BOOKCOMP, Inc. — John Wiley & Sons / Page 1137 / 2nd Proofs / Heat Transfer Handbook / Bejan BASIC PRINCIPLES 1137 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 [1137], (7) Lines: 264 to 298 ——— 5.52219pt PgVar ——— Normal Page * PgEnds: Eject [1137], (7) K = d 2 φ 3 150(1 − φ) 2 (15.15) b = 1.75(1 − φ) φ 3 d (15.16) Brinkman’s (1947) modification of the Darcy flow model accounts for the transi- tion from Darcy flow to highly viscous flow (without porous matrix), in the limit of extremely high permeability: v = K µ (−∇ρ + ρg) + K∇ 2 v (15.17) The more appropriate way to write Brinkman’s equation is (Nield and Bejan, 1999) ∇P =− µ K v +˜µ∇ 2 v (15.18) which is similar to eq. (15.17) without the body force term and multiplied by K/µ. In eqs. (15.17) and (15.18), two viscous terms are evident. The first is the usual Darcy term, and the second is analogous to the Laplacian term that appears in the Navier– Stokes equation. The coefficient ˜µ is an effective viscosity. Brinkman set µ and ˜µ equal to each other, but in general that is not true. The reader is referred to Nield and Bejan (1999) for a critical discussion of the applicability of eq. (15.18). There are situations in which it is convenient to use the Brinkman equation. One such situation is when flows in porous media are compared with those in clear fluids. The Brinkman equation has a parameter K (the permeability) such that the equation reduces to a form of the Navier–Stokes equation as K/L 2 →∞and to the Darcy equation as K/L 2 → 0. Another situation is when it is desired to match solutions in a porous medium and in an adjacent viscous fluid. The two modifications of the Darcy flow model discussed above, the Forchheimer model of eq. (15.13) and the Brinkman model of eq. (15.17), were used simultane- ously by Vafai and Tien (1981) in a study of forced-convection boundary layer heat transfer. In the presence of gravitational acceleration, Vafai and Tien’s momentum equations would read v + bρK µ |v|v = K µ (−∇P + ρg) + K ∇ 2 v (15.19) None of the foregoing models account adequately for the transition from porous medium flow to pure fluid flow as the permeability K increases. Note that in the high-K limit, the terms that survive in eq. (15.17) or (15.19) account for momen- tum conservation only in highly viscous flows in which the effect of fluid inertia is negligible relative to pressure and friction forces. A model that bridges the entire gap between the Darcy–Forchheimer model and the Navier–Stokes equations was proposed by Vafai and Tien (1981): BOOKCOMP, Inc. — John Wiley & Sons / Page 1138 / 2nd Proofs / Heat Transfer Handbook / Bejan 1138 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 [1138], (8) Lines: 298 to 339 ——— 1.72618pt PgVar ——— Long Page * PgEnds: Eject [1138], (8) ν K v + b|v|v =− Dv Dt − 1 ρ ∇P + ν ∇ 2 v + g (15.20) As the permeability K increases, the left-hand side vanishes and gives way to the complete vectorial Navier–Stokes equation for Newtonian constant-property flow. The state of the art in the development of flow models and new questions about the older models are discussed in Nield and Bejan (1999). 15.2.3 Energy Conservation Consider now the first law of thermodynamics for flow through a porous medium. For simplicity, assume that the medium is isotropic and that radiative effects, viscous dissipation, and the work done by pressure changes are negligible. For most cases, it is acceptable to assume that there is local thermal equilibrium so that T s = T f = T , where T s and T f are the temperatures of the solid and fluid phases, respectively. Also assume that heat conduction in the solid and fluid phases takes place in parallel so that there is no net heat transfer from one phase to the other. Taking averages over an elemental volume of the medium give, for the solid phase (Nield and Bejan, 1999), (1 − φ)(ρc) s ∂T s ∂t = (1 −φ)∇·(k s ∇T s ) + (1 −φ)q s (15.21) and for the fluid phase, φ(ρc p ) f ∂T f ∂t + (ρc p ) f v ·∇T f = φ∇·(k f ∇T f ) + φq f (15.22) Here the subscripts s and f refer to the solid and fluid phases, respectively, c is the specific heat of the solid, c p is the specific heat at constant pressure of the fluid, k is the thermal conductivity, and q (W/m 3 ) is the heat generation rate per unit volume. In writing eqs. (15.21) and (15.22) it has been assumed that the surface porosity is equal to the porosity. For example, −k s ∇T s is the conductive heat flux through the solid, and thus ∇·(k s ∇T s ) is the net rate of heat conduction into a unit volume of the solid. In eq. (15.21), this appears multiplied by the factor (1 − φ), which is the ratio of the cross-sectional area occupied by solid to the total cross-sectional area of the medium. The other two terms in eq. (15.21) also contain the factor (1 −φ), because this is the ratio of volume occupied by solid to the total volume of the element. In eq. (15.22) there also appears a convective term, due to the seepage velocity. We recognize that v ·∇T f is the rate of change of temperature in the elemental volume due to the convection of fluid into it, so this, multiplied by (ρc p ) f , must be the rate of change of thermal energy, per unit volume of fluid, due to the convection. Note further that in writing eq. (15.22), use has been made of the Dupuit–Forchheimer relationship (Nield and Bejan, 1999), v = φV. Setting T s = T f = T and adding eqs. (15.21) and (15.22) yields (ρc) m ∂T ∂t + (ρc) f v ·∇T =∇·(k m ∇T)+q m (15.23) BOOKCOMP, Inc. — John Wiley & Sons / Page 1139 / 2nd Proofs / Heat Transfer Handbook / Bejan BASIC PRINCIPLES 1139 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 [1139], (9) Lines: 339 to 402 ——— 3.88634pt PgVar ——— Long Page * PgEnds: Eject [1139], (9) where (ρc) m = (1 −φ)(ρc) s + φ(ρc p ) f (15.24) k m = (1 −φ)k s + φk f (15.25) q m = (1 −φ)q s + φq f (15.26) are, respectively, the overall heat capacity per unit volume, overall thermal conduc- tivity, and overall heat production per unit volume of the medium. Equation (15.24) may also be written as (ρc) m = (ρc p ) f σ (15.27) where σ is the heat capacity ratio, σ = φ + (1 + φ) (ρc) s (ρc p ) f (15.28) The energy balance of eq. (15.24) becomes σ ∂T ∂t + v ·∇T =∇·(α m ∇T)+ q m (ρc p ) f (15.29) where α m is the thermal diffusivity of the fluid saturated porous medium, α m = k m (ρc p ) f (15.30) If the assumption of local thermal equilibrium is abandoned, accountmustbetaken for the local heat transfer between solid and fluid. Equations (15.21) and (15.22) are replaced by (1 − φ)(ρc) s ∂T ∂t = (1 −φ)∇·(k s ∇T s ) + (1 −φ)q s + h(T f − T s ) (15.31) φ(ρc p ) f ∂T f ∂t + (ρc p ) f v ·∇T = φ∇·(k f ∇T)+φq f + h(T s − T f ) (15.32) where h is the heat transfer coefficient. A critical aspect of using this approach lies in the determination of the appropriate value of h. Experimental values of h are found in an indirect manner and methods are reviewed by Nield and Bejan (1999). In situations where the fluid that saturates the porous structure is a mixture of two or more chemical species, the equation that expresses the conservation of species is (Bejan, 1995) φ ∂C ∂t + v ·∇C = D ∇ 2 C +˙m i (15.33) BOOKCOMP, Inc. — John Wiley & Sons / Page 1140 / 2nd Proofs / Heat Transfer Handbook / Bejan 1140 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 [1140], (10) Lines: 402 to 452 ——— -0.95587pt PgVar ——— Long Page PgEnds: T E X [1140], (10) In this equation C is the concentration of i, expressed in kilograms of i per unit volume of porous medium; D is the mass diffusivity of i through the porous medium with the fluid mixture in it; and ˙m i is the number of kilograms of i produced by a chemical reaction per unit time and per unit volume of porous medium. 15.3 CONDUCTION Pure thermal diffusion is the mechanism for heat transfer when there is no motion through the pores of the solid structure. The conservation of energy is described by eq. (15.29), in which the convection term is absent: σ ∂T ∂t =∇·(α m ∇T)+ q (ρc p ) f (15.34) Except for the heat capacity ratio σ that appears on the left side, eq. (15.34) is the same as the equation for time-dependent conduction through a solid (Bejan, 1993). This means that the mathematical methods developed for conduction in solids (Bejan, 1993; Carslaw and Jaeger, 1959) apply to porous media saturated with stagnant fluid. For example, the thermal penetration depth due to time-dependent conduction into a semiinfinite porous medium without fluid motion is of order (α m t/σ) 1/2 . The reader is directed to Chapter 3 in this book for additional mathematical solutions for key configurations. The overall thermal conductivity of