The electronic controls of a medical imaging device are housed in a compartment such that it dissipates heat to the ambient from a flat vertical plate surface, 15 cm wide and 20 cm high, to which fins are attached so as to enhance the heat transfer and improve the cooling of the electric systems. If 10 fins made of aluminum with a rectangular cross section are attached that extend over the full vertical height of the plate surface, where each fin is 0.15 cm thick with equal spacing of 1.5 cm along the width of the plate, determine the optimum fin height and the rate of heat transfer to the surrounding air. The room air temperature is at 20°C and the base temperature of the plate surface to which the fins are attached should not exceed 70°C.
GIVEN
? An electronic device with vertical aluminum fins in air
? Air temperature (T?) = 20°C
? Vertical Height of fin (H) = 0.2 m
? Width of surface (w) = 0.15 m
? Fin thickness(t) = 0.0015 m
? Maximum base temperatures: Tb= 70°C
? Fin spacing (s)= 1.5 cm=0.015 m
FIND
(a) Optimum fin height perpendicular to surface.
(b) Rate of heat transfer to surrounding air.
ASSUMPTIONS
? The fins are at a uniform temperature equal to the base temperature
PROPERTIES AND CONSTANTS
From Appendix 2, Table 28, for dry air at the mean temperatures for each of the base temperatures
From Appendix 2, Table 12, the thermal conductivity of aluminum in the range of around 70°C
(ka) = 240 (W/(m K))
(a) The boundary layer thickness on a vertical flat plate is given by Equation (8.11b)
Let, L be the optimum fin height.
The fin spacing (s) must be twice the boundary thickness at the top of the fin (x = H) to avoid
boundary layer interference
The average Nusselt number of the fins is given by Equation (8.12b)
The rate of heat transfer is
From fin analysis in chapter 2
Now measuring efficiency for each value of L from Figure (2.21) and substituting the values of L and corresponding nf in above two equation.
Thus the required fin legth is about 12.3 cm.
(b) The rate of heat transfer to the surrounding from total fins is given by Q=
366.5 nf =230 W
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