% _________________________________________________________________________ % To study heat flow in one-dimensional system (insulated rod) % ie finite difference solution to parabolic equation % By Mahesha MG, MIT % Date: 02/05/2013 % _________________________________INPUT___________________________________ clc; clear; k1=input('Enter thermal conductivity of the material of rod: '); cp=input('Enter specific heat of the material of rod: '); rho=input('Enter density of the material of rod: '); l=input('Enter length of the rod: '); h=input('Enter step length: '); Tl=input('Enter temperature at left side of the rod: '); Tr=input('Enter temperature at right side of the rod: '); tmax=input('Enter tmax: '); f1=input('Enter initial condition (a valid MATLAB expression in x): ','s'); % ___________________________INITIAL CONDITION_____________________________ k=k1/(cp*rho); tou=h^2/(2*k); %time step r=(tou*k)/h^2; x=0:h:l; temp=eval(f1); temp(1)=Tl; temp(size(x))=Tr; temp2=temp; temp1=zeros(size(temp)); temp1(1)=Tl; temp1(size(x))=Tr; t0=0; % _________________________FINITE DIFFERENCE METHOD________________________ while t0<tmax t0=t0+tou; for i=2:size(x')-1 temp1(i)=0.5*(temp2(i-1)+temp2(i+1)); end temp2=temp1; temp=[temp;temp1]; end temp %temperature at different interior points with time t=0:tou:tmax; surf(x,t,temp) xlabel('length in m') ylabel('Time in s') zlabel('Temperature in C') % _________________________SAMPLE INPUT AND RESULT_________________________ % Enter thermal conductivity of the material of rod: 2 % Enter specific heat of the material of rod: 1 % Enter density of the material of rod: 1 % Enter length of the rod: 4 % Enter step length: 1 % Enter temperature at left side of the rod: 0 % Enter temperature at right side of the rod: 0 % Enter tmax: 1.5 % Enter initial condition (a valid MATLAB expression in x): 50*(4-x) % % temp = % % 0 150.0000 100.0000 50.0000 0 % 0 50.0000 100.0000 50.0000 0 % 0 50.0000 50.0000 50.0000 0 % 0 25.0000 50.0000 25.0000 0 % 0 25.0000 25.0000 25.0000 0 % 0 12.5000 25.0000 12.5000 0 % 0 12.5000 12.5000 12.5000 0 % Problem taken from Numerical Methods by E Balagurusamy % _________________________________________________________________________
