Conduction, heat problem, variational formulation

Biot and Daughaday (B6) have improved an earlier application by Citron (C5) of the variational formulation given originally by Biot for the heat conduction problem which is exactly analogous to the classical dynamical scheme. In particular, a thermal potential V, a dissipation function D, and generalized thermal force Qi are defined which satisfy the Lagrangian heat flow equation... 

When the radial variation of temperature must be taken into account, the problem assumes an entirely different character. Each of the equations is now a partial differential equation, and both radial and axial profiles must be calculated a mesh or network of radial and axial lines is set up, and the temperature and composition are calculated for each intersection. A great deal of work has been done on the formulation of difference equations for solving the related diffusion or heat-conduction equations most of this has been directed towards the case in which there is only one dependent variable and in which the source is a linear function of that variable. Although the results obtained for one dependent variable are only partially applicable to the multiple-variable problem,... 

In Section S-3 we considered one-dimensional heat conduction and assumed heat conduction in other directions to be negligible. Many heat transfer problems encountered in practice can be approximated as being one-dimensional, but this is not always the case. Sometimes we need to consider heat transfer in other directions as well when the variation of temperature in other directions is significani. In this section we consider the numerical formulation and solution of two-dimensional steady lieat conduclion in rectangular coordinates using the finite difference method. The approach presented below can be extended to three-dimensional cases. 

The finite difference formulation of transieiii heat conduction problems is based on an energy balance that also accounts for tire variation of the energy content of the volume element during a time interval At. The heat transfer and heat generation terms are expressed at the previous time. step fin the explicit method, and at the new time step i I 1 in the implicit method. For a general node III, the finite difference formulations are expressed as... 

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