BOOKCOMP, Inc. — John Wiley & Sons / Page 412 / 2nd Proofs / Heat Transfer Handbook / Bejan 412 FORCED CONVECTION: INTERNAL FLOWS 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 [412], (18) Lines: 688 to 732 ——— 3.9784pt PgVar ——— Normal Page * PgEnds: Eject [412], (18) Nu 0−x = 2.236x −1/3 ∗ x ∗ ≤ 0.001 2.236x −1/3 ∗ + 0.90 0.001 <x ∗ ≤ 0.01 8.235 + 0.0364/x ∗ x ∗ ≥ 0.01 (5.40) The Nusselt number definitions are Nu x = q D h k [ T 0 (x) − T m (x) ] Nu 0−x = q D h k ∆T avg where ∆T avg is furnished by eq. (5.36). It is worth repeating that x ∗ represents the dimensionless longitudinal coordinate in the thermal entrance region, eq. (5.31), which in the case of the parallel-plate channel becomes x ∗ = x/D h Re Dh · Pr All of the results compiled in this section hold in the limit Pr →∞. 5.4.3 Thermally and Hydraulically Developing Flow When the Prandtl number is not much greater than 1, especially when X T and X are comparable, the temperature and velocity profiles develop together, at the same longitudinal location x from the duct entrance. This is the most general case, and heat transfer results for many duct geometries have been developed numerically by a number of investigators. Their work is reviewed by Shah and London (1978). Here, a few leading examples of analytical correlations of the numerical data are shown. Figure 5.7 shows a sample of the finite-Pr data available for a round tube with an isothermal wall. The recommended analytical expressions for the local (Shah and Bhatti, 1987) and overall (Stephan, 1959) Nusselt numbers in the range 0.1 ≤ Pr ≤ 1000 in parallel-plate channels are Nu x = 7.55 + 0.024x −1.14 ∗ 0.0179Pr 0.17 x −0.64 ∗ − 0.14 1 + 0.0358Pr 0.17 x −0.64 ∗ 2 (5.41) Nu 0−x = 7.55 + 0.024x −1.14 ∗ 1 + 0.0358Pr 0.17 x −0.64 ∗ (5.42) The pressure drop over the hydrodynamically developing length x,or∆P = P(0) − P(x), can be calculated using (Shah and London, 1978) ∆P 1 2 ρU 2 = 13.74(x + ) 1/2 + 1.25 + 64x + − 13.74(x + ) 1/2 1 + 0.00021(x + ) −2 (5.43) BOOKCOMP, Inc. — John Wiley & Sons / Page 413 / 2nd Proofs / Heat Transfer Handbook / Bejan OPTIMAL CHANNEL SIZES FOR LAMINAR FLOW 413 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 [413], (19) Lines: 732 to 792 ——— 1.0923pt PgVar ——— Normal Page PgEnds: T E X [413], (19) On the right-hand side, x + is the dimensionless coordinate for the hydrodynamic entrance region, x + = x/D Re D (5.44) which also appears on the abscissa of Fig. 5.3. Note the difference beween X + and x ∗ . Equation (5.43) can be used instead of the (C f ) 0−x ·Re D curve shown in Fig. 5.3 by noting the force balance ∆P πD 2 4 = τ 0−x πDx or ∆P 1 2 ρU 2 = 4x + C f 0−x · Re D (5.45) Figures 5.8 and 5.9 show, respectively, several finite-Pr solutions for the local Nusselt number in the entrance region of a tube with uniform heat flux. A closed-form expression that covers both the entrance and fully developed regions was developed by Churchill and Ozoe (1973): Nu x 4.364 1 + (Gz/29.6) 2 1/6 = 1 + Gz/19.04 1 + (Pr/0.0207) 2/3 1/2 1 + (Gz/29.6) 2 1/3 3/2 1/3 (5.46) where Nu x = q D k [ T 0 (x) − T m (x) ] The Graetz number is defined as Gz = π 4x ∗ = π 4 Re D · Pr x/D Equation (5.46) agrees within 5% with numerical data for Pr = 0.7 and Pr = 10 and has the correct asymptotic behavior for large and small Gz and Pr. 5.5 OPTIMAL CHANNEL SIZES FOR LAMINAR FLOW The geometry of the boundary layers in the entrance region of a duct (Fig. 5.6) brings with it a fundamental property of great importance in numerous and diverse heat BOOKCOMP, Inc. — John Wiley & Sons / Page 414 / 2nd Proofs / Heat Transfer Handbook / Bejan 