T(N+1)=T(N-1);
    plot(T);
    hold on
end
hold off
for j=1:N
    disp(T(j,1));
end

I just corrected the code like this and it gave me the temperature results as the analytic solutions. I can't thank you enough, is there anything else I have to correct in my code.

Cubii4 2022-11-30

Thank you very much I really appreciated Your help. Can you write it as an answer so I can accept it.

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采纳的回答

Torsten 2022-11-30

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1866658-finite-difference-temperature-distribution-with-tdma#answer_1116453

移动：Torsten 2022-11-30

在 MATLAB Online 中打开

You solve the system iteratively. But you were told to solve it with the Thomas-Algorithm. So apply this algorithm to the matrix A below.

k = 20;
T_inf = 20;
h = 10;
D = 0.04;
L = 0.16;
T_at_wall = 200;
N = 6;
dL = L/(N-1);
A = zeros(N+1);
b = zeros(N+1,1);
A(1,1) = 1.0;
b(1) = T_at_wall;
for i = 2:N
    A(i,i-1) = k/dL^2;
    A(i,i) = -(2*k/dL^2 + 4/D*h);
    A(i,i+1) = k/dL^2;
    b(i) = -4/D*h*T_inf;
end
A(N+1,N-1) = -1.0;
A(N+1,N+1) = 1.0;
b(N+1) = 0;
T = A\b;
T = T(1:N)
T = 6×1
  200.0000
  171.3794
  150.5095
  136.3216
  128.0894
  125.3914