Comparison of MFM and MMM models

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The major advantage of MFM over MMM is that MFM has the potential to capture the details of each phase, including interface. Furthermore, the nonequilibrium thermodynamics for temperature and species concentration in each phase can be obtained by MFM. Another benefit is the potential to account for the pore size distribution in a porous media and model how the saturation of each pore is not equivalent. The pore size distribution is not well researched in the literature. This is why the Leverette function of capillary pressure is commonly used, even though this function applies to soil engineering and probably is not the best choice for other applications.

In the MFM model, the momentum equation can be directly inserted into the continuity equation, therefore eliminating the intrinsic velocity of each phase. It also can be directly inserted into the energy and species equations. Also, the phase saturation of phase j is known if the saturation of phase k is known. Similarly, only $N - 1$ species equations are needed in each phase because the species balance is maintained through continuity. Therefore, the main flow variables in the MFM model are

${{s}_{k}},{{p}_{j}},{{\left\langle {{\omega }_{k,i}} \right\rangle }^{k}},{{\left\langle {{\omega }_{j,i}} \right\rangle }^{j}},{{\left\langle {{T}_{k}} \right\rangle }^{k}},{{\left\langle {{T}_{j}} \right\rangle }^{j}}$

The number of equations and the number of unknowns using the MFM model for two phases when the energy equation is needed is

$5+2\times \left( N-1 \right)$

These equations are: two continuity equations, three energy equations, and N-1 species equations for each phase. Applying the appropriate boundary conditions, this is a well posed problem. In the MMM model, applying the momentum equation directly into the continuity equation to eliminate the velocity, the main flow variables are:

$\bar{\rho },\bar{p},{{\bar{\omega }}_{i}},\bar{h}$

The variables are the mixture density, the mixture pressure, $N - 1$ species mass fraction, and the temperature, i.e., the number of $\text{variables is }3+\left( N-1 \right)$ .

The main equations are one mixture continuity equation, $N - 1$ mixture mass fraction equations, and one energy equation, i.e., the number of $\text{equations}=2+\left( N-1 \right)$ .

One more equation needs to be solved in order for this approach to be well posed, since there is one more dependent variable than equations. The first approach to this problem is to solve the continuity equation for one of the phases. Using the MMM model variables, the continuity equation for phase k is:

$\frac{\partial }{\partial t}\left( \varepsilon {{s}_{k}}{{\rho }_{k}} \right)+\nabla \cdot \left( {{\lambda }_{k}}\bar{\rho }\mathbf{\bar{V}} \right)=-\nabla \cdot {{\mathbf{J}}_{k}}+{{{\dot{m}}'''}_{k}} \qquad \qquad(1)$

The other approach, which is suggested by Wang and Cheng (1996) and used in Wang and Wang (2003) to model fuel cells, is to solve species equations for all N species, and not solve the continuity equation for phase k. It is very important to note that if this approach is taken, the mixture species mass fraction as well the species mass fraction in each phase must add up to unity.

$\sum\limits_{i=1}^{N}{{{{\bar{\omega }}}_{i}}}=\sum\limits_{i=1}^{N}{{{\left\langle {{\omega }_{k,i}} \right\rangle }^{k}}=\sum\limits_{i=1}^{N}{{{\left\langle {{\omega }_{j,i}} \right\rangle }^{j}}=1}} \qquad \qquad(2)$

If these criteria are met, the saturation calculated by equation (23) from Multiphase Mixture Model (MMM) ${{s}_{k}}=\frac{{{\rho }_{j}}\left( {{{\bar{\omega }}}_{i}}-{{\left\langle {{\omega }_{j,i}} \right\rangle }^{j}} \right)}{{{\rho }_{k}}\left( {{{\bar{\omega }}}_{i}}-{{\left\langle {{\omega }_{k,i}} \right\rangle }^{k}} \right)+{{\rho }_{j}}\left( {{\left\langle {{\omega }_{j,i}} \right\rangle }^{k}}-{{{\bar{\omega }}}_{i}} \right)}$ is correct. Assuming the relations in eq. (2) are upheld, than the species equation for N^th component can be written in terms of all the other components.

