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Transport in Porous Media

, Volume 96, Issue 2, pp 305–318 | Cite as

Effect of Conduction in Bottom Wall on Bénard Convection in a Porous Enclosure with Localized Heating and Lateral Cooling

  • A. Alhashash
  • H. Saleh
  • I. HashimEmail author
Article

Abstract

Darcy-Bénard convection in a square porous enclosure with a localized heating from below and lateral cooling is studied numerically in the present paper. A finite-thickness bottom wall is locally heated, the top wall is kept at a lower temperature than the bottom wall temperature, and the lateral walls are cooled. The finite difference method has been used to solve the dimensionless governing equations. The analysis in the undergoing numerical investigation is performed in the following ranges of the associated dimensionless groups: the heat source length—\({0.2\leq H \leq 0.9}\), the wall thickness—\({0.05\leq D \leq 0.4}\), the thermal conductivity ratio—\({0.8\leq K_{\rm r} \leq 9.8}\), and the Biot number—\({0.1\leq Bi \leq 1.1}\). It is observed that the heat transfer rate could increase with increasing heat source lengths, thermal conductivity ratio, and cooling intensity. There exists a critical wall thickness for a high wall conductivity below which the increasing wall thickness increases the heat transfer rate and above which the increasing wall thickness decreases the heat transfer rate.

Keywords

Conjugate heat transfer Natural convection Localized heating Darcy’s law 

List of symbols

Bi

Biot number

dD

Wall thickness, dimensionless wall thickness

g

Gravitational acceleration

h, H

Heater size, dimensionless heater size

ha

Atmospheric convective heat transfer coefficient

K

Permeability of the porous medium

Kr

Thermal conductivity ratio

kp

Effective thermal conductivity of porous medium

kw

Thermal conductivity of wall

k

Thermal conductivity

\({\ell}\)

Width and height of cavity

\({\overline{Nu}}\)

Average Nusselt number

Pr

Prandtl number

Ra

Rayleigh number

T

Temperature

u,v

Velocity components in the x- and y-directions

U,V

Dimensionless velocity components in the X- and Y-directions

x,y & X, Y

Space coordinates & dimensionless space coordinates

Greek symbols

αm

Thermal diffusivity

β

Thermal expansion coefficient

\({\Theta}\)

Dimensionless temperature

\({\nu}\)

Kinematic viscosity

\({\psi \& \Psi}\)

Stream function & dimensionless stream function

Subscript

c

Cold

h

Hot

p

Porous

w

Wall

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Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversiti Kebangsaan MalaysiaBangi SelangorMalaysia
  2. 2.Solar Energy Research InstituteUniversiti Kebangsaan MalaysiaBangi SelangorMalaysia

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