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CHAPTER 15 Porous Media
ADRIAN BEJAN
Department of Mechanical Engineering and Materials Science Duke University
Durham,North Carolina
15.1 Introduction 15.2 Basic principles 15.2.1 Mass conservation 15.2.2 Flow models 15.2.3 Energy conservation 15.3 Conduction
15.4 Forced convection 15.4.1 Plane wall with constant temperature 15.4.2 Sphere and cylinder
15.4.3 Concentrated heat sources 15.4.4 Channels filled with porous media 15.4.5 Compact heat exchangers as porous media 15.5 External natural convection
15.5.1 Vertical walls 15.5.2 Horizontal walls 15.5.3 Sphere and horizontal cylinder 15.5.4 Concentrated heat sources 15.6 Internal natural convection 15.6.1 Enclosures heated from the side 15.6.2 Cylindrical and spherical enclosures 15.6.3 Enclosures heated from below 15.6.4 Penetrative convection 15.7 Other configurations
Nomenclature References
15.1 INTRODUCTION
Heat and mass transfer through saturated porous media is an important development and an area ofvery rapid growth in contemporary heat transfer research Although the mechanics offluid flow through porous media has preoccupied engineers and physicists for more than a century, the study of heat transfer has reached the status
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ofa separate field ofresearch during the last three decades (Nield and Bejan, 1999)
It has also become an established topic in heat transfer education, where it became a part ofthe convection course in 1984 (Bejan, 1984, 1995) The reader is directed to
a growing number ofmonographs that outline the fundamentals (Scheiddeger, 1957;
Bear, 1972; Bejan, 1987; Kaviany, 1995; Ingham and Pop, 1998, 2002; Vafai, 2000;
Pop and Ingham, 2001; Bejan et al., 2004)
Porous media and transport are becoming increasingly important in heat exchanger analysis and design It was pointed out in Bejan (1995) that the gradual miniatur-ization ofheat transfer devices leads the flow toward lower Reynolds numbers and brings the designer into a domain where dimensions are considerably smaller and structures considerably more complex than those covered by the single-configuration correlations developed historically for large-scale heat exchangers The race toward small scales and large heat fluxes in the cooling ofelectronic devices is the strongest manifestation of this trend It is fair to say that the reformulation of heat exchanger analysis and design as the basis ofporous medium flow principles is the next area of growth in heat exchanger theory for small-scale applications
The objective ofthis chapter is to provide a concise and effective review ofsome ofthe most basic results on heat transfer through porous media This coverage is
an updated and condensed version ofa review presented first in Bejan (1987) More detailed and tutorial alternatives were developed subsequently in Bejan (1995, 1999) and Nield and Bejan (1999), to which the interested reader is directed
15.2 BASIC PRINCIPLES
The description ofheat and fluid flow through a porous medium saturated with fluid (liquid or gas) is based on a series ofspecial concepts that are not found in the pure-fluid heat transfer Examples are the porosity and the permeability of the porous medium, and the volume-averaged properties ofthe fluid flowing through the porous medium The porosity ofthe porous medium is defined as
φ =void volume contained in porous medium sample
total volume ofporous medium sample (15.1) The engineering heat transfer results assembled in this chapter refer primarily to fluid-saturated porous media that can be modeled as nondeformable, homogeneous, and isotropic In such media, the volumetric porosityφ is the same as the area ratio (void
area contained in the sample cross section)/(total area ofthe sample cross section)
Representative values are shown in Table 15.1
The phenomenon ofconvection through the porous medium is described in terms ofvolume-averaged quantities such as temperature, pressure, concentration, and velocity components Each volume-averaged quantity (ψ) is defined through the
operation
ψ = 1
V
v
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TABLE 15.1 Properties of Common Porous Materials
Porosity, Permeability, Surface per unit
Black slate powder 0.57–0.66 4.9 × 10−10–1.2 × 10−9 7× 103–8.9 × 103
Brick 0.12–0.34 4.8 × 10−11–2.2 × 10−9
Catalyst (Fischer–Tropsch, granules only)
Cigarette filters 0.17–0.49
Concrete (bituminous) — 1× 10−9–2.3 × 10−7
Concrete (ordinary mixes) 0.02–0.07 Copper powder
(hot-compacted)
0.09–0.34 3.3 × 10−6–1.5 × 10−5
Granular crushed rock 0.45 Hair (on mammals) 0.95–0.99
Leather 0.56–0.59 9.5 × 10−10–1.2 × 10−9 1.2 × 104–1.6 × 104
Limestone (dolomite) 0.04–0.10 2× 10−11–4.5 × 10−10
Sandstone (“oil sand”) 0.08–0.38 5× 10−12–3× 10−8
Silica powder 0.37–0.49 1.3 × 10−10–5.1 × 10−10 6.8 × 103–8.9 × 103
Soil 0.43–0.54 2.9 × 10−9–1.4 × 10−7
Spherical packings (well shaken)
0.36–0.43
Source: Data from Nield and Bejan (1999), Scheidegger (1957), and Bejan and Lage (1991).