414 FORCED CONVECTION: INTERNAL FLOWS 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 [414], (20) Lines: 792 to 800 ——— 0.927pt PgVar ——— Normal Page PgEnds: T E X [414], (20) transfer applications. When the objective is to use many channels in parallel, for the purpose of cooling or heating a specified volume that is penetrated by these channels, there is an optimal channel geometry such that the heat transfer rate integrated over the volume is maximal. Alternatively, because the volume is fixed, use of a bundle of channels of optimal sizes means that the heat transfer density (the average heat transfer per unit volume) is maximal. In brief, the optimal channel size (D, in Fig. 5.10 must be such that the thermal boundary layers merge just as the streams leave the channels. The streamwise length of the given volume (L) matches the scale of X T . In this case most of the space oc- cupied by fluid is busy transferring heat between the solid walls and the fluid spaces. This geometric optimization principle is widely applicable, as this brief review shows. The reason is that every duct begins with an entrance region, and from the constructal point of view of deducing flow architecture by maximizing global performance sub- ject to global constaints (such as the volume) (Bejan, 2000), the most useful portion of the duct is its entrance length. This principle was first stated for bundles of vertical channels with natural convection (Bejan, 1984; Bar-Cohen and Rohsenow, 1984). It was extended to bundles of channels with forced convection by Bejan and Sciubba (1992). It continues to generate an expanding class of geometric results, which was Figure 5.10 Two-dimensional volume that generates heat andis cooled by forced convection. (From Bejan, 2000.) BOOKCOMP, Inc. — John Wiley & Sons / Page 415 / 2nd Proofs / Heat Transfer Handbook / Bejan OPTIMAL CHANNEL SIZES FOR LAMINAR FLOW 415 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 [415], (21) Lines: 800 to 831 ——— 0.3511pt PgVar ——— Normal Page PgEnds: T E X [415], (21) reviewed in Kim and Lee (1996) and Bejan (2000). In this section some of the forced convection results are reviewed. The opportunity for optimizing internal channel geometry becomes evident if Fig. 5.10 is regarded as a single-stream heat exchanger intended for cooling electronics. The global thermal conductance of the electronics is the ratio between the total rate of heat generation in the package (q) and the maximum excess temperature registered in the host spots (T max − T 0 ). The entrance temperature of the coolant is T 0 . Designs with more components and circuitry installed in a given volume are desirable as is a larger q. This can be accommodated by increasing the ceiling temperature T max (usually limited by the design of electronics) and by increasing the conductance ratio q/(T max − T min ). The latter puts the design on a course of geometry optimization, because the conductance is dictated by the flow geometry. The degree of freedom in the design is the channel size D, on the number of channels, n = H/D. The existence of an optimal geometry is discovered by designing the stack of Fig. 5.10 in its two extremes: a few large spacings, and many small spacings. When D is large, the total heat transfer surface is small, the global thermal resistance is high, and consequently, each heat-generating surface is overheated. When D is small, the coolant cannot flow through the package. The imposed heat generation rate can be removed only by allowing the entire volume to overheat. An optimal D exists in between and is located by intersecting the large D and small D asymptotes. For the two-dimensional parallel-plate stack of Fig. 5.10, where Pr ≥ 1 and the pressure differnce ∆P is fixed, the optimal spacing is (Bejan and Sciubba, 1992) D opt L 2.7Be −1/4 (5.47) where Be = ∆PL 2 /µα is the specified pressure drop number, which Bhattacharjee and Grosshandler (1988) and Petrescu (1994) have termed the Bejan number. Equation (5.47) underestimates (by only 12%) the value obtained by optimizing the stack geometry numerically. Fur- thermore, eq. (5.47) is robust, because it holds for both uniform-flux and isothermal plates. Equation (5.47) holds even when the plate thickness is not negligible relative to the plate-to-plate spacing (Mereu et al., 1993). The maximized thermal conductance q max /(T max − T 0 ), or maximized heat transfer rate per unit volume that corresponds to eq. (5.47), is q max HLW 0.60 k L 2 (T max − T 0 )Be 1/2 (5.48) where W is the volume dimension in the direction perpendicular to the plane of Fig. 5.10. Equation (5.48) overestimates (by 20%) the heat transfer density maximized numerically (Bejan and Sciubba, 1992). The pressure drop number Be = ∆PL 2 /µα is important, because in the field of internal forced convection it plays the same role that the Rayleigh number plays BOOKCOMP, Inc. — John Wiley & Sons / Page 416 / 2nd Proofs / Heat Transfer Handbook / Bejan 416 FORCED CONVECTION: INTERNAL FLOWS 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 [416], (22) Lines: 831 to 874 ——— 4.87715pt PgVar ——— Normal Page PgEnds: T E X [416], (22) for internal natural convection (Petrescu, 1994). Specifically, the equivalent of eq. (5.47) for natural convection through a H -tall stack of vertical D-wide channels with aPr≥ 1 fluid is D opt H 2.3Ra −1/4 (5.49) where the Rayleigh number is specified (Bejan, 1984): Ra = gβH 3 (T max − T 0 ) αν Equations (5.47) and (5.49) make the Be–Ra analogy evident. The Rayleigh number is the dimensionless group that indicates the slenderness of vertical thermal boundary layers in laminar natural convection, Ra ≈ H δ T 4 where H is the wall height and δ T is the thermal boundary layer thickness. Campo and Li (1996) considered the related problem where the parallel-plate channels are heated asymmetrically: for example, with adiabatic or nearly adiabatic regions. Campo (1999) used the intersection of asymptotes method in the optimiza- tion of the stack of Fig. 5.10, where the plates are with uniform heat flux. His results confirmed the correctness and robustness of the initial result of eq. (5.47). The geo- metric optimization of round tubes with steady and periodic flows and Pr 1 fluids was performed by Rocha and Bejan (2001). Related studies are reviewed in Kim and Lee (1996). The optimal internal spacings belong to the specified volume as a whole, with its purpose and constraints, not to the individual channel. The robustness of this conclusion becomes clear when we look at other elemental shapes for which optimal spacings have been determined. A volume heated by an array of staggered plates in forced convection (Fig. 5.11a) is characterized by an internal spacing D that scales with the swept length of the volume (Fowler et al., 1997): D opt L 5.4Pr −1/4 Re L L b −1/2 (5.50) In this relation the Reynolds number is Re L = U ∞ L/ν. The range in which this cor- relation was developed based on numerical simulations and laboratory experiments is Pr = 0.72, 10 2 ≤ Re L ≤ 10 4 , and 0.5 ≤ Nb/L ≤ 1.3, where N is the number of plate surfaces that face one elemental channel; that is, N = 4 in Fig. 5.11a. Similarly, when the elements are cylinders in crossflow as in Fig. 5.11b, the opti- mal spacing S is influenced the most by the longitudinal dimension of the volume. The optimal spacing was determined based on the method of intersecting the asymptotes BOOKCOMP, Inc. — John Wiley & Sons / Page 417 / 2nd Proofs / Heat Transfer Handbook / Bejan OPTIMAL CHANNEL SIZES FOR LAMINAR FLOW 417 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 [417], (23) Lines: 874 to 876 ——— 1.097pt PgVar ——— Normal Page * PgEnds: Eject [417], (23) Figure 5.11 Forced-convection channels for cooling a heat-generating volume: (a) array of staggered plates; (b) array of horizontal cylinders; (c) square pins with impinging flow. (From Bejan, 2000.) (Stanescu et al., 1996; Bejan, 2000). The asymptotes were derived from the large vol- ume of empirical data accumulated in the literature for single cylinders in crossflow (the large-S limit) and for arrays with many rows of cylinders (the small-S limit). In the range 10 4 ≤ ˜ P ≤ 10 8 , 25 ≤ H/D ≤ 200, and 0.72 ≤ Pr ≤ 50, the optimal spacing is correlated to within 5.6% by BOOKCOMP, Inc. — John Wiley & Sons / Page 418 / 2nd Proofs / Heat Transfer Handbook / Bejan 418 FORCED CONVECTION: INTERNAL FLOWS 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 [418], (24) Lines: 876 to 919 ——— 0.32347pt PgVar ——— Long Page PgEnds: T E X [418], (24) S opt D 1.59 (H/D) 0.52 ˜ P 0.13 · Pr 0.24 (5.51) where ˜ P is an alternative dimensionless pressure drop number based on D: namely, ˜ P = ∆PD 2 /µν. When the free stream velocity U is specified (instead of ∆P ), eq. (5.51) may be transformed by noting that with ∆P approximately equal to ∆P ≈ ρU 2 ∞ /2: S opt D 1.70 (H/D) 0.52 Re 0.26 D · Pr 0.24 (5.52) This correlation is valid in the range 140 ≤ Re D ≤ 14,000, where Re D = UD/ν. The minimized global thermal resistance that corresponds to this optimal spacing is T D − T ∞ qD/kLW 4.5 Re 0.90 D · Pr 0.64 (5.53) where T D is the cylinder temperature and q is the total rate of heat transfer from the HLW volume to the coolant (T ). If the cylinders are arranged such that their centers form equilateral triangles as in Fig. 5.11b, the total number of cylinders present in the bundle is HW/ (S + D) 2 cos 30° . This number and the contact area based on it may be used to deduce from eq. (5.53) the volume-averaged heat transfer coefficient between the array and the stream. Fundamentally, these results show that there always is an optimal spacing for cylin- ders (or tubes) in crossflow heat exchangers when compactness is an objective. This was demonstrated numerically by Matos et al. (2001), who optimized numerically assemblies of staggered round and elliptic tubes in crossflow. Matos et al. (2001) also found that the elliptic tubes perform better relative to round tubes: The heat transfer density is larger by 13%, and the overall flow resistance is smaller by 25%. Optimal spacings emerge also when the flow is three-dimensional, as in an array of pin fins with impinging flow (Fig. 5.11c). The flow is initially aligned with the fins, and later makes a 90° turn to sweep along the base plate and across the fins. The optimal spacings are correlated to within 16% by Ledezma et al. (1996): S opt L 0.81Pr −0.25 · Re −0.32 L (5.54) which is valid in the range 0.06 <D/L≤ 0.14, 0.28 ≤ H/L ≤ 0.56, 0.72 ≤ Pr ≤ 7, 10 ≤ Re D ≤ 700, and 90 ≤ Re L ≤ 6000. Note that the spacing S opt is controlled by the linear dimension of the volume L. The corrseponding minimum global thermal resistance between the array and the coolant is given within 20% by T D − T ∞ q/kH (D/L) 0.31 1.57Re 0.69 L · Pr 0.45 (5.55) The global resistance refers to the entire volume occupied by the array (HL 2 ) and is the ratio between the fin-coolant temperature difference (T D −T ) and the total heat BOOKCOMP, Inc. — John Wiley & Sons / Page 419 / 2nd Proofs / Heat Transfer Handbook / Bejan TURBULENT DUCT FLOW 419 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 [419], (25) Lines: 919 to 952 ——— 0.79907pt PgVar ——— Long Page PgEnds: T E X [419], (25) current q generated by the HL 2 volume. In the square arrangement of Fig. 5.11c the total number of fins is L 2 /(S +D) 2 . Application of the same geometric optimization method to ducts with turbulent flow is reviewed in Section 5.8. 5.6 TURBULENT DUCT FLOW 5.6.1 Time-Averaged Equations The analysis of turbulent duct flow and heat transfer is traditionally presented in terms of time-averaged quantities, which are denoted by a bar superscript. For example, the longitudinal velocity is decomposed as u(r,t)=¯u(r) +u (r,t), where ¯u is the time- averaged velocity and u is the fluctuation, or the time-dependent difference between u and ¯u. In the cylindrical coordinates (r, x) of the round tube shown in Fig. 5.12, the time-averaged equations for the conservation of mass, momentum, and energy are ∂ ¯u ∂x + 1 r ∂ ∂r (r ¯v) = 0 (5.56) ¯u ∂ ¯u ∂x +¯v ∂ ¯u ∂r =− 1 ρ d ¯ P dx + 1 r ∂ ∂r r(ν + M ) ∂ ¯ T ∂r (5.57) ¯u ∂ ¯ T ∂x +¯v ∂ ¯ T ∂r = 1 r ∂ ∂r r(α + H ) ∂ ¯ T ∂r (5.58) These equations have been simplified based on the observation that the duct is a slen- der flow region; the absence of second derivatives in the longitudinal direction may be noted. Note also that d ¯ P/dx, which means that ¯ P(r,x) ¯ P(x). The momentum eddy diffusivity M and the thermal eddy diffusivity H are defined by − ρu v = ρ M ∂ ¯u ∂r and − ρc p v T = ρc p H ∂ ¯ T ∂r (5.59) where u ,v , and T are the fluctuating parts of the longitudinal velocity, radial veloc- ity, and local temperature. The eddy diffusivities augment significantly the transport Figure 5.12 Distribution of apparent shear stress in fully developed turbulent flow. BOOKCOMP, Inc. — John Wiley & Sons / Page 420 / 2nd Proofs / Heat Transfer Handbook / Bejan 420 FORCED CONVECTION: INTERNAL FLOWS 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 [420], (26) Lines: 952 to 1002 ——— 3.76613pt PgVar ——— Normal Page * PgEnds: Eject [420], (26) effect that would occur in the presence of molecular diffusion alone, that is, based on ν for momentum and α for thermal diffusion. 5.6.2 Fully Developed Flow The entrance region for the development of the longitudinal velocity profile and the temperature profile is about 10 times the tube diameter, X D 10 X T D (5.60) This criterion is particularly valid for fluids with Pr values of order 1 (air and water). For ducts with other cross sections, D is the smaller dimension of the cross section. The lengths X and X T are considerably shorter than their laminar-flow counterparts when Re D ≥ 2000. Downstream of x = (X,X T ) the flow is fully developed, and ¯v = 0 and ∂ ¯u/∂x = 0. The left side of eq. (5.57) is zero. For the quantity in brackets, the apparent (or total) shear stress notation is employed: τ app = ρ(ν + M ) ∂ ¯u ∂y (5.61) where y is measured away from the wall, y = r 0 −r, as indicated in Fig. 5.12. The two contributions to τ app , ρv∂ ¯u/∂y and ρ M ∂ ¯u/∂y, are the molecular shear stress and the eddy shear stress, respectively. Note that τ app = 0aty = 0. The momentum equation, eq. (5.57), reduces to d ¯ P dx = 1 r ∂ ∂r (rτ app ) (5.62) where both sides of the equation equal a constant. By integrating eq. (5.62) from the wall to the distance y in the fluid, and by using the force balance