example 4.3 script
%%HTProjEX4point3
clear
clc
% given values
k = 25; % W/m-K
dx = 0.001; % m
h_outer = 1000; % W/m^2-K
h_inner = 200; % W/m^2-K
T_outer = 1700; % K
T_inner = 400; % K
% blank matrix A and constant vector C
A = zeros(21,21);
C = zeros(21,1);
% row then column
% node 1
A(1,1) = -(2+(h_outer*dx/k));
A(1,2) = 1;
A(1,7) = 1;
C(1) = -(h_outer*dx/k)*T_outer;
% node 2
A(2,1) = 1;
A(2,2) = -2*((h_outer*dx/k) + 2);
A(2,3) = 1;
A(2,8) = 2;
C(2) = -2*(h_outer*dx/k)*T_outer;
% node 3
A(3,2) = 1;
A(3,3) = -2*((h_outer*dx/k) + 2);
A(3,4) = 1;
A(3,9) = 2;
C(3) = -2*(h_outer*dx/k)*T_outer;
% node 4
A(4,3) = 1;
A(4,4) = -2*((h_outer*dx/k) + 2);
A(4,5) = 1;
A(4,10) = 2;
C(4) = -2*(h_outer*dx/k)*T_outer;
% node 5
A(5,4) = 1;
A(5,5) = -2*((h_outer*dx/k) + 2);
A(5,6) = 1;
A(5,11) = 2;
C(5) = -2*(h_outer*dx/k)*T_outer;
% node 6
A(6,5) = 1;
A(6,6) = -(2+(h_outer*dx/k));
A(6,12) = 1;
C(6) = -2*(h_outer*dx/k)*T_outer;
% node 7
A(7,1) = 1;
A(7,7) = -4;
A(7,8) = 2;
A(7,13) = 1;
C(7) = 0;
% node 8
A(8,2) = 1;
A(8,7) = 1;
A(8,8) = -4;
A(8,9) = 1;
A(8,14) = 1;
C(8) = 0;
% node 9
A(9,3) = 1;
A(9,8) = 1;
A(9,9) = -4;
A(9,10) = 1;
A(9,15) = 1;
C(9) = 0;
% node 10
A(10,4) = 1;
A(10,9) = 1;
A(10,10) = -4;
A(10,11) = 1;
A(10,16) = 1;
C(10) = 0;
% Node 11
A(11,5) = 1;
A(11,10) = 1;
A(11,11) = -4;
A(11,12) = 1;
A(11,17) = 1;
% node 12
A(12,6) = 1;
A(12,11) = 2;
A(12,12) = -4;
A(12,18) = 1;
C(12) = 0;
% node 13
A(13,7) = 1;
A(13,13) = -4;
A(13,14) = 2;
A(13,19) = 1;
C(13) = 0;
% node 14
A(14,8) = 1;
A(14,13) = 1;
A(14,14) = -4;
A(14,15) = 1;
A(14,20) = 1;
C(14) = 0;
% node 15
A(15,9) = 2;
A(15,14) = 2;
A(15,15) = -2*(3+(h_inner*dx/k));
A(15,16) = 1;
A(15,21) = 1;
C(15) = -2*(h_inner*dx/k)*T_inner;
% node 16
A(16,10) = 2;
A(16,15) = 1;
A(16,16) = -2*((h_inner*dx/k) + 2);
A(16,17) = 1;
C(16) = -2*(h_inner*dx/k)*T_inner;
% node 17
A(17,11) = 2;
A(17,16) = 1;
A(17,17) = -2*((h_inner*dx/k) + 2);
A(17,18) = 1;
C(17) = -2*(h_inner*dx/k)*T_inner;
% node 18
A(18,12) = 1;
A(18,17) = 1;
A(18,18) = -(2+(h_inner*dx/k));
C(18) = -(h_inner*dx/k)*T_inner;
% node 19
A(19,13) = 1;
A(19,19) = -2;
A(19,20) = 1;
% node 20
A(20,14) = 1;
A(20,19) = 1;
A(20,20) = -4;
A(20,21) = 2;
% node 21
A(21,15) = 1;
A(21,20) = 1;
A(21,21) = -(2+(h_inner*dx/k));
C(21) = -2*(h_inner*dx/k)*T_inner;
% solve for nodal temperatures
T = A \ C;
% results
disp('Nodal temperatures for Example 4.3 (K):')
for n = 1:21
fprintf('T%-2.0f = %8.2f K\n', n, T(n))
end
example 5.11
light bulb
STEADY STATE
%%V3NodeChad
clear
clc
% given values
k_fil = 30; % W/m-K
k_air = 0.026; % W/m-K
k_glass = 1.4; % W/m-K
r = 2.286*10^-5; %m
L = 0.5334; %m
% P = 57;
% Volume = pi*(r^2)*L;
% q_dot = P/Volume;
b = .01; % m
dx = 0.002; % m
V_gen = 3 b(dx)^2; % nodes 1–3 volume (unit depth)
q_dot = 40 / V_gen; % use bulb power, NOT filament volume
%q_dot = 0;
h_conv = 10; % W/m^2-K (example)
epsilon_glass = 0.90; % assumption, choose your value
sigma = 5.67e-8; % W/m^2-K^4
T_surface = 295; % K
T_surroundings = 295; % K
T_base = T_surroundings;
dy = dx;
h_rad = epsilon_glass 5.67e-8 (T_surface + T_surroundings) * (T_surface^2 + T_surroundings^2);
