BOOKCOMP, Inc. — John Wiley & Sons / Page 1157 / 2nd Proofs / Heat Transfer Handbook / Bejan INTERNAL NATURAL CONVECTION 1157 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 [1157], (27) Lines: 1004 to 1004 ——— -4.073pt PgVar ——— Short Page PgEnds: T E X [1157], (27) T c T c T c T c T c T c T c T c T h T h T h T h T h T h T h T h r o r o r o r i r i r i d i d i d 1 d 1 d N d N r o T h T c g H H H g g g g g g g ()g ()d ()a ()h ()e ()b ()i ()f ()c l LL x g H 0 0 y 0 0 0 qЉ Partition Insulated Insulated Insulated Figure 15.5 Natural convection heat transfer in confined porous media heated from the side: (a) rectangular enclosure; (b) rectangular enclosure with a horizontal partial partition; (c) rectangular enclosure with a vertical full partition midway; (d) rectangular enclosure made up of N vertical sublayers of different K and α;(e) rectangular enclosure made up of N horizontal sublayers of different K and α;(f ) horizontal cylindrical enclosure; (g) horizontal cylindrical annulus with axial heat flow; (h) horizontal cylindrical or spherical annulus with radial heat flow; (i) vertical cylindrical annulus with radial heat flow. BOOKCOMP, Inc. — John Wiley & Sons / Page 1158 / 2nd Proofs / Heat Transfer Handbook / Bejan 1158 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 [1158], (28) Lines: 1004 to 1022 ——— 0.20607pt PgVar ——— Normal Page PgEnds: T E X [1158], (28) Vertical boundary layers Horizontal boundary layers L x T 0Tall system II III High Ra regime H Ra H IV Shallow system T y H 0 I Conduction Conduction Conduction 10 Ϫ2 10 Ϫ1 110 2 10 4 10 6 1 10 H/L 10 2 Figure 15.6 Four heat transfer regimes for natural convection in an enclosed porous layer heated from the side. (From Bejan, 1984.) (Bejan, 1984), that is, four ways to calculate the overall heat transfer rate q = H 0 q dy. These are summarized in Fig. 15.6. • Regime I: the pure conduction regime, defined by Ra H 1. In this regime, q is approximately equal to the pure conduction estimate k m H(T h − T c )/L. • Regime II: the conduction dominated regime in tall layers, defined by H/L 1 and (L/H )Ra 1/2 H 1. In this regime, the heat transfer rate scales as q ≥ k m H(T h − T c )/L. • Regime III: the convection-dominated regime (or high-Rayleigh-number regime), defined by Ra −1/2 H <H/L<Ra 1/2 H . In this regime, q scales as k m (T h −T c )Ra 1/2 H . BOOKCOMP, Inc. — John Wiley & Sons / Page 1159 / 2nd Proofs / Heat Transfer Handbook / Bejan INTERNAL NATURAL CONVECTION 1159 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 [1159], (29) Lines: 1022 to 1078 ——— -0.58969pt PgVar ——— Normal Page PgEnds: T E X [1159], (29) • Regime IV: the convection-dominated regime in shallow layers, defined by H/L 1 and (H/L)Ra 1/2 H 1. Here the heat transfer rate scales as q ≤ k m (T h − T c )Ra 1/2 H . Considerable analytical, numerical, and experimental work has been done to esti- mate more accurately the overall heat transfer rate q or the overall Nusselt number: Nu = q kH(T h − T c )/L (15.98) Note that unlike the single-wall configurations of Section 15.5.1, in confined layers of thickness L the Nusselt number is defined as the ratio of the actual heat transfer rate to the pure conduction heat transfer rate. An analytical solution that covers the four heat transfer regimes smoothly is (Bejan and Tien, 1978) Nu = K 1 + 1 120 K 3 1 Ra H H L 2 (15.99) where K 1 (H/L, Ra H ) is obtained by solving the system 1 120 δ e · Ra 2 H · K 3 1 H L 3 = 1 − K 1 = 1 2 K 1 H L 1 δ e − δ e (15.100) This result is displayed in chart form in Fig. 15.7, along with numerical results from Hickox and Gartling (1981). The asymptotic values of this solution are Nu ∼ 0.508 L H Ra 1/2 H as Ra H →∞ (15.101) 1 + 1 120 Ra H H L 2 as H L → 0 (15.102) The heat transfer in the convection-dominated regime III is represented well by eq. (15.101) or by alternative solutions developed solely for regime III: for example (Weber, 1975), Nu = 0.577 L H Ra 1/2 H (15.103) Equation (15.103) overestimates experimental and numerical data from three inde- pendent sources (Bejan, 1979) by only 14%. More refined estimates for regime III were developed in Bejan (1979) and Simpkins and Blythe (1980), where the propor- tionality factor between Nu and ( L/H ) Ra 1/2 H is replaced by a function of both H/L and Ra