Finite Differencing Transient Conduction

Hello,

I'm attempting to conduct a finite difference approach to a transient conduction problem.

• Rows are time steps
• Columns are Length

My initial condition is:

• At time = 0, Temp Is uniform at 573K (except for at L=0 where it's 673K).

My Boundary Condition is:

• Temp = 673K at L = 0, constant
• at L = L, no heat transfer, insulated

My descritization loop appears below.

I can populate T0 with the intial and boundary conditions above, but am running into a problem in the finitie difference for loop.

Any help is appreciated.

clc 
clear all
close all

ri = 0.0125;
ro = 0.0375;
R = ro-ri;

t=2000;

k = 50; %W/mK
cp = 1750; %W s /kg K 
rho = 1500; %kg/m3 
alpha = k /(rho*cp); % m2/s

dr = 50;
dt = 1;

FO = alpha* dt/(dr^2);

T0 = zeros(t+1,51);
T1 = zeros(t+1,51);

for i = 1:t
    T0(i,1) = 673;
    T0 = T1;
end

for j = 2:t
    for i = 1
    T0(i,j) = 573;
    T0 = T1;
    end
end


for j = 3:t
    for i = 2:t
        T0(i,j) = (dr/R)*FO*T1(i+1,j)-(dr/R)*FO*T1(i,j)+FO*T1(i-1,j)-2*FO*T1(i,j)+FO*T1(i+1,j);
    end
    T0 = T1;
end

[Torsten]

Why do both loops run over t ?

And why do you divide by R = ro-ri and not be r ?

Two of many other errors in your code.

Seems you better stick to the "pdepe" solution.

Answer 1

[VBBV]

May be the central and forward difference schemes are not correct, but the following change will avoid the code errors

clc 
clear all
close all

ri = 0.0125;
ro = 0.0375;
R = ro-ri;

t=2000;

k = 50; %W/mK
cp = 1750; %W s /kg K 
rho = 1500; %kg/m3 
alpha = k /(rho*cp); % m2/s

dr = 50;
dt = 1;

FO = alpha* dt/(dr^2);

T0 = zeros(t+1,51);
T1 = zeros(t+1,51);

for i = 1:t+1
    T0(i,1) = 673;   
end

for j = 1:t+1    
    T0(1,j) = 573;      
end
 T1 = T0;

for j = 2:t
    for i = 2:t
        % T0(i,j) = FO*((dr/R)*(T1(i+1,j)-T1(i,j))+T1(i-1,j)-(2*T1(i,j)+T1(i+1,j)));
        T0(i,j) = T1(i,j)+ (dr/R)*FO*T1(i+1,j)-(dr/R)*FO*T1(i,j)+(FO*T1(i-1,j)-2*FO*T1(i,j)+FO*T1(i+1,j));
    end
    T1 = T0;
end
T0, T1

Answer 2

[Walter Roberson]

T0 = zeros(t+1,51);
T1 = zeros(t+1,51);

Okay, both zeros

for i = 1:t
    T0(i,1) = 673;

You replace a particular element of T0 with 673. On the first iteration, T0 will now have 673 in row 1 column 1, and zeros everywhere else.

    T0 = T1;

And there you overwrite all of T0 with the zeros stored in T1. If that were your intent then you could be more efficient by just skipping the loop entirely since you already st T0 to all 0.