BOOKCOMP, Inc. — John Wiley & Sons / Page 1167 / 2nd Proofs / Heat Transfer Handbook / Bejan INTERNAL NATURAL CONVECTION 1167 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 [1167], (37) Lines: 1221 to 1243 ——— 0.71606pt PgVar ——— Normal Page * PgEnds: Eject [1167], (37) Figure 15.10 Asymptotes of the function Nu (Ra H , Pr p ) for convection in a porous layer heated from below. (From Bejan, 1995.) In this formulation Nu is a function of two groups, Ra H and Pr p , in which Pr p accounts for the transition from Darcy to Forchheimer flow (Fig. 15.10). In this formulation the Darcy flow result of eq. (15.115) becomes Nu Pr p ∼ 1 40 Ra H Pr p 40 < Ra H < Pr p (15.118) The experimental data for convection in the entire regime spanned by the asymptotes given by eqs. (15.116) and (15.118) are correlated by Nu = Ra H 40 n + c Ra H · Pr p 1/2 n 1/n (15.119) BOOKCOMP, Inc. — John Wiley & Sons / Page 1168 / 2nd Proofs / Heat Transfer Handbook / Bejan 1168 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 [1168], (38) Lines: 1243 to 1277 ——— 0.71902pt PgVar ——— Normal Page PgEnds: T E X [1168], (38) where n =−1.65 and c = 1896 are determined empirically based on measurements reported by many independent sources. The effects of fluid inertia and other depar- tures from Darcy flow are discussed in detail in Nield and Bejan (1999). The correlations of eqs. (15.115)–(15.119) refer to layers with length/height ra- tios considerably greater than 1. They apply when the length (lateral dimension L perpendicular to gravity) of the confined system is greater than the horizontal length scale of a single convective cell (i.e., greater than H · Ra −1/2 H ) according to the scale analysis of Bejan (1984). Natural convection studies have also been reported for porous layers confined in rectangular parallelpipeds heated from below, horizontal circular cylinders, and horizontal annular cylinders. The general conclusion is that the lateral walls have a convection-suppression effect. For example, in a circular cylinder of diameter D and height H (Fig. 15.9b), in the limit D H the critical condition for the onset of convection is (Bau and Torrance, 1982) Ra H = 13.56 H D 2 (15.120) In inclined porous layers that deviate from the horizontal position through an angle φ (Fig. 15.9c), convection sets in at Rayleigh numbers that satisfy the criterion (Combarnous and Bories, 1975) Ra H > 39.48 cos φ (15.121) where it is assumed that the boundaries are isothermal and impermeable. The average heat transfer rate at high Rayleigh numbers can be estimated by Nu = 1 + ∞ s=1 k s 1 − 4π 2 s 2 Ra H cos φ (15.122) where k s = 0ifRa H cos φ < 4π 2 s 2 and k s = 2ifRa H cos φ ≥ 4π 2 s 2 . In a porous medium confined in a wedge-shaped (or attic-shaped) space cooled from above (Fig. 15.9d), convection consisting of a single counterclockwise cell ex- ists even in the limit Ra H → 0, because in this direction the imposed heating is not purely vertical. The same observation holds for Fig. 15.9c. Numerical solutions of transient high-Rayleigh-number convection in wedge-shaped layers show the pres- ence of a B ´ enard-type instability at high enough Rayleigh numbers (Poulikakos and Bejan, 1983b). When H/L = 0.2, the instability occurs above Ra H 620. It was found that this critical Rayleigh number increases as H/L increases. The onset of convection in the layer of Fig.15.9a saturated with water near the state of maximum density has been studied using linear stability analysis (Sun et al., 1970) and time-dependent numerical solutions of the complete governing equations (Blake et al., 1984). In both studies, the condition for the onset of convection is reported graphically or numerically for a discrete series of cases. The numerical results (Blake BOOKCOMP, Inc. — John Wiley & Sons / Page 1169 / 2nd Proofs / Heat Transfer Handbook / Bejan INTERNAL NATURAL CONVECTION 1169 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 [1169], (39) Lines: 1277 to 1325 ——— 1.58115pt PgVar ——— Normal Page * PgEnds: Eject [1169], (39) et al., 1984) for layers with T c = 0°C and 5°C ≤ T h ≤ 8°C suggested the following criterion for the onset of convection: KgH α m ν > 1.25 × 10 5 exp exp(3.8 − 0.446T h ) (15.123) In this expression T h must be expressed in °C. Finite-amplitude heat and fluid flow results for Rayleigh numbers Kg γ(T h −T c ) 2 H/α m ν of up to 10 4 (i.e., about 50 times