Transport in Porous Media

, Volume 106, Issue 1, pp 201–220 | Cite as

Fluid Flow and Heat Transfer with Phase Change and Local Thermal Non-equilibrium in Vertical Porous Channels

  • F. LindnerEmail author
  • Ch. Mundt
  • M. Pfitzner


This article investigates the multiphase flow in a porous channel, heated at one section of the wall. For non-isothermal flow with phase-change, a finite-volume solver in MATLAB® is used with a two-phase mixture model with local thermal non-equilibrium , i.e., allowing for different solid- and fluid temperatures locally. The effect of gravity is examined for aiding and opposing flow. The effects of Stanton number of evaporation, Rayleigh number, Péclet number and Biot number on fluid flow and heat transfer for steady case are discussed. For these characteristic numbers and aiding flow, the largest influence on the local shift of the biphasic zone is with Stanton number, Péclet number and Biot number. The minimal saturation in the domain is correlated with increasing Stanton number, Péclet number or Biot number. The Rayleigh number, however, has an opposite effect on minimal saturation. The displacement effect of generated vapor is discussed for high Stanton numbers, Péclet numbers and Biot numbers. Also, we show that for sufficiently high heat input, opposing flow yields large differences in position of the biphasic zone and minimal saturation. Applicability of local thermal equilibrium for cases with phase change is shown to introduce a large error, not only for small Biot numbers or large Reynolds numbers.


Porous medium Multiphase flow Mixture model  Local thermal non-equilibrium Heat exchanger 

List of Latin Symbols


Biot number \((-)\)


Isobaric specific heat capacity (J/kg K)


Capillary diffusion coefficient (m\(^{2}\)/s)


Hindrance function \((-)\)

\(\mathbf g \)

Gravity vector (m/s\(^{2}\))


Specific enthalpy (J/kg)


Enthalpy (J/m\(^{3}\))


Leverett J-Function \((-)\)

\(\mathbf j \)

Diffusive mass flux (kg/s m\(^{2}\))


Heat conductivity (W/m K)

\(\mathbf K \)

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


Relative permeability \((-)\)


Length of channel (m)


Lengths of unheated wall (m)


Characteristic length (m)


Coordinate normal to wall (m)


Nusselt number \((-)\)


Pressure (Pa)


Péclet number \((-)\)


Prandtl number \((-)\)


Heat flux for boundary condition (W/m\(^{2}\))


Heat flux from solid to fluid (W/m\(^{3}\))


Reynolds number \((-)\)


Liquid saturation \((-)\)


Porous Stanton number of evaporation \((-)\)


Temperature (K)

\(\mathbf u \)

Velocity vector (m/s)


Velocity in x (m/s)


Velocity in y (m/s)


Width of channel (m)

\(x, y\)

Coordinates (m)

Greek Symbols

\(\alpha _{sf}\)

Specific surface of pores (1/m)

\(\alpha \)

Diffusion coefficient (m\(^{2}\)/s)

\(\beta \)

Volumetric thermal expansion coefficient (1/K)

\(\varepsilon \)

Porosity \((-)\)

\(\gamma _h\)

Advection correction coefficient \((-)\)

\(\varGamma _h\)

Effective diffusion coefficient (m\(^{2}\)/s)

\(\lambda \)

Relative mobility \((-)\)

\(\mu \)

Dynamic viscosity (Pa s)

\(\nu \)

Kinematic viscosity (m\(^{2}\)/s)

\(\rho \)

Density (kg/m\(^{3}\))

\(\sigma \)

Surface tension (N/m)





Phase \(i\)












At dryout state












At saturation state


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Aerospace Engineering, Institute for ThermodynamicsUniversity of the Federal Armed Forces MunichNeubibergGermany

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