Advertisement

Transport in Porous Media

, Volume 112, Issue 1, pp 167–187 | Cite as

Phase-Change Transpiration Cooling in a Porous Medium: Determination of the Liquid/Two-Phase/Vapor Interfaces as a Problem of Eigenvalues

  • M. Peralta
  • F. Méndez
  • O. BautistaEmail author
Article

Abstract

In this work, we carry out a theoretical analysis of the transpiration cooling with a liquid coolant phase change in a porous medium. The evaporation of the liquid inside the porous medium causes the appearance of three regions: a liquid region, a two-phase region and a vapor region. This kind of physical problem has been widely studied in the specialized literature and the main contributions are based on the separated phase model or by using the two-phase mixture model. Here, we propose a new model that permits to numerically determine the thickness of the three regions that are formed inside of the porous medium. To analyze this phenomenon, the governing equations were appropriately nondimensionalized, where suitable Péclet numbers for each region appear such that these numbers represent eigenvalues for the mathematical model, and when they are obtained, the relative position of the interfaces between each region is found.

Keywords

Transpiration cooling Porous media Coolant phase change Eigenvalues 

List of Symbols

Symbol definition

a

Parameter defined in Eq. (14)

b

Parameter defined in Eq. (14)

Bo

Bond number, \(K\rho _{\mathrm{l}}g{/}\sigma \)

\(c_{\mathrm{pl}}\)

Effective specific heat of liquid

\(c_{\mathrm{pv}}\)

Effective specific heat of vapor

Ca

Capillary number, \(\dot{m}_{0}\nu _{\mathrm{l}} / \sigma \)

Da

Darcy number, \((\varepsilon K)^{1/2}/\hbox {H}\)

\(d_{\mathrm{p}}\)

Pore diameter

g

Acceleration of gravity

H

Thickness of the porous matrix

\(h_{\mathrm{lv}}\)

Latent heat of evaporation

J(s)

Leverett’s function

\(Ja_{\mathrm{lv}}\)

Jakob number for the two-phase region, \(c_{\mathrm{pl}} T_{\mathrm{sat}} /h_{\mathrm{lv}}\)

Ja

Jakob number, \(c_{\mathrm{pl}} (T_{\mathrm{sat}}-T_{0}) /h_{\mathrm{lv}}\)

K

Absolute permeability

\(K_{\mathrm{rl}}\)

Relative permeability of the liquid phase

\(K_{\mathrm{rv}}\)

Relative permeability of the vapor phase

\(k_{\mathrm{lv}}\)

Effective thermal conductivity in the two-phase region

\(k_{\mathrm{l}}\)

Thermal conductivity of the liquid phase

\(k_{\mathrm{v}}\)

Thermal conductivity of the vapor phase

\(k_{\mathrm{s}}\)

Thermal conductivity of the solid

\(k_{\alpha e}\)

Effective thermal conductivity in the liquid and vapor regions

\(k^{*}\)

Dimensionless effective thermal conductivity

\(L_{\mathrm{l}}\)

Unknown length of the liquid region

\(\bar{L}_{\mathrm{l}}\)

Dimensionless length of the liquid region, \(L_{\mathrm{l}}/\hbox {H}\)

\(L_{\mathrm{lv}}\)

Unknown length of the two-phase region

\(\bar{L}_{\mathrm{lv}}\)

Dimensionless length of the two-phase region, \(L_{\mathrm{lv}}/\hbox {H}\)

\(L_{\mathrm{v}}\)

Unknown length of the vapor region

\(\bar{L}_{\mathrm{v}}\)

Dimensionless length of the vapor region, \(L_{\mathrm{v}}/\hbox {H}\)

Le

Modified capillary Lewis number

\(\dot{m}_{0}\)

Coolant flow rate

\(\dot{m}_{\mathrm{l}}\)

Mass flux of liquid in the two-phase region

\(\dot{m}_{\mathrm{v}}\)

Mass flux of vapor in the two-phase region

M

Molar mass

\(p_{\mathrm{l}}\)

Pressure of the liquid phase

\(p_{\mathrm{v}}\)

Pressure of the vapor phase

\(Pe_{\mathrm{l}}\)

Péclet number of liquid region

\(Pe_{\mathrm{v}}\)

Péclet number of vapor region

\(Pe_{\mathrm{c}}\)

Péclet number of the system

\((Pe_{\mathrm{v}})_{i+1}\)

Péclet number used in the iteration process

\((Pe_\mathrm{v})_{i}\)