T_inf = T_surroundings;
h_outer = 1000; % W/m^2-K
h_inner = 200; % W/m^2-K
T_outer = 1700; % K
T_inner = 400; % K
% blank matrix A and constant vector C
% row x column
A = zeros(15,15);
C = zeros(15,1);
% node 1
A(1,1) = -(3*k_fil+k_air);
A(1,2) = k_fil;
A(1,7) = k_air;
C(1) = -2*k_fil*T_base - q_dot*(dx)^2 ;
% node 2
A(2,1) = k_fil;
A(2,2) = -(2*k_fil + k_air);
A(2,3) = k_fil;
A(2,8) = k_air;
C(2) = -q_dot*(dx)^2;
% node 3
A(3,2) = k_fil;
A(3,3) = -(k_fil + 2*k_air);
A(3,4) = k_air;
A(3,9) = k_air;
C(3) = -q_dot*(dx)^2;
% node 4
A(4,3) = k_fil;
A(4,4) = -(k_air + k_glass + k_fil);
A(4,5) = k_glass;
A(4,10) = k_air;
C(4) = 0;
% node 5
A(5,4) = k_air;
A(5,5) = -(k_air + 2*k_glass);
A(5,6) = k_glass;
A(5,11) = k_glass;
C(5) = 0;
% node 6
A(6,5) = k_glass;
A(6,6) = -(k_glass + 2*(h_conv + h_rad)*dx);
C(6) = -2*dx*(h_conv + h_rad)*T_inf;
% node 7
A(7,1) = k_fil;
A(7,7) = -(k_fil + 3*k_air + k_glass);
A(7,8) = k_air;
A(7,12) = k_glass;
C(7) = -2*k_air*T_base;
% node 8
A(8,2) = k_fil;
A(8,7) = k_air;
A(8,8) = -(2*k_air + k_fil + k_glass);
A(8,9) = k_air;
A(8,13) = k_glass;
C(8) = 0;
% node 9
A(9,3) = k_fil;
A(9,8) = k_air;
A(9,9) = -(2*k_air + k_fil + k_glass);
A(9,10) = k_air;
A(9,14) = k_glass;
C(9) = 0;
% node 10
A(10,4) = k_air;
A(10,9) = k_air;
A(10,10) = -(k_glass + 3*k_air);
A(10,11) = k_air;
A(10,15) = k_glass;
C(10) = 0;
% Node 11
A(11,5) = k_glass;
A(11,10) = k_air;
A(11,11) = -(k_air + k_glass + 2*(h_conv + h_rad)*dy);
C(11) = -2*(h_conv + h_rad)*dy*T_inf;
% node 12
A(12,7) = k_air;
A(12,12) = -(3*k_air + k_glass + (h_conv+h_rad)*dx);
A(12,13) = k_glass;
C(12) = -(h_conv + h_rad)*dy*T_inf - (2*k_air*T_base);
% node 13
A(13,8) = k_air;
A(13,12) = k_glass;
A(13,13) = -(k_air + 2*k_glass + (h_conv + h_rad)*dx);
A(13,14) = k_glass;
C(13) = -(h_conv + h_rad)*dy*T_inf;
% node 14
A(14,9) = k_air;
A(14,13) = k_glass;
A(14,14) = -(k_air + 2*k_glass + (h_conv + h_rad)*dx);
A(14,15) = k_glass;
C(14) = -(h_conv + h_rad)*dy*T_inf;
% node 15
A(15,10) = k_air;
A(15,14) = k_glass;
A(15,15) = -(k_glass + k_air + 2*(h_conv + h_rad)*dy);
C(15) = -2*((h_conv + h_rad)*dy)*T_inf;
% solve for nodal temperatures
T = A \ C;
% Temperatures for each Node
disp('Nodal temperatures for Light Bulb (K):')
for n = 1:15
fprintf('T%-2.0f = %8.2f K\n', n, T(n))
end
% ===== IMPROVED TEMPERATURE CONTOUR PLOT =====
% Create nodal layout
Tplot = zeros(3,6);
Tplot(3,1:4) = [T(12) T(13) T(14) T(15)];
Tplot(2,1:5) = [T(7) T(8) T(9) T(10) T(11)];
Tplot(1,1:6) = [T(1) T(2) T(3) T(4) T(5) T(6)];
% Original grid
[x,y] = meshgrid(1:6,1:3);
% Fine grid for interpolation
[xq,yq] = meshgrid(linspace(1,6,100), linspace(1,3,100));
% Interpolate (stable method)
Tinterp = griddata(x, y, Tplot, xq, yq, 'linear');
% Plot
figure
contourf(xq,yq,Tinterp,30,'LineColor','none')
colorbar
colormap(jet)
caxis([290 500]) % lock to physical range
title('Temperature Distribution (K)')
xlabel('Node Position (x)')
ylabel('Node Position (y)')
set(gca,'YDir','normal')
axis equal
grid on
hold on
contour(xq,yq,Tinterp,15,'k')
hold off
TRASIENT - EXPLICENT