Volume-averaging method

Another important factor in diffusion measurements that is often encountered in NMR experiments is the effect of time on diffusion coefficients. For example, Kinsey et al. [195] found water diffusion coefficients in muscles to be time dependent. The effects of diffusion time can be described by transient closure problems within the framework of the volume averaging method [195,285]. Other methods also account for time effects [204,247,341]. 

The effective diffusivity depends on the statistical distribution of the pore transport coefficients W j. The derivation shows that the semi-empirical volume-averaging method can only be regarded as an approximation to a more complex dynamic behavior which depends non-locally on the history of the system. Under certain circumstances the long-time (t —> oo) diffusivity will not depend on t (for further details, see [191]). In such a case, the usual Pick diffusion scenario applies. The derivation presented above can, with minor revisions, be applied to the problem of flow in porous media. When considering the heat conduction problem, however, some new aspects have to be taken into accoimt, as heat is transported not only inside the pore space, but also inside the solid phase. 

Roos M., Batawi E., Hamisch U., Hocker Th., 2003. Efficient simulation of fuel cell stacks with the volume averaging method. Journal of Power Sources 118, 86-95. 

Similar dependences for describing the viscosity of emulsions on the basis of the volume averaging method were derived by Mellema and Willemse [60]. They took into account the contribution to the effective viscosity of the interphase tension and obtained a general expression, different from (52) ... 

It is noted that the requirement of proper separation of scales represents the main drawback of the volume averaging method. The constitutive equations used generally depend strongly on this assumption which is hardly ever fulfilled performing simulations of laboratory, pilot and industrial scale reactor units. 

In a similar manner as for the volume averaging method in sect 3.4.1, the stress terms are normally rewritten introducing interfacial mean quantities weighted by the interfacial mass transfer rate (i.e., in the volume averaging method the interfacial area averaged stresses are not weighted by the mass transfer rate). The interfacial mean pressure weighted by the interfacial mass transfer rate per unit surface area becomes ... 

To relate the classical mixture theory to the more familiar volume averaging method we may assume that the mixture CV, which is larger than a phase element but smaller than the characteristic domain dimension, coincides with the averaging volume used in the volume averaging approach. 

The conventional formulas for the interfacial momentum transfer due to phase change are defined in sect 3.3 for the different averaging methods commonly applied in chemical reactor analysis. The modeling concepts usually adopted for the different averages are mathematically similar, so we choose to present a representative interfacial momentum transfer term in the framework of the volume averaging method described in sect 3.4.1. 

Under the volume averaging method, the macroscopic superficial velocity (also called the Darcy velocity) of the fluid over the representative elementary volume can be expressed by taking into account the tortuosity r and the porosity < ) as... 

Here a typical treatment in the model using a volume-averaging method is described to illustrate how this approach works and what sensible solntions in the computation can be derived from a model with temporal and spatial resolntions. Let Nj be the average flux of species i in the pore solution when averaged over the cross-sectional area of the electrode. Thus, for a plane surface, of normal unit vector n cutting the porous solid, n-N represents the amount of species i crossing this plane in the solution phase, but referred to the projected area of the entire plane rather than to the area of an individual phase. The superficial current density I2 in the pore phase is due to the movanent of charged solutes ... 

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