Initial guessed Péclet number

\(\Delta Pe_\mathrm{v}\)

Increment to the corrected Péclet number

\(p_{\mathrm{c}}\)

Capillary pressure

\(q''\)

Heat flux

\(q''_{\mathrm{v}}\)

Heat flux defined in Eq. (24)

Q

Parameter defined in Eq. (38)

R

Universal gas constant

s

Dimensionless liquid saturation, \((s_{\mathrm{l}}-s_{\mathrm{li}})/(1-s_{\mathrm{li}})\)

\(s_{\mathrm{l}}\)

Liquid saturation

\(s_{\mathrm{li}}\)

Irreducible liquid saturation

T

Absolute temperature

\(T_{\mathrm{sat}}\)

Saturation temperature

\(T_{\mathrm{top}}\)

Temperature at the upper surface

\(T_{\mathrm{l}}\)

Temperature of the liquid region

\(T_{\mathrm{lv}}\)

Temperature of the two-phase region

\(T_{\mathrm{v}}\)

Temperature of the vapor region

\(T_{0}\)

Temperature of the coolant flow at the bottom of the porous matrix

\(T_{\mathrm{lv}}\)

Temperature in the two-phase region

\(T_{\mathrm{int}}\)

Temperature of the interface between the two-phase and vapor region

\(\Delta T_{\mathrm{v}}\)

Temperature drop in the vapor region

\(u_{\mathrm{l}}\)

Superficial velocity of the liquid phase

\(u_{\mathrm{v}}\)

Superficial velocity of the vapor phase

y

Transversal coordinate at the system

Y

Dimensionless transversal coordinate of the vapor region

z

Dimensionless transversal coordinate of the two-phase region

Greek Letters

\(\alpha \)

Represents the liquid or vapor phases, \(\alpha = \hbox {l, v}\)

\(\bar{\alpha }\)

Ratio of vapor to liquid density

\(\beta \)

Dimensionless parameter defined in Eq. (43)

\(\gamma \)

Dimensionless parameter defined in Eq. (31)

\(\bar{\gamma }\)

Dimensionless parameter defined in Eq. (31)

\(\varepsilon \)

Porosity of the porous medium

\(\eta \)

Dimensionless transversal coordinate for the liquid region

\(\theta _{\mathrm{l}}\)

Dimensionless temperature for the liquid region

\(\theta _{\mathrm{lv}}\)

Dimensionless temperature for the two-phase region

\(\theta _{\mathrm{v}}\)

Dimensionless temperature for the vapor region

\(\mu _{\mathrm{l}}\)

Dynamic viscosity of the fluid

\(\nu _{\mathrm{l}}\)

Kinematic viscosity for the liquid phase

\(\nu _{\mathrm{v}}\)

Kinematic viscosity for the vapor phase

\(\bar{\nu }\)

Ratio of kinematic viscosities of liquid and vapor

\(\Omega \)

Dimensionless parameter defined in Eq. (31)

\(\xi \)

Dimensionless parameter defined in Eq. (38)

\(\rho _{\mathrm{l}}\)

Density of the liquid phase

\(\rho _{\mathrm{v}}\)

Density of the vapor phase

\(\sigma \)

Surface tension

\(\phi \)

Dimensionless parameter defined in Eq. (38)

Subscripts

c

Capilar

e

Effective

int

Refers to the interface between the two-phase region and vapor region

0

Coolant conditions

l

Conditions at the liquid phase

lv

Conditions at the two-phase region

r

Relative

s

Conditions at the solid phase

sat

Conditions of saturation

top

Refers to the upper surface

v

Conditions at vapor phase

Notes

Acknowledgments

This work has been supported by the research grants no. CB-2013/220900 of Consejo Nacional de Ciencia y Tecnología (CONACyT) and 20150919 of SIP-IPN at Mexico. M. Peralta acknowledges the support of UNAM program of doctoral fellowship, and to the CONACyT program for a postdoctoral fellowship (12525) granted to spend a year as a postdoctoral researcher at SEPI-ESIME IPN. The authors also wish to express their thanks to the very competent Reviewers for the valuable comments and suggestions

