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  1. Home/
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  3. Week 5.1 - Mid term project - Solving the steady and unsteady 2D heat conduction problem

Week 5.1 - Mid term project - Solving the steady and unsteady 2D heat conduction problem

OBJECTIVE: To solve Steady and Unsteady state 2D Heat Conduction Equation by using the point iterative techniques such as Jacobi, Gauss Seidel & Successive over-relaxation for both implicit and explicit schemes using MATLAB. Problem statement: Computation area is to be a square. Assume nx = ny [No. of…

  • HTML
  • MATLAB

  • Kishoremoorthy SP

    updated on 27 Apr 2023

OBJECTIVE:

To solve Steady and Unsteady state 2D Heat Conduction Equation by using the point iterative techniques such as Jacobi, Gauss Seidel & Successive over-relaxation for both implicit and explicit schemes using MATLAB.

Problem statement:

  1. Computation area is to be a square.
  2. Assume nx = ny [No. of points on x-direction is equal to no. of points in y-direction]
  3. We assume nx = ny = 10
  4. Boundary Conditions are
    Left      = 400k
    Right    = 600k
    Top       = 800k 
    Bottom = 900k
  5. Apart from the boundary conditions, all other nodes are given an initial temperature of 300k.
  6. Various methods to be employed in solving the equation are given below:

    1. Various States: Steady-state & Unsteady-state Condition
    2. Various Schemes: Jacobi, Gauss-seidel & Successive over-relaxation
    3. Various Methods to be used in Unsteady-state: IMPLICIT & EXPLICIT

By using Point iterative method, solve the 2D Heat Conduction Equation and plot the Temperature Contour for all the three iterative methods namely Jacobi, Gauss-Siedel & SOR for Steady and Unsteady conditions under IMPLICIT scheme also additionally solving the EXPLICIT scheme for Unsteady Condition.

THEORY:

The heat equation is a partial differential equation that describes how the distribution of some quantity (such as heat) evolves over time in a solid medium, as it spontaneously flows from places where it is higher towards places where it is lower.

For a function 

{\displaystyle u(x,y,z,t)}of three spatial variables (x,y,z) (see Cartesian coordinate system) and the time variable t, the heat equation is

                                     {\displaystyle {\frac {\partial u}{\partial t}}=\alpha \left({\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}}\right)}

where, 

\alpha  is a real coefficient called the diffusivity of the medium.

 

Types of Heat Conduction schemes

This transfer of Heat energy may occur under two conditions

  • Steady-state condition
  • Unsteady/Transient state condition

Unsteady state conditions are a precursor to steady state conditions. No system exists initially under steady state conditions. Some time must pass, after heat transfer is initiated, before the system reaches steady state. During that period of transition the system is under unsteady state conditions.

 

Time Loop Logic:

Boundary Condition Map:

Here,

TL  =  Left boundary

TT = Top boundary

TP= Middle/Point boundary

TB = Bottom boundary

TR = Right boundary

 

Governing Equations

In order to solve the 2D Heat Conduction problem, the governing equations for respective conditions with respect to cartesian coordinates should be discretized to enter that in a code which is given under MAIN PROGRAM below.

 

MAIN PROGRAM:

Below there is a sequence of seven individual programs, starting with IMPLICIT Scheme of Steady & Unsteady-state conditions for three iterative methods Jacobi, Gauss-siedel & SOR followed by EXPLICIT Scheme for Unsteady/Transient Conditions.

IMPLICIT Scheme - Steady-state Condition Program:

Steady-state Equation discretization:

In steady-state conduction, the amount of heat entering any region of an object is equal to the amount of heat coming out (if this were not so, the temperature would be rising or falling, as thermal energy was tapped or trapped in a region).

                   

Jacobi Iteration

close
clear 
clc

%Solving the Steady state 2D heat conduction (Jacobi Method)

%Input Parameters
%Number of grid points
nx = 10;
ny = nx;

x = linspace(0,1,nx);
y = linspace(0,1,ny);

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

%Absolute error criteria > tolerance
error = 9e9;
tolerance = 1e-4;

%Defining Boundary conditions
T_L = 400;
T_T = 600;
T_R = 800;
T_B = 900;

T = 300*ones(nx,ny);

T(2:ny - 1, 1)  = T_L;
T(2:ny - 1, nx) = T_R;
T(1, 2:nx - 1)  = T_T;
T(ny, 2:nx - 1) = T_B;