H . This alternative is illustrated in Fig. 15.8. For expedient engineering cal- culations of heat transfer dominated by convection, Fig. 15.7 is recommended for shallow layers (H/L < 1), and Fig. 15.8 for square and tall layers (H/L 1) in the boundary layer regime, Ra −1/2 H <H/L<Ra 1/2 H . BOOKCOMP, Inc. — John Wiley & Sons / Page 1160 / 2nd Proofs / Heat Transfer Handbook / Bejan 1160 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 [1160], (30) Lines: 1078 to 1087 ——— 0.27701pt PgVar ——— Normal Page PgEnds: T E X [1160], (30) 20 100 1 10 10 H — L Ra H Bejan and Tien (1978): / =0 HL Hickox and Gartling (1981) / = 0.5 0.2 0.1 HL Nu 0.1 0.2 0.5 1 Figure 15.7 Total heat transfer rate through an enclosed porous layer heated from the side. (From Bejan and Tien, 1978.) In the field of thermal insulation engineering, a more appropriate model for heat transfer in the configuration of Fig. 15.5a is the case where the heat flux q is distributed uniformly along the two vertical sides of the porous layer. In the high- Rayleigh-number regime (regime III), the overall heat transfer rate is given by (Bejan, 1983a) Nu = 1 2 L H 4/5 · Ra ∗2/5 H (15.104) where Ra ∗ H = KgβH 2 q /α m νk m . The overall Nusselt number is defined as in eq. (15.98), where T h −T c is the height-averaged temperature difference between the two sides of the rectangular cross section. Equation (15.104) holds in the high-Rayleigh- number regime Ra ∗−1/3 H <H/L<Ra ∗1/3 H . Impermeable partitions (flow obstructions) inserted in the confined porous me- dium can have a dramatic effect on the overall heat transfer rate across the enclosure (Bejan, 1983b). With reference to the two-dimensional geometry of Fig. 15.5b,in the convection-dominated regimes III and IV the overall heat transfer rate decreases BOOKCOMP, Inc. — John Wiley & Sons / Page 1161 / 2nd Proofs / Heat Transfer Handbook / Bejan INTERNAL NATURAL CONVECTION 1161 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 [1161], (31) Lines: 1087 to 1089 ——— -0.23695pt PgVar ——— Normal Page PgEnds: T E X [1161], (31) Figure 15.8 Heat transfer rate in regime III through a porous layer heated from the side. (From Bejan, 1979.) steadily as the length l of the horizontal partition approaches L, that is, as the partition divides the porous layer into two shorter layers. The horizontal partition has practi- cally no effect in regimes I and II, where the overall heat transfer rate is dominated by conduction. If the partition is oriented vertically (Fig. 15.5c), in the convection- dominated regime the overall heat transfer rate is approximately 40% of what it would have been in the same porous medium without the internal partition. The nonuniformity of permeability and thermal diffusivity can have a dominating effect on the overall heat transfer rate (Poulikakos and Bejan, 1983b). In cases where the properties vary so that the porous layer can be modeled as a sandwich of vertical sublayers of different permeability and diffusivity (Fig. 15.5d), an important parame- ter is the ratio of the peripheral sublayer thickness (d 1 ) to the thermal boundary layer thickness (δ T,1 ) based on the properties of the d 1 sublayer (note that δ T,1 scales as H · Ra −1/2 H,1 , where the Rayleigh number Ra H,1 = K 1 gβH(T h − T c )/α 1 ν and where the subscript 1 represents the properties of the d 1 sublayer). If d 1 > δ T,1 , the heat transfer through the left side of the porous system of Fig. 15.5d is impeded by a thermal resistance of order δ T,1 /k 1 H . If the sublayer situated next to the right wall (d N ) has exactly the same properties as the d 1 sublayer, and if δ T,1 <(d 1 ,d N ), the overall heat transfer rate in the convection-dominated regime can be estimated using BOOKCOMP, Inc. — John Wiley & Sons / Page 1162 / 2nd Proofs / Heat Transfer Handbook / Bejan 1162 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 [1162], (32) Lines: 1089 to 1128 ——— 0.45009pt PgVar ——— Normal Page PgEnds: T E X [1162], (32) eq. (15.101), in which both Nu and Ra H are based on the properties of the peripheral layers. When the porous-medium inhomogeneity may be modeled as a sandwich of N horizontal sublayers (Fig. 15.5e), the scale of the overall Nusselt number in the convection-dominated regime can be evaluated as (Poulikakos and