Further Developments of New Methods in Heat-Flow Analysis

Abstract

SUMMARY Lagrangian methods in heat-flow problems and transport phenomena were introduced by the writer in some previous work. The present paper develops further one particular aspect of the method,-i.e., the elimination of “ignorable coordinates.” This is accomplished by a special choice of generalized coordinates, each of which is constituted by an arbitrary temperature distribution and an “associated flow field.” The latter is a vector field which is derived from the corresponding scalar field by a variational method. The procedure is valid for a certain class of nonlinear problems, provided we replace the temperature by the heat content as the unknown. It is shown that for normal coordinates derivation of the associated flow field is immediate. The use of normal coordinates and their associated flow fields is illustrated by an example. Introduction of Dirac functions and associated flow fields yields a procedure which constitutes a generalization of the classical formulation by Green’s functions and integral equations. This is illustrated by application to onedimensional problems of heating of a homogeneous or composite slab and directly verified by classical methods in the Appendix. N A PREVIOUS PUBLICATION~~~ haveintroducednew methods in heat-flow analysis. These methods have a two-fold basis; first, a new formulation of the thermodynamics of irreversible processes, and, second, the application of Lagrangian techniques to the mathematical analysis itself. The earlier developments were carried out in references 2, 3, and 4. We have recently reviewed in more detail in reference 5 the basic thermodynamic concepts and principles. The formulation of dissipative phenomena into a variational language can be achieved in many ways.$ The particular method which we have chosen is different from the classical variational approaches. It appears to be the most general and fits into a unified thermodynamic theory embracing a large category of physical phenomena, which leads to equations of the same type as in Lagrangian mechanics. A particularly useful concept which has been introduced is the generalized thermodynamic force by a @inciple of virtual work. In the case of thermal problems this results in a representation of the physical system by means of a vectorial flow field and to the use of a generalized thermal force to represent the externally applied tem