%Calculating Average temperature at corners
T(1,1) = (T_T + T_L)/2;
T(nx,ny) = (T_R + T_B)/2;
T(1,ny) = (T_T + T_R)/2;
T(nx,1) = (T_L + T_B)/2;

%Assigning orginal values to T
T_intial = T;
T_old = T;


%Calculation of 2D steady heat conduction EQUATION by JACOBI method
iterative_solver = 1;

if iterative_solver == 1
    jacobi_iteration = 1;
    
    while(error > tolerance)
        
        for i = 2:nx - 1
            
            for j = 2:ny - 1
                T(i,j) = 0.25*(T_old(i-1,j) + T_old(i+1,j) + T_old(i,j-1) + T_old(i,j+1));
            end
            
        end
        
        error = max(max(abs(T_old - T)));
        T_old = T;
        jacobi_iteration = jacobi_iteration + 1;  
        
    end
end

%Plotting
figure(1)
contourf(x,y,T)
clabel(contourf(x,y,T))
colorbar
colormap(jet)
set(gca, 'ydir', 'reverse')

xlabel('X-Axis')
ylabel('Y-Axis')
title(sprintf('No. of Jacobi Iterations(IMPLICIT) = %d', jacobi_iteration));
pause(0.03);
grid on

Gauss-Siedel Iteration

close all
clear all
clc

%Solving the Steady state 2D heat conduction (Gauss-seidel Method)

%Input Parameters
%Number of grid points
nx = 10;
ny = nx;

x = linspace(0,1,nx);
y = linspace(0,1,ny);

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

%Absolute error criteria > tolerance
error = 9e9;
tolerance = 1e-4;

%Defining Boundary conditions
T_L = 400;
T_T = 600;
T_R = 800;
T_B = 900;

T = 300*ones(nx,ny);

T(2:ny - 1, 1)  = T_L;
T(2:ny - 1, nx) = T_R;
T(1, 2:nx - 1)  = T_T;
T(ny, 2:nx - 1) = T_B;

%Calculating Average temperature at corners
T(1,1) = (T_T + T_L)/2;
T(nx,ny) = (T_R + T_B)/2;
T(1,ny) = (T_T + T_R)/2;
T(nx,1) = (T_L + T_B)/2;

%Assigning orginal values to T
T_intial = T;
T_old = T;


%Calculation of 2D steady heat conduction EQUATION by Gauss-Seidel method
iterative_solver = 2;

if iterative_solver == 2
    Gauss_seidel_iteration = 1;
    
    while(error > tolerance)
        
        for i = 2:nx - 1
            
            for j = 2:ny - 1
                T(i,j) = 0.25*(T(i-1,j) + T_old(i+1,j) + T(i,j-1) + T_old(i,j+1));
            end
            
        end
        
        error = max(max(abs(T_old - T)));
        T_old = T;
        Gauss_seidel_iteration = Gauss_seidel_iteration + 1;  
        
    end
end

%Plotting
figure(1)
contourf(x,y,T)
clabel(contourf(x,y,T))
colorbar
colormap(jet)
set(gca, 'ydir', 'reverse')

xlabel('X-Axis')
ylabel('Y-Axis')
title(sprintf('No. of Gauss-Seidel Iterations(IMPLICIT) = %d', Gauss_seidel_iteration));
pause(0.03);
grid on

Successive over-relaxation

close all
clear all
clc

%Solving the Steady state 2D heat conduction (Successive over-relaxation Method)

%Input Parameters
%Number of grid points
nx = 10;
ny = nx;

x = linspace(0,1,nx);
y = linspace(0,1,ny);

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

%Absolute error criteria > tolerance
error = 9e9;
tolerance = 1e-4;
omega = 1.2;

%Defining Boundary conditions
T_L = 400;
T_T = 600;
T_R = 800;
T_B = 900;

T = 300*ones(nx,ny);

T(2:ny - 1, 1)  = T_L;
T(2:ny - 1, nx) = T_R;
T(1, 2:nx - 1)  = T_T;
T(ny, 2:nx - 1) = T_B;

%Calculating Average temperature at corners
T(1,1) = (T_T + T_L)/2;
T(nx,ny) = (T_R + T_B)/2;
T(1,ny) = (T_T + T_R)/2;
T(nx,1) = (T_L + T_B)/2;

%Assigning orginal values to T
T_intial = T;
T_old = T;


