# Homogeneous model

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- | The multi-fluid model presented above is obtained by performing | + | The multi-fluid model presented above is obtained by performing [[Averaging approaches|phase averaging]] as defined in |

- | ==== | + | <center><math>{{\left\langle {{\Phi }_{k}} \right\rangle }^{k}}=\frac{1}{\Delta {{V}_{k}}}\int_{\Delta {{V}_{k}}}{{{\Phi }_{k}}dV}</math> and <math>\left\langle {{\Phi }_{k}} \right\rangle =\frac{1}{\Delta V}\int_{\Delta {{V}_{k}}}{{{\Phi }_{k}}dV}</math></center> |

+ | |||

+ | 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 [[Averaging approaches|phase averaging]] is given in | ||

+ | |||

+ | <center><math>\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}}}</math> | ||

+ | </center> | ||

+ | |||

+ | 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 multifluid model is expressed by eq. <math>\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}}} </math> from [[Multi-fluid model]]. Summing the continuity equations for all <math>\Pi </math> phases together, one obtains | The continuity equation for phase k in the multifluid model is expressed by eq. <math>\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}}} </math> from [[Multi-fluid model]]. Summing the continuity equations for all <math>\Pi </math> phases together, one obtains |

## Revision as of 13:25, 18 July 2010

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

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

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

## Contents |

## Continuity Equation

The continuity equation for phase k in the multifluid model is expressed by eq. from Multi-fluid model. Summing the continuity equations for all Π phases together, one obtains

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 eq. from Multi-fluid model, the continuity equation becomes

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

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

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

#### Momentum Equation

The momentum equation for phase k in the multi-fluid model is expressed in eq. from Multi-fluid model. By adding together the momentum equations for all Π phases, one obtains

The stress tensor of the multiphase mixture is

The summation of all interphase forces must be zero since , i.e.,

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

where

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 multifluid model, eq. from Multi-fluid model, one obtains

The mass average enthalpy of the multiphase mixture is

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

It is usually not zero although .

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

#### Species

Summing the equations for conservation of species mass, eq. , for all phases yields

By applying eq. from Multi-fluid model to the mass density of the *i*^{th} component, one obtains

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

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