Skip to main content
SpringerLink
Log in

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

  • Published:
Transport in Porous Media Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Abbreviations

\(Bi\) :

Biot number \((-)\)

\(c\) :

Isobaric specific heat capacity (J/kg K)

\(D\) :

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

\(f\) :

Hindrance function \((-)\)

\(\mathbf g \) :

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

\(h\) :

Specific enthalpy (J/kg)

\(H\) :

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

\(J\) :

Leverett J-Function \((-)\)

\(\mathbf j \) :

Diffusive mass flux (kg/s m\(^{2}\))

\(k\) :

Heat conductivity (W/m K)

\(\mathbf K \) :

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

\(k_r\) :

Relative permeability \((-)\)

\(L\) :

Length of channel (m)

\(l_1,l_2\) :

Lengths of unheated wall (m)

\(\fancyscript{L}\) :

Characteristic length (m)

\(n\) :

Coordinate normal to wall (m)

\(Nu\) :

Nusselt number \((-)\)

\(p\) :

Pressure (Pa)

\(Pe\) :

Péclet number \((-)\)

\(Pr\) :

Prandtl number \((-)\)

\(\dot{q}\) :

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

\(\dot{q}_{sf}\) :

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

\(Re\) :

Reynolds number \((-)\)

\(s\) :

Liquid saturation \((-)\)

\(St_v\) :

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

\(T\) :

Temperature (K)

\(\mathbf u \) :

Velocity vector (m/s)

\(u\) :

Velocity in x (m/s)

\(v\) :

Velocity in y (m/s)

\(W\) :

Width of channel (m)

\(x, y\) :

Coordinates (m)

\(\alpha _{sf}\) :

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)

\(f\) :

Fluid

\(i\) :

Phase \(i\)

\(k\) :

Kinetic

\(l\) :

Liquid

\(s\) :

Solid

\(v\) :

Vapor

boil:

boiling

dryout:

At dryout state

eff:

Effective

entry:

Entry

in:

Input

out:

Output

ref:

Reference

sat:

At saturation state

References

  • Bau, H.H., Torrance, K.: Boiling in low-permeability porous materials. Int. J. Heat. Mass. Transf. 25(1), 45–55 (1982)

    Article  Google Scholar 

  • Clark, J., Rohsenow, W.: Local boiling heat transfer to water at low reynolds numbers and high pressures. Trans. ASME 76, 554–562 (1954)

    Google Scholar 

  • Dehghan, M., Valipour, M., Saedodin, S.: Perturbation analysis of the local thermal non-equilibrium condition in a fluid-saturated porous medium bounded by an iso-thermal channel. Transp. Porous Media 102(2), 139–152 (2014). doi:10.1007/s11242-013-0267-2

    Article  Google Scholar 

  • Easterday, O., Wang, C., Cheng, P.: A numerical and experimental study of two-phase flow and heat transfer in a porous formation with localized heating from below. In: Proceedings of ASME Heat Transfer and Fluid Engineering Divisions (1995)

  • He, F., Wang, J., Xu, L., Wang, X.: Modeling and simulation of transpiration cooling with phase change. Appl. Therm. Eng. 58, 173–180 (2013)

    Article  Google Scholar 

  • Kaviany, : Principles of Heat Transfer in Porous Media, 2nd edn. Springer, Berlin (1999)

    Google Scholar 

  • Li, H., Leong, K., Jin, L., Chai, J.: Three-dimensional numerical simulation of fluid flow with phase change heat transfer in an asymmetrically heated porous channel. Int. J. Therm. Sci. 49(12), 2363–2375 (2010a)

    Article  Google Scholar 

  • Li, H., Leong, K., Jin, L., Chai, J.: Transient behavior of fluid flow and heat transfer with phase change in vertical porous channels. Int. J. Heat. Mass. Transf. 53(23–24), 5209–5222 (2010b)

    Article  Google Scholar 

  • Li, H., Leong, K., Jin, L., Chai, J.: Transient two-phase flow and heat transfer with localized heating in porous media. Int. J. Therm. Sci. 49(7), 1115–1127 (2010c)

    Article  Google Scholar 

  • Lie, K.A., Krogstad, S., Ligaarden, I.S., Natvig, J.R., Nilsen, H.M., Skaflestad, B.: Open-source matlab implementation of consistent discretisations on complex grids. Comput. Geosci. 16, 297–322 (2012)

    Article  Google Scholar 

  • Lindner, F., Nuske, P., Helmig, R., Mundt, C., Pfitzner, M.: Transpiration cooling with local thermal non-equilibrium: Model comparison in multiphase flow in porous media. (to be published) (2014).

