2D Heat Conduction

This Problem is solved in two steps:

1.Steady State Conduction through Explicit Method:

2.UnSteady State Conduction through Explicit and Implicit Method:

Energy Equation in 2D state is given  by:

`(delT)/(delt)+(del(uT))/(delx)+(del(vT))/(dely)=alpha((del^2T)/(delx^2)+(del^2T)/(dely^2))+q`

Where First term on the Left side indicates Unsteady term or Time Derivative.

Second term on the left side indicates Convective Term.

where as first term on the right side indicates Diffussion Term

Second term on the Rigth side indicates Source Term.

For Solving 2D heat Conduction:

Above Equation Reduces to 

`((del^2T)/(delx^2)+(del^2T)/(dely^2))=0`

Using Finite Difference Method discretizing the above equation we get:

1.Steady State Conduction through Explicit Method:

Code for the Steady State Heat Conduction using Explicit Method is given as:

%Steady State Heat Conduction using Explicit method
close all
clc

%inputs
l=input('Enter the length of the domain: ');
b=input('Enter the Width of the domain: ');

nx=input('Enter the Number of nodes in x direction: ');
ny=input('Enter the number of nodes in y direction: ');

solver=input('Enter the Number for the solver method: \n Gauss Jacobi= 1 \n Gauss seidel= 2  \n SOR= 3 \n Your Choice of the Solver: ');

%specifying the grid 
x=linspace(0,l,nx);
y=linspace(0,b,ny);

dx=x(2)-x(1);
dy=y(2)-y(1);

omega=1.25;
%Initialisation
T=298*ones(nx,ny);
Told=T;


error=9e9;
tol=1e-5;


%Boundary Conditions

T(:,1)=400;
T(:,end)=800;
T(end,:)=600;
T(1,:)=900;

%EdgeRefinement
T(1,1)=(900+400)/2;
T(1,end)=(900+800)/2;
T(end,1)=(400+600)/2;
T(end,end)=(800+600)/2;

%defining Constants
k=(2/dx^2)+(2/dy^2);

[xx,yy]=meshgrid(x,y); 

%Jacobi Method

if (solver==1)
    
    tic
    
   counter_1=1;
   
    while(error > tol)
for i=2:nx-1
    for j=2:ny-1
       
        Term(1)=(Told(i-1,j)+Told(i+1,j))/(k*dx^2);
        Term(2)=(Told(i,j+1)+Told(i,j-1))/(k*dy^2);
        T(i,j)=Term(1)+Term(2);
    
    end
end
error=max(max(abs(Told-T)));
Told=T;
counter_1=counter_1+1;
    end
timecounter_1=toc;
[C,h]=contourf(xx,yy,T);
colorbar
colormap(jet)
clabel(C,h);
figure(1)
title(sprintf('2d Steady State Heat Conduction \n Method:Jacobi Explicit \n No of iterations=%d \n Time: %f s',counter_1,timecounter_1));
end

%Gauss Seidel Method
if (solver==2)
    tic
    counter=1;
    while(error > tol)
for i=2:nx-1
    for j=2:ny-1
       
        Term(1)=(T(i-1,j)+T(i+1,j))/(k*dx^2);
        Term(2)=(T(i,j+1)+T(i,j-1))/(k*dy^2);
        T(i,j)=Term(1)+Term(2);
    
    end
end
error=max(max(abs(Told-T)));
Told=T;
counter=counter+1;
    end
 timecounter=toc;
figure(1)
[C,h]=contourf(xx,yy,T);
colorbar
colormap(jet)
clabel(C,h);
title(sprintf('2d Steady State Heat Conduction \n Method:Gauss Seidel Explicit \n No of iterations= %d \n Time: %f s',counter,timecounter));
end
   

%Successive Over Relaxation method
if (solver==3) 
    tic
    counter_2=1;
    while(error > tol)
for i=2:nx-1
    for j=2:ny-1
       
        Term(1)=(T(i-1,j)+T(i+1,j));
        Term(2)=(T(i,j+1)+T(i,j-1));
        T(i,j)=(1-omega)*Told(i,j)+(omega/4)*(Term(1)+Term(2));
    
    end
end
error=max(max(abs(Told-T)));
Told=T;
counter_2=counter_2+1;
    end
    timecounter_2=toc;
figure(1)
[C,h]=contourf(xx,yy,T);
colorbar
colormap(jet)
clabel(C,h);
title(sprintf('2d Steady State Heat Conduction \n Method:Successive Over Relaxation Explicit \n No of iterations= %d \n Time: %f s',counter_2,timecounter_2));
end
   

Plot for Jacobi Method:

Inputs:

Output:

Plot for Gauss Seidel Method:

Inputs:

Output:

Plot for Successive Over Relaxation Method:

Inputs:

Output:

2.Unsteady State Conduction through Explicit and Implicit Method:

A.Explicit Method:

Code for the UnSteady State Heat Conduction using Explicit Method is given as:

