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

AIM:

• To solve the 2D heat conduction problem for the steady and unsteady state using Jacobi, gauss seidel, and successive over-relaxation iterative techniques.

THEORY:

• Conduction is a mode of heat transfer, generally takes place in solids due to temperature differences associated with free electron and lattice vibration.
• A generalized conduction equation is given

• Thermal diffusivity is a thermophysical property of a material defined as the ratio of thermal conductivity of the material and its thermal capacity.
• Since the problem is 2D and there is no heat generation is considered, therefore those terms are neglected. So the equation represents the unsteady 2D heat conduction (parabolic).

For steady-state, time derivative is not considered, their fore below equation represents the 2D steady-state Heat conduction (elliptic).

• Difference equations for the steady-state and unsteady-state 2D conduction equation are derived to obtain the numerical solution.

Steady State 2D heat conduction
Transient 2D heat conduction (explicit)
Transient 2D heat conduction (implicit)

• Here iterative methods are used to solve the difference equation, Iterative methods are numerical techniques used to converge the solution(accurately) of a problem with some initial guess. 
• Here Jacobi, gauss-seidel, and successive over-relaxation techniques are used.

MATLAB CODE:

• There is 2 Matlab code written separately for steady and unsteady conditions.
• But for both conditions inputs are the same
• while programming it is assumed that, the domain is the surface of area (1mX1m). Discretized into 10 nodes both horizontally and vertically.
• Constant values like alpha and time step 'data, by keeping the explicit condition in mind.
• relaxation factor omega is also assumed while programming only.

Inputs

Steady-State 2D Heat conduction

• First inputs are defined into variables and the temperature array is created.
• Boundary conditions are introduced into the temperature array, and at corners, boundary values are averaged.
• Constants like alpha, omega, tolerance are defined to run the iteration.
• Input must be provided by the user, to choose the iterative method.
• The if-else loop is used to solve the equation using the selected(input from the user) iterative technique.
• while loop is used to run the iterative technique till the required accuracy is achieved.
• Two for loop is used to calculate the temperature values at every node at each iterative step.

  clear all
  close all 
  clc
  
  % Domain and dicretization length
  number_node = 10;
  domain_length = 1;
  
  %value of x and y at nodes in a discretization plane
  x = linspace(0,domain_length,number_node);
  y = linspace(0,domain_length,number_node);
  dx = domain_length/(number_node-1);
  dy = domain_length/(number_node-1);
  
  
  % Let define a temperature value at all nodes with initial guess 300k
  T = 300*ones(number_node,number_node);
  
  % Initia boundary conditions at left and right
  T(1:end,1) = 400;
  T(1:end,end) = 800;
  
  % Initia boundary conditions at bottom and top
  T(end,1:end) = 900;
  T(1,1:end) = 600;
  
  %average values of T at corner boundaries
  T(1,1) = 500;
  T(end,1) = 650;
  T(1,end) = 700;
  T(end,end) = 850;
  
  %copy of Temperature to compute to compute
  T_old = T;
  
  %constant
  K = (2*(dx^2+dy^2))/(dx^2*dy^2);
  omegha = 1.2;
  
  % to enter while loop initialize error value randomly
  error = 10;
  tol = 1e-4;
  no_iteration = 0;
  
  method = input('Enter(1) for jacobi,(2) for gauss siediel,(3) for sos');
  if method == 1
      while(error>tol)
          for i = 2:number_node-1
              for j = 2:number_node-1
              term1 = (T_old(i+1,j)+T_old(i-1,j))/dx^2;
              term2 = (T_old(i,j+1)+T_old(i,j-1))/dy^2;
              T(i,j) = (term1+term2)/K;
              end
          end
          error = max(max(abs(T_old-T)));
          
          % to upadte the previous velcity with new velocity
          %got from the iteration
          T_old = T;
          no_iteration = no_iteration+1;
          
      end
  elseif method == 2
      while(error>tol)
          for i = 2:number_node-1
              for j = 2:number_node-1
              term1 = (T_old(i+1,j)+T(i-1,j))/dx^2;
              term2 = (T_old(i,j+1)+T(i,j-1))/dy^2;
              T(i,j) = (term1+term2)/K;
              end
          end
          error = max(max(abs(T_old-T)));
          
          % to upadte the previous velcity with new velocity
          %got from the iteration
          T_old = T;
          no_iteration = no_iteration+1;
      end
      
  else
      while(error>tol)
          for i = 2:number_node-1
              for j = 2:number_node-1
              term1 = (T_old(i+1,j)+T(i-1,j))/dx^2;
              term2 = (T_old(i,j+1)+T(i,j-1))/dy^2;
              T(i,j) = ((1-omegha)*T_old(i,j))+(omegha*((term1+term2)/K));
              end
          end
          error = max(max(abs(T_old-T)));
          
          % to upadte the previous velcity with new velocity
          %got from the iteration
          T_old = T;
          no_iteration = no_iteration+1;
          
      end
  end
  
  % to visualize the result 
  
  [X,Y] = meshgrid(x,y);
  contourf(X,Y,T)
  colorbar
  set(gca,'ydir','reverse')
  ​

• Unsteady-State 2D Heat conduction

  • Inputs are defined into variables and the temperature array is created.
  • Boundary conditions are introduced into the temperature array, and at corners, boundary values are averaged.
  • Constants like alpha, omega, tolerance are defined to run the iteration.
  • Here a user must provide, one or two inputs.
  • The first input is to whether solve the problem using an explicit approach or an implicit approach.
  • If an implicit approach is selected, then the user must provide another input, to select the iterative technique.
  • The if-else loop is used to solve the equation using the selected(input from the user) approach and iterative technique.
  • For loop is used to run the iteration for a large number of times to reach steady-state conditions.
  • while loop is used to run the iterative technique till the required accuracy is achieved.
  • Two for loop is used to calculate the temperature values at every node at each iterative step.
  • Finally, a temperature distribution graph is obtained using 'conturf' command.
• 

• Steady-State analysis

  • Jacobi

  

  • Gauss-Siedel

  

  • Successive over relaxation

  

  Transient analysis

  • Out of a number of iterations,1500 iterations is due to time step
  • Jacobi

  

  • Gauss-Siedel

  

  • Successive over-relaxation

  

  • Explicit
  • 1500 iterations are due to time step
  • According to CFL stability criteria, K1+K2<0>