Multi-Fluid Model (MFM)

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In this model, there are numerous liquid/vapor interfacial regions within each elemental pore; therefore, the average change of these interfacial characteristics is considered as a model for this type of problem. The transport equations for a single-component substance coexisting in the vapor phase, v, and the liquid phase, \ell , are given. In these equations, the liquid and vapor are considered to be in local thermal equilibrium. However, the liquid and vapor phases are in local thermal non-equilibrium with the solid. Whitaker (1977) developed governing equations for the periodic dispersion of phases within a porous zone, as seen in Fig. 1 from Multiphase Transport, The continuity equation, (4.178), for the vapor and liquid phases become

\frac{\partial }{\partial t}\left( {{\varepsilon }_{v}}{{\rho }_{v}} \right)+\nabla \cdot \left( {{\rho }_{v}}\left\langle {{\mathbf{V}}_{v}} \right\rangle  \right)=\left\langle {{{{\dot{m}}'''}}_{\ell v}} \right\rangle \qquad \qquad(1)
\frac{\partial {{\varepsilon }_{\ell }}}{\partial t}+\nabla \cdot \left\langle {{\mathbf{V}}_{\ell }} \right\rangle =-\frac{\left\langle {{{{\dot{m}}'''}}_{\ell v}} \right\rangle }{{{\rho }_{\ell }}} \qquad \qquad(2)

where the density in the liquid phase is assumed constant. The volumetric mass transfer term, \left\langle {{{{\dot{m}}'''}}_{\ell v}} \right\rangle , with units of kg / m3s, represents the evaporation rate when it has a positive value and the condensation rate when it has a negative value. When gravity is the only body force, the momentum equation (4.190), for the vapor and liquid phases becomes

  & {{\rho }_{v}}\left[ \frac{1}{{{\varepsilon }_{v}}}\frac{\partial \left\langle {{\mathbf{V}}_{v}} \right\rangle }{\partial t}+\frac{\left\langle {{\mathbf{V}}_{v}} \right\rangle }{{{\varepsilon }_{v}}}\cdot \nabla \left( \frac{\left\langle {{\mathbf{V}}_{v}} \right\rangle }{{{\varepsilon }_{v}}} \right) \right] \\ 
 & =-\nabla {{\left\langle {{p}_{v}} \right\rangle }^{v}}+\frac{{{\mu }_{v}}}{{{\varepsilon }_{v}}}{{\nabla }^{2}}\left\langle {{\mathbf{V}}_{v}} \right\rangle -\frac{{{\mu }_{v}}}{{{\mathbf{K}}_{v}}}\left\langle {{\mathbf{V}}_{v}} \right\rangle -\frac{{{C}_{f}}}{{{\mathbf{K}}^{{1}/{2}\;}}}{{\rho }_{v}}\left| \left\langle {{\mathbf{V}}_{v}} \right\rangle  \right|\left\langle {{\mathbf{V}}_{v}} \right\rangle +{{\rho }_{v}}\mathbf{g} \\ 
\end{align} \qquad \qquad(3)
  & {{\rho }_{\ell }}\left[ \frac{1}{{{\varepsilon }_{\ell }}}\frac{\partial \left\langle {{\mathbf{V}}_{\ell }} \right\rangle }{\partial t}+\frac{\left\langle {{\mathbf{V}}_{\ell }} \right\rangle }{{{\varepsilon }_{\ell }}}\cdot \nabla \left( \frac{\left\langle {{\mathbf{V}}_{\ell }} \right\rangle }{{{\varepsilon }_{\ell }}} \right) \right] \\ 
 & =-\frac{{{\mu }_{\ell }}}{{{\mathbf{K}}_{\ell }}}\left\langle {{\mathbf{V}}_{\ell }} \right\rangle +\frac{{{\mu }_{\ell }}}{{{\varepsilon }_{\ell }}}{{\nabla }^{2}}\left\langle {{\mathbf{V}}_{\ell }} \right\rangle -\left[ {{C}_{\varepsilon }}\nabla {{\varepsilon }_{\ell }}+{{C}_{T}}\nabla {{\left\langle {{T}_{f}} \right\rangle }^{f}}+\nabla {{\left\langle {{p}_{v}} \right\rangle }^{v}} \right]+{{\rho }_{\ell }}\mathbf{g} \\ 
\end{align} \qquad \qquad(4)

