Interfacial momentum transfer

It should be highlighted that equation (47) holds for solid particles. In the case of liquid particles, e.g. with emulsions, the convective diffusion process is very different due to interfacial momentum transfer which gives rise to a different velocity profile. Consequently, convective diffusion to/from a liquid particle is more effective than that for a solid particle. Starting again from equation (43),... 

The interfacial momentum transfer due to phase change is defined by ... 

It has become customary to rewrite the interfacial momentum transfer term M.J in terms of the interfacially averaged pressure pk)Ai and shear stresses Ck)Ai of phase k to separate the mean field effects from local effects [233, 194, 112, 54, 56, 153, 154, 193]. In the volume averaging approach the interfacial area averaged pressure is defined by ... 

Besides, by making use of (3.130), introducing the interfacial momentum transfer term given by (3.149), and defining the generalized drag term by ... 

For wall bounded flows the wall interaction forces have a similar origin as the interfacial momentum transfer terms [129]. These terms are assessed in further details in sect 3.4.6. 

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. 

In fluid mechanics the interfacial momentum transfer due to phase change (3.148) is normally expressed in terms of interfacial mass flux weighted quantities... 

For catalytic solid surfaces in packed beds and porous materials the speed of displacement of the interface is assumed to be zero, vi-Hk =0. For this reason the interfacial momentum transfer due to phase change can be reduced to ... 

This relation can be reformulated adopting one out of several possible modeling approaches. The conventional continuum mechanical approach for rewriting the interfacial momentum transfer terms for dispersed flows was outlined in sect 3.4.3. Hence, an alternative approach for calculating the interfacial momentum transfer terms based on kinetic or probabilistic theories, as proposed by Simonin and co-workers, is examined in this section. 

He Simonin [65] argued that to find a relation for the drag force acting on a single sphere in a suspension, the velocity field of the undisturbed flow is needed. They derived a momentum equation for an undisturbed flow based on probabilistic arguments. Based on the momentum equations for the disturbed and undisturbed flow, they derived an expression for the interfacial momentum transfer. The interfacial momentum transfer term was thus decomposed as follows ... 

The interfacial momentum transfer terms for the disturbed flow are approximated by the steady drag force ... 

The resulting decomposition of the interfacial momentum transfer term is equivalent to the conventional closure outlined in sect 3.4.3, and adopted by several investigations on gas solids flow [64, 65, 39, 108]. Nevertheless, as for the conventional formulation, several simplifying assumptions are invoked in... 

The surface tension force was neglected so the interfacial momentum transfer terms satisfy ... 

The first three terms on the right-hand side of Eq. (5.6) account for the diffusion of turbulence dissipation, the mean flow velocity gradient production term, and the homogeneous dissipation term. The last group of terms in Eq. (5.6) describes the effect of interfacial momentum transfer on the production of turbulence dissipation. The time constant is given by the following empirical correlation ... 

In order to close the two-fluid model, constitutive equations are required for (i) stresses (Table 4.2), (ii) internal heat transfer (Table 4.3), (iii) internal mass transfer (Table 4.4), (iv) interfacial heat transfer (Table 4.5), (v) interfacial momentum transfer (Table 4.6), and (vi) solid phase collision pressure (Table 4.7). To solve the mathematical model, the finite volume discretization technique was employed. 

Table 4.6 Constitutive equations for interfacial momentum transfer ...

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