Using your results from Problem 4.8, find the heat flow at the base of the fin for the following conditions:
Use a grid spacing of 0.5 cm.
GIVEN
A fin with variable cross-sectional area and perimeter
FIND
(a) Heat flow rate for conditions given above
and the perimeter at any node is
Heat transfer at the fin root is
The difference equation as derived in
The boundary conditions can be written as
This can be written in the form of a tridiagonal matrix, per Equation (4.10) where the coefficients of
the matrix are
This set of equations can be easily solved using the matrix inversion function of a spreadsheet
The above problem is also solved using matlab. The matlab code for the problem is given below
L=0.05; % in m
N=11;
delx=L/(N-1); % in m
h=110; % W/(m^2 K)
k=34; % W/(m K)
T(1)=93; % in celsius
T(2:N)=0; % in celsius
Tinf=27; % in celsius
for l=1:1:N
A(l)=3.23*10^-4*(1-sinh((l-1)*delx/L)/3);
P(l)=A(l)^0.5;
end
for j=1:1:3000
for i=N-1:-1:2
Tf=T;
T(i)=(((delx^2)*P(i)*h*Tinf/k)+(T(i+1)*A(i+1))+(T(i-
1)*A(i)))/(A(i)+A(i+1)+((delx^2)*P(i)*h/k));
T(N)=T(N-1);
end;
count=0;
for i=1:N
if abs(Tf(i)-T(i))<=10^-4
count=count+1;
end
end
if count==N
break
end
end
q=(T(1)-T(2))*A(1)*k/delx % in Watt
From the above MATLAB program we get q=4.884 Watt
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