Bejan, 1983a) Nu ∼ 2 −3/2 Ra 1/2 H,1 L H N i=1 k i k 1 K i d i α 1 K 1 d 1 α i 1/2 (15.105) where both Nu and Ra H,1 are based on the properties of the d 1 sublayer (Fig. 15.5e). The correlation of eq. (15.105) was tested via numerical experiments in two-layer systems. The heat transfer results reviewed in this section are based on the idealization that the surface that surrounds the porous medium is impermeable. With reference to the two-dimensional geometry of Fig. 15.5a, the heat transfer through a shallow porous layer with one or both end surfaces permeable is anticipated analytically in Bejan and Tien (1978). Subsequent laboratory measurements and numerical solutions for Ra H values up to 120 validate the theory (Haajizadeh and Tien, 1983). Natural convection in cold water saturating the porous-medium configuration of Fig. 15.5a was considered in Poulikakos (1984). Instead of the linear approximation of eq. (15.60), this study used the parabolic model ρ m − ρ = γρ m (T − T m ) 2 (15.106) where γ ≈ 8.0 ×10 −6 K −2 and T m = 3.98°C for pure water at atmospheric pressure. The parabolic density model is valid in the temperature range 0 to 10°C. In the convection-dominated regime Nu 1, the scale analysis (Bejan, 1995) leads to the Nusselt number correlation (Bejan, 1987) Nu = c 3 L/H Ra −1/2 γh + c 4 Ra −1/2 γc (15.107) where Ra γh = KgγH(T h − T m ) 2 /α m ν, Ra γc = KgγH(T m − T c ) 2 /α m ν, and where the Nusselt number is defined in eq. (15.98). For the convection-dominated regime, the numerical study (Poulikakos, 1984) tabulated results primarily for the case T c = 0°C,T h = 7.96°C; using these data for cases in which T c and T h are symmetrically positioned around T m (i.e., when Ra γh = Ra γc ), the scaling-correct correlation in eq. (15.107) takes the form (Bejan, 1987) Nu ≈ 0.26 L H Ra 1/2 γh (15.108) In other words, the two constants that appear in eq. (15.107) satisfy the relationship c 3 ≈ 0.26(1 + c 4 ). More experimental data for the high-Rayleigh-number regime in vertical layers with Ra γh = Ra γc are needed to determine c 3 and c 4 uniquely. BOOKCOMP, Inc. — John Wiley & Sons / Page 1163 / 2nd Proofs / Heat Transfer Handbook / Bejan INTERNAL NATURAL CONVECTION 1163 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 [1163], (33) Lines: 1128 to 1156 ——— 9.79126pt PgVar ——— Normal Page PgEnds: T E X [1163], (33) 15.6.2 Cylindrical and Spherical Enclosures The convection occurring in a porous medium confined in a horizontal cylinder with disk-shaped ends at different temperatures (Fig. 15.5f ) has features similar to the problem of Fig. 15.5a. A parametric solution for the horizontal cylinder problem is reported in Bejan and Tien (1978). The corresponding phenomenon in a porous medium in the shape of a horizontal cylinder with annular cross section (Fig. 15.5g) is documented in Bejan and Tien (1979). A pivotal important geometric configuration in thermal insulation engineering is a horizontal annular space filled with fibrous or granular insulation (Fig. 15.5h). In this configuration the heat transfer is radial between the concentric cylindrical surfaces of radii r i and r o , unlike in the earlier sketch (Fig. 15.5g), where the cylindrical surfaces were insulated and the heat transfer was axial. Experimental measurements and numerical solutions for the overall heat transfer in the configuration of Fig. 15.5h have been reported in Caltagirone (1976) and Burns and Tien (1979). These results were correlated based on scale analysis (Bejan, 1987) in the range 1.19 ≤ r o /r i ≤ 4 and the results are correlated by Nu = q actual q conduction ≈ 0.44Ra 1/2 r i ln(r o /r i ) 1 + 0.916(r i /r o ) 1/2 (15.109) where Ra r i = Kgβr i (T h −T c )/α m ν and q conduction = 2πk m (T h −T c )/ ln(r o /r i ). This correlation is valid in the convection-dominated limit, Nu1. Porous media confined to the space formed between two concentric spheres are also an important component in thermal insulation engineering. Figure 15.5h can be interpreted as a vertical cross section through the concentric-sphere arrangement. Nu- merical heat transfer solutions for discrete values of Rayleigh number and radius ratio are reported graphically in Burns and Tien (1979). Using the scale analysis method (Bejan, 1984, 1995) the data that