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Conjugate Natural Convection in a Porous Enclosure with Non-Uniform Heat Generation

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Abstract

Effects of a conductive wall on natural convection in a square porous enclosure having internal heating at a rate proportional to a power of temperature difference is studied numerically in this article. The horizontal heating is considered, where the vertical walls heated isothermally at different temperatures while the horizontal walls are kept adiabatic. The Darcy model is used in the mathematical formulation for the porous layer and finite difference method is applied to solve the dimensionless governing equations. The governing parameters considered are the Rayleigh number (0 ≤ Ra ≤ 1000), the internal heating and the local exponent parameters (0 ≤ γ ≤ 5), (1 ≤ λ ≤ 3), the wall to porous thermal conductivity ratio (0.44 ≤ Kr ≤ 9.9) and the ratio of wall thickness to its width (0.02 ≤ D ≤ 0.5). The results are presented to show the effect of these parameters on the fluid flow and heat transfer characteristics. It is found a strong internal heating can generate significant maximum fluid temperature more than the conductive solid wall. Increasing value thermal conductivity ratio and/or decreasing the thickness of solid wall can increase the maximum fluid temperature. It is also found that at very low Rayleigh number, the heat transfer across the porous enclosure remain stable for any values of the thermal conductivity ratio.

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Abbreviations

d, D :

Wall thickness, dimensionless wall thickness

g :

Gravitational acceleration

G :

Measure of the internal heat generation

K :

Permeability of the porous medium

Kr :

Thermal conductivity ratio

k p :

Effective thermal conductivity of porous medium

k w :

Thermal conductivity of wall

ℓ :

Width and height of enclosure

\({\overline{Nu}}\) :

Average Nusselt number

Ra :

Rayleigh number

T :

Temperature

u, v :

Velocity components in the x- and y-directions

x, y and X, Y :

Space coordinates and dimensionless space coordinates

α :

Effective thermal diffusivity

β :

Thermal expansion coefficient

γ :

Internal heating parameter

λ :

Local heating exponent

ψ, Ψ:

Stream function, dimensionless stream function

Θ:

Dimensionless temperature

ν :

Kinematic viscosity

c:

Cold

h:

Hot

max:

Maximum

p:

Porous

w:

Wall

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Authors and Affiliations

  1. School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600, UKM, Bangi, Selangor, Malaysia

    H. Saleh

  2. Centre for Modelling & Data Analysis, School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600, UKM, Bangi, Selangor, Malaysia

    I. Hashim

Authors
  1. H. Saleh
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  2. I. Hashim
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Correspondence to H. Saleh.

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Saleh, H., Hashim, I. Conjugate Natural Convection in a Porous Enclosure with Non-Uniform Heat Generation. Transp Porous Med 94, 759–774 (2012). https://doi.org/10.1007/s11242-012-0023-z

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