%Calculation of 2D steady heat conduction EQUATION by Successive over-relaxation method
iterative_solver = 3;

if iterative_solver == 3
    Successive_Over_Relaxation_iteration = 1;
    
    while(error > tolerance)
        
        for i = 2:nx - 1
            
            for j = 2:ny - 1
                T(i,j) = omega*(0.25*(T(i-1,j) + T_old(i+1,j) + T(i,j-1) + T_old(i,j+1))) + (1 - omega)*T_old(i,j);
            end
            
        end
        
        error = max(max(abs(T_old - T)));
        T_old = T;
        Successive_Over_Relaxation_iteration = Successive_Over_Relaxation_iteration + 1;  
        
    end
end

%Plotting
figure(1)
contourf(x,y,T)
clabel(contourf(x,y,T))
colorbar
colormap(jet)
set(gca, 'ydir', 'reverse')

xlabel('X-Axis')
ylabel('Y-Axis')
title(sprintf('No. of Successive Over-Relaxation Iterations(IMPLICIT) = %d', Successive_Over_Relaxation_iteration));
pause(0.03);
grid on

 Output: 

  • Jacobi
  • Gauss-Siedel Iteration

  • Successive over-relaxation

IMPLICIT Scheme Unsteady/Transient-state Condition Program:

Unsteady-state Equation discretization:

Non-steady-state situations appear after an imposed change in temperature at a boundary as well as inside of an object, as a result of a new source or sink of heat suddenly introduced within an object, causing temperatures near the source or sink to change in time.

Jacobi Iteration code

close all
clear all
clc

%Solving the Unsteady state 2D heat conduction by Jacobi Method(IMPLICIT SCHEME)

%Input Parameters
%Number of grid points
nx = 10;
ny = nx;
nt = 1400;

x = linspace(0,1,nx);
y = linspace(0,1,ny);

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

%Absolute error criteria > tolerance
error = 9e9;
tolerance = 1e-4;
dt = 1e-3;

%Defining Boundary conditions
T_L = 400;
T_T = 600;
T_R = 800;
T_B = 900;

T = 300*ones(nx,ny);

T(2:ny - 1, 1)  = T_L;
T(2:ny - 1, nx) = T_R;
T(1, 2:nx - 1)  = T_T;
T(ny, 2:nx - 1) = T_B;

%Calculating Average temperature at corners
T(1,1) = (T_T + T_L)/2;
T(nx,ny) = (T_R + T_B)/2;
T(1,ny) = (T_T + T_R)/2;
T(nx,1) = (T_L + T_B)/2;

%Assigning orginal values to T
T_old = T;
T_intial = T;

%Calculation of 2D steady heat conduction EQUATION by Successive over-relaxation method
k1 = 1.1*(dt/(dx^2));
k2 = 1.1*(dt/(dy^2));

Jacobi_iteration = 1;
for k = 1:nt
    error = 9e9; 
    
    while(error > tolerance)
        
        for i = 2:nx - 1
            
            for j = 2:ny - 1
                    
                Term_1 = 1/(1 + (2*k1) + (2*k2));
                Term_2 = k1*Term_1;
                Term_3 = k2*Term_1;
                H = (T_old(i-1,j)) + (T_old(i+1,j));
                V = (T_old(i,j+1)) + (T_old(i,j-1));
                    
                T(i,j) = (T_intial(i,j)*Term_1) + (H*Term_2) + (V*Term_3);
   
            end
            
        end
        
        error = max(max(abs(T_old - T)));
        T_old = T;
        Jacobi_iteration = Jacobi_iteration + 1;  
        
    end
    
    T_intial = T;
    
    %Plotting
    figure(1)
    contourf(x,y,T)
    clabel(contourf(x,y,T))
    colorbar
    colormap(jet)
    set(gca, 'ydir', 'reverse')

    xlabel('X-Axis')
    ylabel('Y-Axis')
    title(sprintf('No. of Unsteady Jacobi Iterations(EXPLICIT) = %d', Jacobi_iteration));
    pause(0.03)
end

Gauss-Siedel Iteration Code

close all
clear all
clc

%Solving the Unsteady state 2D heat conduction by Gauss Seidel Method(IMPLICIT SCHEME)

%Input Parameters
%Number of grid points
nx = 10;
ny = nx;
nt = 1400;

x = linspace(0,1,nx);
y = linspace(0,1,ny);

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

%Absolute error criteria > tolerance
error = 9e9;
tolerance = 1e-4;
dt = 1e-3;