%Unsteady State Heat Conduction using Explicit method
close all
clc

%inputs
l=input('Enter the length of the domain: ');
b=input('Enter the Width of the domain: ');

nx=input('Enter the Number of nodes in x direction: ');
ny=input('Enter the number of nodes in y direction: ');

t=input('Enter the Time: ');

solver=input('Enter the Number for the Solver method: \n Gauss Jacobi= 1 \n Gauss seidel= 2  \n SOR= 3 \n Your Choice of the Solver: ');

%specifying the grid 
x=linspace(0,l,nx);
y=linspace(0,b,ny);

dx=x(2)-x(1);
dy=y(2)-y(1);

%Initialisation
T=298*ones(nx,ny);
Told=T;


error=9e9;
tol=1e-5;


%Boundary Conditions

T(:,1)=400;
T(:,end)=800;
T(end,:)=600;
T(1,:)=900;

%EdgeRefinement
T(1,1)=(900+400)/2;
T(1,end)=(900+800)/2;
T(end,1)=(400+600)/2;
T(end,end)=(800+600)/2;


%Input for time step
dt=0.01;
alpha=1e-4;
k1=(alpha*dt)/dx^2;
k2=(alpha*dt)/dy^2;
tic;
[xx,yy]=meshgrid(x,y); 
nt=t/dt;

c=alpha*dt*((1/dx^2)+(1/dy^2));

%Jacobi Method
if (solver==1)
    tic;
    counter_1=1;
    while(error > tol)
for k=1:nt
for i=2:nx-1
    for j=2:ny-1
       
        Term(1)=Told(i,j);
        Term(2)=k1*(Told(i-1,j)-2*Told(i,j)+Told(i+1,j));
        Term(3)=k2*(Told(i,j+1)-2*Told(i,j)+Told(i,j-1));
        T(i,j)=Term(1)+Term(2)+Term(3);
    
    end
end
error=max(max(abs(Told-T)));
Told=T;
counter_1=counter_1+1;
end
    end
timecounter_1=toc;
[C,h]=contourf(xx,yy,T);
colorbar
colormap(jet)
clabel(C,h);
figure(1)
title(sprintf('2d Unsteady State heat Conduction \n Method: Jacobi Explicit \n No of iterations=%d \n Time: %f s \n CFL No: %d',counter_1,timecounter_1,c));
end

%Gauss Seidel Method

if (solver==2)
    tic;
    counter=1;
    while(error > tol)
for k=1:nt
for i=2:nx-1
    for j=2:ny-1
       
        Term(1)=Told(i,j);
        Term(2)=k1*(T(i-1,j)-2*T(i,j)+Told(i+1,j));
        Term(3)=k2*(Told(i,j+1)-2*T(i,j)+T(i,j-1));
        T(i,j)=Term(1)+Term(2)+Term(3);
    
    end
end
error=max(max(abs(Told-T)));
Told=T;
counter=counter+1;
end
    end
    timecounter=toc;
figure(1)
[C,h]=contourf(xx,yy,T);
colorbar
colormap(jet)
clabel(C,h);
title(sprintf('2d Unsteady State heat Conduction \n Method: Gauss Seidel Explicit  \n No of iterations= %d \n Time: %f s',counter,timecounter));
end


%Successive over Relaxation Method

if (solver==3)
    tic
    counter_2=1;
    while(error > tol)
for k=1:nt
for i=2:nx-1
    for j=2:ny-1
       
        Term(1)=Told(i,j);
        Term(2)=k1*(T(i-1,j)-2*T(i,j)+Told(i+1,j));
        Term(3)=k2*(Told(i,j+1)-2*T(i,j)+T(i,j-1));
        T(i,j)=(1-1.2)*Told(i,j)+1.2*(Term(1)+Term(2)+Term(3));
    
    end
end
error=max(max(abs(Told-T)));
Told=T;
counter_2=counter_2+1;
end
Tprev=T;
    end
    timecounter_2=toc;
figure(1)
[C,h]=contourf(xx,yy,T);
colorbar
colormap(jet)
clabel(C,h);
title(sprintf('2d Unsteady State heat Conduction\n Method: Successive Over Relaxation Explicit \n No of iterations= %d \n Time: %f s',counter_2,timecounter_2));
end

Plot for Jacobi Method:

Inputs:

Output:

Plot for Gauss Seidel Method:

Inputs:

Output:

Plot for Successive Over Relaxation Method:

Inputs:

Output:

B.Implicit Method:

Code for the UnSteady State Heat Conduction using Implicit Method is given as:

close all
clc

%inputs
l=input('Enter the length of the domain: ');
b=input('Enter the Width of the domain: ');

nx=input('Enter the Number of nodes in x direction: ');
ny=input('Enter the number of nodes in y direction: ');

t=input('enter the time: ');

solver=input('Enter the Number for the solver method: \n Gauss Jacobi=1 \n Gauss seidel=2  \n SOR=3 \n Your Choice of the Solver: ');

