Multiphase Mixture Model (MMM)

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The general procedure to relate the MFM to the MMM is to sum each equation over all of the phases. The introductions of the mass-averaged density and velocity, as well as the relative mobility, are given where appropriate. The MMM was initially developed by Wang and Cheng (1996) and was applied to model multiphase flow in fuel cells.

The continuity equation summed over two phases, k and j, is:

\frac{\partial }{\partial t}\left[ \varepsilon \left( {{s}_{k}}{{\rho }_{k}}+{{s}_{j}}{{\rho }_{j}} \right) \right]+\nabla \cdot \left[ \varepsilon \left( {{s}_{k}}{{\rho }_{k}}{{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}}+{{s}_{j}}{{\rho }_{j}}{{\left\langle {{\mathbf{V}}_{j}} \right\rangle }^{j}} \right) \right]=0 \qquad \qquad(1)

Defining mixture density and the mixture velocity

\bar{\rho }={{s}_{k}}{{\rho }_{k}}+{{s}_{j}}{{\rho }_{j}} \qquad \qquad(2)
\bar{\rho }\mathbf{\bar{V}}=\varepsilon {{s}_{k}}{{\rho }_{k}}{{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}}+\varepsilon {{s}_{j}}{{\rho }_{j}}{{\left\langle {{\mathbf{V}}_{j}} \right\rangle }^{j}} \qquad \qquad(3)

the continuity equation can be rewritten as

\frac{\partial }{\partial t}\left( \varepsilon \bar{\rho } \right)+\nabla \cdot \left( \bar{\rho }\mathbf{\bar{V}} \right)=0 \qquad \qquad(4)

The momentum equation can be summed over both phases.

\varepsilon {{s}_{k}}{{\rho }_{k}}{{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}}+\varepsilon {{s}_{j}}{{\rho }_{j}}{{\left\langle {{\mathbf{V}}_{j}} \right\rangle }^{j}}=-K\left[ \frac{{{K}_{rk}}}{{{\nu }_{k}}}\left( \nabla {{p}_{k}}-{{\rho }_{k}}\mathbf{g} \right)+\frac{{{K}_{rj}}}{{{\nu }_{j}}}\left( \nabla {{p}_{j}}-{{\rho }_{j}}\mathbf{g} \right) \right] \qquad \qquad(5)

A capillary pressure that relates the pressure in phase k to the pressure in phase j is introduced:

{{p}_{c}}={{p}_{j}}-{{p}_{k}} \qquad \qquad(6)

The capillary pressure is often expressed as the Leverette function (Leverette, 1940). This function relates the capillary pressure to the wetting phase saturation, sw . A phase is said to wet the porous material if the contact angle that phase makes with it is less than 90{}^\circ . Therefore, if {{\theta }_{k}}<90{}^\circ , the wetting phase is phase k, sw = sk


{{\theta }_{j}}<90{}^\circ , then phase j is the wetting phase, sw = sj .

{{p}_{c}}=\sigma \cos \theta {{\left( \frac{\varepsilon }{K} \right)}^{1/2}}\left[ 1.417\left( 1-{{s}_{w}} \right)-2.120{{\left( 1-{{s}_{w}} \right)}^{2}}+1.263{{\left( 1-{{s}_{w}} \right)}^{3}} \right] \qquad \qquad(7)

This function was developed to describe the capillary pressure in soils engineering; however, its use has been extended to other technology such as fuel cells. The reason for the over usage of the Leverette function is the lack of functions to describe the capillary pressure for other types of porous media. With the definition of the mixture velocity and the capillary pressure, the mixture momentum equation can be rewritten as:

\bar{\rho }\mathbf{\bar{V}}=-K\left[ \left( \frac{{{K}_{rk}}}{{{\nu }_{k}}}+\frac{{{K}_{rj}}}{{{\nu }_{j}}} \right)\nabla {{p}_{j}}-\frac{{{K}_{rk}}}{{{\nu }_{k}}}\nabla {{p}_{c}}-\left( \frac{{{\rho }_{k}}{{K}_{rk}}}{{{\nu }_{k}}}+\frac{{{\rho }_{j}}{{K}_{rj}}}{{{\nu }_{j}}} \right)\mathbf{g} \right] \qquad \qquad(8)

Note that the capillary pressure gradient can be reduced into its relative components.