%Defining Boundary conditions
T_L = 400;
T_T = 600;
T_R = 800;
T_B = 900;

T = 300*ones(nx,ny);

T(2:ny - 1, 1)  = T_L;
T(2:ny - 1, nx) = T_R;
T(1, 2:nx - 1)  = T_T;
T(ny, 2:nx - 1) = T_B;

%Calculating Average temperature at corners
T(1,1) = (T_T + T_L)/2;
T(nx,ny) = (T_R + T_B)/2;
T(1,ny) = (T_T + T_R)/2;
T(nx,1) = (T_L + T_B)/2;

%Assigning orginal values to T
T_old = T;
T_intial = T;

%Calculation of 2D steady heat conduction EQUATION by Successive over-relaxation method
k1 = 1.1*(dt/(dx^2));
k2 = 1.1*(dt/(dy^2));

Gauss_Seidel_iteration = 1;
for k = 1:nt
    error = 9e9; 
    
    while(error > tolerance)
        
        for i = 2:nx - 1
            
            for j = 2:ny - 1
                    
                Term_1 = 1/(1 + (2*k1) + (2*k2));
                Term_2 = k1*Term_1;
                Term_3 = k2*Term_1;
                H = (T(i-1,j)) + (T_old(i+1,j));
                V = (T(i,j+1)) + (T_old(i,j-1));
                    
                T(i,j) = (T_intial(i,j)*Term_1) + (H*Term_2) + (V*Term_3);
   
            end
            
        end
        
        error = max(max(abs(T_old - T)));
        T_old = T;
        Gauss_Seidel_iteration = Gauss_Seidel_iteration + 1;  
        
    end
    
    T_intial = T;
    
    %Plotting
    figure(1)
    contourf(x,y,T)
    clabel(contourf(x,y,T))
    colorbar
    colormap(jet)
    set(gca, 'ydir', 'reverse')

    xlabel('X-Axis')
    ylabel('Y-Axis')
    title(sprintf('No. of Unsteady Gauss-Seidel Iterations(IMPLICIT) = %d', Gauss_Seidel_iteration));
    pause(0.03)
end

Successive over-relaxation Code:

close all
clear all
clc

%Solving the Unsteady state 2D heat conduction by Successive over-relaxation Method(IMPLICIT SCHEME)

%Input Parameters
%Number of grid points
nx = 10;
ny = nx;
nt = 1400;

x = linspace(0,1,nx);
y = linspace(0,1,ny);

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

%Absolute error criteria > tolerance
error = 9e9;
tolerance = 1e-4;
dt = 1e-3;
omega = 1.1;

%Defining Boundary conditions
T_L = 400;
T_T = 600;
T_R = 800;
T_B = 900;

T = 300*ones(nx,ny);

T(2:ny - 1, 1)  = T_L;
T(2:ny - 1, nx) = T_R;
T(1, 2:nx - 1)  = T_T;
T(ny, 2:nx - 1) = T_B;

%Calculating Average temperature at corners
T(1,1) = (T_T + T_L)/2;
T(nx,ny) = (T_R + T_B)/2;
T(1,ny) = (T_T + T_R)/2;
T(nx,1) = (T_L + T_B)/2;

%Assigning orginal values to T
T_old = T;
T_intial = T;
T_end = T;

%Calculation of 2D steady heat conduction EQUATION by Successive over-relaxation method
k1 = 1.1*(dt/(dx^2));
k2 = 1.1*(dt/(dy^2));

Successive_over_relaxation_iteration = 1;
for k = 1:nt
    error = 9e9; 
    
    while(error > tolerance)
        
        for i = 2:nx - 1
            
            for j = 2:ny - 1
                    
                Term_1 = 1/(1 + (2*k1) + (2*k2));
                Term_2 = k1*Term_1;
                Term_3 = k2*Term_1;
                H = (T(i-1,j)) + (T_old(i+1,j));
                V = (T(i,j+1)) + (T_old(i,j-1));
                    
                T(i,j) = (T_intial(i,j)*Term_1) + (H*Term_2) + (V*Term_3);
                T_end(i,j) = ((1-omega)*T_old(i,j)) + (T(i,j)*omega);
                
            end
            
        end
        
        error = max(max(abs(T_old - T)));
        T_old = T_end;
        Successive_over_relaxation_iteration = Successive_over_relaxation_iteration + 1;  
        
    end
    
    T_intial = T_end;
    
