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  1. Home/
  2. Himanshu Chavan/
  3. Solving the steady and unsteady 2D heat conduction equation

Solving the steady and unsteady 2D heat conduction equation

Aim: To solve a heat conduction equation of a 2D plate surrounded by constant temperature by the Finite Difference Method. Here we will consider both the Steady-state and Transient state(Unsteady) case and compare which iterative method gives better results with fewer number iteration. Also, we will understand the effect…

    • Himanshu Chavan

      updated on 02 Jul 2021

    Aim: To solve a heat conduction equation of a 2D plate surrounded by constant temperature by the Finite Difference Method. Here we will consider both the Steady-state and Transient state(Unsteady) case and compare which iterative method gives better results with fewer number iteration. Also, we will understand the effect of CFL numbers in the case of transient cases.

     

    Assumptions:

    1. Absolute error criteria is 1e-4

    2. Consider an equal number of grid points or nodes along the X-axis and Y-axis

    3. No internal heat generation

    4. Boundary conditions are as follows 

    • Top Boundary = 600 k
    • Bottom Boundary = 900 k
    • Left Boundary = 400 k
    • Right Boundary = 800k

     

     

    Governing Equation:∂2T∂x2+∂2T∂y2+∂2T∂z2+qk=ρcpk∂T∂t∂2T∂x2+∂2T∂y2+∂2T∂z2+qk=ρcpk∂T∂t 

     

    Here.

     q= Rate of Heat Generation

     k= Thermal Conductivity

    ρρ= Density

    cpcp=Specific heat capacity at constant pressure

    ρcpρcp=volumetric heat capacity

     

    Considering 2D case and no heat generation constant pressure

    ∂2T∂x2+∂2T∂y2=1α∂T∂t∂2T∂x2+∂2T∂y2=1α∂T∂t

    ∂Tdt=α[∂2T∂x2+∂2T∂y2]∂Tdt=α[∂2T∂x2+∂2T∂y2]

     

    Numerical Discretization:

     

    1. Steady-State Case:

    In the case of steady-state, the time derivative term of temperature becomes zero. So the governing equation becomes :

    ∂2T∂x2+∂2T∂y2=0∂2T∂x2+∂2T∂y2=0

    Now Discretizing each of the terma by finite difference method we get,

     

    ∂2T∂x2=Tni,j+1+2Tni,j+Tni,j−1(Δx)2∂2T∂x2=Ti,j+1n+2Ti,jn+Ti,j-1n(Δx)2

    ∂2T∂y2=Tni+1,j+2Tni,j+Tni−1,j(Δy)2∂2T∂y2=Ti+1,jn+2Ti,jn+Ti-1,jn(Δy)2

     

    After rearranging the terms we get,

    Ti,j=Ti+1,j+Ti−1,j+Ti,j+1+Ti,j−14Ti,j=Ti+1,j+Ti-1,j+Ti,j+1+Ti,j-14

    This simultaneous set of eqaution can be solved using the followiing iterative method:

    • Jacobi:TP=ToldT+T_(oldB)+ToldL+ToldRTP=ToldT+T_(oldB)+ToldL+ToldR
    • Gauss -Seidal:TP=TT+ToldB+TL+ToldRTP=TT+ToldB+TL+ToldR
    • Successive Order Reduction:TpSOR=(1−w)Tpold+wTpGSTpSOR=(1-w)Tpold+wTpGS

    Here the subscript old refers to the value in the previous iteration and the subscripts T, B R and L refders to Top, Bottom,Right and Left respectively. 

     

    2. Transient State Case:

    In the case of the transient case the time derivative term of temperature is considered. So the governing equation becomes:

     

    α(∂2T∂x2+∂2T∂y2)=∂T∂tα(∂2T∂x2+∂2T∂y2)=∂T∂t

    Here,

    α=kρcp=1.4m2sα=kρcp=1.4m2s(considered arbitarily) 

     

    Now Discretizing each of the terms by finite difference method we get,

    ∂T∂t=Tn+1l−TnlΔt∂T∂t=Tln+1-TlnΔt

    Now we get two cases, Explicit and Implicit based on the discretization of the remaining two terms.

    Considering the previous time step will give rise to explicit terms and considering the same time step will give rise to the implicit case.