References

  1. Baggio, P., Bonacina, C., Scherefler, B.A.: Some considerations on modeling heat and mass transfer in porous media. Transp. Porous Media 28(3), 233–251 (1997)CrossRefGoogle Scholar
  2. Bau, H.H., Torrance, K.: Boiling in low-permeability porous materials. Int. J. Heat Mass Transf. 25, 45–55 (1982)CrossRefGoogle Scholar
  3. Bridge, L., Bradean, R., Ward, M.J., Wetton, B.R.: The analysis of a two-phase zone with condensation in a porous medium. J. Eng. Math. 45(3–4), 247–268 (2003)CrossRefGoogle Scholar
  4. Faghri, A., Zhang, Y.: Transport Phenomena in Multiphase Systems. Academic Press, New York (2006)Google Scholar
  5. He, F., Wang, J., Xu, L., Wang, X.: Modeling and simulation of transpiration cooling with phase change. Appl. Therm. Eng. 58, 173–180 (2013)CrossRefGoogle Scholar
  6. Hoffman, J.D.: Numerical Methods for Engineers and Scientists. Marcel Dekker Inc., New York (2001)Google Scholar
  7. Kaviany, M.: Principles of Heat and Mass Transfer. Springer, New York (1995)Google Scholar
  8. Kukreti, A.R., Rajapaksa, Y.: A numerical model for simulating two-phase flow through porous media. Appl. Math. Model. 13, 268–281 (1989)CrossRefGoogle Scholar
  9. Leverett, M.: Capillary behavior in porous solids. Trans. AIME 142(1), 152–169 (1941)CrossRefGoogle Scholar
  10. Olivella, S., Gens, A.: Vapour transport in low permeability unsaturated soils with capillary effects. Transp. Porous Media 40, 219–241 (2000)CrossRefGoogle Scholar
  11. Rubin, A., Schweitzer, S.: Heat transfer in porous media with phase change. Int. J. Heat Mass Transf. 15, 43–60 (1972)CrossRefGoogle Scholar
  12. Rutqvist, J., Zheng, L., Chen, F., Liu, H.H., Birkholzer, J.: Modeling of coupled thermo-hydro-mechanical processes with links to geochemistry associated with bentonite-backfilled repository tunnels in clay formations. Rock Mech. Rock Eng. 47, 167–186 (2014)CrossRefGoogle Scholar
  13. Shi, J.X., Wang, J.H.: A numerical investigation of transpiration cooling with liquid coolant phase change. Transp. Porous Media 87, 703–716 (2011)CrossRefGoogle Scholar
  14. Teplitskii, Y.S., Kovenskii, V.I.: Concerning the filtrational cooling of a heat-releasing granular bed in the presence of a first-order phase transition. J. Eng. Phys. Thermophys. 80(2), 311–321 (2007)CrossRefGoogle Scholar
  15. Udell, K.S.: Heat transfer in porous media considering phase change and capillarity—heat pipe effect. Int. J. Heat Mass Transf. 28, 485–494 (1985)CrossRefGoogle Scholar
  16. Udell, K.S.: Heat transfer in porous media heated from above with evaporation condensation, and capillary effects. Trans. ASME J. Heat Transf. 105(3), 485–492 (1983)CrossRefGoogle Scholar
  17. Wang, C.Y., Beckermann, C., Fan, C.: Numerical study of boiling and natural convection in capillary porous media using the two-phase mixture model. Numer. Heat Trans. A Appl. 26(4), 375–398 (1994)CrossRefGoogle Scholar
  18. Wang, C.Y.: A fixed-grid numerical algorithm for two-phase flow and heat transfer in porous media. Numer. Heat Tranf. B Fundam. 32(1), 85–105 (1997)CrossRefGoogle Scholar
  19. 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 (1993a)Google Scholar
  20. Wang, C.Y., Beckermann, C.: A two-phase mixture model of liquid–gas flow and heat transfer in capillary porous media-II. Application to pressure-driven boiling flow adjacent to a vertical heated plate. Int. J. Heat Mass Transf. 36(11), 2759–2768 (1993b)Google Scholar
  21. Wei, K., Wang, J., Mao, M.: Model discussion of transpiration cooling with boiling. Transp. Porous Media 94(1), 303–318 (2012)CrossRefGoogle Scholar
  22. Wetton, B.R., Bridge, L.J.: A mixture formulation for numerical capturing of a two-phase/vapor interface in a porous medium. J. Comput. Phys. 225(2), 2043–2068 (2007)CrossRefGoogle Scholar
  23. Xin, C., Rao, Z., You, X., Song, Z., Han, D.: Numerical investigation of vapor–liquid heat and mass transfer in porous media. Energy Convers. Manag. 78, 1–7 (2014)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.SEPI-ESIME AzcapotzalcoInstituto Politécnico NacionalMexicoMexico
  2. 2.Departamento de Termofluidos, Facultad de IngenieríaUNAMMexicoMexico

Personalised recommendations