Transcribed Text
Application to Steady-State Heat Conduction.
Use function for LU decomposition and substitution in order to solve a problem involving 1-D
steady-state heat conduction. Consider a slender bar of length L. On one end (x = 0) it is subjected
to a fixed temperature, To, while at the other end (. = L) it is insulated. There is an
external heat source, with heat per unit length, dc(I). The governing equation for the temperature
distribution, T(x), is
d²T
dT
T(0) = To,
(L) = O,
(4)
da
where 6 is the thermal conductivity. You should obtain a linear set of equations, AT = b, for the
temperatures at the nodes. Your routines must set up the problem-including setting up the matrix A
for the discrete set of equations-and solve it and plot results.
In your study, set L - 1 m and To = 500 K. The thermal conductivity is N = 40W/(m K).
Discretize the problem on a grid with 200 points. The heat source has the form
(5)
Use 40 = 1000W/m². Try two different values for 20 (where both values are between 0 and L) and
two different values of OT (I suggest a = 0.5 and a = 0.05). Remember that the point of using LU
decomposition is that this only needs to be performed once for a given matrix A. Only the forward
and backward substitution need to be repeated when the right-hand side changes.
This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.
%Application to steady state heat conduction
%Clearing the screen, workspace and closing all plots
clear
clc
close all
%%
%Defining parametres
L=1; %L=1m
T0=500; %T0=500K
k=40; %k=40W/(m*K)
points=200;
q0=1000; %q0=1000W/m^2
sigma1=0.5; %sigma=0.5
sigma2=0.05; %sigma=0.05
x01=0.3; %x=0.3m
x02=0.8; %x=0.8m
n=points;
%%...