a porous medium depends in a complex fash- ion on the geometry of the medium (Nield and Bejan, 1999; Nield, 1991). If the heat conduction in the solid and fluid phases occurs in parallel, the overall conductivity k A is the weighted arithmetic mean of the conductivities k s and k f : k A = (1 −φ)k s + φk f (15.35) On the other hand, if the structure and orientation of the porous medium is such that the heat conduction takes place in series, with all the heat flux passing through both solid and fluid, the overall conductivity k H is the weighted harmonic mean of k s and k f : 1 k H = 1 − φ k s + φ k f (15.36) In general, k A and k H will provide upper and lower bounds, respectively, on the actual overall conductivity k m . It is always true that k H ≤ k A , with equality if and only if k s = k f . For practical purposes, a rough-and-steady estimate for k m is provided by k G , the weighted geometric mean of k s and k f , defined by k G = k 1−φ s k φ f (15.37) This provides a good estimate as long as k s and k f are not too different from each other (Nield, 1991). More complicated correlation formulas for the conductivity of packed beds have been proposed. Experiments by Prasad et al. (1989) showed that eqs. BOOKCOMP, Inc. — John Wiley & Sons / Page 1141 / 2nd Proofs / Heat Transfer Handbook / Bejan FORCED CONVECTION 1141 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 [1141], (11) Lines: 452 to 495 ——— 1.54228pt PgVar ——— Long Page PgEnds: T E X [1141], (11) (15.35)–(15.37) gave reasonably good results provided that k f was not significantly greater than k s . Additional thermal conductivity models are discussed in Nield and Bejan (1999). 15.4 FORCED CONVECTION 15.4.1 Plane Wall with Constant Temperature The newer results developed for heat transfer through porous media refer to forced and natural convection. The heat transfer results listed next refer to a uniform unidi- rectional seepage flow through a homogeneous and isotropic porous medium. They are based on the idealization that the solid and fluid phases are locally in thermal equilibrium. Consider the uniform flow (u, T ∞ ) parallel to a solid wall heated to a constant temperature T w , as shown in Fig. 15.1b. The boundary layer solution for the local Nusselt number is available analytically (Bejan, 1995): Nu x = q T w − T ∞ x k m = 0.564Pe 1/2 x (15.38) where Pe x is the P ´ eclet number based on the local longitudinal position, Pe x = ux/α m . The heat flux q and the heat transfer coefficient q /(T w − T ∞ ) decrease as x −1/2 . The overall Nusselt number based on the heat flux, q averaged from x = 0 to a given wall length x = L is Nu L = q T w − T ∞ L k m = 1.128Pe 1/2 L (15.39) where the overall P ´ eclet number is Pe L = uL/α m . The total heat transfer rate through the wall is q = q L. Related boundary layer solutions are reviewed in Nield and Bejan (1999). The local Nusselt number for boundarylayerheattransfer near awall with constant heat flux is also available in closed form (Bejan, 1995): Nu x = q T w (x) −T ∞ x k m = 0.886Pe 1/2 x (15.40) where Pe x = ux/α m . The temperature difference T w (x) − T ∞ increases as x 1/2 . The overall Nusselt number that is based on the average wall temperature T w (specifically, the temperature averaged from x = 0tox = L is Nu L = q T w − T ∞ L k m = 1.329Pe 1/2 L (15.41) where Pe L = uL/α m . Equations (15.38)–(15.41) are valid when the respective P ´ eclet numbers are greater than 1 in an order-of-magnitude sense. Mass transfer counter- parts to these heat transfer formulas are obtained through the notation transformation BOOKCOMP, Inc. — John Wiley & Sons / Page 1142 / 2nd Proofs / Heat Transfer Handbook / Bejan 1142 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 [1142], (12) Lines: 495 to 545 ——— 3.25816pt PgVar ——— Normal Page PgEnds: T E X [1142], (12) Nu → Sh,q → j ,T → C,k m → D, and α m → D, where Sh, j ,C, and D are the Sherwood number, mass flux, species concentration, and mass diffusivity constant. From a fluid mechanics standpoint, these results are valid if the flow is parallel, that is, in the Darcy regime or Darcy–Forchheimer regime through a homogeneous and isotropic porous medium. The special effect of the flow resistance exerted by the solid wall was documented numerically in Vafai and Tien (1981). 