Heat Transfer Through a Porous Medium
In this chapter we focus on the equation that expresses the first law of thermodynamics in a porous medium. We start with a simple situation in which the medium is isotropic and where radiative effects, viscous dissipation, and the work done by pressure c
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Heat Transfer Through a Porous Medium
2.1
Energy Equation: Simple Case
In this chapter we focus on the equation that expresses the first law of thermodynamics in a porous medium. We start with a simple situation in which the medium is isotropic and where radiative effects, viscous dissipation, and the work done by pressure changes are negligible. Very shortly we shall assume that there is local thermal equilibrium so that Ts ¼ Tf ¼ T, where Ts and Tf are the temperatures of the solid and fluid phases, respectively. Here we also assume that heat conduction in the solid and fluid phases takes place in parallel so that there is no net heat transfer from one phase to the other. More complex situations will be considered in Sect. 6.5. The fundamentals of heat transfer in porous media are also presented in Bejan et al. (2004) and Bejan (2004a). Taking averages over an elemental volume of the medium we have, for the solid phase, ð1 ’ÞðrcÞ s
@Ts ¼ ð1 ’Þr ðks rTs Þ þ ð1 ’Þq000 s @t
(2.1)
and, for the fluid phase, ’ ðr cP Þ f
@Tf þ ðr cP Þ f v rTf ¼ ’r ðkf r Tf Þ þ ’ q000 f : @t
(2.2)
Here the subscripts s and f refer to the solid and fluid phases, respectively, c is the specific heat of the solid, cP is the specific heat at constant pressure of the fluid, k is the thermal conductivity, and q000 (W/m3) is the heat production per unit volume. In writing Eqs. (2.1) and (2.2) we have assumed that the surface porosity is equal to the porosity. This is pertinent to the conduction terms. For example, ks∇Ts is the conductive heat flux through the solid, and thus ∇·(ks∇Ts) is the net rate of heat conduction into a unit volume of the solid. In Eq. (2.1) this appears multiplied by
D.A. Nield and A. Bejan, Convection in Porous Media, DOI 10.1007/978-1-4614-5541-7_2, # Springer Science+Business Media New York 2013
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2 Heat Transfer Through a Porous Medium
the factor (1 ’), which is the ratio of the cross-sectional area occupied by solid to the total cross-sectional area of the medium. The other two terms in Eq. (2.1) also contain the factor (1 ’) because this is the ratio of volume occupied by solid to the total volume of the element. In Eq. (2.2) there also appears a convective term, due to the seepage velocity. We recognize that V·∇Tf is the rate of change of temperature in the elemental volume due to the convection of fluid into it, so this, multiplied by (rcP)f, must be the rate of change of thermal energy, per unit volume of fluid, due to the convection. Note further that in writing Eq. (2.2) use has been made of the Dupuit-Forchheimer relationship v ¼ ’V. Setting Ts ¼ Tf ¼ T and adding Eqs. (2.1) and (2.2) we have ðr cÞm
@T þ ðr cÞ f v rT ¼ r ðkm rTÞ þ q000 m; @t
(2.3)
where ðr cÞm ¼ ð1 ’Þ ðr cÞ s þ ’ ðr cP Þ f ;
(2.4)
km ¼ ð1 ’Þ ks þ ’ kf ;
(2.5)
000 000 q000 m ¼ ð1 ’Þ qs þ ’ qf
(2.6)
are, respectively, the overall heat capacity per unit volume, overall thermal conductivity, and overall heat production per unit volume of the medium.
2.2 2.2.1
Energy Equation: Extensions to
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