  • MATLAB: Matlab release 2013a. The MathWorks Inc, Natick (2013)

  • Mondal, P.: Thermodynamically consistent limiting forced convection heat transfer in a asymmetrically heated porous channel: An analytical study. Transp. Porous Media 100, 17–37 (2013)

    Article  Google Scholar 

  • Patankar, S.: Numerical Heat Transfer and Fluid Flow. Series in Computational Methods in Mechanics and Thermal Sciences. Taylor & Francis, New York (1980)

    Google Scholar 

  • Rohsenow, W.: A method of correlating heat transfer data for surface boiling of liquids. Trans. ASME 74, 969–976 (1952)

    Google Scholar 

  • Shi, J., Wang, J.: A numerical investigation of transpiration cooling with liquid coolant phase change. Transp. Porous Media 87, 703–716 (2011)

    Article  Google Scholar 

  • Wakao, N., Kaguei, S., Funazkri, T.: Effect of fluid dispersion coefficients on particle-to-fluid heat transfer coefficients in packed beds: correlation of nusselt numbers. Chem. Eng. Sci. 34, 325–336 (1979)

    Article  Google Scholar 

  • Wang, C.Y., et al.: Multiphase flow and heat transfer in porous media. In: Hartnett Jr, J.P. (ed.) Advances in Heat Transfer, vol. 30, pp. 93–196. Academic Press, New York (1997)

    Google Scholar 

  • Wang, C.: Multi-dimensional modeling of steam injection into porous media. In: Proceedings 2nd European Thermal Sciences and 14th UIT National Heat Transfer Conference, Rome (1996)

  • Wang, C.: A fixed-grid numerical algorithm for two-phase flow and heat transfer in porous media. Numer. Heat Transf. Part B 32(1), 85–105 (1997)

    Article  Google Scholar 

  • Wang, C., Cheng, P.: A multiphase mixture model for multiphase, multicomponent transport in capillary porous media. part i: model development. Int. J. Heat. Mass. Transf. 39(17), 3607–3618 (1996)

    Article  Google Scholar 

  • Wang, C., Beckermann, C., Fan, C.: Numerical study of boiling and natural convection in capillary porous media using the two-phase mixture model. Numer. Heat Transf. 26, 375–398 (1994)

    Article  Google Scholar 

  • Wang, C.Y., Beckermann, C.: A two-phase mixture model of liquid-gas flow and heat transfer in capillary porous media—i. formulation. Int. J. Heat. Mass. Transf. 36(11), 2747–2758 (1993)

    Article  Google Scholar 

  • Wang, J., Shi, J.: Discussion of boundary conditions of transpiration cooling problems using analytical solution of ltne model. ASME J. Heat Transf. 130(1), 014504 (2008)

    Article  Google Scholar 

  • Wang, J., Wang, H.: A discussion of transpiration cooling problems through an analytical solution of local thermal nonequilibrium model. J. Heat Transf. 128, 1093–1098 (2006)

    Article  Google Scholar 

  • Wei, K., Wang, J., Mao, M.: Model discussion of transpiration cooling with boiling. Transp. Porous Media 94, 303–318 (2012)

    Article  Google Scholar 

  • Xue, G.: Numerical methods for multiphysics, multiphase, and multicomponent models for fuel cells. PhD thesis, The Pennsylvania State University (2008)

  • Yuki, K., Abei, J., Hashizume, H., Toda, S.: Numerical investigation of thermofluid flow characteristics with phase change against high heat flux in porous media. J. Heat Transf. 130(1), 012602 (2008)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Aerospace Engineering, Institute for Thermodynamics, University of the Federal Armed Forces Munich, Werner-Heisenberg-Weg 39, 85577, Neubiberg, Germany

    F. Lindner, Ch. Mundt & M. Pfitzner

Authors
  1. F. Lindner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ch. Mundt
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Pfitzner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. Lindner.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lindner, F., Mundt, C. & Pfitzner, M. Fluid Flow and Heat Transfer with Phase Change and Local Thermal Non-equilibrium in Vertical Porous Channels. Transp Porous Med 106, 201–220 (2015). https://doi.org/10.1007/s11242-014-0396-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11242-014-0396-2

Keywords

Search

Navigation