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

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## 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

*d*,*D*Wall thickness, dimensionless wall thickness

*g*Gravitational acceleration

*h*,*H*Heater size, dimensionless heater size

*h*_{a}Atmospheric convective heat transfer coefficient

*K*Permeability of the porous medium

*K*_{r}Thermal conductivity ratio

*k*_{p}Effective thermal conductivity of porous medium

*k*_{w}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

*α*_{m}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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