This program is a thermal Finite Element Analysis (FEA) solver for steady state heat transfer across 2D plates. The program numerically solves the steady state conduction problem using the Finite Difference Method. After the results are calculated, the program displays a color contour plot of the temperature of the plate after a given time interval.
Results are shown for a 2D fin with specified boundry conditions.
The model assumes 2D conduction with constant properties throughout the simulation. The model takes advantage of the vertical line of thermal symmetry at x = 0.020 m by adding an adiabatic boundary condition along the line of symmetry. The model assumes a specified convection boundary condition on x = 0 and y = Ly. The model assumes constant temperature boundary condition along the fin base.
Contour plot of fin temperature (deg C) for 2D steady-state conduction with Lx = 0.020 m, Ly = 0.200 m, Nx = 20, Ny = 200, fin base at 200 deg C, ambient air at 100 deg C, h = 500 W/m2K, and k = 50 W/mK.
Download the MATLAB program.
% Program:
% Steady_Conduction_With_Finite_Differences.m
% Steady-state 2D conduction solver using finite difference method.
%
% Description:
% Numerically solves the steady-state two dimensional conduction problem
% using the finite difference method and plots color contour plot. Assumes
% steady-state 2D conduction with constant properties. Takes advantage of
% vertical line of thermal symmetry at x = 0.020 m by adding adiabatic BC
% on line of symmetry. Implements specified convection BC on x = 0 and
% y = Ly. Implements constant temperature BC along fin base.
%
% Variable List:
% T = Temperature (deg. Celsius)
% T1 = Boundary condition temperature 1 (deg. Celsius)
% T2 = Boundary condition temperature 2 (deg. Celsius)
% Tinf = Ambient fluid temperature (deg. Celsius)
% theta = Non-dimensionalized temperature difference = (T-T1)/(T2-T1)
% Lx = Plate length in x-direction (m)
% Ly = Plate length in y-direction (m)
% AR = Aspect ratio of Ly / Lx to ensure dx = dy
% h = Convection coefficient (W/m^2K)
% k = Thermal conductivity (W/mK)
% Bi = Finite-difference Biot number
% Nx = Number of increments in x-direction
% Ny = Number of increments in y-direction
% dx = Increment size in x-direction (m)
% dy = Increment size in y-direction (m)
% dT = Temperature step between contours
% tol = Maximum temperature difference for convergence (deg. Celsius)
% pmax = Maximum number of iterations
% Told = Stores temperature matrix for previous time step
% diff = Maximum difference in T between iterations (deg. Celsius)
% i = Current column
% j = Current row
% p = Current iteration
% v = Sets temperature levels for contours
% x = Create x-distance node locations
% y = Create y-distance node locations
% Nc = Number of contours for plot
clear, clc % Clear command window and workspace
Lx = .020; % Plate half-length in x-direction (m)
Ly = .200; % Plate length in y-direction (m)
Nx = 14; % Number of increments in x-direction
AR = Ly/Lx; % Aspect ratio of Ly /Lx to ensure dx = dy
Ny = AR*Nx; % Number of increments in y-direction
dx = Lx/Nx; % Increment size in x-direction (m)
dy = Ly/Ny; % Increment size in y-direction (m)
T1 = 200; % BC temperature at end of fin (deg. Celsius)
T2 = 100; % BC temperature at base of fin (deg. Celsius)
Tinf = T2; % Ambient fluid temperature (deg. Celsius)
h = 500; % Convection coefficient (W/m^2K)
k = 50; % Thermal conductivity (W/m^2K)
Bi = h*dx/k; % Finite-difference Biot number
T = T1*ones(Nx+1,Ny+1); % Initialize T matrix to T1 everywhere
T(1:Nx+1,1) = T1; % Initialize base of fin to T1 BC
tol = 10^-6; % Max temp delta to converge (deg. Celsius)
pmax = 10*10^6; % Maximum number of iterations
x = 0:dx:Lx; % Create x-distance node locations
y = 0:dy:Ly; % Create y-distance node locations
for p = 1:pmax % Loop through iterations
Told = T; % Store previous T array as Told for later
for j = 2:Ny % Loop through rows
for i = 2:Nx % Loop through interior columns
% Calculates convection BC along left side
if i == 2
T(1,j) = (2*T(2,j)+T(1,j-1)+T(1,j+1)+2*Bi*Tinf)/(4+2*Bi);
end
% Calculates interior node temperatures
T(i,j) = .25*(T(i-1,j)+T(i+1,j)+T(i,j-1)+T(i,j+1));
% Calculates adiabatic BC along right side
if i == Nx
T(Nx+1,j) = .25*(2*T(Nx,j)+T(Nx+1,j-1)+T(Nx+1,j+1));
end
end
end
for i = 2:Nx % Loop through interior columns at top row
% Calculates top left corner, conv/conv BC's
if i == 2
T(1,Ny+1) = (T(1,Ny)+T(2,Ny+1)+2*Bi*Tinf)/(2+2*Bi);
end
% Calculates convection BC along top
T(i,Ny+1) = (2*T(i,Ny)+T(i-1,Ny+1)+T(i+1,Ny+1)+2*Bi*Tinf)...
/(4+2*Bi);
% Calculates top right, conv/adiabatic BC's
if i == Nx
T(Nx+1,Ny+1) = (T(Nx+1,Ny)+T(Nx,Ny+1)+Bi*Tinf)/(2+Bi);
end
end
diff = max(max(abs(T - Told))); % Max difference between iterations
fprintf('Iter = %8.0f - Dif. = %10.6f deg. C\n', p, diff);
if (diff < tol) % Exit iteration loop because of convergence
break
end
end
fprintf('Number of iterations = \t %8.0f \n\n', p) % Print time steps
if (p == pmax) % Warn if number of iterations exceeds maximum
disp('Warning: code did not converge')
fprintf('\n')
end
disp('Temperatures in brick in deg. C = ')
for j = Ny+1:-1:1 % Loop through each row in T
fprintf('%7.1f', T(:,j)) % Print T for current row
fprintf('\n')
end
Nc = 50; % Number of contours for plot
dT = (T2 - T1)/Nc; % Temperature step between contours
v = T1:dT:T2; % Sets temperature levels for contours
colormap(jet) % Sets colors used for contour plot
contourf(x*100, y*100, T',v, 'LineStyle', 'none')
colorbar % Adds a scale to the plot
axis equal tight % Makes the axes have equal length
title('Contour Plot of Temperature in deg. C')
xlabel('x (cm)')
ylabel('y (cm)')
pause(0.001) % Pause between time steps to display graph