\nabla {{p}_{c}}=\frac{\partial {{p}_{c}}}{\partial {{s}_{k}}}\nabla {{s}_{k}}+\frac{\partial {{p}_{c}}}{\partial {{\sigma }_{jk}}}\sum\limits_{i=1}^{N-1}{\frac{\partial {{\sigma }_{jk}}}{\partial {{\left\langle {{\omega }_{k,i}} \right\rangle }^{k}}}\nabla {{\left\langle {{\omega }_{k,i}} \right\rangle }^{k}}}+\frac{\partial {{p}_{c}}}{\partial {{\sigma }_{jk}}}\frac{\partial {{\sigma }_{jk}}}{\partial T}\nabla T \qquad \qquad(9)

Now the mixture kinematic viscosity, \bar{\nu } , and relative mobility, λk , are introduced.

\bar{\nu }={{\left( \frac{{{K}_{rk}}}{{{\nu }_{k}}}+\frac{{{K}_{rj}}}{{{\nu }_{j}}} \right)}^{-1}} \qquad \qquad(10)
{{\lambda }_{k}}=\frac{{{K}_{rk}}}{{{\nu }_{k}}}\bar{\nu } \qquad \qquad(11)

The momentum equation, eq. (8), can be rewritten as:

\bar{\rho }\mathbf{\bar{V}}=-\frac{K}{{\bar{\nu }}}\left[ \nabla {{p}_{j}}-{{\lambda }_{k}}\nabla {{p}_{c}}-\left( {{\lambda }_{k}}{{\rho }_{k}}+{{\lambda }_{j}}{{\rho }_{j}} \right)\mathbf{g} \right] \qquad \qquad(12)

A definition of the mixture pressure is introduced so that:

\nabla \bar{p}=\nabla {{p}_{j}}-{{\lambda }_{k}}\nabla {{p}_{c}} \qquad \qquad(13)

A density correction factor, γρ , is also introduced.

{{\gamma }_{\rho }}=\frac{1}{{\bar{\rho }}}\left( {{\rho }_{k}}{{\lambda }_{k}}+{{\rho }_{j}}{{\lambda }_{j}} \right) \qquad \qquad(14)

The momentum equation (12) can be written in the form

\bar{\rho }\mathbf{\bar{V}}=-\frac{K}{{\bar{\nu }}}\left[ \nabla \bar{p}-{{\gamma }_{\rho }}\rho \mathbf{g} \right] \qquad \qquad(15)

In order to simplify the MMM derivation for the species and energy equations, a diffusive phase-mass flux is introduced. This term is analogous to the diffusion mass flux in multicomponent mixtures, but refers to each phase, rather than each component in a phase. This value relates the actual mass flux of phase k to the mixture mass flux.

{{\mathbf{j}}_{k}}=\varepsilon {{s}_{k}}{{\rho }_{k}}{{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}}-{{\lambda }_{k}}\bar{\rho }\mathbf{\bar{V}} \qquad \qquad(16)

From this relation, it can be shown that the diffusive phase-mass flux is:

{{\mathbf{j}}_{k}}=-{{\mathbf{j}}_{j}}=\frac{{{\lambda }_{k}}{{\lambda }_{j}}}{{\bar{\nu }}}\left[ K\nabla {{p}_{c}}+\left( {{\rho }_{k}}-{{\rho }_{j}} \right)\mathbf{g} \right] \qquad \qquad(17)

This relation will be useful, and will be used henceforth. The mixture species equation for species i is obtained by adding the species i equation for phase k and phase j.