correspond to the convection-dominated regime (Nu 1.5) are correlated within 2% by the scaling-correct expression (Bejan, 1987) Nu = q actual q conduction = 0.756Ra 1/2 r i 1 − r i /r o 1 + 1.422(r i /r o ) 3/2 (15.110) where Ra r i = Kgβr i (T h −T c )/α m ν and q conduction = 4πk m (T h −T c )/(r −1 i −r −1 o ).In terms of the Rayleigh number based on the insulation thickness Ra r o −r i = Kgβ(r o − r i )(T h − T c )/α m ν, the correlation (15.110) becomes Nu = 0.756Ra 1/2 r o −r i r i /r o − (r i /r o ) 2 1/2 1 + 1.422(r i /r o ) 3/2 (15.111) In this form, the Nusselt number expression has a maximum in r i /r o (at r i /r o = 0.3). Heat transfer by natural convection through an annular porous insulation oriented vertically (Fig. 15.5i) was investigated numerically (Havstad and Burns, 1982) and experimentally (Prasad et al., 1985). For systems where both vertical cylindrical BOOKCOMP, Inc. — John Wiley & Sons / Page 1164 / 2nd Proofs / Heat Transfer Handbook / Bejan 1164 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 [1164], (34) Lines: 1156 to 1187 ——— 0.74707pt PgVar ——— Normal Page PgEnds: T E X [1164], (34) surfaces may be modeled as isothermal (T h and T c ), results were correlated with the five-constant empirical formula (Havstad and Burns, 1982) Nu = 1 + a 1 r i r o 1 − r i r o a 2 · Ra a 4 r o H r o a 5 exp −a 3 r i r o (15.112) where a 1 = 0.2196,a 2 = 1.334,a 3 = 3.702,a 4 = 0.9296, and a 5 = 1.168 and where Ra r o = Kg βr o (T h − T c )/α m ν. The Nusselt number is defined as Nu = q actual /q conduction , where q conduction = 2πk m H(T h − T c )/ ln(r o /r i ). The correlation of eq. (15.112) fits the numerical data in the range 1 ≤ H/r o ≤ 20, 0 < Ra r o < 150, 0 <r i /r o ≤ 1, and 1 < Nu < 3. In the boundary layer convection regime (at high Rayleigh and Nussselt numbers), the scale analysis of this two-boundary-layer problem suggests the following scaling law (Bejan, 1987): Nu = c 1 ln(r o /r i ) c 2 + r o /r i r o H Ra 1/2 H (15.113) where Ra H = KgβH(T h −T c )/α m ν. Experimental data in the convection-dominated regime Nu 1 are needed to determine the constants c 1 and c 2 (note that Havstad and Burns’s data are for moderate Nusselt numbers 1 < Nu < 3, i.e., for cases where pure conduction plays an important role). There is a large and still-growing volume of additional results for enclosed porous media heated from the side, for example, with application to cavernous bricks and walls for buildings (Lorente et al., 1996, 1998; Lorente 2002; Lorente and Bejan, 2002; Vasile et al., 1998). 15.6.3 Enclosures Heated from Below The most basic configuration of a confined porous layer heated in the vertical direc- tion is shown in Fig. 15.9a. An important difference between heat transfer in this configuration and heat transfer in confined layers heated from the side is that in Fig. 15.9a convection occurs only when the imposed temperature difference or heating rate exceeds a certain, finite value. Recall that in configurations such as Fig. 15.5a, convection is present even in the limit of vanishingly small temperature differences (Fig. 15.6). Assume that the fluid saturating the porous medium of Fig. 15.9a expands upon heating (β > 0). By analogy with the phenomenon of B ´ enard convection in a pure fluid, in the convection regime the flow consists of finite-sized cells that become more slender and multiply discretely as the destabilizing temperature difference T h − T c increases. If T h − T c does not exceed the critical value necessary for the onset of convection, the heat transfer mechanism through the layer of thickness H is that of pure thermal conduction. If β > 0 and the porous layer is heated from above (i.e., if T h and T c change places in Fig. 15.9a), the fluid remains stably stratified and the heat transfer is again due to pure thermal conduction: q = k m L(T h − T c )/H . The onset of convection in an infinitely long porous layer heated from below as examined on the basis of linearized hydrodynamic stability analysis (Nield and Bejan, 1999; Horton and Rogers, 1945; Lapwood, 1948). For fluid layers confined between BOOKCOMP, Inc. — John Wiley & Sons / Page 1165 / 2nd Proofs / Heat Transfer Handbook / Bejan INTERNAL NATURAL CONVECTION 1165 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 [1165], (35) Lines: 1187 to 1187 ——— * 65.754pt PgVar ——— Normal Page PgEnds: T E X [1165], (35) T c T c T c T c T c C t T c T c T c T c T c T c T c T h T h T h T h T h T h T h T h C b T h T h T h T h L y L x L y L y L y L r o L x T c T h T c Temp. distr. g L H H H H H H H g g g gg g g ()g ()d ()a ()h ()e ()b ()i ()f ()c Permeable end L D L x y g H Permeable end Figure 15.9 Natural convection heat transfer in confined porous layers heated from below (a–d), and due to penetrative flows (e–i): (a) rectangular enclosure; (b) vertical cylindrical en- closure; (c) inclined rectangular enclosure; (d) wedge-shaped enclosure; (e) vertical cylindrical enclosure; ( f ) horizontal rectangular enclosure; (g) semi-infinite porous medium bounded by a horizontal surface with alternate zones of heating and cooling; (h) shallow rectangular en- closure heated and cooled from one vertical wall; (i) slender rectangular enclosure heated and cooled from one vertical wall. BOOKCOMP, Inc. — John Wiley & Sons / Page 1166 / 2nd Proofs / Heat Transfer Handbook / Bejan 1166 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 [1166], (36) Lines: 1187 to 1221 ——— 1.74217pt PgVar ——— Normal Page * PgEnds: Eject [1166], (36) impermeable and isothermal horizontal walls, it was found that convection is possible if the Rayleigh number based on height exceeds the critical value Ra H = Kg βH(T h − T c ) α m ν = 4π 2 = 39.48 (15.114) A much simpler closed-form analysis based on constructal theory (Nelson and Bejan, 1998; Bejan, 2000) predicted the critical Rayleigh number 12π = 37.70, which approaches within 5% the hydrodynamic stability result. For a history of the early theoretical and experimental work on the onset of B ´ enard convection in porous media, and for a rigorous generalization of the stability analysis to convection driven by combined buoyancy effects, the reader is directed to Nield (1968), where it is shown that the critical Rayleigh number for the onset of convection in infinitely shallow layers depends to a certain extent on the heat and fluid flow conditions imposed along the two horizontal boundaries. Of practical interest in heat transfer engineering is the heat transfer rate at Rayleigh numbers that are higher than critical. There has been a considerable amount of an- alytical, numerical, and experimental work devoted to this issue. Reviews of these advances may be found in Nield and Bejan (1999) and Cheng (1978). Constructal theory anticipates the entire curve relating heat transfer to Rayleigh number (Nelson and Bejan, 1998; Bejan, 2000). The scale analysis of the convection regime with Darcy flow (Bejan, 1984) con- cludes that the Nusselt number should increase linearly with the Rayleigh number, whence the relationship Nu ≈ 1 40 Ra H for Ra H > 40 (15.115) This linear relationship is confirmed by numerical heat transfer calculations at large Rayleigh numbers in Darcy flow (Kimura et al., 1986). The experimental data com- piled in Cheng (1978) show that the scaling law of eq. (15.115) serves as an upper bound for some of the high-Ra H experimental data available in the literature. Most of the data show that in the convection regime Nu increases as Ra n H , where n becomes progressively smaller than 1 as Ra H increases. This behavior is anticipated by the constructal-theory solution (Nelson and Bejan, 1998; Bejan, 2000). The expo- nent n ∼ 1 2 revealed by data at high Rayleigh numbers was anticipated based on a scale analysis of convection rolls in the Forchheimer regime (Bejan, 1995): Nu Pr p ∼ Ra H Pr p 1/2 Ra H > Pr p (15.116) where Pr p is the porous-medium Prandtl number for the Forchheimer regime (Bejan, 1995), Pr p = H ν bKα m (15.117) . number is defined as the ratio of the actual heat transfer rate to the pure conduction heat transfer rate. An analytical solution that covers the four heat transfer regimes smoothly is (Bejan and. 15.6 Four heat transfer regimes for natural convection in an enclosed porous layer heated from the side. (From Bejan, 1984.) (Bejan, 1984), that is, four ways to calculate the overall heat transfer. Total heat transfer rate through an enclosed porous layer heated from the side. (From Bejan and Tien, 1978.) In the field of thermal insulation engineering, a more appropriate model for heat transfer