BOOKCOMP, Inc. — John Wiley & Sons / Page 1127 / 2nd Proofs / Heat Transfer Handbook / Bejan REFERENCES 1127 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1127], (99) Lines: 2443 to 2483 ——— * 15.0pt PgVar ——— Custom Page (7.0pt) PgEnds: T E X [1127], (99) Use of Internally Grooved Rough Surfaces and Twisted Tapes, Heat Transfer Jpn. Res., 13(4), 19–32. Vachon, R. I., Nix, G. H., Tanger, G. E., and Cobb, R. O. (1969). Pool Boiling Heat Transfer from Teflon-Coated Stainless Steel, J. Heat Transfer, 91, 364–370. Van Der Meer, T. H., and Hoogenedoorn, C. J. (1978). Heat Transfer Coefficients for Viscous Fluids in a Static Mixer, Chem. Eng. Sci., 33, 1277–1282. Van Rooyen, R. S., and Kr ¨ oger, D. G. (1978). Laminar Flow Heat Transfer in Internally Finned Tubes with Twisted-Tape Inserts, in Heat Transfer 1978, Vol. 2, Hemisphere Publishing, Washington, DC, pp. 577–581. van Stralen, S. J. D. (1959). 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BOOKCOMP, Inc. — John Wiley & Sons / Page 1130 / 2nd Proofs / Heat Transfer Handbook / Bejan 1130 HEAT TRANSFER ENHANCEMENT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [Last Page] [1130], (102) Lines: 2560 to 2593 ——— 34.119pt PgVar ——— Normal Page PgEnds: T E X [1130], (102) Yilmaz, S., Palen, J. W., and Taborek, J. (1981). Enhanced Surfaces as Single Tubes and Tube Bundles, in Advances in Heat Transfer 1981, R. L. Webb, et al., eds., ASME-HTD-18, ASME, New York, pp. 123–129. Young, R. X., and Hummel, R. L. (1995). Improved Nucleate Boiling Heat Transfer, Chem. Eng. Prog. Symp. Ser., 61(59), 264–470. Yudin, V. F. (1982). Teploobmen Poperechnoorebrenykh Trub (Heat Transfer of Crossfinned Tubes), Mashinostroyeniye Publishing House, Leningrad, Russia. Zatell, V. A. (1973). Method of Modifying a Finned Tube for Boiling Enhancement, U.S. patent 3,768,290. Zhang, J., and Manglik, R. M. (2003). 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BOOKCOMP, Inc. — John Wiley & Sons / Page 1131 / 2nd Proofs / Heat Transfer Handbook / Bejan 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [First Page] [1131], (1) Lines: 0 to 82 ——— 0.34608pt PgVar ——— Normal Page PgEnds: T E X [1131], (1) 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 of very rapid growth in contemporary heat transfer research. Although the mechanics of fluid flow through porous media has preoccupied engineers and physicists for more than a century, the study of heat transfer has reached the status 1131 BOOKCOMP, Inc. — John Wiley & Sons / Page 1132 / 2nd Proofs / Heat Transfer Handbook / Bejan 1132 POROUS MEDIA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1132], (2) Lines: 82 to 108 ——— -2.55106pt PgVar ——— Long Page * PgEnds: Eject [1132], (2) of a separate field of research 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 of the convection course in 1984 (Bejan, 1984, 1995). The reader is directed to a growing number of monographs 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 of heat 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 of electronic 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 of porous medium flow principles is the next area of growth in heat exchanger theory for small-scale applications. The objective of this chapter is to provide a concise and effective review of some of the most basic results on heat transfer through porous media. This coverage is an updated and condensed version of a 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 of heat and fluid flow through a porous medium saturated with fluid (liquid or gas) is based on a series