Transport in Porous Media

, Volume 69, Issue 3, pp 343–357

Onset of convection in a horizontal porous channel with uniform heat generation using a thermal nonequilibrium model

  • Ali Nouri-Borujerdi
  • Amin R. Noghrehabadi
  • D. Andrew S. Rees
Original Paper

DOI: 10.1007/s11242-006-9076-1

Cite this article as:
Nouri-Borujerdi, A., Noghrehabadi, A.R. & Rees, D.A.S. Transp Porous Med (2007) 69: 343. doi:10.1007/s11242-006-9076-1

Abstract

This paper considers the onset of free convection in a horizontal fluid-saturated porous layer with uniform heat generation. Attention is focused on cases where the fluid and solid phases are not in local thermal equilibrium, and where two energy equations describe the evolution of the temperature of each phase. Standard linearized stability theory is used to determine how the criterion for the onset of convection varies with the inter-phase heat transfer coefficient, H, and the porosity-modified thermal conductivity ratio, γ. We also present asymptotic solutions for small values of H. Excellent agreement is obtained between the asymptotic and the numerical results.

Keywords

Local thermal non-equilibriumInstabilityNatural convectionPorous mediaInternal heat generation

Nomenclature

C

Specific heat

Da

Darcy number

h

Inter-phase heat transfer coefficient

H

Nondimensional inter-phase heat transfer coefficient

g

Gravity

k

Wavenumber of the disturbance

K

Permeability

L

Depth of the convection layer

LTE

Local thermal equilibrium

LTNE

Local thermal nonequilibrium

P

Pressure

q′′′

Rate of heat generation

Q

Overall rate of heat generation

Ra

Darcy–Rayleigh number (Eq. 6)

t

Time

T

Temperature

u,v

Horizontal and vertical velocities

x,y

Horizontal and vertical Cartesian coordinates

Greek symbols

 

α

Diffusivity ratio

γ

Porosity-scaled conductivity ratio

β

Coefficient of cubical expansion

ρ

Density

σ

Heat capacity ratio

μ

Dynamic viscosity

λ

Constant

ν

Kinematic viscosity

\(\varepsilon\)

Porosity

\(\varsigma\)

Constant denoting internal heating contribution

\(\psi\)

Streamfunction

\(\psi\)

Streamfunction disturbance

\(\theta,\phi\)

Scaled fluid and solid temperatures

\(\theta,\phi\)

Fluid and solid temperature disturbances

ω

Amplification rate of disturbances

Superscripts and subscripts

 

Dimensional

basic

Basic state

PM

Porous media

f

Fluid

s

Solid

0

Wall temperature

k-derivative

y-derivative

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • Ali Nouri-Borujerdi
    • 1
  • Amin R. Noghrehabadi
    • 1
    • 2
  • D. Andrew S. Rees
    • 2
  1. 1.School of Mechanical EngineeringSharif University of TechnologyTehranIran
  2. 2.Department of Mechanical EngineeringUniversity of BathBathUK