# Numerical simulation of three-dimensional natural convection inside a heat generating anisotropic porous medium

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## Abstract

Natural convection in anisotropic heat generating porous medium enclosed inside a rectangular cavity has been studied. A 3D finite volume based code is developed using the Darcy approximation and validated using experimental results of natural convection around an enclosed rod bundle. Subsequently, detailed simulation is carried out for a cavity, filled with orthotropic porous medium. The effects of heat generation, geometry and anisotropy are studied. Anisotropy is found to be of significant importance for both maximum value and distribution of temperature.

## Keywords

Porous Medium Natural Convection Rayleigh Number Rectangular Cavity Permeability Ratio## Nomenclature

- A
Aspect ratio \( \left( {\frac{{L_z }} {L}} \right) \)

*g*Acceleration due to gravity, m s

^{−2}*k*Thermal conductivity, W m

^{−1}K^{−1}*K*Permeability, m

^{2}*K*^{*}Permeability ratio \( \left( {\frac{{K_z }} {{K_x }}} \right) \)

*L*Length in

*x*-direction, m*p*Pressure, N m

^{−2}*P*Non-dimensional pressure, \( P = \frac{p} {{\mu \alpha _x /K_x }} \)

- \( q''' \)
Internal heat generation rate, W m

^{−3}- Ra
^{*} Modified Rayleigh number, \({\text{Ra}}^* = {\text{ }}\frac{{\rho g\beta \left( {q'''L^2 /k_x } \right)K_z L}} {{\nu \alpha _x }}\)

*T*Temperature, K

*T*_{cw}Temperature of cold wall, K

*T*_{f}, \( T_\infty \)Constant cooling fluid (air) temperature, K

*u*Velocity in

*x*-direction, m s^{−1}*v*Velocity in

*y*-direction, m s^{−1}*w*Velocity in

*z*-direction, m s^{−1}*U*Non-dimensional velocity in

*X*-direction, \( U = \frac{u} {{\alpha _x /L}}, \)*V*Non-dimensional velocity in

*Y*-direction, \( V = \frac{v} {{\alpha _x /L}}, \)*W*Non-dimensional velocity in

*W*-direction, \( W = \frac{w} {{\alpha _x /L}} \)*x*,*y*,*z*Co-ordinates in dimensional form, m

*X*,*Y*,*Z*Co-ordinates in non-dimensional form \( \left( {X = \frac{x} {L},\,Y = \frac{y} {L},\,Z = \frac{z} {L}} \right) \)

## Greek Symbols

- K*
Permeability ratio \( \left( {\frac{{K_z }} {{K_x }}} \right) \)

- λ
Thermal conductivity ratio \( \left( {\frac{{k_z }} {{k_x }}} \right) \)

- ρ
Density, kg m

^{−3}- α
Thermal diffusivity, m

^{2}s^{−1}- β
Coefficient of thermal expansion, K

^{–1}- μ
Viscosity of fluid, N s m

^{−2}- ν
Kinematic viscosity, m

^{2}s^{−1}- θ
Non-dimensional temperature \( \left( {\theta = \frac{{T - T_f }} {{q^{'''} L^2 /k_x }}} \right) \)

## References

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