# Homogeneous model

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The multi-fluid model presented above is obtained by performing phase averaging as defined in

${{\left\langle {{\Phi }_{k}} \right\rangle }^{k}}=\frac{1}{\Delta {{V}_{k}}}\int_{\Delta {{V}_{k}}}{{{\Phi }_{k}}dV}$ and $\left\langle {{\Phi }_{k}} \right\rangle =\frac{1}{\Delta V}\int_{\Delta {{V}_{k}}}{{{\Phi }_{k}}dV}$

If spatial averaging is performed for all phases within a multiphase control volume, the homogeneous (or mixture) model can be obtained. The relationship between volume averaging and phase averaging is given in

$\left\langle \Phi \right\rangle =\sum\limits_{k=1}^{\Pi }{\left\langle {{\Phi }_{k}} \right\rangle }=\sum\limits_{k=1}^{\Pi }{{{\varepsilon }_{k}}{{\left\langle {{\Phi }_{k}} \right\rangle }^{k}}}$

which indicates that the homogeneous model can be obtained by summing the individual phase equations of the multi-fluid model.

## Continuity Equation

The continuity equation for phase k in the multi-fluid model model is expressed as

$\frac{\partial }{\partial t}\left( {{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}} \right)+\nabla \cdot \left( {{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}{{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}} \right)=\sum\limits_{j=1(j\ne k)}^{\Pi }{{{{{\dot{m}}'''}}_{jk}}}$

Summing the continuity equations for all Π phases together, one obtains

$\frac{\partial }{\partial t}\left( \sum\limits_{k=1}^{\Pi }{{{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}} \right)+\nabla \cdot \sum\limits_{k=1}^{\Pi }{{{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}{{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}}}=\sum\limits_{k=1}^{\Pi }{\sum\limits_{j=1(j\ne k)}^{\Pi }{{{{{\dot{m}}'''}}_{jk}}}}\qquad \qquad(1)$

The right-hand side of equation (1) must be zero because the total mass of all phases produced by phase change must equal the total mass of all phases consumed by phase change. Considering this fact and $\left\langle {{\rho }_{k}} \right\rangle ={{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}$ (see multi-fluid model), the continuity equation becomes

$\frac{\partial \left\langle \rho \right\rangle }{\partial t}+\nabla \cdot \sum\limits_{k=1}^{\Pi }{{{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}{{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}}}=0\qquad \qquad(2)$

The bulk velocity of the multiphase mixture is the mass-averaged velocity of all the individual phases:

$\mathbf{\tilde{V}}=\frac{1}{\left\langle \rho \right\rangle }\sum\limits_{k=1}^{\Pi }{{{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}{{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}}}\qquad \qquad(3)$

Substituting eq. (3) into eq. (2), the final form of the continuity equation for a multiphase mixture is

$\frac{\partial \left\langle \rho \right\rangle }{\partial t}+\nabla \cdot \left( \left\langle \rho \right\rangle \mathbf{\tilde{V}} \right)=0\qquad \qquad(4)$

It can be seen that eq. (4) has the same form as the local continuity equation (2.51), except that the volume-averaged density and velocity are used in eq. (4), where $\left\langle \rho \right\rangle =\sum\limits_{k=1}^{\Pi }{{{\varepsilon }_{k}}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}.$

## Momentum Equation

The momentum equation for phase k in the multi-fluid model is expressed as

\begin{align} & \frac{\partial }{\partial t}\left( {{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}{{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}} \right)+\nabla \cdot \left( {{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}{{\left\langle {{\mathbf{V}}_{k}}{{\mathbf{V}}_{k}} \right\rangle }^{k}} \right) \\ & =\nabla \cdot \left( {{\varepsilon }_{k}}{{\left\langle {{{\mathbf{{\tau }'}}}_{k}} \right\rangle }^{k}} \right)+{{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}{{\mathbf{X}}_{k}}+\sum\limits_{j=1(j\ne k)}^{\Pi }{\left( \left\langle {{\mathbf{F}}_{jk}} \right\rangle +\left\langle {{{{\dot{m}}'''}}_{jk}} \right\rangle {{\left\langle {{\mathbf{V}}_{k,I}} \right\rangle }^{k}} \right)} \\ \end{align}

By adding together the momentum equations for all Π phases, one obtains

\begin{align} & \frac{\partial }{\partial t}\left( \sum\limits_{k=1}^{\Pi }{{{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}{{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}}} \right)+\nabla \cdot \left( \sum\limits_{k=1}^{\Pi }{{{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}{{\left\langle {{\mathbf{V}}_{k}}{{\mathbf{V}}_{k}} \right\rangle }^{k}}} \right) \\ & =\nabla \cdot \sum\limits_{k=1}^{\Pi }{{{\varepsilon }_{k}}{{\left\langle {{{\mathbf{{\tau }'}}}_{k}} \right\rangle }^{k}}}+\sum\limits_{k=1}^{\Pi }{\left( {{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}} \right)}\mathbf{X}+\sum\limits_{k=1}^{\Pi }{\sum\limits_{j=1(j\ne k)}^{\Pi }{\left( \left\langle {{\mathbf{F}}_{jk}} \right\rangle +\left\langle {{{{\dot{m}}'''}}_{jk}} \right\rangle {{\left\langle {{\mathbf{V}}_{k,I}} \right\rangle }^{k}} \right)}} \\ \end{align}\qquad \qquad(5)