15.4.2 Sphere and Cylinder Consider the thermal boundary layer region around a sphere or around a circular cylinder that is perpendicular to the uniform flow with volume-averaged velocity u. As indicated in Fig. 15.1c, the sphere or cylinder radius is r 0 and the surface temper- ature is T w . The distributions of heat flux around the sphere and cylinder in Darcy flow were determined in Cheng (1982). With reference to the angular coordinate θ defined in Fig. 15.1c, the local peripheral Nusselt numbers are, for the sphere, Nu θ = 0.564 ur 0 θ α m 1/2 3 2 θ 1/2 sin 2 θ 1 3 cos 3 θ − cos θ + 2 3 −1/2 (15.42) and for the cylinder, Nu θ = 0.564 ur 0 θ α m 1/2 (2θ) 1/2 sin θ(1 − cos θ) −1/2 (15.43) The P ´ eclet number is based on the swept arc r 0 θ: namely, Pe θ = ur 0 θ/α m . The local Nusselt number is defined as Nu θ = q T w − T ∞ r 0 θ k m (15.44) Equations (15.42) and (15.43) are valid when the boundary layers are distinct (thin), that is, when the bondary layer thickness r 0 ·Pe −1/2 θ is smaller than the radius r 0 . This requirement can also be written as Pe 1/2 θ 1orNu θ 1. The conceptual similarity between the thermal boundary layers of the cylinder and the sphere (Fig. 15.1c) and that of the flat wall (Fig. 15.1b) is illustrated further by Nield and Bejan’s (1999) correlation of the heat transfer results for these three configurations. The heat flux averaged over the area of the cylinder and sphere, q , can be calculated by averaging the local heat flux q expressed by eqs. (15.42)–(15.44). The results are for the sphere, Nu D = 1.128Pe 1/2 D (15.45) and for the cylinder, Nu D = 1.015Pe 1/2 D (15.46) BOOKCOMP, Inc. — John Wiley & Sons / Page 1143 / 2nd Proofs / Heat Transfer Handbook / Bejan FORCED CONVECTION 1143 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 [1143], (13) Lines: 545 to 589 ——— -2.67775pt PgVar ——— Normal Page * PgEnds: Eject [1143], (13) In these expressions, the Nusselt and P ´ eclet numbers are based on the diameter D = 2r 0 : Nu D = q T w − T ∞ D k m Pe D = uD α m (15.47) 15.4.3 Concentrated Heat Sources In the region downstream from the hot sphere or cylinder of Fig. 15.1c, the heated fluid forms a thermal wake whose thickness increases as x 1/2 . This behavior is il- lustrated in Fig. 15.1d and e, in which x measures the distance downstream from the heat source. Seen from the distant wake region, the embedded sphere appears as a point source (Fig. 15.1d), while the cylinder perpendicular to the uniform flow (u, T ∞ ) looks like a line source (Fig. 15.1e). Consider the two-dimensional frame attached to the line source q in Fig. 15.1e. The temperature distribution in the wake region, T (x,y),is T (x,y) − T ∞ = 0.282 q k m α m ux 1/2 e −uy 2 /4α m x (15.48) This shows that the width of the wake increases as x 1/2 , while the temperature excess on the centerline [ T(x,0) −T ∞ ] decreases as x −1/2 . The corresponding solution for the temperature distribution T(x,r)in the round wake behind the point source q of Fig. 15.1d is T(x,r)− T ∞ = q 4πk m x e −ur 2 /4α m x (15.49) In this case, the excess temperature on the wake centerline decreases as x −1 , that is, more rapidly than on the centerline of the two-dimensional wake. Equations (15.48) and (15.49) are valid when the wake region is slender, in other words, when Pe x 1. When this P ´ eclet number condition is not satisfied, the temperature field around the source is dominated by the effect of thermal diffusion, not convection. In such cases, the effect of the heat source is felt in all directions, not only downstream. In the limit where the flow (u,T ∞ ) is so slow that the convection effect can be neglected, the temperature distribution can be derived by the classical methods of pure conduction. A steady-state temperature field can exist only around the point source (Bejan, 1993), T(r)− T ∞ = q 4πk m r (15.50) The pure-conduction temperature distribution around the line source remains time- dependent. When the time t is sufficiently long so that (x 2 + y 2 )/(4α m t) 1, the excess temperature around the line source is approximated by T (x,y, t) − T ∞ q 4πk m ln 4α m t σ(x 2 + y 2 ) − 0.5772 (15.51) BOOKCOMP, Inc. — John Wiley & Sons / Page 1144 / 2nd Proofs / Heat Transfer Handbook / Bejan 1144 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 [1144], (14) Lines: 589 