  & \frac{\partial }{\partial t}\left( \varepsilon {{s}_{k}}{{\rho }_{k}}{{\left\langle {{\omega }_{k,i}} \right\rangle }^{k}}+\varepsilon {{s}_{j}}{{\rho }_{j}}{{\left\langle {{\omega }_{j,i}} \right\rangle }^{j}} \right)+\nabla \cdot \left( \varepsilon {{s}_{k}}{{\rho }_{k}}{{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}}{{\left\langle {{\omega }_{k,i}} \right\rangle }^{k}}+\varepsilon {{s}_{j}}{{\rho }_{j}}{{\left\langle {{\mathbf{V}}_{j}} \right\rangle }^{j}}{{\left\langle {{\omega }_{j,i}} \right\rangle }^{j}} \right) \\ 
 & =-\nabla \cdot \left\langle {{\mathbf{J}}_{k,i}}+{{\mathbf{J}}_{j,i}} \right\rangle +{{{{\dot{m}}'''}}_{k,i}}+{{{{\dot{m}}'''}}_{j,i}} \\ 
\end{align} \qquad \qquad(18)

From eq. (18) from Multi-Fluid Model (MFM) \sum\limits_{k=1}^{\Pi }{{{{{\dot{m}}'''}}_{k,i}}}={{{\dot{m}}'''}_{i}}, the summation of the species production over all the phases is the species production due to chemical reaction. The mixture mass fraction is:

\bar{\rho }{{\bar{\omega }}_{i}}={{s}_{k}}{{\rho }_{k}}{{\left\langle {{\omega }_{k,i}} \right\rangle }^{k}}+{{s}_{j}}{{\rho }_{j}}{{\left\langle {{\omega }_{j,i}} \right\rangle }^{j}} \qquad \qquad(19)

The correction factor for species advection, γi , is introduced.

{{\gamma }_{i}}=\frac{1}{{{{\bar{\omega }}}_{i}}}\left( {{\lambda }_{k}}{{\left\langle {{\omega }_{k,i}} \right\rangle }^{k}}+{{\lambda }_{j}}{{\left\langle {{\omega }_{j,i}} \right\rangle }^{j}} \right) \qquad \qquad(20)

The mixture species equation is:

  & \frac{\partial }{\partial t}\left( \varepsilon \bar{\rho }{{{\bar{\omega }}}_{i}} \right)+\nabla \cdot \left( {{\gamma }_{i}}\bar{\rho }\mathbf{\bar{V}}{{{\bar{\omega }}}_{i}} \right) \\ 
 & =-\nabla \cdot \left\langle {{\mathbf{J}}_{k,i}}+{{\mathbf{J}}_{j,i}} \right\rangle -\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)+{{{{\dot{m}}'''}}_{i}} \\ 
\end{align} \qquad \qquad(21)

It should be noted that eqs. (4), (15), and (21) are also applicable to multiphase and not necessary to two-phase as developed here.

It is important to point out that the mixture species equation still contains the species mass fraction of each phase. Therefore, the species mass fraction in phase k and phase j must be related to the mixture mass fraction through thermodynamic equilibrium. Expanding all the terms in eq. (4.297) yields:

\left( {{s}_{k}}{{\rho }_{k}}+{{s}_{j}}{{\rho }_{j}} \right){{\bar{\omega }}_{i}}={{s}_{k}}{{\rho }_{k}}{{\left\langle {{\omega }_{k,i}} \right\rangle }^{k}}+{{s}_{j}}{{\rho }_{j}}{{\left\langle {{\omega }_{j,i}} \right\rangle }^{j}} \qquad \qquad(22)

Since there are two phases presented, the phase saturations add to unity. Therefore, the phase saturation of phase k can be calculated by:

{{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)} \qquad \qquad(23)

It is important to note that when the saturation of phase k is calculated in this manner, one phase continuity equation is not solved; instead, all N species equations are solved. For more discussion on this approach to the calculation of the phase saturation, refer to the comparison of MFM and MMM models below.

The last transport equation that must be examined is the energy equation. The energy equation of both phases are added together plus an unsteady and heat conduction term represents the influence of the solid matrix on the energy equation.