The stress tensor of the multiphase mixture is

$\left\langle {\mathbf{{\tau }'}} \right\rangle =\sum\limits_{k=1}^{\Pi }{{{\varepsilon }_{k}}{{\left\langle {{\mathbf{\tau }}_{k}} \right\rangle }^{k}}}=-\left\langle p \right\rangle \mathbf{I}+\mu \left[ \nabla \mathbf{\tilde{V}}+\nabla {{{\mathbf{\tilde{V}}}}_{{}}}^{T} \right]-\frac{2}{3}\mu (\nabla \cdot \mathbf{\tilde{V}})\mathbf{I}\qquad \qquad(6)$

The summation of all interphase forces must be zero since $\left\langle {{\mathbf{F}}_{jk}} \right\rangle =-\left\langle {{\mathbf{F}}_{kj}} \right\rangle$, i.e.,

$\sum\limits_{k=1}^{\Pi }{\sum\limits_{j=1(j\ne k)}^{\Pi }{{}}\left\langle {{\mathbf{F}}_{jk}} \right\rangle }=0\qquad \qquad(7)$

Considering eqs. (3), (6) and (7), the momentum equation becomes

$\frac{\partial }{\partial t}\left( \left\langle \rho \right\rangle \mathbf{\tilde{V}} \right)+\nabla \cdot \left( \sum\limits_{k=1}^{\Pi }{{{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}{{\left\langle {{\mathbf{V}}_{k}}{{\mathbf{V}}_{k}} \right\rangle }^{k}}} \right)=\nabla \cdot \left\langle {\mathbf{{\tau }'}} \right\rangle +\left\langle \rho \right\rangle \mathbf{X}+{{\mathbf{{\dot{M}}'''}}_{I}}\qquad \qquad(8)$

where

${{\mathbf{{\dot{M}}'''}}_{I}}=\sum\limits_{k=1}^{\Pi }{\sum\limits_{j=1(j\ne k)}^{\Pi }{{}}\left\langle {{{{\dot{m}}'''}}_{jk}} \right\rangle }{{\left\langle {{\mathbf{V}}_{k,I}} \right\rangle }^{k}}\qquad \qquad(9)$

Equation (9) represents the momentum production rate due to interaction between different phases along their separating interfaces. It must be specified according to the combination of phases in the multiphase system that is under consideration.

## Energy Equation

By summing the energy equations for all Π phases in the multi-fluid model:

\begin{align} & \frac{\partial }{\partial t}\left( {{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}{{\left\langle {{h}_{k}} \right\rangle }^{k}} \right)+\nabla \cdot \left( {{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}{{\left\langle {{\mathbf{V}}_{k}}{{h}_{k}} \right\rangle }^{k}} \right)=-\nabla \cdot \left\langle {{{\mathbf{{q}''}}}_{k}} \right\rangle +\left\langle {{{{q}'''}}_{k}} \right\rangle \\ & +{{\varepsilon }_{k}}\frac{D{{\left\langle {{p}_{k}} \right\rangle }^{k}}}{Dt}+\nabla \left\langle {{\mathbf{V}}_{k}} \right\rangle :\left\langle {{\mathbf{\tau }}_{k}} \right\rangle +\sum\limits_{j=1(j\ne k)}^{\Pi }{\left[ \left\langle {{{{q}'''}}_{jk}} \right\rangle +{{{{\dot{m}}'''}}_{jk}}{{\left\langle {{h}_{k,I}} \right\rangle }^{k}} \right]} \\ \end{align}

one obtains

\begin{align} & \frac{\partial }{\partial t}\left( \sum\limits_{k=1}^{\Pi }{{{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}{{\left\langle {{h}_{k}} \right\rangle }^{k}}} \right)+\nabla \cdot \left( \sum\limits_{k=1}^{\Pi }{{{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}{{\left\langle {{\mathbf{V}}_{k}}{{h}_{k}} \right\rangle }^{k}}} \right)=-\nabla \cdot \left( \sum\limits_{k=1}^{\Pi }{\left\langle {{{\mathbf{{q}''}}}_{k}} \right\rangle } \right)+\sum\limits_{k=1}^{\Pi }{\left\langle {{{{q}'''}}_{k}} \right\rangle } \\ & +\sum\limits_{k=1}^{\Pi }{{{\varepsilon }_{k}}\frac{D{{\left\langle {{p}_{k}} \right\rangle }^{k}}}{Dt}}+\sum\limits_{k=1}^{\Pi }{\nabla {{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}}:{{\varepsilon }_{k}}{{\left\langle {{\mathbf{\tau }}_{k}} \right\rangle }^{k}}}+\sum\limits_{k=1}^{\Pi }{\sum\limits_{j=1(j\ne k)}^{\Pi }{\left\langle {{{{q}'''}}_{jk}} \right\rangle }}+\sum\limits_{k=1}^{\Pi }{\sum\limits_{j=1(j\ne k)}^{\Pi }{{{{{\dot{m}}'''}}_{jk}}{{\left\langle {{h}_{k,I}} \right\rangle }^{k}}}} \\ \end{align}\qquad \qquad(10)