The first two terms in the bracket on the right-hand side of eq. (4) represent the pressure gradient due to the change in capillary pressure, pc, as a function of volume fraction of liquid and temperature. The two coefficients in eq. (4) are defined as

{{C}_{\varepsilon }}=\frac{\partial {{p}_{c}}}{\partial {{\varepsilon }_{\ell }}} \qquad \qquad(5)
{{C}_{T}}=\frac{\partial {{p}_{c}}}{\partial {{\left\langle {{T}_{\ell v}} \right\rangle }^{\ell v}}} \qquad \qquad(6)

where the capillary pressure is defined as

{{p}_{c}}=\frac{2{{\sigma }_{\ell v}}}{{{r}_{eff}}}={{\left\langle {{p}_{v}} \right\rangle }^{v}}-{{\left\langle {{p}_{\ell }} \right\rangle }^{\ell }} \qquad \qquad(7)

and reff is the effective radius of curvature of the interface between the vapor and the liquid. For more information on interfacial phenomena, the reader is referred to Chapter 5. The effective radius is a function of the volume fraction of the liquid and the pore geometry. The energy equation for the fluid (liquid and vapor) phase, where {{\left\langle {{T}_{\ell }} \right\rangle }^{\ell }}={{\left\langle {{T}_{v}} \right\rangle }^{v}}={{\left\langle {{T}_{f}} \right\rangle }^{f}}, is

\left( {{\varepsilon }_{\ell }}{{\rho }_{\ell }}{{c}_{p}}_{\ell }+{{\varepsilon }_{v}}{{\left\langle {{\rho }_{v}} \right\rangle }^{v}}{{c}_{p}}_{v} \right)\frac{\partial {{\left\langle {{T}_{f}} \right\rangle }^{f}}}{\partial t}+\left( {{\rho }_{\ell }}{{c}_{p}}_{\ell }{{\mathbf{v}}_{\ell }}+{{\left\langle {{\rho }_{v}} \right\rangle }^{v}}{{c}_{p}}_{v}{{\mathbf{v}}_{v}} \right)\cdot \nabla {{\left\langle {{T}_{f}} \right\rangle }^{f}}

=\nabla \cdot \left[ \left( {{\varepsilon }_{\ell }}{{k}_{eff,\ell }}+{{\varepsilon }_{v}}{{k}_{eff,v}} \right)\nabla {{\left\langle {{T}_{\ell v}} \right\rangle }^{\ell v}} \right]+{{h}_{sm,\ell }}\left( {{\left\langle {{T}_{sm}} \right\rangle }^{sm}}-{{\left\langle {{T}_{f}} \right\rangle }^{f}} \right)\frac{\Delta {{A}_{sm,\ell }}}{\Delta V}

+{{h}_{sm,v}}\left( {{\left\langle {{T}_{sm}} \right\rangle }^{sm}}-{{\left\langle {{T}_{f}} \right\rangle }^{f}} \right)\frac{\Delta {{A}_{sm,v}}}{\Delta V} \qquad \qquad(8)

where {{h}_{c,sm,\ell }} and hc,sm,v are convective heat transfer coefficients from solid-matrix to liquid and vapor phases, respectively. The energy equation for the solid matrix is