whereψ is the actual value ofthe quantity at a point inside the sample volume V
Alternatively, the volume-averaged quantity equals the value ofthat quantity averaged over the total volume occupied by the porous medium The volume sample is called
representative elementary volume (REV) The length scale ofthe REV is much larger
than the pore scale but considerably smaller than the length scale ofthe macroscopic flow domain
15.2.1 Mass Conservation
The principle ofmass conservation or mass continuity applied locally in a small region ofthe fluid-saturated porous medium is
Dρ
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T w
T w
T w
u T, ⬁
u
u T, ⬁
u g, x
w g, z
v, g y
r0
( )e
( )c
( )a
( )f
( )d
( )b
q⬙
q⬙
q⬘
y
y
y ror
r
y
x
x
x
z
Heated Insulated
T x, y( )
T x, r( )
D
⬃
q
Figure 15.1 Configurations for forced-convection heat transfer: (a) Cartesian coordinate system; (b) boundary layer development over a flat surface in a porous medium; (c) boundary layer development around a cylinder or sphere embedded in a porous medium; (d) point heat source in a porous medium; (e) horizontal line source in a porous medium; ( f ) duct filled with
porous medium
whereD/Dt is the material derivative operator:
D
Dt =
∂
∂t + u
∂
∂x + v
∂
∂y + w
∂
and where v (u, v, w) is the volume-averaged velocity vector (Fig 15.1a) For
ex-ample, the volume-averaged velocity component u in the x direction is equal to
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φu p, whereu pis the average velocity through the pores In many single-phase flows through porous media, the density variations are small enough so that theDρ/Dt
term may be neglected in eq (15.3) The incompressible flow model has been in-voked in the development ofthe majority ofthe analytical and numerical results re-viewed in this section The incompressible flow model should not be confused with the incompressible-substance model encountered in thermodynamics (Bejan, 1997)
15.2.2 Flow Models
The most frequently used model for volume-averaged flow through a porous medium
is the Darcy flow model (Nield and Bejan, 1999; Bejan, 1995) According to this model, the volume-averaged velocity in a certain direction is directly proportional to the net pressure gradient in that direction,
u = K
µ
−∂P
∂x
(15.5)
In three dimensions and in the presence ofa body acceleration vector g= (g x , gy, gz)
(Fig 15.1a), the Darcy flow model is
The proportionality factor K in Darcy’s model is the permeability ofthe porous
medium The units ofK are m2 In general, the permeability is an empirical constant that can be determined by measuring the pressure drop and the flow rate through a column-shaped sample ofporous material, as suggested by eq (15.5) The perme-ability can also be estimated from simplified models of the labyrinth formed by the interconnected pores Modeling the pores as a bundle ofparallel capillary tubes of radiusr0yields (Bejan, 1995)
K = πr04
8
N
whereN is the number oftubes counted on a cross section ofarea A Modeling the
pores as a stack ofparallel capillary fissures ofwidthB and fissure-to-fissure spacing
a + b yields the permeability formula (Bejan, 1995)
Modeling the porous medium as a collection ofsolid spheres ofdiameterd, Kozeny
obtained the relationship (Nield and Bejan, 1999)
K ∼ d2φ3
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A more applicable version ofeq (15.9) was obtained by considering a bed ofparticles
or fibers with an effective average particle or fiber diameterD p The hydraulic radius theory ofCarman-Kozeny leads to the relationship (Nield and Bejan, 1999)
2
p2φ3
where
D p2=
∞
0 D3
p h(D p )dD p
∞
0 D2
andh(Dp) is the density function for the distribution of diameters Dp The constant
180 in eq (15.10) was obtained by seeking a best fit with experimental results Equa-tion (15.10) gives satisfactory results for media that consist ofparticles ofapproxi-mately spherical shape and whose diameters fall within a narrow range The equation
is often not valid in the cases of particles that deviate strongly from the spherical shapes, broad particle-size distributions and consolidated media Nevertheless, it is widely used because it seems to be the best simple expression available Additional limitations to the use ofeq (15.10) and alternate statistical models leading to Darcy’s law are reviewed in Nield and Bejan (1999)
The Darcy flow model is valid in circumstances where the order ofmagnitude of
the local pore Reynolds number, based on the local volume-averaged speed (u2 +
v2+ w2)1/2andK1/2, is smaller than 1 (Ward, 1964) At pore Reynolds numbers of
order 1 and greater, the measured relationship between pressure gradient and volume-averaged velocity is correlated by Forchheimer’s modification ofDarcy’s model of
eq (15.5) (Nield and Bejan, 1999):
−∂P
∂x =
µ
The termbρu2accounts for the increasingly important role played by fluid inertia In three dimensions and in the presence ofbody acceleration, the Forchheimer modifi-cation ofthe Darcy flow model is
v+bρKµ |v|v = Kµ(−∇P + ρg) (15.13) Experimental measurements (Ward, 1964) suggest that as the local pore Reynolds number exceeds the order of10, Forchheimer’s constantb approaches asymptotically
the value
Extensive measurements involving gas flow through columns ofpacked spheres, sand, and pulverized coal (Ergun, 1952) led to the following correlations forK and b: