Forced Convection
The fundamental question in heat transfer engineering is to determine the relationship between the heat transfer rate and the driving temperature difference. In nature, many saturated porous media interact thermally with one another and with solid surface
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Forced Convection
The fundamental question in heat transfer engineering is to determine the relationship between the heat transfer rate and the driving temperature difference. In nature, many saturated porous media interact thermally with one another and with solid surfaces that confine them or are embedded in them. In this chapter we analyze the basic heat transfer question by looking only at forced-convection situations, in which the fluid flow is caused (forced) by an external agent unrelated to the heating effect. First we discuss the results that have been developed based on the Darcy flow model and later we address the more recent work on the non-Darcy effects. We end this chapter with a review of current engineering applications of the method of forced convection through porous media. Some fundamental aspects of the subject have been discussed by Lage and Narasimhan (2000), and the topic has been reviewed by Lauriat and Ghafir (2000).
4.1
Plane Wall with Prescribed Temperature
Perhaps the simplest and most common heat transfer arrangement is the flow parallel to a flat surface that borders the fluid-saturated porous medium. With reference to the two-dimensional geometry defined in Fig. 4.1, we recognize the equations governing the conservation of mass, momentum (Darcy flow), and energy in the flow region of thickness dT: @u @v þ ¼ 0; @x @y u¼
K @P ; m @x
v¼
(4.1) K @P ; m @y
D.A. Nield and A. Bejan, Convection in Porous Media, DOI 10.1007/978-1-4614-5541-7_4, # Springer Science+Business Media New York 2013
(4.2)
69
70
4 Forced Convection T∞
Uniform flow U∞, T∞
y
Porous medium
v δT
u
x=0 Unheated wall section
Tw
x
Heated wall section
Heat flux variation, q ⬙ ~ x −1/2
Fig. 4.1 Parallel flow near an isothermal wall (Bejan 1984)
u
@T @T @2T þv ¼ am 2 : @x @y @y
(4.3)
Note the boundary-layer-approximated right-hand side of Eq. (4.3), which is based on the assumption that the region of thickness dT and length x is slender (dT x). The fluid mechanics part of the problem statement [namely, Eqs. (4.1) and (4.2)] is satisfied by the uniform parallel flow u ¼ U;
v ¼ 0:
(4.4)
The constant pressure gradient that drives this flow (dP/dx ¼ mU1/K) is assumed known. The heat transfer rate between the surface at temperature Tw and the saturated porous medium at far-field temperature T1 can be determined in several ways. The scale analysis begins with writing DT ¼ TwT1 so that the order-of-magnitude counterpart of Eq. (4.3) becomes U1
DT DT am 2 : x dT
(4.5)
4.1 Plane Wall with Prescribed Temperature
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From this we can determine the thickness of the thermal boundary layer dT xPex 1=2
(4.6)
in which Pex is the Pe´clet number based on U1 and x: Pex ¼
U1 x : am
(4.7)
For the local heat flux q00 we note the scale q00 ~ km DT/dT or the corresponding local Nusselt number Nux ¼
q00 x Pex 1=2 : DT km
(4.8)
Figure 4.1 qualitatively illustrates the main characteristics of the heat transfer region, namely, the boundary-layer thickness that increases as x1/2 and the heat flux that
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