Chapter 12, Problem 12.133P

An annular fin of thickness t is used as a radiator to dissipate heat for a space power system. The fin is insulated on the bottom and may be exposed to solar irradiation $G_s$ . The fin is coated with a diffuse, spectrally selective material whose spectral reflectivity is specified

Heat is conducted to the fin through a solid rod of radius $r$ , and the exposed upper surface of the fin radiates to free space, which is essentially at absolute zero temperature.

(a) If conduction through the rod maintains a fin base temperature of $T\left(r_i\right)=T_b=400k$ and the fin efficiency is 100%, what is the rate of heat dissipation for a fin of radius $r_o=0.5\text{ m}$ ? Consider two cases, one for which the radiator is exposed to the sun with $G_s=1000{\text{ W/m}}^2$ and the other with no exposure $\left(G_s=0\right)$ .

(b) In practice, the fin efficiency will be less than 100% and its temperature will decrease with increasing radius. Beginning with an appropriate control volume, derive the differential equation that determines the steady-state, radial temperature distribution in the fin. Specify appropriate boundary conditions.