greater than critical) are also reported in Blake et al. (1984). Nuclear safety considerations have led to the study of natural convection in hori- zontal saturated porous layers (Fig. 15.9a) heated volumetrically at a rate q . Bound- ary conditions and observations regarding the onset of convection and overall Nusselt numbers are presented in Nield and Bejan (1999). It is found that convection sets in at internal Rayleigh numbers (Kulacki and Freeman, 1979) Ra I = k m β α m ν f KgH 3 q 2k m (15.124) in the range 33 to 46, where the subscript f indicates properties of the fluid alone. Top and bottom surface temperature measurements in the convection-dominated regime are adequately represented by (Buretta and Berman, 1976) q H 2 2k m (T h − T c ) ≈ 0.116Ra 0.573 I (15.125) where T h and T c are the resulting bottom and top temperatures if q is distributed throughout the layer of Fig. 15.9a. The empirical correlation (11.25) is based on experiments that reach into the high Ra I range of 10 3 to 10 4 . 15.6.4 Penetrative Convection In this section attention is focused on buoyancy-driven flows that penetrate the en- closed porous medium only partially. This class of convection phenomena have been categorized as penetrative flows (Bejan, 1984). With reference to the two vertical cylindrical configurations sketched in Fig. 15.9e, if the cylindrical space is slender enough, the flow penetrates vertically to the distance L y = 0.085r o · Ra r o (15.126) where Ra r o = Kg βr o (T h −T c )/α m ν and L y <H. The overall convection heat transfer rate through the permeable horizontal end is q = 0.255r o k m (T h − T c )Ra r o (15.127) A similar partial penetration mechanism is encountered in the two horizontal geometries of Fig. 15.9f. The length of lateral penetration, L x , and the convection heat transfer rate in the two-dimensional geometry, q (W/m), are (Bejan, 1984, 1995) BOOKCOMP, Inc. — John Wiley & Sons / Page 1170 / 2nd Proofs / Heat Transfer Handbook / Bejan 1170 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 [1170], (40) Lines: 1325 to 1364 ——— -0.3278pt PgVar ——— Short Page PgEnds: T E X [1170], (40) L x = 0.16H ·Ra 1/2 H (15.128) q = 0.32k m (T h − T c )Ra 1/2 H (15.129) where Ra h = Kg β(T h − T c )H/α m ν and L x <L. In a semi-infinite porous medium bounded from below or from above by a horizon- tal surface with alternating zones of heating and cooling (Fig. 15.9g), the buoyancy- driven flow penetrates vertically to a height or depth approximately equal to λ ·Ra 1/2 λ , where Ra λ = Kg βλ(T h − T c )/α m ν and λ is the distance between a heated zone and an adjacent cooled zone. Numerical and graphic results are reported in Poulikakos and Bejan (1984a) for the Ra λ range 1 to 100. There are many other circumstances in which penetrative flows can occur, steady and transient (Nield and Bejan, 1999). For example, in a porous medium heated and cooled along the same vertical wall of height H , the incomplete penetration can be either horizontal (Fig. 15.9h) or vertical (Fig. 15.9i) (Poulikakos and Bejan, 1984b). In the case of incompletehorizontal penetration, the penetration length and convective heat transfer rate scale as L x ∼ H ·Ra 1/2 H (15.130) q ∼ k m (T h − T c )Ra 1/2 H (15.131) The derivation of these order-of-magnitude results can also be found in Bejan (1995). They are valid if Ra 1/2 H < L/H and Ra H 1. The corresponding scales of incomplete vertical penetration (Fig. 15.9i) are L y ∼ H L H 2/3 · Ra −1/3 H (15.132) q ∼ k m (T h − T c ) L H · Ra H 1/3 (15.133) and are valid if Ra 1/2 H > L/H and Ra 1/2 H >H/L. The penetrative flows of Fig. 15.9h and i occur when the heated section T h is situated above the cooled section T c . When the positions of T h and T c are reversed, the buoyancy-driven flow fills the entire space H ×L. 15.7 OTHER CONFIGURATIONS Research on heat and mass transfer in porous media has grown impressively during the past two decades, beyond the fundamental results highlighted in Sections 15.1 and 15.6. The most up-to-date review of the current state of the literature on heat transfer in porous media is provided in the latest edition of Nield and Bejan’s (1999) BOOKCOMP, Inc. — John Wiley & Sons / Page 1171 / 2nd Proofs / Heat Transfer Handbook / Bejan OTHER CONFIGURATIONS 1171 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 [1171], (41) Lines: 1364 to 1405 ——— 1.39511pt PgVar ——— Short