to 632 ——— 1.29828pt PgVar ——— Short Page * PgEnds: Eject [1144], (14) 15.4.4 Channels Filled with Porous Media Consider now the forced-convection heat transfer in a channel or duct packed with a porous material as in Fig. 15.1f. In the Darcy flow regime the longitudinal volume- averaged velocity u is uniform over the channel cross section. When the temperature field is fully developed, the relationship between the wall heat flux q and the local temperature difference (T w −T b ) is analogous to the relationship for fully developed heat transfer to slug flow through a channel without a porous matrix (Bejan, 1995). The temperature T b is the mean or bulk temperature of the stream that flows through the channel, T b = 1 A A TdA (15.52) in which A is the area of the channel cross section. In cases where the confining wall is a tube with the internal diameter D, the relation for fully developed heat transfer can be expressed as a constant Nusselt number (Rohsenow and Choi, 1961): Nu D = q (x) T w − T b (x) D k = 5.78 (tube, T w = constant) (15.53) q T w (x) −T b (x) D k m = 8 (parallel plates, q = constant) (15.54) When the porous matrix is sandwiched between two parallel plates with the spacing D, the corresponding Nusselt numbers are (Rohsenow and Hartnett, 1973) Nu D = q (x) T w − T b (x) D k m = 4.93 (parallel plates,T w = constant) (15.55) q T w (x) −T b (x) D k m = 6 (parallel plates,q = constant) (15.56) The forced-convection results of eqs. (15.53)–(15.56) are valid when the temper- ature profile across the channel is fully developed (sufficiently far from the entrance x = 0). The entrance length, or length needed for the temperature profile to become fully developed, can be estimated by noting that the thermal boundary layer thick- ness scales as (α m x/u) 1/2 . Setting (α m x/u) 1/2 ∼ D, the thermal entrance length x T ∼ D 2 u/α m is obtained. Inside the entrance region 0 <x<x T , the heat transfer is impeded by the forced-convection thermal boundary layers that line the channel walls, and can be calculated approximately using eqs. (15.38)–(15.41). One important application of the results for a channel packed with a porous ma- terial is in the area of heat transfer augmentation. The Nusselt numbers for fully de- veloped heat transfer in a channel without a porous matrix are given by expressions similar to eqs. (15.53)–(15.56) except that the saturated porous medium conductivity k m is replaced by the thermal conductivity of the fluid alone, k f . The relative heat transfer augmentation effect is indicated approximately by the ratio BOOKCOMP, Inc. — John Wiley & Sons / Page 1145 / 2nd Proofs / Heat Transfer Handbook / Bejan FORCED CONVECTION 1145 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 [1145], (15) Lines: 632 to 658 ——— -0.27193pt PgVar ——— Short Page PgEnds: T E X [1145], (15) h x (with porous matrix) h x (without porous matrix) ∼ k m k f (15.57) in which h x is the local heat transfer coefficient q /(T w −T b ). In conclusion, a signif- icant heat transfer augmentation effect can be achieved by using a high-conductivity matrix material, so that k m is considerably greater than k f . Key results for forced convection in porous media have also been developed for tree networks of cracks (constructal theory) (Bejan, 2000), time-dependent heating, annular channels, stepwise changes in wall temperature, local thermal nonequilib- rium, and other external flows (such as over a cone or wedge). These are also reviewed in Nield and Bejan (1999), which also describes improvement in porous-medium modeling that account for fluid inertia, thermal dispersion, boundary friction, non- Newtonian fluids, and porosity variation. The concepts of heatfunctions and heatlines were introduced for the purpose of visualizing the true path of the flow of energy through a convective medium (Bejan, 1995). The heatfunction accounts simultaneously for the transfer of heat by conduc- tion and convection at every point in the medium. The heatlines are a generalization of the flux lines used routinely in the field of conduction. The concept of heatfunction is a spatial generalization of the concept of the Nusselt number, that is, a way of indi- cating the magnitude of the heat transfer rate through any unit surface drawn through any point on the convective medium. The heatline method was extended recently to several configurations of convection through fluid-saturated porous media (Morega and Bejan, 1994). 