Intrinsic phase average

Note that the intrinsic phase average and the overall phase average are related by the poros ity or volume fraction of the phases by [423]... 

The average molar flux across some domain in terms of the intrinsic phase average concentration or the phase average concentration is [427]... 

It is more convenient to use an intrinsic phase average for the pressure drop because it is measured that way experimentally. Replacing (Pr) by er(PrY and using the fact that [16],... 

We have rewritten the contribution due to viscous dissipation and chemical reaction in terms of their intrinsic phase averages because they are a more appropriate average for these quantities. For a stationary porous media the first surface integral on the left-hand side does not come into play and the second integral is zero due to the no slip boundary condition. For a moving porous media the two integrals cancel each other. 

Volume average or phase average of a quantity (..Intrinsic phase average over the resin (... ) f Intrinsic phase average over the fiber... 

If t is a constant, then (i /() = ef ie, where e( = VeIV. In dealing with the single-phase flow, the phase average suffices, otherwise the intrinsic phase average is used. 

Hence, this model for the characteristic interstitial velocity as given by equation 74 gives a higher value than the intrinsic phase average interstitial velocity or the interstitial velocity based on the Dupuit s assumption, which is given by... 

Besides the phase average, it is also important to introduce the intrinsic average, which is defined by... 

Hence, the phase average can be related to the intrinsic average by the volume fraction of phase k, c, as... 

Usually, the intrinsic average reflects the real physical property or quantity such as density and velocity, while the phase average gives a pseudoproperty or quantity based on the selection of control volume. Phase averages are used to construct the continuum of each phase to which Eulerian description can be applied. 

Introducing the phase k volume fraction (3.120) and the intrinsic volume average operator (3.118), we get ... 

That is, the interfacial heat transfer per unit volume is modeled as the product of the interfacial area concentration and a mean normal interfacial flux which is proportional to the difference between the interfacial average and the intrinsic volume average of the temperature of phase k. 

Here stands for a-phase volume fraction Sa = Va/V), tf/a represents an arbitrary tensor-valued function, i//aY is the intrinsic a-phase average quantity normalized with respect to its own volume Va, is an arbitrary tensor-valued function representing a surface flux, (Qa) represents the value averaged over the volume V, cJaY represents the intrinsic a-phase average of the tensor indicating the source term, ( ) is the dispersion vector, and is the interfacial area in the averaging volume for the a and (i phases. 

The development of macroscopic equations that involve solid concentrations use the intrinsic volume average over the solid phase in REV, defined as. 

Polymer Crystalline density (Per) Noncrystalline density (Pne) Phase average refractive indices Intrinsic birefringence (X 10 ) ... 

Continuum models remove the difficulties associated with the statistical sampling of phase space, but they do so at the cost of losing molecular-level detail. In most continuum models, dynamical properties associated with the solvent and with solute-solvent interactions are replaced by equilibrium averages. Furthermore, the choice of where the primary subsystem ends and the dielectric continuum begins , i.e., the boundary and the shape of the cavity containing the primary subsystem, is ambiguous (since such a boundary is intrinsically nonphysical). Typically this boundary is placed on some sort of van der Waals envelope of either the solute or the solute plus a few key solvent molecules. 

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