The mass average enthalpy of the multiphase mixture is

$\tilde{h}=\frac{1}{\left\langle \rho \right\rangle }\sum\limits_{k=1}^{\Pi }{{{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}{{\left\langle {{h}_{k}} \right\rangle }^{k}}}\qquad \qquad(11)$

The fifth term on the right-hand side of eq. (10) is for summation of all interphase heat transfer and it must be zero. The last term on the right-hand side of eq. (10) accounts for contribution of interphase phase change energy flux due to phase change; it can be defined as

$\sum\limits_{k=1}^{\Pi }{\sum\limits_{j=1(j\ne k)}^{\Pi }{{{{{\dot{m}}'''}}_{jk}}{{\left\langle {{h}_{k,I}} \right\rangle }^{k}}}}={{{q}'''}_{I}}\qquad \qquad(12)$

It is usually not zero although $\sum\limits_{k=1}^{\Pi }{\sum\limits_{j=1(j\ne k)}^{\Pi }{{{\dot{m}}'''}}}=0$.

Considering eqs. (11) and (12), the energy equation (10) becomes

$\frac{\partial }{\partial t}\left( \left\langle \rho \right\rangle \tilde{h} \right)+\nabla \cdot \left( \sum\limits_{k=1}^{\Pi }{{{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}{{\left\langle {{\mathbf{V}}_{k}}{{h}_{k}} \right\rangle }^{k}}} \right)=-\nabla \cdot \left\langle {\mathbf{{q}''}} \right\rangle +\frac{D\left\langle p \right\rangle }{Dt}+\left\langle {{q}'''} \right\rangle +\nabla \mathbf{\tilde{V}}:\left\langle \mathbf{\tau } \right\rangle +{{{q}'''}_{I}}\qquad \qquad(13)$

## Species

Summing the equations for conservation of species mass for multi-fluid model,

$\frac{\partial \left( {{\varepsilon }_{k}}{{\left\langle {{\rho }_{k,i}} \right\rangle }^{k}} \right)}{\partial t}+\nabla \cdot \left( {{\varepsilon }_{k}}{{\left\langle {{\rho }_{k,i}} \right\rangle }^{k}}{{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}} \right)=-\nabla \cdot \left\langle {{\mathbf{J}}_{k,i}} \right\rangle +\left\langle {{{{\dot{m}}'''}}_{k,i}} \right\rangle +\sum\limits_{j=1(j\ne k)}^{\Pi }{{{{{\dot{m}}'''}}_{jk,i}}}$

for all phases yields

\begin{align} & \frac{\partial }{\partial t}\left( \sum\limits_{k=1}^{\Pi }{{{\varepsilon }_{k}}{{\left\langle {{\rho }_{k,i}} \right\rangle }^{k}}} \right)+\nabla \cdot \left( \sum\limits_{k=1}^{\Pi }{{{\varepsilon }_{k}}{{\left\langle {{\rho }_{k,i}} \right\rangle }^{k}}{{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}}} \right) \\ & =-\nabla \cdot \left( \sum\limits_{k=1}^{\Pi }{\left\langle {{\mathbf{J}}_{k,i}} \right\rangle } \right)+\sum\limits_{k=1}^{\Pi }{\left\langle {{{{\dot{m}}'''}}_{k,i}} \right\rangle }+\sum\limits_{k=1}^{\Pi }{\sum\limits_{j=1(j\ne k)}^{\Pi }{{{{{\dot{m}}'''}}_{jk,i}}}} \\ \end{align}\qquad \qquad(14)

By applying $\left\langle {{\rho }_{k}} \right\rangle ={{\varepsilon }_{k}}{{\left\langle {{\rho }_{k}} \right\rangle }^{k}}$ (see multi-fluid model) to the mass density of the ith component, one obtains

$\left\langle {{\rho }_{i}} \right\rangle =\sum\limits_{k=1}^{\Pi }{{{\varepsilon }_{k}}{{\left\langle {{\rho }_{k,i}} \right\rangle }^{k}}}\qquad \qquad(15)$

In accordance with the conservation of mass, the mass source (or sink) of the ith component due to phase change in all phases must add up to zero, i.e.,

$\sum\limits_{k=1}^{\Pi }{\sum\limits_{j=1(j\ne k)}^{\Pi }{{{{{\dot{m}}'''}}_{jk,i}}}}=0\qquad \qquad(16)$

Substituting eqs. (15) and (16) into eq. (14), and using the mass-averaged velocity defined in eq. (3), the conservation of species mass becomes

$\frac{\partial \left\langle {{\rho }_{i}} \right\rangle }{\partial t}+\nabla \cdot \left\langle {{\rho }_{i}} \right\rangle \mathbf{\tilde{V}}=-\nabla \cdot {{\mathbf{J}}_{i}}+{{{\dot{m}}'''}_{i}}\qquad \qquad(17)$

## References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.