     

    Explicit Case:

    ∂2T∂x2=Tni,j+1+2Tni,j+Tni,j−1(Δx)2∂2T∂x2=Ti,j+1n+2Ti,jn+Ti,j-1n(Δx)2

    ∂2T∂y2=Tni+1,j+2Tni,j+Tni−1,j(Δy)2∂2T∂y2=Ti+1,jn+2Ti,jn+Ti-1,jn(Δy)2

     

    After calculation consider Δx=ΔyΔx=Δy   and k1=k2

    Tp=Tpold[1−4k]+k(TTold+TBold+TLold+TRold)Tp=Tpold[1-4k]+k(TTold+TBold+TLold+TRold)

     

    Implicit Case:

    ∂2T∂x2=Tn+1i,j+1+2Tn+1i,j+Tn+1i,j−1Δx2∂2T∂x2=Ti,j+1n+1+2Ti,jn+1+Ti,j-1n+1Δx2

    ∂2t∂y2=Tn+1i+1,j+2Tn+1i,j+Tn+1i−1,j(Δy)2∂2t∂y2=Ti+1,jn+1+2Ti,jn+1+Ti-1,jn+1(Δy)2

    Now just like the staeadt state equations this set of implicit case can be solved by the following iterative methods.

    Here TPprevTPprev= values of TpTpin the previous time step

    • jacobi
    • gauss- seidal
    • Successive Over Relaxation.

    User-Defined Functions:

    1. linear_eqn_solver_jacobi

    function linear_eqn_solver_jacobi(T,Told,Tprev,nx,ny,nt,x,y,k1,k2,tol,type)
%%Transient Implicit (Jacobi)

if type == 11
    tic;
    iteration = 1;
    for k= 1:nt   %time loop
       error=9e9;
         while(error > tol)  % Convergence loop
              for i= 2:nx-1   % Space loop
                  for j= 2:ny-1
                      term1 = (1+(2*k1)+(2*k2))^-1;
                      V = Told(i-1,j) + Told(i+1,j);
                      H = Told(i,j-1) + Told(i,j+1);
                      T(i,j) = (Tprev(i,j)*term1) + (k1*term1*H) + (k2*term1*V);
                  end
               end % end of space loop
              error = max(max(abs(Told-T)));
              Told =T;
              iteration = iteration +1;
          end % end of convergence loop
          Tprev =T;
    end % end of time loop
    time = toc;
    
    f1=figure(1)
    contourf(x,y,T,'--','ShowText', 'on')
    colorbar
    colormap('jet')
    text=sprintf("2D Transient Heat Conduction-Implicite-Jacobi  No. of iteration %f  Time of simulation %f Sec",iteration,time);
    title(text)
    set(gca,'YDIR','reverse')
    
    
    
    %Steady State (Jacobi)
    
elseif type== 21
    iteration =1;
    tic;
    error=9e9;
    while (error> tol)  %Convergence loop
        for i= 2:nx-1   %Space Loop
            for j=2:ny-1
                term1 = Told(i,j-1) +Told(i,j+1);
                term2 = Told(i-1,j) + Told(i+1,j);
                T(i,j) = (1/4)*(term1+term2);
            end
        end  % End of space loop
        error=max(max(abs(Told-T)));
        Told=T;
        iteration =iteration+1;
    end  % End of convergence loop
    time=toc;
    
    
    f1=figure(1)
    contourf(x,y,T,'--','ShowText','on')
    colorbar
    colormap('jet')
    text = sprintf("2D Steady State Heat Conduction - Jacobi  No. of iteration %f Time of Simulation %f Sec",iteration,time);
    title(text)
    set(gca,'YDIR','reverse')
 end
end

     

    2.linear_eqn_solver_Gs

    function linear_eqn_solver_Gs(T,Told,Tprev,nx,ny,nt,x,y,k1,k2,tol,type)

%Transient Implicit(Gauss seidal)

if type == 12
    iteration = 1;
    tic;
    for k=1:nt  %Time loop
        error = 9e9;
        while(error > tol) % convergence loop
            for i=2:nx-1  %Space Loop
                for j=2:ny-1
                    term1 = (1+(2*k1)+(2*k2))^-1;
                    V = T(i-1,j) + Told(i+1,j);
                    H = T(i,j-1)+ Told(i,j+1);
                    T(i,j) = (Tprev(i,j)*term1)+(k1*term1*H)+(k2*term1*V);
                end
            end %end of space loop
            error =max(max(abs(Told-T)));
            Told=T;
            iteration = iteration+1;
        end % End of convergence loop
        Tprev= T;
    end% Enf of time loop
    time= toc;
    
    f2 = figure(2)
    contourf(x,y,T,'--','ShowText','on')
    colorbar
    colormap('jet')
    text =sprintf("2D Transient Heat Conduction - Implicit - GS  NO. of iteration %f  Time of simulation %f Sec", iteration, time);
    title(text)
    set(gca,'YDIR','reverse')
    
