D 2 • MATHEMATICAL FORMULATION 



chamber walls. In regard to steady state problems which are important 

 in cooled units, it is merely noted that, on the one hand, their solutions 

 correspond in general to limiting cases of transient heat flow for long 

 times; on the other hand their solutions may be obtained by special, 

 elementary methods abundantly illustrated in the literature [1,2,3,4,5]. 



In the paragraphs below we summarize the general mathematical 

 formulation of the heat conduction problem with emphasis on those 

 features of the solution which have a special bearing on the particular 

 problems presented in this section. 



D,2. Mathematical Formulation. For our purposes it is sufficient 

 to consider the differential equation of heat conduction in an isotropic 

 two-dimensional domain G without internal heat sources or sinks [6] : 



In this equation T is the temperature, p, c, and k are the density, specific 

 heat, and thermal conductivity of the medium in domain G, while x, y, 

 and t are, respectively, the two rectangular coordinates and the time. 

 As stated in Art. 1, we shall consider heat transfer only by conduction 

 and approximate radiation described by the Newtonian form with a heat 

 transfer coefficient h independent of the temperature or time. Thus 



k^ + h(x, y)[T - T^] = Q on the boundary r of the domain G (2-2) 



where, under the conditions stated, either Tg = flame or gas temperature 

 when h ^ or otherwise h = 0. A derivative with respect to the (outer) 

 normal to the boundary is represented by dT/dn. 

 The initial condition is 



T(x, y, to) = Fix, y) 



where, without loss of generality, we may write 



to = 



Actually we are interested primarily in the cases where F(x, y) = const, 

 corresponding to ambient temperature. The conductivity k, the density p, 

 and the specific heat c may be functions of position as well as of the tem- 

 perature T. The boundary condition may be even more general, as in the 

 case of surface melting. In that case a term involving the heat energy 

 absorbed in the change of state will also appear in the boundary con- 

 dition, while the boundary T itself will be changing with time (cf. [7]). 



It is seen that the general differential equation and boundary con- 

 ditions are nonlinear and, except for some special cases, cannot be dis- 

 cussed in any general manner. We shall discuss below at some length 



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