  & \left( {{\varepsilon }_{sm}}{{\rho }_{sm}}{{c}_{p,sm}} \right)\frac{\partial {{\left\langle {{T}_{sm}} \right\rangle }^{sm}}}{\partial t}=\nabla \cdot \left( {{\varepsilon }_{sm}}{{k}_{eff,sm}}\nabla {{\left\langle {{T}_{sm}} \right\rangle }^{sm}} \right) \\ 
 & -{{h}_{sm,\ell }}\left( {{\left\langle {{T}_{sm}} \right\rangle }^{sm}}-{{\left\langle {{T}_{f}} \right\rangle }^{f}} \right)\frac{\Delta {{A}_{sm,\ell }}}{\Delta V}-{{h}_{sm,v}}\left( {{\left\langle {{T}_{sm}} \right\rangle }^{sm}}-{{\left\langle {{T}_{f}} \right\rangle }^{f}} \right)\frac{\Delta {{A}_{sm,v}}}{\Delta V} \\ 
\end{align} \qquad \qquad(9)

The volume fractions of all the phases must satisfy the following constraint:

{{\varepsilon }_{\ell }}+{{\varepsilon }_{v}}+{{\varepsilon }_{sm}}=1 \qquad \qquad(10)

The equation of state for the vapor phase, assuming it is an ideal gas, is

{{\left\langle {{p}_{v}} \right\rangle }^{v}}={{\left\langle {{\rho }_{v}} \right\rangle }^{v}}{{R}_{g}}{{\left\langle {{T}_{f}} \right\rangle }^{f}} \qquad \qquad(11)

The thermodynamic relation for saturation pressure is

{{\left\langle {{p}_{v}} \right\rangle }^{v}}={{p}_{0}}\exp \left\{ -\left[ \frac{2{{\sigma }_{\ell v}}}{{{r}_{eff}}{{\rho }_{\ell }}{{R}_{g}}{{\left\langle {{T}_{f}} \right\rangle }^{f}}}+\frac{{{h}_{lv}}}{{{R}_{g}}}\left( \frac{1}{{{\left\langle {{T}_{f}} \right\rangle }^{f}}}-\frac{1}{{{T}_{0}}} \right) \right] \right\} \qquad \qquad(12)

where T0 is saturation temperature corresponding to a reference pressure p0.

The above governing equations for multiphase transport in porous media are based upon the assumption that the liquid and vapor phases are in thermal equilibrium. Gray (2000) addressed macroscale equilibrium conditions for liquid-gas two-phase flow in porous media. Duval et al. (2004) used the method of volume averaging to derive a three-temperature macroscopic model that considered local thermal nonequilibrium among the three phases. A closed form of the evaporation rate at the macroscopic level can be obtained depending on the macroscopic temperatures and the effective properties. Many multiphase problems have a clear liquid/vapor interface within the porous zone. These problems can be dealt with by solving the phases as separate solutions with boundary conditions between the phases.

The basic equations for the MFM model can also be laid out in terms of phase saturation. The volume averaged continuity equation for phase k in terms of phase saturation is:

\frac{\partial }{\partial t}\left( \varepsilon {{s}_{k}}{{\rho }_{k}} \right)+\nabla \cdot \left( \varepsilon {{s}_{k}}{{\rho }_{k}}{{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}} \right)={{{\dot{m}}'''}_{k}} \qquad \qquad(13)

The porosity of the porous medium is \varepsilon , the phase saturation is sk , the intrinsic phase average velocity is {{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}} and the volumetric mass production due to phase change from all other phases to phase k is {{{\dot{m}}'''}_{k}} . The summation of the phase saturation is unity, \sum\limits_{k=1}^{\Pi }{{{s}_{k}}}=1 . The phase saturation is the fraction of a pore occupied by phase k; therefore, the product of the phase saturation and the porosity is the total volume fraction of that phase, {{\varepsilon }_{k}} .

{{\varepsilon }_{k}}=\varepsilon {{s}_{k}} \qquad \qquad(14)

The summation of the total volume fraction of each phase and the solid matrix is unity, \sum\limits_{k=1}^{\Pi }{{{\varepsilon }_{k}}}+{{\varepsilon }_{sm}}=1 . The momentum equation in a porous medium for phase k is Darcian if inertia and macroscopic shear effects (shear stresses between pores) are neglected.

