Internal Natural Convection: Heating from the Side
Enclosures heated from the side are most representative of porous systems that function while oriented vertically, as in the insulations for buildings, industrial cold-storage installations, and cryogenics. As in the earlier chapters, we begin with the mo
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Internal Natural Convection: Heating from the Side
Enclosures heated from the side are most representative of porous systems that function while oriented vertically, as in the insulations for buildings, industrial cold-storage installations, and cryogenics. As in the earlier chapters, we begin with the most fundamental aspects of the convection heat transfer process when the flow is steady and in the Darcy regime. Later, we examine the special features of flows that deviate from the Darcy regime, flows that are time dependent, and flows that are confined in geometries more complicated than the two-dimensional rectangular space shown in Fig. 7.1. Some of the topics of this chapter have been reviewed by Oosthuizen (2000).
7.1 7.1.1
Darcy Flow Between Isothermal Sidewalls Heat Transfer Regimes
Consider the basic scales of the clockwise convection pattern maintained by the side-to-side heating of the porous medium defined in Fig. 7.1. In accordance with the homogeneous porous medium model, we begin with the equations for the conservation of mass, Darcy flow, and the conservation of energy in the H L space: @u @v þ ¼ 0; @x @y
(7.1)
K @P ; m @x
(7.2)
u¼
D.A. Nield and A. Bejan, Convection in Porous Media, 331 DOI 10.1007/978-1-4614-5541-7_7, # Springer Science+Business Media New York 2013
332
7 Internal Natural Convection: Heating from the Side
Fig. 7.1 Two-dimensional rectangular porous layer held between differently heated sidewalls (Bejan 1984)
Insulated wall H g Warm wall Th
v
δ
Cold wall Tc
u
Porous medium
y
δ
0 0
v¼
x
K @P þ rg ; m @y
2 @T @T @ T @2T u þv ¼ am þ : @x @y @x2 @y2
L
(7.3)
(7.4)
Note that in contrast to the system used in Sect. 5.1, the y axis is now vertically upward. By eliminating the pressure P between (7.2) and (7.3) and by invoking the Boussinesq approximation r ffi r0[1 b(T T0)] in the body force term rg of (7.3), we obtain a single equation for momentum conservation: @u @v Kgb @T ¼ : @y @x v @x
(7.5)
In this equation n is the kinematic viscosity m/r0, which is assumed constant along with the other properties, the permeability K, the coefficient of volumetric thermal expansion b, and the porous medium thermal diffusivity am ¼ km/(rcP)f. The three equations (7.1), (7.4), and (7.5) hold in the entire domain H L subject to the boundary conditions indicated in the figure. The four walls are impermeable, and the side-to-side temperature difference is Th Tc ¼ DT. Of special interest are the scales of the vertical boundary layers of thickness d
7.1 Darcy Flow Between Isothermal Sidewalls
333
and height H. In each d H region, the order-of-magnitude equivalents of (7.1), (7.4), and (7.5) are Mass: u y ; d H
(7.6)
DT DT DT DT ;v u am 2 ; am 2 ; d H H d
(7.7)
u v Kgb DT ; : H d v d
(7.8)
Energy:
Momentum:
To begin with, the mass balance (7.6) shows that the two scales on the left-hand side of (7.7) are of the same order, namely vDT/H. On the right-hand side of (7.7), the second scale can be neglected in favor of the first,
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