15.4.5 Compact Heat Exchangers as Porous Media An important application of the formalism of forced convection in porous media is in the field of heat exchanger simulation and design. Heat exchangers are a century- old technology based on information and concepts stimulated by the development of large-scale devices. The modern emphasis on heat transfer augmentation, and the more recent push toward miniaturization in the cooling of electronics, have led to the development of compact devices with much smaller features than in the past. These devices operate at lower Reynolds numbers, where their compactness and small dimensions (“pores”) make them candidates for modeling as saturated porous media. Such modeling promises to revolutionize the nomenclature and numerical simulation of the flow and heat transfer through heat exchangers (Bejan et al., 2004). To illustrate this change, consider Zhukauskas’s (1987) classical chart for the pressure drop in crossflow through arrays of staggered cylinders (e.g., Fig. 9.38 in Bejan, 1993). The four curves drawn on this chart for the transverse pitch/cylinder diameter ratios 1.25, 1.5, 2, and 2.5 can be made to collapse into a single curve (Bejan and Morega, 1993), as shown in Fig. 15.2. The technique consists of treating the bundle as a fluid-saturated porous medium and using the volume-averaged velocity U, the pore Reynolds number UK 1/2 /ν on the abscissa, and the dimensionless pressure gradient group (∆P /L)K 1/2 /ρU 2 on the ordinate. BOOKCOMP, Inc. — John Wiley & Sons / Page 1146 / 2nd Proofs / Heat Transfer Handbook / Bejan 1146 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 [1146], (16) Lines: 658 to 662 ——— 1.097pt PgVar ——— Normal Page PgEnds: T E X [1146], (16) Figure 15.2 Porous medium representation of the classical pressure-drop data for flow through staggered cylinders and stacks of parallel plates. (From Bejan and Morega, 1993). The method of presentation of Fig. 15.2 deserves to be extended to other heat exchanger geometries. Another reason for pursuing this direction is that the heat and fluid flow process can be simulated numerically much more easily if the heat ex- changer is replaced at every point by a porous medium with volume-averaged properties. Another important application of porous media concepts in engineering is in the optimization of the internal spacings of heat exchangers subjected to overall volume constraints. Packages of electronics cooled by forced convection are examples of heat exchangers that must function in fixed volumes. The design objective is to install as many components (i.e., heat generation rate) as possible, while the maximum temperature that occurs at a point (hot spot) inside the given volume does not exceed a specified limit. A very basic trade-off exists with respect to the number of components installed (Bejan and Sciubba, 1992) regarding the size of the pores through which the coolant flows. This trade-off is evident when the two extremes are imagined, these extremes being numerous components (small pores) and few components (large spacings). When the components and pores are numerous and small, the package functions as a heat-generating porous medium. When the installed heat generation rate is fixed, the hot-spot temperature increases as the spacings become smaller, because in this limit the coolant flow is being shut off gradually. In the opposite limit, the hot- spot temperature increases again because the heat transfer contact area decreases as the component size and spacing become larger. At the intersection of these two . Mass transfer counter- parts to these heat transfer formulas are obtained through the notation transformation BOOKCOMP, Inc. — John Wiley & Sons / Page 1142 / 2nd Proofs / Heat Transfer Handbook. k f . The relative heat transfer augmentation effect is indicated approximately by the ratio BOOKCOMP, Inc. — John Wiley & Sons / Page 1145 / 2nd Proofs / Heat Transfer Handbook / Bejan FORCED. The total heat transfer rate through the wall is q = q L. Related boundary layer solutions are reviewed in Nield and Bejan (1999). The local Nusselt number for boundarylayerheattransfer near