Multiphase mixture model

In eq 51, the first term represents a convection term, and the second comes from a mass flux of water that can be broken down as flow due to capillary phenomena and flow due to interfacial drag between the phases. The velocity of the mixture is basically determined from Darcy s law using the properties of the mixture. The appearance of the mixture velocity is a big difference between this approach and the others, and it could be a reason the permeability is higher for simulations based on the multiphase mixture model. 

The overall gain of the multiphase mixture model approach above is that the two-phase flow is still considered, but the simulations have only to solve pseudo-one-phase equations. Problems can arise if the equations are not averaged correctly. Also, the pseudo-one-phase treatment may not allow for pore-size distribution and mixed wettability effects to be considered. Furthermore, the multiphase mixture model predicts much lower saturations than those of Natarajan and Nguyen - and Weber and Newman even though the limiting current densities are comparable. However, without good experimental data on relative permeabilities and the like, one cannot say which approach is more valid. 

V. Gurau. Response to Comment on A look at the multiphase mixture model for PEM fuel cell simulations . Electrochem. Solid State Lett., 12 S4-S6, 2009. 

MULTIPHASE MODEL OF MIXTURE (MULTIPHASE MIXTURE MODEL)... 

The mathematical model of two-phase flow in PEMFC is often based on the multiphase mixture model (MMM) developed by Wang et al. (1996), Chang et al. (1996) and Wang et al. (1997). The key idea of this model is to focus not on the level of separate phases, but on the level of a multiphase mixture such as mass-averaged mixture velocity. Hence the model need not to track phase interface separating one from two-phase region. The developed formulation based on the MMM is as follows (Wang et al., 2001 and You et al., 2002). In GDL, continuity equation is... 

Cheng, P. and Wang, C.Y., A multiphase mixture model for multiphase, multicomponent transport in capillary porous media-2, Numerical simulation of the transport of organic compounds in the subsurface, Int. J. Heat and Mass... 

In problems in which the dispersed phase momentum equations can be approximated and reduced to an algebraic relation the mixture model is simpler to solve than the corresponding multi-fluid model, however this model reduction requires several approximate constitutive assumptions so important characteristics of the flow can be lost. Nevertheless the simplicity of this form of the mixture model makes it very useful in many engineering applications. This approximate mixture model formulation is generally expected to provide reasonable predictions for dilute and uniform multiphase flows which are not influenced by any wall effects. In these cases the dispersed phase elements do not significantly affect the momentum and density of the mixture. Such a situation may occur when the dispersed phase elements are very small. There are several concepts available for the purpose of relating the dispersed phase velocity to the mixture velocity, and thereby reducing the dispersed... 

This form of the mixture model is called the drift flux model. In particular cases the flow calculations is significantly simplified when the problem is described in terms of drift velocities, as for example when ad is constant or time dependent only. However, in reactor technology this model formulation is restricted to multiphase cold flow studies as the drift-flux model cannot be adopted simulating reactive systems in which the densities are not constants and interfacial mass transfer is required. 

Maiminen M, Taivassalo V, Kallio S (1996) On the mixture model for multiphase flow. Technical Research Center of Finland VIT Publications, Espoo. 

The statistical description of multiphase flow is developed based on the Boltzmann theory of gases [37, 121, 93, 11, 94, 58, 61]. The fundamental variable is the particle distribution function with an appropriate choice of internal coordinates relevant for the particular problem in question. Most of the multiphase flow modeling work performed so far has focused on isothermal, non-reactive mono-disperse mixtures. However, in chemical reactor engineering the industrial interest lies in multiphase systems that include multiple particle t3q)es and reactive flow mixtures, with their associated effects of mixing, segregation and heat transfer. 

In this chapter we study empirical methods of analysis in the context of process units for separating mixtures, just as we studied mathematical modeling in the context of a desalinator. The physical basis for most separations is equilibrium thermodynamics of multiphase mixtures. Universal laws, and even constrained laws, are sometimes not available. Instead, one s analysis is based on empirical data, which is often presented graphically. 

Gas sparging can be modeled using the Eulerian multiphase model or the algebraic slip mixture model. For the Eulerian multiphase model, two sets of momentum equations are used, and the same comments regarding the slip velocity between phases apply, although the issue is not as critical. That is, the velocity data used for the gas phase could be corrected slightly from the liquid-phase velocities but need not be because the gas phase has so little inertia compared to the liquid phase. When the algebraic slip mixture model is used, separate boundary conditions are not required for the individual phases, so a correction of the velocity data is not required. 

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