An electronic device is to be cooled in air at 20°C by an array of equally spaced vertical rectangular fins as shown in the sketch below. The fins are made of aluminum and their average temperature, Ts, is 100°C.

Question

Estimate (a) The optimum spacing, s

(b) The number of fins

(c) The rate of heat transfer from one fin

(d) The total rate of heat dissipation

(e) Is the assumption of a uniform fin temperature justified?

GIVEN

? Electronic device with vertical aluminum fins in air

? Air temperature (T?) = 20°C

? Average fin temperature (Ts) = 100°C

FIND

(a) The optimum spacing (s)

(b) The number of fins

(c) The rate of heat transfer from one fin

(d) The total rate of heat dissipation

(e) Is the assumption of a uniform fin temperature justified?

ASSUMPTIONS

? Steady state

? Uniform fin temperature

? The air is still

? Heat transfer from the top and bottom of the fins is negligible

? The heat transfer coefficient on the wall area between the fins is approximately the same as on the fins

PROPERTIES AND CONSTANTS

From Appendix 2, Table 12

For aluminum: Thermal conductivity (kal) = 239 W/(m K) at 100°C

From Appendix 2, Table 28, for dry air at the mean temperature of 60°C

Answer 1

The Grashof number for the fins, based on vertical height of the fin (L) is



Therefore, the Rayleigh number is



(a) The optimum fin spacing (s) is given by Equation (8.56a)



(b) Let n = the number of fins on the device, then



40 fins will fit on the device with optimum spacing.

(c) The average heat transfer coefficient over a fin is given in Table 8.1



The rate of heat transfer from a single fin is



(d) The total rate of heat dissipation is the sum of the heat transfer from the fins and the heat transfer from the wall area between the fins



(e) From Table 2.1, if the heat transfer from the tips of the fins is neglected, the temperature

distribution along each fin is



The temperature change along the fin is



therefore, the assumption of an isothermal fin is justified.