of eq. (5.12), one can show that τ app decreases linearly from τ 0 at the wall to zero on the centerline, τ app = τ 0 1 − y r 0 Sufficiently close to the wall, where y r 0 , the apparent shear stress is nearly constant, τ app 0. The mixing-length analysis that produced the law of the wall for the turbulent boundary layer applies near the tube wall. Measurements confirm that the time-averaged velocity profile fits the law of the wall, u + = 2.5lny + + 5.5 (5.63) where 2.5 and 5.5 are curve-fitting constants, and u + = ¯u u ∗ y + = u ∗ y ν u ∗ = τ 0 ρ 1/2 (5.64) BOOKCOMP, Inc. — John Wiley & Sons / Page 421 / 2nd Proofs / Heat Transfer Handbook / Bejan TURBULENT DUCT FLOW 421 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 [421], (27) Lines: 1002 to 1050 ——— 0.0161pt PgVar ——— Normal Page PgEnds: T E X [421], (27) The group u ∗ is known as the friction velocity. The major drawback of the τ app approximation is that the velocity profile deduced from it, eq. (5.63), has a finite slope at the centerline. An empirical profile that has zero slope at the centerline and matches eq. (5.64) as y + → 0 is that of Reichardt (1951): u + = 2.5ln 3(1 + r/r 0 ) 2 1 + 2(r/r 0 ) 2 y + + 5.5 (5.65) The friction factor is defined by eq. (5.11) and is related to the friction velocity (τ 0 /ρ) 1/2 : τ 0 ρ 1/2 = U f 2 1/2 (5.66) An analysis based on a velocity curve fit where, instead of eq. (5.63), u + is propor- tional to (y + ) 1/7 (Prandtl, 1969) leads to f 0.078Re −1/4 D (5.67) where Re D = UD/ν and D = 2r 0 . Equation (5.67) agrees with measurements up to Re D = 8 × 10 4 . An empirical relation that holds at higher Reynolds numbers in smooth tubes (Fig. 5.13) is f 0.046Re −1/5 D 2 × 10 4 < Re D < 10 6 (5.68) An alternative that has wider applicability is obtained by using the law of the wall, eq. (5.63), instead of Prandtl’s 1 7 power law, u + ∼ (y + ) 1/7 . The result is (Prandtl, 1969) 1 f 1/2 = 1.737 ln Re D f 1/2 − 0.396 (5.69) which agrees with measurements for Re D values up to O(10 6 ). The heat transfer literature refers to eq. (5.69) as the K ´ arm ´ an–Nikuradse relation (Kays and Perkins, 1973); this relation is displayed as the lowest curve in Fig. 5.13. This figure is known as the Moody chart (Moody, 1944). The laminar flow line in Fig. 5.13 is for a round tube. The figure shows that the friction factor in turbulent flow is considerably greater than that in laminar flow in the hypothetical case that the laminar regime can exist at such large Reynolds numbers. For fully developed flow through ducts with cross sections other than round, the K ´ arm ´ an–Nikuradse relation of eq. (5.69) still holds if Re D is replaced by the Reynolds number based on hydraulic diameter, Re D h . Note that for a duct of noncircular cross section, the time-averaged τ 0 is not uniform around the periphery of the cross section; hence, in the friction factor definition of eq. (5.11), τ 0 is the perimeter-averaged wall shear stress. . / Heat Transfer Handbook / Bejan 412 FORCED CONVECTION: INTERNAL FLOWS 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 [412],. 13.74(x + ) 1/2 + 1.25 + 64x + − 13.74(x + ) 1/2 1 + 0.00021(x + ) −2 (5 .43) BOOKCOMP, Inc. — John Wiley & Sons / Page 413 / 2nd Proofs / Heat Transfer Handbook / Bejan OPTIMAL CHANNEL SIZES FOR LAMINAR FLOW 413 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 [413],. property of great importance in numerous and diverse heat BOOKCOMP, Inc. — John Wiley & Sons / Page 414 / 2nd Proofs / Heat Transfer Handbook / Bejan 414 FORCED CONVECTION: INTERNAL FLOWS 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 [414],