of special 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 of the fluid flowing through the porous medium. The porosity of the porous medium is defined as φ = void volume contained in porous medium sample total volume of porous 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 porosity φ is the same as the area ratio (void area contained in the sample cross section)/(total area of the sample cross section). Representative values are shown in Table 15.1. The phenomenon of convection through the porous medium is described in terms of volume-averaged quantities such as temperature, pressure, concentration, and velocity components. Each volume-averaged quantity (ψ) is defined through the operation ψ= 1 V v ψ dV (15.2) BOOKCOMP, Inc. — John Wiley & Sons / Page 1133 / 2nd Proofs / Heat Transfer Handbook / Bejan BASIC PRINCIPLES 1133 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1133], (3) Lines: 108 to 162 ——— 0.78207pt PgVar ——— Long Page * PgEnds: Eject [1133], (3) TABLE 15.1 Properties of Common Porous Materials Porosity, Permeability, Surface per unit Material φ K(cm 2 ) Volume (cm −1 ) Agar–agar — 2 ×10 −10 –4.4 ×10 −9 Black slate powder 0.57–0.66 4.9 ×10 −10 –1.2 ×10 −9 7 × 10 3 –8.9 × 10 3 Brick 0.12–0.34 4.8 ×10 −11 –2.2 ×10 −9 Catalyst (Fischer–Tropsch, granules only) 0.45 — 5.6 × 10 5 Cigarette — 1.1 × 10 −5 Cigarette filters 0.17–0.49 Coal 0.02–0.12 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 Corkboard — 2.4 ×10 −7 –5.1 × 10 −7 Fiberglass 0.88–0.93 — 560–770 Granular crushed rock 0.45 Hair (on mammals) 0.95–0.99 Hair felt — 8.3 ×10 −6 –1.2 ×10 −5 Leather 0.56–0.59 9.5 × 10 −10 –1.2 ×10 −9 1.2 ×10 4 –1.6 × 10 4 Limestone (dolomite) 0.04–0.10 2 ×10 −11 –4.5 ×10 −10 Sand 0.37–0.50 2 ×10 −7 –1.8 × 10 −6 150–220 Sandstone (“oil sand”) 0.08–0.38 5 × 10 −12 –3 × 10 −8 Silica grains 0.65 Silica powder 0.37–0.49 1.3 ×10 −10 –5.1 × 10 −10 6.8 × 10 3 –8.9 × 10 3 Soil 0.43–0.54 2.9 × 10 −9 –1.4 ×10 −7 Spherical packings (well shaken) 0.36–0.43 Wire crimps 0.68–0.76 3.8 × 10 −5 –1 × 10 −4 29–40 Source: Data from Nield and Bejan (1999), Scheidegger (1957), and Bejan and Lage (1991). where ψ is the actual value of the quantity at a point inside the sample volume V . Alternatively, the volume-averaged quantity equals the value of that quantity averaged over the total volume occupied by the porous medium. The volume sample is called representative elementary volume (REV). The length scale of the REV is much larger than the pore scale but considerably smaller than the length scale of the macroscopic flow domain. 15.2.1 Mass Conservation The principle of mass conservation or mass continuity applied locally in a small region of the fluid-saturated porous medium is Dρ Dt + ρ∇·v = 0 (15.3) BOOKCOMP, Inc. — John Wiley & Sons / Page 1134 / 2nd Proofs / Heat Transfer Handbook / Bejan 1134 POROUS MEDIA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1134], (4) Lines: 162 to 181 ——— 0.21704pt PgVar ——— Normal Page PgEnds: T E X [1134], (4) T w T w T w uT, ϱ uT, ϱ uT, ϱ u uT, ϱ ug, x wg, z v, g y r 0 ()e ()c ()a ()f ()d ()b qЉ qЉ qЈ y y yror r y x x x x x z HeatedInsulated Tx,y() Tx,r() D ϳ 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. where D/Dt is the material derivative operator: D Dt = ∂ ∂t + u ∂ ∂x + v ∂ ∂y + w ∂ ∂z (15.4) 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 BOOKCOMP, Inc. — John Wiley & Sons / Page 1135 / 2nd Proofs / Heat Transfer Handbook / Bejan BASIC PRINCIPLES 1135 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1135], (5) Lines: 181 to 220 ——— 0.99022pt PgVar ——— Normal