Light-emitting diodes or LEDs are currently perhaps the most energy-efficient lighting systems. Finned surface heat sinks are used to cool high-intensity LED lighting that are used for spot and/or track lighting systems. A typical circular pin-fin heat sink is shown in the figure, and it is desirable that the fin-base temperature be less than 115°C to ensure efficient performance and longer life of the LED lamp. The fins are made of cold-forged, high-conductivity aluminum (k = 210 W/m K). Each pin has a diameter of 4 mm and a length of 40 mm. If the surrounding air is at 22°C, and it has a convection heat transfer coefficient of ch = 10 W/(m2 K), determine the temperature distribution in the fin, considering convection from the fin tip and the heat transfer rate from the fin. Model it
as a one-dimensional system, use a minimum of 9 nodes (including the ones at the base and tip) or more in your numerical scheme, and determine the effect of extra nodes. Compare the results with the one-dimensional fin analysis of Chapter 2. Also, if 90 such fins are evenly distributed on a 50-mm-diameter circular base, what is the maximum heat transfer rate that is dissipated by this heat sink so that the base temperature is less than 115°C? In a typical LED lamp, approximately 70% of the electric power (or wattage) is dissipated as heat (the remainder 30% is useful light). Determine the maximum power of the lamp, in watts, for which this heat sink is used.
GIVEN
• Light emitting diodes
• Fin base temperature (T0)= 1150C
• Surrounding air temperature (T?)= 220C
• Convection heat transfer coefficient ( ch )= 10 W/(m2 K)
• Thermal conductivity (k)=210 W/(m K)
• Fin diameter (d)= 4 mm = 0.004 m
• Fin length (L)= 40 mm =0.04 m
• Number of nodes (N)=9
FIND (a) Temperature distribution in the fin considering convection from fin tip
(b) Heat transfer rate from the fin.
(c) Maximum heat transfer rate dissipated by 90 such evenly distributed fins
(d) Maximum power of lamp in watts.
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
For the control volume at node i = N, an energy balance gives
In the tridiagonal form this becomes
The heat transferred through the fin is given by
We can solve the above problem by MATLAB by discretization. The boundary conditions for the
discretization are
L=0.04; % in m
N=9;
delx=L/(N-1); % in m
d=0.004; % in m
h=10; % W/(m^2 K)
k=210; % W/(m K)
T(1)=115; % Celsius
T(2:N)=0; % Celsius
Tinf=22; % Celsius
A=pi*d^2/4; % m^2
P=pi*d; % m
for j=1:1:3000
for i=N-1:-1:2
Tf=T;
T(i)=(((delx^2)*P*h*Tinf/(k*A))+T(i+1)+T(i-1))/(2+((delx^2)*P*h/(k*A)));
T(N)=(T(N-
1)+(h*delx/(k*A)*(0.5*P*delx+A)*Tinf))/(1+(h*delx/(k*A)*(0.5*P*delx+A)))
;
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))*A*k/delx+0.5*h*delx*P*(T(1)-Tinf) % in Watts
(a) The following graph is obtained for temperature distribution along the length of fin.
following graph is obtained for temperature distribution along the length of fin.
(b) The heat transfer through the fin is calculated from above program using the
expression
For the exact solution, entry 4 with For the exact solution, entry 4 with
Which is very close to the heat transfer rate obtained by discretization.
(c) If 90 such fins are evenly distributed the heat transfer through 90 fins is given by
(d) If P is the total wattage of the LED lamp, maximum heat dissipated by the lamp is 0.7 P. Thus
0.7*P=42.13 Watt
P= 60 Watt
Thus the wattage of the lamp is 60 Watt.
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