The effect of thermal dispersion on free convection film condensation on a vertical plate with a thin porous layer

  • M. Asbik
  • B. Zeghmati
  • H. Louahlia-Gualous
  • W. M. Yan
Original Paper

Abstract

An analytical solution to the problem of condensation by natural convection over a thin porous substrate attached to a cooled impermeable surface has been conducted to determine the velocity and temperature profiles within the porous layer, the dimensionless thickness film and the local Nusselt number. In the porous region, the Darcy–Brinkman–Forchheimer (DBF) model describes the flow and the thermal dispersion is taken into account in the energy equation. The classical boundary layer equations without inertia and enthalpyterms are used in the condensate region. It is found that due to the thermal dispersion effect, the increasing of heat transfer is significant. The comparison of the DBF model and the Darcy–Brinkman (DB) one is carried out.

Keywords

Condensation Natural convection Porous layer Thermal dispersion Darcy–Brinkman–Forchheimer model Analytical method 

Nomenclature

A

Dimensionless flow inertia term ( \(A=\lambda \varepsilon Re_K H^{\ast 2}\sqrt{Da} )\)

Cp

Specific heat of fluid at constant pressure (J kg−1 K−1)

Da

Darcy number ( \(Da=K \mathord{\left/ {\vphantom {K {L^2}}} \right. \kern-\nulldelimiterspace} {L^2}\))

F

Flow inertia term

g

Gravitational acceleration (m s−2)

hfg

latent heat of vaporization (J kg−1)

H

Thickness of porous coating (m)

Ja

Jakob number

k

Thermal conductivity (W m−1 K−1)

K

Permeability (m2)

L

Length of plate (m)

Nux

Local Nusselt number

Pe

Peclet number

Pr

Prandtl number

ReK

Reynolds number based on \(\sqrt{K}\)

T

Temperature (K)

u

x-component velocity (m s−1)

ur

Reference velocity (m s−1)

x

Vertical coordinate (m)

y

Horizontal coordinate (m)

Greek Symbols

 

γ

Dimensionless dispersion parameter

δ

Film condensation thickness (m)

ɛ

Porosity

λ

Dimensionless inertia parameter

μ

Dynamic viscosity (kg m−1 s−1)

ν

Kinematic viscosity (m2 s−1)

θ

Dimensionless temperature

ρ

Density (kg m−3)

Subscripts—Superscripts

 

ar

Arbitrary

d

Dispersion

eff

Effective

\(\ell\)

Liquid

i

Porous layer-pure condensate interface

p

Porous layer

s

Saturation

v

Vapor

w

Wall

*

Dimensionless quantity

References

  1. Asbik M., Ouldhadda D., Zeghmati B., Khmou A., Il Idrissi A. (2000). Forced Convection laminar film condensation of downward flowing vapor on a single horizontal elliptic cylinder or a bank of elliptical tubes. Numer. Heat. Tr. A. 37:511–544CrossRefGoogle Scholar
  2. Asbik, M., Chaynane, R., Zeghmati, B., Bresson, J.: Etude analytique de la condensation en film laminaire en convection forcée sur la paroi d’une plaque poreuse verticale: effet de la dispersion thermique, Actes du congrès de la Société Française des Thermiciens (SFT2002), pp. 381–386 (2002)Google Scholar
  3. Asbik M., Chaynane R., Boushaba H., Zeghmati B., Khmou A. (2003). Analytical investigation of forced convection film condensation on a vertical porous-layer coated surface. Heat. Mass Transfer 40:143–155CrossRefGoogle Scholar
  4. Carbonell R.G., Whitaker S. (1983). Dispersion in pulsed system-II. Theoretical development for passive dispersion in porous media. Chem. Eng. Sci. 38:1795–1832CrossRefGoogle Scholar
  5. Chaynane R., Asbik M., Boushaba H., Zeghmati B., et Khmou A. (2004). Etude de la condensation en film laminaire en convection forcée d’une vapeur pure et saturée sur la paroi poreuse d’une plaque inclinée. Revue de Mécanique et Industrie. 5(4):381–391CrossRefGoogle Scholar
  6. Cheng P., Hsu C.T. (1986). Applications of Van Driest’s Mixing Length theory to transverse thermal dispersion in forced convective flow through a packed bed. Int. Comm. Heat Transfer 13:613–625CrossRefGoogle Scholar
  7. Hong J.T., Tien C.L. (1987). Analysis of thermal dispersion effect on vertical-plate natural convection in porous media. Int. J. Heat Mass Tran. 30:143–150CrossRefGoogle Scholar
  8. Kaviany M. (1995). Principles of Heat Transfer in Porous Media, 2nd Edn. pp. 561–572. Springer-Verlag, New YorkGoogle Scholar
  9. Kuznetsov A.V., Cheng L., Xiong M. (2002). Effects of thermal dipersion and turbulence in forced convection in a composite parallel-plate channel: investigation of constant wall heat flux and constant wall temperature cases. Numer. Heat Tr. A. 42:365–383CrossRefGoogle Scholar
  10. Lai F.C., Kulacki F.A. (1989). Thermal dispersion effect of Non-Darcy convection on horizontal surface in saturated porous media. Int. J. Heat Mass Tran. 32:971–976CrossRefGoogle Scholar
  11. Nakayam A., Kuwahara F. (2005). Algebraic model for thermal dispersion heat flux within porous media, AICHE J. 51(10):2859–2864Google Scholar
  12. Nield D.A., Béjan A. (1999). Convection in Porous Media, 2nd Edn. pp. 343–344. Springer-Verlag, New YorkGoogle Scholar
  13. Renken K.J, Soltykiewicz D.J, Poulikakos D. (1989). A study of laminar film condensation on a vertical surface with a porous coating. Int. Commun. Heat Mass. 16:181–192CrossRefGoogle Scholar
  14. Renken K.J., Aboye M. (1993). Experiments on film condensation promotion within thin inclined porous coatings. Int. J. Heat Mass Tran. 36:1347–1355CrossRefGoogle Scholar
  15. Renken K.J., Mueller C.D. (1993). Measurements of enhanced film condensation utilizing a porous metallic coating. J. Thermophys. Heat Tr. 7:148–152Google Scholar
  16. Renken K.J., Meechan K. (1995). Impact of thermal dispersion during forced convection condensation in a thin porous/fluid composite system. Chem. Eng. Commun. 131:189–205Google Scholar
  17. Renken K.J., Raich M.R. (1996). Forced convection steam condensation experiments within thin porous coatings. Int. J. Heat Mass Tran. 39:2937–2945CrossRefGoogle Scholar
  18. Shekarriz A., Plumb O.A. (1989). Enhancement of film condensation using porous fins. J. Thermophys. Heat Tr. 3(3):309–314CrossRefGoogle Scholar
  19. Woodruff D.W., Westwater J.W. (1981). Steam condensation on various gold surface. J. Heat Transf. 103:685–692CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  • M. Asbik
    • 1
  • B. Zeghmati
    • 2
  • H. Louahlia-Gualous
    • 3
  • W. M. Yan
    • 4
  1. 1.Faculté des Sciences et TechniquesGroupe de Thermodynamique Energétique (G.T.E)Boutalamine ErrachidiaMorocco
  2. 2.Laboratoire de Mathématiques et Physique des Systèmes,—Groupe de Mécanique Energétique, (MEPS—GME)Université de Perpignan Via DomitiaPerpignan CedexFrance
  3. 3.Département CRESTFEMTO ST, UTBM-UFC, UMR CNRS 6174BelfortFrance
  4. 4.Department of Mechatronic EngineeringHuafan UniversityTaipeiRepublic of China

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