Energy local volume averaged equation

Local Volume Averaging. The local volume-averaging treatment leading to the coupling between the energy equation for each phase is formulated by Carbonell and Whitaker [81] and is given in Zanotti and Carbonell [82], Levee and Carbonell [83], and Quintard et al. [84]. Their development for the transient heat transfer with a steady flow is reviewed here. Some of the features of their treatment are discussed first. 

Inclusion of the pore-level (or particle-based) hydrodynamics along with the appropriate volume averaging allows for the inclusion of the local variation of D and D into the energy equations. In principle, these variations can only be included if the change from the bulk value to zero at the surface takes place over several representative elementary volumes. Otherwise it will not be in accord with the volume averaging. 

A local thermodynamic state is determined as elementary volumes at individual points for a nonequilibrium system. These volumes are small such that the substance in them can be treated as homogeneous and contain a sufficient number of molecules for the phenomenological laws to be applicable. This local state shows microscopic reversibility that is the symmetry of all mechanical equations of motion of individual particles with respect to time. In the case of microscopic reversibility for a chemical system, when there are two alternative paths for a simple reversible reaction, and one of these paths is preferred for the backward reaction, the same path must also be preferred for the forward reaction. Onsager s derivation of the reciprocal rules is based on the assnmption of microscopic reversibUity. The reversibility of molecular behavior gives rise to a kind of symmetry in which the transport processes are coupled to each other. Although a thermodynamic system as a whole may not be in equUibrium, the local states may be in local thermodynamic equilibrium, all intensive thermodynamic variables become functions of position and time. The local equilibrium temperature is defined in terms of the average molecular translational kinetic energy within the small local volume. 

One consequence of the continuum approximation is the necessity to hypothesize two independent mechanisms for heat or momentum transfer one associated with the transport of heat or momentum by means of the continuum or macroscopic velocity field u, and the other described as a molecular mechanism for heat or momentum transfer that will appear as a surface contribution to the macroscopic momentum and energy conservation equations. This split into two independent transport mechanisms is a direct consequence of the coarse resolution that is inherent in the continuum description of the fluid system. If we revert to a microscopic or molecular point of view for a moment, it is clear that there is only a single class of mechanisms available for transport of any quantity, namely, those mechanisms associated with the motions and forces of interaction between the molecules (and particles in the case of suspensions). When we adopt the continuum or macroscopic point of view, however, we effectively spht the molecular motion of the material into two parts a molecular average velocity u = (w) and local fluctuations relative to this average. Because we define u as an instantaneous spatial average, it is evident that the local net volume flux of fluid across any surface in the fluid will be u n, where n is the unit normal to the surface. In particular, the local fluctuations in molecular velocity relative to the average value (w) yield no net flux of mass across any macroscopic surface in the fluid. However, these local random motions will generally lead to a net flux of heat or momentum across the same surface. 

Assuming that under the specified conditions the local rate of energy dissipation can be replaced by the average rate P/V (power consumption per unit stirred volume) for the whole tank, equations 5 and 8 give... 

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