$\begin{align} & \frac{\partial }{\partial t}\left( \varepsilon \bar{\rho } \right)-\sum\limits_{i=1}^{N-1}{\frac{\partial }{\partial t}\left( \varepsilon \bar{\rho }{{{\bar{\omega }}}_{i}} \right)}+\nabla \cdot \left( \bar{\rho }\mathbf{\bar{V}} \right)+\sum\limits_{i=1}^{N-1}{\nabla \cdot \left( {{\gamma }_{i}}\bar{\rho }\mathbf{\bar{V}}{{{\bar{\omega }}}_{i}} \right)}= \\ & \sum\limits_{i=1}^{N-1}{\nabla \cdot \left\langle {{\mathbf{J}}_{k,i}}+{{\mathbf{J}}_{j,i}} \right\rangle }+\sum\limits_{i=1}^{N-1}{\nabla \cdot \left( {{\mathbf{j}}_{k}}\left( {{\left\langle {{\omega }_{k,i}} \right\rangle }^{k}}-{{\left\langle {{\omega }_{j,i}} \right\rangle }^{j}} \right) \right)}-\sum\limits_{i=1}^{N-1}{{{{{\dot{m}}'''}}_{i}}} \\ \end{align} \qquad \qquad(3)$

This equation is simply the continuity equation subtracted by N–1 species equations. Mathematically, this equation is not a new (independent) equation, and therefore solving for the N^th species equation still leaves the system of equations ill-posed. This leads us to using the continuity equation for phase k to complete the system of equations.

The formulation of MMM reduces the total number of equations, which, at first glance, would seem to decrease the computational time. However, when looking at the number of terms that these equations incorporate, it is not very different from the MFM. For example, examining the species equation in the MFM model for $Π$ phases and $D x$ dimensions, and comparing it to the MMM model for the same number of phases and dimensions can show how many terms are solved (Table 1; Rice 2006a).

For two-phase, two-dimensional systems, the MFM model solves a total of 10 terms, while the MMM model solves a total of 11 terms. So the computational time difference between the MMM and the MFM models are probably not as dramatic as it first appears. Since the total number of terms solved is very close, the computational time is probably limited by the convergence rate more than anything else.

Table 1: Comparison of MFM and MMM, Total Terms Computed in Species Equation

Model	1^st Order Time	1^st Order space (Advection)	2^nd Order Space (Diffusion)	Total
MFM	\Pi	$Π D x$	$Π D x$	$\Pi \left( 2{D_x}+1 \right)$
MMM	1	$\left( 1+ \Pi \right){D_x}$	$Π D x$	${D_x} \left( 2 \Pi +1 \right)+1$

Unsaturated flow theory comes from MFM, with additional assumptions. In unsaturated flow theory (UFT), only the liquid phase is considered, and the gas pressure is assumed to be equal to the hydrostatic pressure of the gas. If the capillary pressure is assumed to be only a function of liquid saturation, and there is no mass transfer considered, then Richard’s equation is valid. Richard’s equation is:

$\frac{\partial }{\partial t}\left( \varepsilon {{s}_{k}}{{\rho }_{k}} \right)+\nabla \cdot \left[ K\frac{{{K}_{rk}}}{{{\nu }_{k}}}\left( \frac{d{{p}_{c}}}{ds}\nabla {{s}_{k}}+\left( {{\rho }_{k}}-{{\rho }_{j}} \right)\mathbf{g} \right) \right]=0 \qquad \qquad(4)$

References

Rice, J. 2006a, Personal Communication, University of Connecticut, Storrs, CT.

Wang, C.Y. and Cheng, P., 1996, “A Multiphase Mixture Model for Multiphase Multicomponent Transport in Capillary Porous Media Part I: Model Development,” International Journal of Heat and Mass Transfer, Vol. 39, pp. 3607-3618.

Wang, Z.H. and Wang, C.Y. 2003, “Mathematical Modeling of Liquid-Feed Direct Methanol Feed Cells,” Journal of The Electrochemical Society, Vol. 140, pp. A508-A519.

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Comparison of MFM and MMM models

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