\varepsilon {{s}_{k}}{{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}}=-\frac{K{{K}_{rk}}}{{{\mu }_{k}}}\left( \nabla {{p}_{k}}-{{\rho }_{k}}\mathbf{g} \right) \qquad \qquad(15)

Equation (15) can be directly inserted into the continuity equation to get a Laplace-type equation for the pressure of phase k. The volume-averaged species equation for the species mass fraction, {{\left\langle {{\omega }_{k,i}} \right\rangle }^{k}} , is:

\frac{\partial }{\partial t}\left( \varepsilon {{s}_{k}}{{\rho }_{k}}{{\left\langle {{\omega }_{k,i}} \right\rangle }^{k}} \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}} \right)=-\left\langle \nabla \cdot {{\mathbf{J}}_{k,i}} \right\rangle +{{{\dot{m}}'''}_{k,i}} \qquad \qquad(16)

The diffusive mass flux of species i is {{\mathbf{J}}_{k,i}} , and the species generation rate due to phase change or chemical reaction is {{{\dot{m}}'''}_{k,i}} . The products of chemical reactions in phase k are also in phase k; therefore the sum of the species generation term is simply the mass production due to phase change:

\sum\limits_{i=1}^{N}{{{{{\dot{m}}'''}}_{k,i}}}={{{\dot{m}}'''}_{k}} \qquad \qquad(17)

Also, the summation of the species generation for one component overall the phases is simply the species generation rate due to chemical reactions, {{{\dot{m}}'''}_{i}} , only because phase change processes do not change the species.

\sum\limits_{k=1}^{\Pi }{{{{{\dot{m}}'''}}_{k,i}}}={{{\dot{m}}'''}_{i}} \qquad \qquad(18)

The final equation is the energy equation for phase k, and it is:

  & \frac{\partial }{\partial t}\left( \varepsilon {{s}_{k}}{{\rho }_{k}}{{\left\langle {{h}_{k}} \right\rangle }^{k}} \right)+\nabla \cdot \left( \varepsilon {{s}_{k}}{{\rho }_{k}}{{\left\langle {{\mathbf{V}}_{k}} \right\rangle }^{k}}{{\left\langle {{h}_{k}} \right\rangle }^{k}} \right) \\ 
 & =-\left\langle \nabla \cdot {{{\mathbf{{q}''}}}_{k,i}} \right\rangle +{{{{\dot{q}}'''}}_{k}}+{{{{\dot{m}}'''}}_{k}}{{\left\langle {{h}_{k}} \right\rangle }^{k}}+{{{{\dot{q}}'''}}_{k,E}} \\ 
\end{align} \qquad \qquad(19)

The second term on the right hand side represents the total volumetric heat transfer rate from all other phases to phase k, and the third term on the right hand side is the heat added through the mass production of phase k through phase change. The last term is the heat generated in phase k to an external heat source such as radiation or electrical current. The summation of the second and third terms over all the phases is zero, because of the interfacial energy balance.

\sum\limits_{k=1}^{\Pi }{\left( {{{{\dot{q}}'''}}_{k}}+{{{{\dot{m}}'''}}_{k}}{{\left\langle {{h}_{k}} \right\rangle }^{k}} \right)}=0 \qquad \qquad(20)


Duval, F., Fichot, F., and Quintard, M., 2004, “A Local Thermal Non-Equilibrium Model for Two-Phase Flows with Phase-Change in Porous Media,” International Journal of Heat and Mass Transfer, Vol. 47, pp. 613-639.

Whitaker, S., 1977, “Simultaneous Heat, Mass and Momentum Transfer in Porous Media: a Theory of Drying,” Advances in Heat Transfer, Vol. 13, pp. 119-203.

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