    
    %%Steady State Gauss Seidal
elseif type ==22
    iteration =1;
    error= 9e9;
    tic;
    while(error > tol ) %convergence loop
        for i=2:nx-1 % Space loop
            for j=2:ny-1
                term1 = T(i,j-1) + Told(i,j+1);
                term2 = T(i-1,j) + Told(i+1,j);
                T(i,j) = (1/4)*(term1 +term2);
            end
        end %end of space loop
        error = max(max(abs(Told-T)));
        Told =T;
        iteration =iteration+1;
    end % End of convergence loop
    time = toc;
    
    f2= figure(2)
    contourf(x,y,T,'--','ShowText','on')
    colormap('jet')
    text = sprintf("2D ASteady State Heat Conduction - GS   No. of iterations %f Time of Simulation %f Sec", iteration,time);
    title(text)
    set(gca,'YDIR','reverse')
   end
end

     

    3.linear_eqn_solver_SOR

    function [iteration] = linear_eqn_solver_SOR(T,Told,Tprev,x,y,nx,ny,nt,k1,k2,tol,type)
%%% Transient Implicit (SOR)
w=1.1;
if type == 13
    iteration =1;
    tic;
    for k =1:nt %time loop
        error=9e9;
        while(error > tol) %convergence loop
            for i=2:nx-1 %space loop
                for j= 2:ny-1
                    term1 = (1+(2*k1)+(2*k2))^-1;
                    V = T(i-1,j) + Told(i+1,j); % Ttop(updated) + Tbottom
                    H = T(i,j-1) + Told(i,j+1); %Tleft(updated) + Tright
                    T_GS(i,j) = (Tprev(i,j)*term1) + (k1*term1*H) + (k2*term1*V);
                    T(i,j) = ((1-w)*Told(i,j)) + (w*T_GS(i,j));
                end
            end % end of space loop
                   error = max(max(abs(Told-T)));
                   Told=T;
                   iteration = iteration+1;
        end %end of convergence loop
        Tprev= T;
    end % end of time loop
     time = toc;
    
     f3 = figure(3)
    contourf(x,y,T,'--','ShowText','on')
    colorbar
    colormap('jet')
    text =sprintf("2D Transient Heat Conduction - Implicit - SOR  NO. of iteration %f  Time of simulation %f Sec", iteration, time);
    title(text)
    set(gca,'YDIR','reverse')
    
    
    %%% Steady State SOR
elseif type == 23
    iteration = 1;
    error=9e9;
    tic;
    while(error > tol) %convergence loop
        for i=2:nx-1 % Space loop
            for j=2:ny-1
                term1 = T(i,j-1) + Told(i,j+1);
                term2 = T(i-1,j) + Told(i+1,j);
                T_GS(i,j) = (1/4)*(term1 + term2);
                T(i,j) = ((1-w)*Told(i,j)) + (w*T_GS(i,j));
            end
        end %end of space loop
        error= max(max(abs(Told-T)));
        Told=T;
        iteration= iteration+1;
    end % end of convergence loop
    time=toc;
    
     f3 = figure(3)
    contourf(x,y,T,'--','ShowText','on')
    colorbar
    colormap('jet')
    text =sprintf("2D Steady State Heat Conduction -SOR NO. of iteration %f Time of simulation %f Sec", iteration, time);
    title(text)
    set(gca,'YDIR','reverse')
end
end

     

    4. linear_eqn_solver_exlpicit

    function linear_eqn_solver_explicit(T,Told,nx,ny,nt,x,y,k1,k2)
 %%Transient Explicit
 iteration= 1;
 tic;
 for k=1:nt %time loop
     for i=2:nx-1 %time loop
         for j=2:ny-1
             term1=(1+(2*k1)+(2*k2))^-1;
             H = Told(i+1,j) + Told(i-1,j);
             V = Told(i,j+1) + Told(i,j-1);
             T(i,j) = (Told(i,j)*term1) + (k1*H) + (k2*V);
         end
     end % end if space loop
     error = max(max(abs(Told-T)));
     Told =T;
     iteration=iteration+1;
 end %end of time loop
 time =toc;
 
 f4=figure(4)
 contourf(x,y, T,'--','ShowText','on')
 colorbar
 colormap('jet')
 text=sprintf("2D Transient Heat Conduction- Explicit- No. of iterations %f Time of Simulation %f Sec",iteration,time);
 title(text)
 set(gca,'YDIR','reverse')
end

