An aluminum pin fin of diameter d = 12 mm, length L = 72 mm, and thermal conductivity k = 204 W m^-1 degree C^-1 is exposed to ambient air at T_infinity = 25 degree C with a convection heat transfer coefficient h = 300 W m^-2 degree C^-1. The left end of the fin is maintained at temperature T_o = 275 degree C. The governing equation of the problem is -d^2 degree/dx^2 + q degree = 0 on 0 < x < L where theta = T – T_infinity is the temperature, and q is given by q = 4h/kd The boundary conditions are theta (0) = T(0) – T_infinity = 250 degree C, (d degree/dx)_x = L + h/k theta (L) = 0 (a) Derive the exact analytic solution for the temperature distribution in the fin. Answer: theta (x) = theta (0) [cos h b(L – x) + (h/bk) sin h b(L – x)]/cos h bL + (h/bk)si nh bL where b = squareroot q. (b) Plot the temperature (primary variable) theta against x. (c) Plot the rate of heat flow (the negative of secondary variable) Q_k = -kAdT/dx against x.
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