Transport in Porous Media

, Volume 112, Issue 2, pp 313–332 | Cite as

Multiphase, Multicomponent Flow in Porous Media with Local Thermal Non-equilibrium in a Multiphase Mixture Model

  • Franz LindnerEmail author
  • Michael Pfitzner
  • Christian Mundt


For multiphase, multicomponent flow, the multiphase mixture model of Wang and Cheng (Int J Heat Mass Trans 39(17):3607–3618, 1996) has been used in the literature. In our self-written solver, based on this mathematical background, we use a modified multiphase mixture model for a simple transpiration cooling application. In this modeling approach, we do not solve for every phase or component separately. Instead, we solve for mixture variables and thus only use a simple set of conservation equations, which is valid in the entire domain. The information of concentration of components and phase saturations is incorporated within the multiphase mixture variables and can be extracted out of them easily with arithmetic equations. It has been shown that this approach is completely equivalent to using the separate phase models. We implemented an augmentation to local thermal non-equilibrium; i.e., we allow for different fluid and solid temperatures locally. We show solutions for stationary cases of evaporation and discuss the effect of variation of heat input, inlet temperature, inlet mass flux and fraction of components.


Porous medium Multicomponent fluid Evaporation  Local thermal non-equilibrium Mixture model 

List of Symbols

Latin Symbols

\(A^\alpha ,B^\alpha ,C^\alpha \)

Coefficients of Antoine equation of component \(\alpha \)

\(a_s \)

Specific surface of pores Open image in new window


Material-fluid parameter for boiling heat flux




Isobaric specific heat capacity Open image in new window


Characteristic size of porous medium \(({\mathrm {m}})\)


Capillary diffusion coefficient Open image in new window


Hindrance function

\(\mathbf g \)

Gravity vector Open image in new window


Specific enthalpy Open image in new window


Volumetric enthalpy Open image in new window


Enthalpy of vaporization Open image in new window


Mean heat transfer coefficient of solid and liquid phase Open image in new window


Mean heat transfer coefficient of solid and vapor phase Open image in new window


Leverett J-function

\(\mathbf j _k\)

Diffusive mass flux of phase k within the multiphase mixture Open image in new window

\(k \)

Heat conductivity Open image in new window


Permeability tensor \(({\mathrm {m}}^2)\)


Relative permeability


Molar mass Open image in new window


Mass \(({\mathrm {kg}})\)


Mass flux Open image in new window


Constant, number of moles \(({\mathrm {mol}})\), index for iterations


Number of cells


Pressure \(({\mathrm {Pa}})\)


Polynomial to approximate H as a function of Open image in new window

\(p^\alpha \)

Partial pressure of component \(\alpha \) \(({\mathrm {Pa}})\)

\(p^{\alpha ,0}\)

Equilibrium vapor pressure of component \(\alpha \,({\mathrm {Pa}})\)


Capillary pressure (phase pressure difference) \(({\mathrm {Pa}})\)


Term in Taylor series expansion for temperature \(({\mathrm {K}})\)


Heat exchange between solid and fluid phase Open image in new window

\(\dot{q}_\text {boil}\)

Additional heat exchange during boiling Open image in new window

\(\dot{q}_\text {total}\)

Net heat flux into the fluid Open image in new window


Heat flux at outlet boundary Open image in new window

\(\mathbf r \)

Right-hand side of discretization system


Saturation of phase k


Short for \(s_l\), saturation of liquid phase


Temperature \((^\circ {\mathrm {C}})\)

\(\mathbf u \)

Velocity vector Open image in new window


Volume \(({\mathrm {m}}^3)\)


Mass fraction


Coordinate \(({\mathrm {m}})\), molar fraction


Nusselt number


Prandtl number

\(\mathcal {L}\)

Characteristic length \(({\mathrm {m}})\)

Greek Symbols

\(\varXi \)

Constant for volumetric enthalpy

\(\phi \)


\(\gamma _h\)

Advection correction coefficient

\(\gamma _\rho \)

Density correction coefficient

\(\lambda \)

Relative mobility

\(\mu \)

Dynamic viscosity \(({\mathrm {Pa}}\,{\mathrm {s}})\)

\(\nu \)

Kinematic viscosity Open image in new window

\(\varOmega \)

Cell length, 1d \(({\mathrm {m}})\)

\(\varrho \)

Density Open image in new window

\(\sigma \)

Surface tension Open image in new window

\(\zeta \)

Measure for global energy conservation


\(\alpha \)

General component


Iteration number


m-th derivation


\(\text {boil}\)


\(\text {bubble}\)

At bubble point

\(\text {dew}\)

At dew point

\(\text {dryout}\)

At dryout state

\(\text {eff}\)





Discrete value

\(\text {in}\)

At inlet


General phase


Liquid phase

\(\text {norm}\)


\(\text {exit}\)

At exit


Solid phase

\(\text {sat}\)

At saturation state

\(\text {total}\)

Total value


Vapor phase



This work was developed during a research project with Institut für Technik Intelligenter Systeme (ITIS GmbH) and Institute for Thermodynamics at University of the Federal Armed Forces, Munich.


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Franz Lindner
    • 1
    Email author
  • Michael Pfitzner
    • 1
  • Christian Mundt
    • 1
  1. 1.Department of Aerospace Engineering, Institute for ThermodynamicsUniversity of the Federal Armed Forces MunichMunichGermany

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