     

     

    MAIN CODE:

    clear all
close all
clc

% Inputs
L=1;
nx=11;
x=linspace(0,L,nx);
dx=x(2)-x(1);

W=1;
ny=nx;
y=linspace(0,W,ny);
dy=y(2)-y(1);


prompt = 'type='; %transient case =1 and Steady State =2
type= input(prompt);

if type ==1
    prompt = 'CFL=';
    CFL = input(prompt); % Fails for Explicit if CFL >0.5
                         % CFL CFLx+CFLy = alpha*dt/dx^2 + alpha*dt/dy^2=2*alpha*dt/dx^2
else
    CFL =0;
end
alpha =1.4;

% Finding dt & nt
dt = CFL*dx^2/(2*alpha);
time=.03*60; % mins
nt = time/dt;

% Error Limit Inputs
error=9e9;
tol=1e-4;


%Boundary Conditions
T=300*ones(nx,ny);
T(:,1) = 400; % left
T(:,end) = 800;% right
T(1,:) = 600; %top
T(end,:) = 900;%bottom
T(1,1) = (400+600)/2; 
T(end,end) = (800+900)/2;
T(end,1) = (400+900)/2;
T(1,end) = (600+800)/2;

Told=T;
Tprev=T;
k1=(alpha*dt/(dx^2));
k2=(alpha*dt/(dy^2));

% Calling the Solvers
if type ==1 % Transient case
    linear_eqn_solver_jacobi(T,Told,Tprev,nx,ny,nt,x,y,k1,k2,tol,11);
    linear_eqn_solver_Gs(T,Told,Tprev,nx,ny,nt,x,y,k1,k2,tol,12);
    linear_eqn_solver_SOR(T,Told,Tprev,x,y,nx,ny,nt,k1,k2,tol,13);
    linear_eqn_solver_explicit(T,Told,nx,ny,nt,x,y,k1,k2); % Transient Explicit
elseif type ==2 % Steady State Case
    linear_eqn_solver_jacobi(T,Told,Tprev,nx,ny,nt,x,y,k1,k2,tol,21);
    linear_eqn_solver_Gs(T,Told,Tprev,nx,ny,nt,x,y,k1,k2,tol,22);
    linear_eqn_solver_SOR(T,Told,Tprev,x,y,nx,ny,nt,k1,k2,tol,23);
end

     

    Algorithm:

    1. Firstly we have defined the size of the 2D rectangle onto which we want to calculate the temperature contour. Then divided the geometry into equal grid points or nodes of equal length and width. After that, we calculated the node size, dx & dy which are equal in magnitude that is dx=dy.
    2. Then we accept the type of simulation we want to run that is whether a steady-state or transient state. Pressing"1" signifies transient state for which we take in another user input for CFL number and pressing "2" means steady state.
    3. From CFL number time step dt is automatically calculated. Here the total simulation time is set to 0.03 minutes and from dt and total time, we calculate the total number of time steps,nt.
    4.  In Step 4 we define the boundary conditions which are as follow:                                               Top boundary = 600 k                                                                                                     Bottom Boundary = 900 k                                                                                                 Left Boundary = 400 k                                                                                                     Right Boundary = 800 k
    5. Then as per the type of simulation, the calculations are being done by the 4 user defined functions namely , linear_eqn_solver_jacobi(),linear_eqn_solver_Gs(),linear_eqn_solver_SOR(),linear_eqn_solver_explicit() and print the graphs along with the iteration number and simulation time.
    6. Among this functions for the first 3 linear_eqn_solver_jacobi(),linear_eqn_solver_Gs(),linear_eqn_solver_SOR() solves the Transient state and Steady state is solved by next 3 mention under the type 2.

    The main difference between these solvers is that the discretized equations are solved by different iterative methods like Jacobi, Gauss-seidel, or Successive overrelaxation methods.

     

    OUTPUT:

    1.Transient-State Analysis:

     

     

     

     

    EXPLICIT

     

    2.STEADY-STATE.

     

     

     

     

    Discussion of Results:

    1. In the case of both implicit Transient and Steady-state SOR converges faster than others. SOR>GS>Jacobi. This can will be understood by the iteration number and simulation time.

    2. Between Implicit and Explicit methods of Transient, there is a limitation of explicit methods on the CFL number. The solutions blow up for CFL greater than 0.5

    3. Between the fastest method of the steady-state and transient that is SOR, steady-state converges faster than transient. 

     

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