  & \frac{\partial }{\partial t}\left( \left( 1-\varepsilon  \right){{\rho }_{sm}}{{\left\langle {{h}_{sm}} \right\rangle }^{sm}}+\varepsilon \left[ {{s}_{k}}{{\rho }_{k}}{{\left\langle {{h}_{k}} \right\rangle }^{k}}+{{s}_{j}}{{\rho }_{j}}{{\left\langle {{h}_{j}} \right\rangle }^{j}} \right] \right)+ \\ 
 & \nabla \cdot \left( \varepsilon \left[ {{s}_{k}}{{\rho }_{k}}{{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}}{{\left\langle {{h}_{k}} \right\rangle }^{k}}+{{s}_{j}}{{\rho }_{j}}{{\left\langle {{\mathbf{V}}_{j}} \right\rangle }^{j}}{{\left\langle {{h}_{j}} \right\rangle }^{j}} \right] \right)=-\nabla \cdot \left\langle {{{\mathbf{{q}''}}}_{k,i}}+{{{\mathbf{{q}''}}}_{j,i}}+{{{\mathbf{{q}''}}}_{sm,i}} \right\rangle +{{{{\dot{q}}'''}}_{E}} \\ 
\end{align} \qquad \qquad(24)

The external heat generation is represented by {{{\dot{q}}'''}_{E}} , and it is the summation of the external heat generation over all of the phases. The mixture enthalpy is

\bar{\rho }\bar{h}={{s}_{k}}{{\rho }_{k}}{{\left\langle {{h}_{k}} \right\rangle }^{k}}+{{s}_{j}}{{\rho }_{k}}{{\left\langle {{h}_{j}} \right\rangle }^{j}} \qquad \qquad(25)

The correction factor for energy advection, γh , is also introduced:

{{\gamma }_{h}}=\frac{1}{{\bar{h}}}\left( {{\lambda }_{k}}{{\left\langle {{h}_{k}} \right\rangle }^{k}}+{{\lambda }_{j}}{{\left\langle {{h}_{j}} \right\rangle }^{j}} \right) \qquad \qquad(26)

An assumption that corresponds with the mixture enthalpy is that each phase is in thermodynamic equilibrium, {{\left\langle {{T}_{k}} \right\rangle }^{k}}={{\left\langle {{T}_{j}} \right\rangle }^{j}} . If Fourier’s law governs the heat conduction and effective thermal conductivity can be used, the energy equation can be rewritten using an effective thermal conductivity and the diffusive phase-mass flux as:

  & \frac{\partial }{\partial t}\left( \left( 1-\varepsilon  \right){{\rho }_{sm}}{{\left\langle {{h}_{sm}} \right\rangle }^{sm}}+\varepsilon \bar{\rho }\bar{h} \right)+\nabla \cdot \left( {{\gamma }_{h}}\bar{\rho }\mathbf{\bar{V}}\bar{h} \right)= \\ 
 & \nabla \cdot \left( {{k}_{eff}}\nabla T \right)-\nabla \cdot \left( {{\mathbf{j}}_{k}}\left( {{\left\langle {{h}_{k}} \right\rangle }^{k}}-{{\left\langle {{h}_{j}} \right\rangle }^{j}} \right) \right)+{{{{\dot{q}}'''}}_{E}} \\ 
\end{align} \qquad \qquad(27)

where the effective thermal conductivity is related to the phase saturation, porosity, geometry of the porous medium (tortuosity, \Im ) and the conductivity of the porous media and both phases, {{k}_{eff}}=f\left( {{k}_{sm}},{{k}_{k}},{{k}_{j}},\varepsilon ,{{s}_{k}},\Im  \right) .

It is important to note that the heat generation and consumption of chemical reactions and latent heat are all embedded in this governing equation.


Leverette, M.C., 1940, “Capillary Behavior in Porous Solids,” Transaction of AIME, Vol. 142, pp. 152-169.

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.

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