%specifying the grid 
x=linspace(0,l,nx);
y=linspace(0,b,ny);

dx=x(2)-x(1);
dy=y(2)-y(1);

%Initialisation
T=298*ones(nx,ny);
Told=T;
Tprev=T;


error=9e9;
tol=1e-5;


%Boundary Conditions

T(:,1)=400;
T(:,end)=800;
T(end,:)=600;
T(1,:)=900;

%EdgeRefinement
T(1,1)=(900+400)/2;
T(1,end)=(900+800)/2;
T(end,1)=(400+600)/2;
T(end,end)=(800+600)/2;


%Input for time step
dt=0.01;
alpha=1e-4;
k1=(alpha*dt)/dx^2;
k2=(alpha*dt)/dy^2;
tic;
[xx,yy]=meshgrid(x,y); 
nt=t/dt;
c=alpha*dt*((1/dx^2)+(1/dy^2));

%Jacobi Method
if (solver==1)
    tic
    counter_1=1;
    while(error > tol)
for k=1:nt
for i=2:nx-1
    for j=2:ny-1
       
        Term(1)=Tprev(i,j);
        Term(2)=k1*(Told(i-1,j)+Told(i+1,j));
        Term(3)=k2*(Told(i,j+1)+Told(i,j-1));
        T(i,j)=(Term(1)+Term(2)+Term(3))/(1+2*k1+2*k2);
    
    end
end
error=max(max(abs(Told-T)));
Told=T;
counter_1=counter_1+1;
error_1=max(max(abs(Told-T)));
Tprev=T;
timecounter_1=toc;

end

    end
[C,h]=contourf(xx,yy,T);
colorbar
colormap(jet)
clabel(C,h);
figure(1)
title(sprintf('2d Unsteady State heat Conduction\n Method:Jacobi Implicit \n No of iterations=%d \n Time: %f s \n CFL No: %d',counter_1,timecounter_1,c));
end

%Gauss Seidel Method

if (solver==2)
    tic
    counter_2=1;
    while(error > tol)
for k=1:nt
for i=2:nx-1
    for j=2:ny-1
       
        Term(1)=Tprev(i,j);
        Term(2)=k1*(T(i-1,j)+T(i+1,j));
        Term(3)=k2*(T(i,j+1)+T(i,j-1));
        T(i,j)=(Term(1)+Term(2)+Term(3))/(1+2*k1+2*k2);
    
    end
end
error=max(max(abs(Told-T)));
Told=T;
counter_2=counter_2+1;
error_1=max(max(abs(Told-T)));
Tprev=T;
timecounter_2=toc;
end
    end
figure(2)
[C,h]=contourf(xx,yy,T);
colorbar
colormap(jet)
clabel(C,h);
title(sprintf('2d Unsteady State heat Conduction\n Method:Gauss Seidel Implicit \n No of iterations=%d \n Time: %f s',counter_2,timecounter_2));
end

%succeesive Over Relaxation
if (solver==3)
    tic
    counter_3=1;
    while(error > tol)
for k=1:nt
for i=2:nx-1
    for j=2:ny-1
       
        Term(1)=Tprev(i,j);
        Term(2)=k1*(T(i-1,j)+T(i+1,j));
        Term(3)=k2*(T(i,j+1)+T(i,j-1));
        T(i,j)=(1-1.3)*(Tprev(i,j))+(1.3*(Term(1)+Term(2)+Term(3))/(1+2*k1+2*k2));
    
    end
end
error=max(max(abs(Told-T)));
Told=T;
counter_3=counter_3+1;
error_1=max(max(abs(Told-T)));
Tprev=T;
timecounter_3=toc;

end

    end
figure(3)
[C,h]=contourf(xx,yy,T);
colorbar
colormap(jet)
clabel(C,h);
title(sprintf('2d Unsteady State heat Conduction\n Method:sor Implicit \n No of iterations=%d \n Time: %f s',counter_3,timecounter_3));
end



Plot for Jacobi Method:

Inputs:

Output:

Plot for Gauss Seidel Method:

Inputs:

Output:

Plot for Succesive Over Relaxation Method:

Inputs:

Output:

Explanation for Steady State Results:

1.From Steady State Heat Conduction, we can see that time taken by Successive Over Relaxation is less than the Jacobi and Gauss seidel method.

2.And it also takes less no of iterations to solve compare to Jacobi & Gauss Seidel method.

2.In short we can conclude that time taken for computation is Jacobi>Gauss Seidel>Successive Over Relaxation Method.

Explanation for Unsteady State Results:

1.From transient solvers we can conclude that explicit method converges faster than implicit method.

2.In both implicit and explicit method, computation time trend follows the same way as it follows in steady state heat conduction results.

3.For the same CFL Number comparing jacobi explicit with jacobi implicit we can say that computation time as well as number of iterations performed are more.

But in a Nutshell we can conclude that Successive Over Relaxation method is the method is the best whether it is steady or unsteady state heat conduction or implicit or explicit method.

 

 

 


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The End