A Problem in Cooling Fin Design

We consider two related one-dimensional steady-state problems of heat conduction/radiation in a cooling fin. In both problems the solution consists of the value of a parameter (length of the fin) and a monotone temperature distribution in the fin. Three boundary conditions are accordingly imposed: temperature at both ends of the fin and heat flux at one end. We show that these problems have solutions only for data satisfying certain constraints, and we obtain sufficient conditions for existence. Good estimates for the required fin length are also provided. Introduction and formulation. Consider a cooling fin (or spine, or an array of stacked fins or spines) projecting into the region 0 < z < a from an object whose edge lies at z = 0. We suppose that the fin is to conduct away from z = 0 a specified amount of heat per unit time and that the object maintains a temperature T at z = 0; thus if U denotes the temperature in the fin we have the boundary conditions % = U\z=o = T. U* 2=0 For the steady state we have U"(z)-<p(z,U(*),U'(z),a) = 0, where ip represents heat radiation from the fin; specifically, ip(z,U,U', a) denotes heat radiation per unit length of fin located at position z E [0, a] when the temperature at z is U, the temperature gradient is U', and the fin occupies the region 0 < z < a. Our goal is to establish the existence of a fin length a such that the temperature at the end z = a of the array is a given value Tq < T. This third boundary condition may come about either because the fin is to be connected to a heat reservoir at temperature To or because it is desired to have the end of the fin at a safely cool temperature. (In the latter case we tacitly assume that the rate of heat loss through this end of the fin is small enough to be ignored, except in the last section.) Moreover, we require a fin length for which Received March 11, 1997. 1991 Mathematics Subject Classification. Primary 34B99.