Equation multiphase mixture

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. 

We propose the balance principles for an immiscible mixture of continua with microstructure in presence of phenomena of chemical reactions, adsorption and diffusion by generalizing previous multiphase mixture [9] and use a new formulation for the balance of rotational momentum. New terms are also included in the energy equations corresponding to work done by respective terms in the micromomentum balances. 

In this section the classical continuum theory of mixtures is reviewed [15]. In this concept the multiphase mixture is treated as a single homogeneous continuum. Thereby the balance principle can be applied to derive conservation laws for the macroscopic pseudo-fluid in analogy to the single phase formulation examined in chap 1. Approximate constitutive equations are postulated for the expected macroscopic behavior of the phases. 

For reactive flows in packed beds a set of cross sectional average balance equations is written for the gas-solid multiphase mixture [3, 5]. 

In en equilibrated, multiphase mixture at temperature Tand pressure P, foreveiy component r, (he fugacity /, must be the same In all phases. Therefore, if a method is available for calculating /, for phase as a function of temperature, pressure, and the phase s composition, (hen it is possible to calculate ail equilibrium ocxnpositjoua at any desired temperature and pressure. When all phases are fluids, such a method is given by an equation of state. 

In studying various processes in multiphase mixtures, the scientists usually assume that the size of inclusions in a mixture (particles, drops, bubbles, the pores in the porous mediums) is much greater than the size of the molecules. This assumption named the continuity hypothesis, allows us to use the mechanics of continuous mediums for description of processes occurring inside or near the separate inclusions. For description of physical properties of phases, such as viscosity, heat conductivity etc., it is possible to use equations and parameters of an appropriate single-phase medium. 

Let us now turn to the energy-equation for multiphase mixture and introduce specific energy as a sum of specific kinetic and internal energies ... 

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... 

First a derivative is given of the equations of change for a pure fluid. Then the equations of change for a multicomponent fluid mixture are given (without proof), and a discussion is given of the range of applicability of these equations. Next the basic equations for a multicomponent mixture are specialized for binary mixtures, which are then discussed in considerably more detail. Finally diffusion processes in multicomponent systems, turbulent systems, multiphase systems, and systems with convection are discussed briefly. 

The full extent and variety of the phase behavior for water-isopropanol-C02 mixtures observed experimentally and calculated with the Peng-Robinson equation of state was not anticipated based on known phase behavior for the constituent binary mixtures or similar ternary mixtures. These results suggest that multiphase behavior for related model surfactant systems could also be complex. Measurements of all the critical endpoint curves, the tricritical points, and secondary critical endpoint for such systems would be tedious and are extremely difficult. However, by coupling limited experimental data with a thermodynamic model based on this cubic equation of state, complex multiphase behavior can be comprehensively described. 

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... 

Joseph DD, Lundgren TS, Jackson R, SaviUe DA (1990) Ensemble Averaged and Mixture Theory Equations for Incompressible Fluid-Particle Suspensions. Int J Multiphase Flow 16 (l) 35-42... 

The pellet temperature equation for the multiphase gas-solid mixture can be expressed as ... 

The pellet temperature equation considering the multiphase gas-solid mixture was expressed on the form (11.29). 

---