Page PgEnds: T E X [1171], (41) book, which contains over 1600 references and in a new book (Bejan et al., 2004). This closing section is a brief review of some of the subfields that have emerged. When the flow is due to a combination of driving forces, buoyancy and imposed pressure differences, the heat transfer characteristics depend greatly on which force dominates. Two papers (Lai et al., 1991; Vargas et al., 1995) review most of what is known on mixed convection in two-dimensional external flow, along walls (vertical, inclined, horizontal) and in wedge-shaped domains. The free and forced-convection effects are governed by the Rayleigh and P ´ eclet numbers, respectively, Ra x = g x βKx(T w − T ∞ ) α m ν (15.134) Pe x = U ∞ x ν (15.135) where x is the position from the leading edge measured along the wall and g x is the acceleration component aligned with x. The graphic presentation of the heat transfer results suggests that, in broad terms, free convection is the dominating effect when Ra x /Pe x >O(1). The corresponding class of mixed convection problems concerning the embedded sphere and horizontal cylinder in a uniform vertical flow was treated in Cheng (1982). Inertial effects were introduced in the modeling of these problems by several authors. The most comprehensive treatment is the unifying analysis (Nakayama and Pop, 1991) for mixed convection based on the Darcy–Forchheimer model, which is valid for plane walls and axisymmetric bodies of arbitrary shape. This unifying treatment and other configurations (internal flow) can also be found in Nield and Bejan (1999). Another combination or competition of driving forces occurs when the local den- sity variations that cause buoyancy are due not only to temperature gradients but also to concentration gradients. This class of phenomena is also known as double-diffusive or thermohaline convection. As an example, consider the onset of convection in a hor- izontal porous layer subjected to heat and mass transfer between the confining bottom and top walls (Fig. 15.9a). An additional buoyancy effect is due to the concentration of constituent i maintained along the bottom wall (C b ) and the top wall (C t ). The linearized density–temperature relation is ρ ≈ ρ b 1 − β(T − T h ) − β C (C − C b ) (15.136) where β C is the concentration expansion coefficient, β C =− 1 ρ ∂ρ ∂C P (15.137) For saturated porous layers confined between impermeable walls with uniform T and C distributions, convection is possible if (Nield, 1968) Ra H + Ra D,H > 39.48 (15.138) BOOKCOMP, Inc. — John Wiley & Sons / Page 1172 / 2nd Proofs / Heat Transfer Handbook / Bejan 1172 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 [1172], (42) Lines: 1405 to 1448 ——— 8.90933pt PgVar ——— Normal Page PgEnds: T E X [1172], (42) where Ra H = Kg βH(T h − T c ) α m ν (15.139) Ra D,H = Kg β C H(C b − C t ) νD (15.140) with D as the mass diffusivity of constituent i through the solution-saturated porous medium. Therefore, because β C can be positive or negative, the effect of mass transfer from below can be, respectively, either to decrease or increase the critical Ra H for the onset of convection. Alternatives to eq. (15.138) for horizontal porous layers subjected to other boundary conditions are presented in Nield and Bejan (1999) and Nield (1968). In the single-wall vertical flow configuration ofFig. 15.3a, the additional buoyancy effect caused by the imposed concentration difference C w − C ∞ can either aid or oppose the familiar flow due to T w − T ∞ (Bejan, 1984, 1995). An important role is played by the buoyancy ratio N = β C (C w − C ∞ ) β(T w − T ∞ ) (15.141) In heat-transfer-driven flows (|N|1), the heat transfer rate is given by eq. (15.66). The overall mass transfer rate can be estimated based on the scaling laws j D(C w − C ∞ ) ∼ Ra 1/2 H · Le 1/2 for Le 1 Ra 1/2 H · Le for Le 1 (15.142) where j /(kg/s · m) is the overall mass transfer rate per unit length and Le is the Lewis number of the solution-saturated porous medium, α m /D. In mass transfer– driven situations (|N|1), the overall mass transfer rate is j D(C w − C ∞ ) = 0.888(Ra H · Le|N|) 1/2 (15.143) for all Lewis numbers. The corresponding overall Nusselt number obeys the scaling laws q k(T w − T ∞ ) ∼ (Ra H |N|) 1/2 for Le 1 Le −1/2 (Ra H |N|) 1/2 for Le 1 (15.144) The order-of-magnitude results of eqs. (15.142) and (15.144) agree within 15% with