The thermodynamic significance of the local volume averaged temperature

Since the temperature is not an additive function, the traditional thermodynamic point of view suggests that the volume integral of the temperature has no precise physical meaning. This observation conflicts with the customary analysis of non-isothermal catalytic reactors, heat pipes, driers, geothermal processes, etc., in which the volume averaged temperature plays a crucial role. In this paper we identify the thermodynamic significance of the volume averaged temperature in terms of a simple two-phase heat transfer process. Given the internal energy as a function of the point temperature and the density e(beta) = F (T-beta, rho(beta)), we show that the volume averaged internal energy is represented by [e(beta)](beta) = F([T-beta](beta), [rho(beta)](beta)), when e(beta) is a linear function of T-beta and rho(beta), or when the traditional length-scale constraints associated with the method of volume averaging are satisfied. When these conditions are not met, higher order terms involving the temperature gradient and the density gradient appear in the representation for [e(beta)](beta).