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Transport in Porous Media

, Volume 98, Issue 1, pp 209–221 | Cite as

Free Convection Heat Transfer From A Sphere In A Porous Medium Using A Thermal Non-equilibrium Model

  • Ching-Yang ChengEmail author
Article

Abstract

This work studies the free convection heat transfer from a sphere with constant wall temperature embedded in a fluid-saturated porous medium using a thermal non-equilibrium model. The governing equations are transformed into boundary-layer partial differential equations by the coordinate transform, and the obtained governing equations are then solved by the cubic spline collocation method. The temperature distributions for fluid and solid phases are shown for different values of the porosity scaled thermal conductivity ratio, the interphase heat transfer parameter, and the streamwise coordinate. The effects of the porosity scaled thermal conductivity ratio and the interphase heat transfer parameter between solid and fluid phases on the local Nusselt numbers for fluid and solid phases are examined. Results show the local Nusset number for the porous medium can be increased by increasing the porosity scaled thermal conductivity ratio. Moreover, the thermal non-equilibrium effect is more significant for low values of the porosity scaled thermal conductivity ratio or the interphase heat transfer parameter.

Keywords

Thermal non-equilibrium model Free convection Porous medium Sphere 

List of Symbols

Variables

\(a\)

Radius of the sphere

\(A\)

Angle made by the outward normal from the sphere with the downward vertical

\(C_p \)

Constant-pressure specific heat

\(f\)

Dimensionless stream function

\(g\)

Acceleration due to gravity

\(h\)

Interphase heat transfer coefficient between solid and fluid phases

\(H\)

Interphase heat transfer parameter between solid and fluid phases

\(K\)

Permeability of the porous medium

\(Nu\)

Local Nusselt number

\(Ra\)

Darcy–Rayleigh number

\(T\)

Temperature

\(u,v\)

Dimensional velocity components along \(x\) and \(y\) axes

\(x,y\)

Dimensional Cartesian coordinates along and normal to the sphere

Greek Symbols

\(\alpha _\mathrm{f}\)

Thermal diffusivity of the fluid

\(\beta \)

Coefficient of thermal expansion

\(\gamma \)

Porosity scaled thermal conductivity ratio

\(\varepsilon \)

Porosity

\(\theta \)

Dimensionless temperature

\(\nu _\mathrm{f} \)

Fluid kinematic viscosity

\(\xi ,\eta \)

Dimensionless coordinates

\(\rho \)

Density

\(\bar{\psi }\)

Stream function

Subscripts

f

Fluid phase

s

Solid phase

w

Condition at wall

\(\infty \)

Condition at infinity

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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringSouthern Taiwan University of Science and TechnologyYungkangTaiwan

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