overall heat and mass transfer calculations based on similarity solutions to the same problem (Bejan and Khair, 1985). The corresponding enclosure problem, where the vertical walls are maintained at different temperatures and concentrations, the BOOKCOMP, Inc. — John Wiley & Sons / Page 1173 / 2nd Proofs / Heat Transfer Handbook / Bejan NOMENCLATURE 1173 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 [1173], (43) Lines: 1448 to 1535 ——— -0.79318pt PgVar ——— Normal Page PgEnds: T E X [1173], (43) heat and mass transfer due to convection driven by combined buoyancy effects was documented in terms of numerical experiments in Trevisan and Bejan (1985, 1986). Another expanding area is the study of conduction and convection in the presence of phase change: melting, solidification, boiling, and condensation. These problems are recommended by specialized applications in diverse fields such as geophysics, manufacturing, small-scale heat exchangers, and spaces filled with fibers coated with energy storage (phase-change) material. These and other applications are treated in detail in the most recent edition of Nield and Bejan (1999). NOMENCLATURE Roman Letter Symbols A cross-sectional area, m 2 a fissure spacing, m B width of stack, m combination of terms, dimensionless Be Bejan number, dimensionless b fissure spacing, m coefficient in Forchheimer’s modification of Darcy’s law, m −1 stratification parameter, dimensionless C constituent concentration, kg/m 3 c specific heat, J/kg ·K c p specific heat at constant pressure, J/kg · K D mass diffusivity, m 2 /s diameter of round tube, m distance between parallel plates, m D p fiber diameter, m D/Dt material derivative operator, s −1 d diameter, m peripheral sublayer thickness, m g gravitational acceleration, m/s 2 H height, m h heat transfer coefficient, W/m 2 · K h m local mass transfer coefficient, m/s j constituent mass flow per unit length, kg/m · s j constituent mass flow per unit area, kg/m 2 · s K permeability, m 2 k thermal conductivity, W/m ·K k A overall average thermal conductivity, W/m ·K k G weighted geometric mean thermal conductivity, W/m ·K k H weighted harmonic mean thermal conductivity, W/m ·K k m porous medium thermal conductivity, W/m · K BOOKCOMP, Inc. — John Wiley & Sons / Page 1174 / 2nd Proofs / Heat Transfer Handbook / Bejan 1174 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 [1174], (44) Lines: 1535 to 1535 ——— 0.0048pt PgVar ——— Normal Page PgEnds: T E X [1174], (44) L length, m L x penetration length, m L y penetration height, m Le Lewis number, dimensionless m mass produced in chemical reaction, kg/m 3 N buoyancy ratio, dimensionless number of horizontal sublayers, dimensionless Nu Nusselt number, dimensionless Nu H Nusselt number based on height, dimensionless Nu L Nusselt number based on wall length, dimensionless Nu θ peripheral Nusselt number, dimensionless Nu x Nusselt number based on local longitudinal position, dimensionless Nu y Nusselt number based on heat flux, dimensionless P pressure, Pa Pe P ´ eclet number, dimensionless Pe L overall P ´ eclet number, dimensionless Pe x P ´ eclet number based on local longitudinal position, dimensionless Pr p porous medium Prandtl number, dimensionless q heat transfer rate, W q heat transfer rate per unit length, W/m q heat transfer rate per unit area, W/m 2 q volumetric heat generation rate, W/m 3 R parameter defined in eq. (15.91), dimensionless r radial coordinate, m spherical coordinate, m r i inner radius, m r o outer radius, m Ra H Rayleigh number based on height and temperature difference, dimensionless Ra ∗ H Rayleigh number based on heat flux, dimensionless Ra I internal Rayleigh number based on volumetric heat generation rate, dimensionless Ra y Darcy-modified Rayleigh number, dimensionless Rayleigh number based on heat flux, dimensionless Ra ∗ ∞,y Rayleigh number for inertial flow based on heat flux, dimensionless Ra γc Rayleigh number for the cold side of a porous medium saturated with fluid near the density maximum, dimensionless Ra γh Rayleigh number for the hot side of a porous medium saturated with fluid near the density maximum, dimensionless S spacing, m BOOKCOMP, Inc. — John Wiley & Sons / Page 1175 / 2nd Proofs / Heat Transfer Handbook / Bejan NOMENCLATURE 1175 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 [1175], (45) Lines: 1535 to 1585 ——— 0.41075pt PgVar ——— Normal Page PgEnds: T E X [1175], (45) Sh Sherwood number, dimensionless t time, s T temperature, K u velocity component in the x direction, m/s V volume, m 3 V volume averaged velocity vector, m/s W width, m w velocity component in the z direction, m/s x Cartesian coordinate, m y Cartesian coordinate, m z Cartesian coordinate, m Greek Letter Symbols α m porous medium thermal diffusivity, m 2 /s α m empirical factor in the density–temperature relation for water, dimensionless β coefficient of thermal expansion, K −1 β C coefficient of concentration expansion, m 3 /kg γ vertical temperature gradient, K/m δ T thermal boundary layer thickness, m η similarity variable, dimensionless λ distance, m µ dynamic viscosity, Pa ·s ν kinematic viscosity, m 2 /s ρ density, kg/m 3 σ capacity ratio, dimensionless τ time, dimensionless φ porosity, dimensionless spherical coordinate, rad angle, rad ψ stream function, m 2 /s spherical coordinate, dimensionless ω wall thickness parameter, dimensionless Subscripts A overall b bottom or bulk c cold f fluid (liquid or gas) phase G geometric mean H harmonic mean h hot i inner L property based on plate length m bulk property BOOKCOMP, Inc. — John Wiley & Sons / Page 1176 / 2nd Proofs / Heat Transfer Handbook / Bejan 1176 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 [1176], (46) Lines: 1585 to 1629 ——— -1.0109pt PgVar ——— Short Page PgEnds: T E X [1176], (46) property of the state of maximum density porous medium o outer opt optimum p constant pressure condition s solid phase w wall x local property ∞ free stream condition REFERENCES Bau, H. H., and Torrance, K. E. (1982). Low Rayleigh Number Thermal Convection in a Vertical Cylinder Filled with Porous Materials and Heated from Below, J. Heat Transfer, 104, 166–172. Bear, J. (1972). Dynamics of Fluids in Porous Media, American Elsevier, New York. Bejan, A. (1978). Natural Convection in an Infinite Porous Medium with a Concentrated Heat Source, J. Fluid Mech., 89, 97–107. Bejan, A. (1979). On the Boundary Layer Regime in a Vertical Enclosure Filled with a Porous Medium, Lett. Heat Mass Transfer, 6, 93–102. Bejan, A. (1983a). The Boundary Layer Regime in a Porous Layer with Uniform Heat Flux from the Side, Int. J. Heat Mass Transfer, 26, 1339–1346. Bejan, A. (1983b). Natural Convection Heat Transfer in a Porous Layer with Internal Flow Obstructions, Int. J. Heat Mass Transfer, 26, 815–822. Bejan, A. (1984). Convection Heat Transfer, Wiley, New York. Bejan, A. (1987). Convective Heat Transfer in Porous Media, in Handbook of Single-Phase Convective Heat Transfer, S. Kakac¸, R. K. Shah, and W. Aung, eds., Wiley, New York. Bejan, A. (1993). Heat Transfer, Wiley, New York. Bejan, A. (1995). Convection Heat Transfer, 2nd ed., Wiley, New York. Bejan, A. (1997). Advanced Engineering Thermodynamics, 2nd ed., Wiley, New York. Bejan, A. (1999). Heat Transfer in Porous Media, in Heat Exchanger Design Update, G. F. Hewitt, ed., Begell House, New York, Vol. 6, Sec. 2.11. Bejan, A. (2000). Shape and Structure, from Engineering to Nature, Cambridge University Press, Cambridge, UK. Bejan, A., and Anderson, R. (1981). Heat Transfer across a Vertical Impermeable Partition Imbedded in a Porous Medium, Int. J. Heat Mass Transfer, 24, 1237–1245. Bejan, A., and Anderson, R. (1983). Natural Convection at the Interface between a Vertical Porous Layer and an Open Space, J. Heat Transfer, 105, 124–129. Bejan, A., and Khair, K. R. (1985). Heat and Mass Transfer by Natural Convection in a Porous Medium, Int. J. Heat Mass Transfer, 28, 909–918. Bejan, A., and Lage, J. L. (1991). Heat Transfer from a Surface Covered with Hair, in Con- vective Heat and Mass Transfer in Porous Media, S. Kakac¸, B. Kilkis, F. A. Kulacki, and F. Arinc, eds., Kluwer Academic, Dordrecht, The Netherlands, pp. 823–845. . Prandtl number, dimensionless q heat transfer rate, W q heat transfer rate per unit length, W/m q heat transfer rate per unit area, W/m 2 q volumetric heat generation rate, W/m 3 R parameter. Uniform Heat Flux from the Side, Int. J. Heat Mass Transfer, 26, 1339–1346. Bejan, A. (1983b). Natural Convection Heat Transfer in a Porous Layer with Internal Flow Obstructions, Int. J. Heat Mass Transfer, . 815–822. Bejan, A. (1984). Convection Heat Transfer, Wiley, New York. Bejan, A. (1987). Convective Heat Transfer in Porous Media, in Handbook of Single-Phase Convective Heat Transfer, S. Kakac¸, R. K. Shah,