A heat sink made of an array of fins that have a straight, rectangular cross section is used to cool an electronic micro-chip module, as schematically shown in the figure. The heat sink (base and fins) is made of copper, and each fin is 3 mm thick and 30 mm long. If the fin-base temperature is 85°C and the cooling air is at 25°C with an average convective heat transfer coefficient of ch =100 W/(m2 K), determine the temperature distribution in one fin and the rate of heat transferred per unit width of the fin. If the fins are made of aluminum instead, what is the change in the rate of heat dissipated by the aluminum fin of same thickness? For the numerical analysis, in both the cases, consider a minimum of 11 nodes from the base to tip and convective heat transfer from the fin tip.
Compare the results with the one- dimensional fin analysis
GIVEN
• Heat sink made of an arry of straight rectangular cross section fins.
• Fin base temperature (T0)= 850C
• Surrounding air temperature (T?)= 250C
• Convection heat transfer coefficient ( ch )= 100 W/(m2 K)
• Fin thickness(t)= 3 mm = 0.003 m
• Fin length (L)= 30 mm =0.03 m
• Number of nodes (N)=11
FIND
(a) Temperature distribution in one fin considering convection from fin tip
(b) Heat transfer rate per unit width of fin
(c) Rate of change in heat dissipated if Aluminum of same thickness is used.
PROPERTIES
• Thermal conductivity of copper(kCu)= 396 W/(m K)
• Thermal conductivity of Aluminum (kAl)=238 W/(m K)
SKETCH
For the node and control volume arrangement shown in the sketch, we have
For the control volume at i = 1, we have a specified temperature, therefore
For the interior control volumes, i = 2, 3, 4, 5,... , 8 an energy balance gives
Writing this in the tridiagonal form
Since W>>t
P=2(W+t) 2W
Cross sectional Area (A)= W*t
For the control volume at node i = N, an energy balance gives
In the tridiagonal form this becomes
The heat transferred through the fin per unit width is given by
We can solve the above problem by MATLAB by discretization. The boundary conditions for the
discretization are
L=0.03; % in m
N=11;
delx=L/(N-1); % in m
t=0.003; % in m
h=100; % W/(m^2 K)
k=396; % W/(m K)
T(1)=85; % Celsius
T(2:N)=0; % Celsius
Tinf=25; % Celsius
for j=1:1:3000
for i=N-1:-1:2
Tf=T;
T(i)=((2*(delx^2)*h*Tinf/(k*t))+T(i+1)+T(i-1))/(2+(2*(delx^2)*h/(k*t)));
T(N)=(T(N-
1)+(h*delx/(k*t)*(delx+t)*Tinf))/(1+(h*delx/(k*t)*(2*delx+t)));
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
for i=1:1:N
x(i)=(i-1)*delx;
end
plot(x,T)
q=(T(1)-T(2))*t*k/delx+h*delx*(T(1)-Tinf) % in Watts
The following temperature distribution is obtained along the length of fin from above program.
The heat transfer per fin is calculated as per equation
When Aluminum fin is used instead of Copper
Replacing thermal conductivity (k) in above program we get
For exact solution entry 4 with For the exact solution use entry 4 with
Which is close to the heat transfer rate obtained by discretization.
You might also like to view...
The Moon's crust is ________ km thick on the Earth side and ________ km thick on the far side
Fill in the blank(s) with correct word
Daisy throws a rock horizontally with a speed of 12.0 m/s from a bridge. It falls 2.28 s before reaching the water below
(a) When the rock hits the water it is traveling with what horizontal component of velocity? (b) What was the total speed just before hitting the water?
Two infinite parallel surfaces carry uniform charge densities of 0.20 nC/m2 and -0.60 nC/m2 . What is the magnitude of the electric field at a point between the two surfaces?
a. 34 N/C b. 23 N/C c. 45 N/C d. 17 N/C e. 90 N/C
Stars only exhibit a Doppler shift if they are moving
A) away from us along the line of sight B) perpendicular to our line of sight C) in a circular motion about us D) toward or away from us along the line of sight