Page * PgEnds: Eject [1135], (5) φu p , where u p is the average velocity through the pores. In many single-phase flows through porous media, the density variations are small enough so that the Dρ/Dt term may be neglected in eq. (15.3). The incompressible flow model has been in- voked in the development of the majority of the 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 of a body acceleration vector g = (g x ,g y ,g z ) (Fig. 15.1a), the Darcy flow model is v = K µ (−∇P +ρg) (15.6) The proportionality factor K in Darcy’s model is the permeability of the porous medium. The units of K are m 2 . 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 of porous 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 of parallel capillary tubes of radius r 0 yields (Bejan, 1995) K = πr 4 0 8 N A (15.7) where N is the number of tubes counted on a cross section of area A. Modeling the pores as a stack of parallel capillary fissures of width B and fissure-to-fissure spacing a +b yields the permeability formula (Bejan, 1995) K = b 3 12(a +b) (15.8) Modeling the porous medium as a collection of solid spheres of diameter d, Kozeny obtained the relationship (Nield and Bejan, 1999) K ∼ d 2 φ 3 (1 − φ) 2 (15.9) BOOKCOMP, Inc. — John Wiley & Sons / Page 1136 / 2nd Proofs / Heat Transfer Handbook / Bejan 1136 POROUS MEDIA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1136], (6) Lines: 220 to 264 ——— 0.4483pt PgVar ——— Normal Page * PgEnds: Eject [1136], (6) A more applicable version of eq. (15.9) was obtained by considering a bed of particles or fibers with an effective average particle or fiber diameter D p . The hydraulic radius theory of Carman-Kozeny leads to the relationship (Nield and Bejan, 1999) K = D 2 p2 φ 3 180(1 − φ) 2 (15.10) where D p2 = ∞ 0 D 3 p h(D p )dD p ∞ 0 D 2 p h(D p )dD p (15.11) and h(D p ) is the density function for the distribution of diameters D p . 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 of particles of approxi- 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 of eq. (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 of magnitude of the local pore Reynolds number, based on the local volume-averaged speed (u 2 + v 2 + w 2 ) 1/2 and K 1/2 , 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 of Darcy’s model of eq. (15.5) (Nield and Bejan, 1999): − ∂P ∂x = µ K u + bρu 2 (15.12) The term bρu 2 accounts for the increasingly important role played by fluid inertia. In three dimensions and in the presence of body acceleration, the Forchheimer modifi- cation of the Darcy flow model is v + bρK µ |v|v = K µ (−∇P +ρg) (15.13) Experimental measurements (Ward, 1964) suggest that as the local pore Reynolds number exceeds the order of 10, Forchheimer’s constant b approaches asymptotically the value b = 0.55K −1/2 (15.14) Extensive measurements involving gas flow through columns of packed spheres, sand, and pulverized coal (Ergun, 1952) led to the following correlations for K and b: . for Use of Enhanced Heat Transfer Surfaces in Heat Exchanger Design, Int. J. Heat Mass Transfer, 24, 715–726. Webb, R. L. (1987).Enhancement of Single-Phase Heat Transfer, in Handbook of Single-Phase Convective. in Heat Transfer 1974, Vol. III, JSME, Tokyo, pp. 250–254. Wang, W. (1987). The Enhancement of Condensation Heat Transfer for Stratified Flow in a Horizontal Tube with Inserted Coil, in Heat Transfer. Vertical Smooth and Ribbed Tubes, in Heat Transfer 1974, Vol. IV, JSME, Tokyo, pp. 275–279. Webb, R. L. (1972). Heat Transfer Having a High Boiling Heat Transfer Coefficient, U.S. patent 3,696,861. Webb,