    %Plotting
    figure(1)
    contourf(x,y,T)
    clabel(contourf(x,y,T))
    colorbar
    colormap(jet)
    set(gca, 'ydir', 'reverse')

    xlabel('X-Axis')
    ylabel('Y-Axis')
    title(sprintf('No. of Unsteady SOR Iterations(IMPLICIT) = %d', Successive_over_relaxation_iteration));
    pause(0.03)
end

Output:

  • Jacobi Iteration
  • Gauss-Siedel Iteration
  • Successive over-relaxation

EXPLICIT Scheme - Unsteady/Transient state Condition Program

Unsteady-state Equation discretization(EXPLICIT):

Explicit Scheme code :

close all
clear all
clc

%Solving the Unsteady state 2D heat conduction (EXPLICIT SCHEME)

%Input Parameters
%Number of grid points
nx = 10;
ny = nx;
nt = 1400;

x = linspace(0,1,nx);
y = linspace(0,1,ny);

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

%Absolute error criteria > tolerance
error = 9e9;
tolerance = 1e-4;
dt = 1e-3;
omega = 1.1;

%Defining Boundary conditions
T_L = 400;
T_T = 600;
T_R = 800;
T_B = 900;

T = 300*ones(nx,ny);

T(2:ny - 1, 1)  = T_L;
T(2:ny - 1, nx) = T_R;
T(1, 2:nx - 1)  = T_T;
T(ny, 2:nx - 1) = T_B;

%Calculating Average temperature at corners
T(1,1) = (T_T + T_L)/2;
T(nx,ny) = (T_R + T_B)/2;
T(1,ny) = (T_T + T_R)/2;
T(nx,1) = (T_L + T_B)/2;

%Assigning orginal values to T
T_intial = T;
T_old = T;


%Calculation of 2D steady heat conduction EQUATION by Successive over-relaxation method
iterative_solver = 1;
k1 = 1.1*(dt/(dx^2));
k2 = 1.1*(dt/(dy^2));

if iterative_solver == 1
    Steady_Explicit = 1;
    for k = 1:nt
        error = 1; 
    
        while(error > tolerance)
        
            for i = 2:nx - 1
            
                for j = 2:ny - 1
                    Term_1 = T_old(i,j);
                    Term_2 = k1*(T_old(i+1,j) - 2*T_old(i,j) + (T_old(i-1,j)));
                    Term_3 = k2*(T_old(i,j+1) - 2*T_old(i,j) + T_old(i,j-1));
                    T(i,j) = Term_1 + Term_2 + Term_3;
                end
            
            end
        
            error = max(max(abs(T_old - T)));
            T_old = T;
            Steady_Explicit = Steady_Explicit + 1;  
        
        end
    end
end

%Plotting
figure(1)
contourf(x,y,T)
clabel(contourf(x,y,T))
colorbar
colormap(jet)
set(gca, 'ydir', 'reverse')

xlabel('X-Axis')
ylabel('Y-Axis')
title(sprintf('No. of Unsteady Iterations(EXPLICIT) = %d', Steady_Explicit));
pause(0.03);

Output:

CONCLUSIONS:

From the above outputs of various methods & comparing their No. of iterations, we can conclude that

  • Jacobi > Gauss-seidel > Successive over-relaxation according to the No. of iterations computed for all methods & Schemes.

  • When comparing Steady state and Unsteady state, it is proven that Steady state is the most successful one as it takes least number of iterations and least time to compute.

  • SOR iteration method is the most effective iteration method as it took least No. of iterations for coverging solutions in both the schemes (EXPLICIT & IMPLICIT) compared to Jacobi & Gauss-siedel.

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Week 11 - Simulation of Flow through a pipe in OpenFoam

Objective:

Aim: Simulate axisymmetric flow in a pipe through foam. Objective: Verify the hydrodynamic length using the numerical result Verify a fully developed flow rate profile with its analytical profile Verify the maximum velocity and pressure drop for fully developed flow Post-process Shear Stress and verify wall shear stress…

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04 May 2023 03:04 PM IST

  • CFD
  • HTML
  • MATLAB
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Week 9 - FVM Literature Review

Objective:

AIM To describe the need for interpolation schemes and flux limiters in Finite Volume Method (FVM).   OBJECTIVE To study and understand What is Finite Volume Method(FVM) Write down the major differences between FDM & FVM Describe the need for interpolation schemes and flux limiters in FVM   INTRODUCTION            …

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03 May 2023 05:47 AM IST

  • CFD
  • HTML
  • STAR-CCM
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