BOOKCOMP, Inc. — John Wiley & Sons / Page 442 / 2nd Proofs / Heat Transfer Handbook / Bejan 442 FORCED CONVECTION: EXTERNAL 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 [442], (4) Lines: 167 to 190 ——— 0.097pt PgVar ——— Normal Page PgEnds: T E X [442], (4) ( ) Flat platea ( ) Rectangular blockc ( ) Cylinderb ( ) Sphered l l h t w w l d d Figure 6.2 Basic configurations of objects. A flat plate of large width (w ) and small thickness (t ) is often used to introduce the concept of boundary layer flow and heat transfer. An infinitely long cylinder ( d) in cross flow is also a geometrically simple object, but the flow develops complexity at high velocities due to separation on the curved surface. Flow passing an infinitely long rectangular body (w h, w ) also involves complexities caused by flow separation at the sharp corners. A sphere in uniform flow is also one of the basic configurations, where axisymmetric flow prevails at low velocities. The orientation of the heated object relative to the fluid flow has a significant influence on heat transfer. Well-studied configurations are parallel flow along a flat plate as in Fig. 6.3a, flows impinging on a plane (Fig. 6.3b), a wedge and a cone (Fig. 6.3c), and the side of a cylinder (Fig. 6.3d). In many studies the impinging flow is assumed to be parallel to the symmetry axis of the object as illustrated by the sketches in Fig. 6.3. The study of impinging heat transfer has as one objective the understanding of heat transfer near the symmetry axis. For the plane and the cylinder, the point on the symmetry axis is called the stagnation point. Figure 6.4 illustrates flows through heated objects placed in regular geometric patterns. These arrays are commonly found in heat exchangers that transmit heat from the external surfaces of tubes or flat plates (strips) to the fluid. In heat exchanger applications, the array of cylinders (Fig. 6.4a and b) is called a tube bank. When the row of objects is deep, the fluid flow develops a repeating pattern after a few rows, that is, a fully developed pattern. The fluid temperature increases toward the end of the row as the fluid absorbs heat from the objects in the upstream rows and the thermal environment for an individual object bears the characteristics of internal flow. The case of a stack of parallel plates (Fig. 6.4c and d) involves an extra feature; that is, the fluid can bypass the plates in the stack. Such a situation is commonly found in the cooling of electronics. BOOKCOMP, Inc. — John Wiley & Sons / Page 443 / 2nd Proofs / Heat Transfer Handbook / Bejan MORPHOLOGY OF EXTERNAL FLOW HEAT TRANSFER 443 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 [443], (5) Lines: 190 to 192 ——— 0.097pt PgVar ——— Normal Page PgEnds: T E X [443], (5) ( ) Parallel flow over a flat platea ( ) Flow impinging on a wedge or a conec ( ) Flow impinging on a plateb ( ) Flow impinging on a cylinderd Figure 6.3 Flow configurations. If the heated object is in contact with the bounding surface, the heat and fluid flow patterns become more complex. Figure 6.5 shows some configurations that are found in industrial equipment, particularly in electronic equipment. A rectangular block on a flat substrate simulates an integrated-circuit (IC) package on a printed wiring board (PWB) in Fig. 6.5a. If a certain number of rectangular packages are mounted on a PWB, with their longer sides extending laterally and leaving narrow edge-to-edge gaps between the packages on the same row, such a package array is modeled by an array of two-dimensional blocks (Fig. 6.5b). Two-dimensional blocks (frequently referred to as ribs) are also provided on a surface to enhance heat transfer. Longitudinal planar fins on a substrate are commonly used as a heat sink for the IC package (Fig. 6.5c). The pin fin array is a common heat sink device for electronic components (Fig. 6.5d). In most of the examples illustrated in Fig.6.5,theflow and the temperature field are three-dimensional. In addition, flow separation and vortex shedding from the objects are common features in the velocity range of practical interest. The contribution made by the substrate is another complicating factor because heat conduction in the substrate is coupled with surface heat transfer on the object as well as the substrate. This coupled process is called conjugate heat transfer. In the case of heat sinks, the substrate is usually made from the same (highly conductive) material as the fins and is in good thermal contact with the fins. The heat source is bonded to the lower side of the substrate, so that the heat flows from the substrate to the fins. These features BOOKCOMP, Inc. — John Wiley & Sons / Page 444 / 2nd Proofs / Heat Transfer Handbook / Bejan 444 FORCED CONVECTION: EXTERNAL 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 [444], (6) Lines: 192 to 192 ——— * 23.854pt PgVar ——— Normal Page PgEnds: T E X [444], (6) ( ) In-line arrangement of cylindersa ( ) Plate stack in free streamc ( ) Staggered arrangement of cylindersb ( ) Staggered array of plates (offset strips) d Figure 6.4 Arrays of objects. ( ) Rectangular block on substratea ( ) Planar fin array on substrate (plate fin heat sink) c ( ) Two-dimensional block arrayb ( ) Pin array on substrate (pin fin heat sink) d Figure 6.5 Objects on substrates. BOOKCOMP, Inc. — John Wiley & Sons / Page 445 / 2nd Proofs / Heat Transfer Handbook / Bejan ANALYSIS OF EXTERNAL FLOW HEAT TRANSFER 445 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 [445], (7) Lines: 192 to 232 ——— -2.75386pt PgVar ——— Normal Page PgEnds: T E X [445], (7) often justify the assumption of an isothermal condition over the substrate, and as a result, conjugate heat transfer does not play a major role. This section concludes with the following observations: 1. Classical analytical solutions are available for two-dimensional cases where the flow is laminar up to the point of transition to turbulent flow or flow separation. For turbulent flows in simple circumstances, approximate analytical solutions based on phenomenological laws of turbulence kinetics are also available. Results for both laminar and turbulent flows are discussed in Sections 6.4 and 6.6. 2. The advent of computational fluid dynamics (CFD) codes has expanded the possibility of finding detailed features of flow and heat transfer in many complex situ- ations. Developments after flow separation (such as vortex shedding) can be studied in detail and three-dimensional situations can be considered, although the computational resource is finite and the scope of analysis in a narrower zone must be limited when the flow involves more complex features. For processes occurring outside the analysis zone, assumptions or approximations are frequently employed. These assumptions in- troduce inaccuracies in the solution, and experiments are required to justify solutions for a certain number of cases. Such experimental verification is called benchmarking, and only benchmarked CFD codes can be used as product design tools. 3. There are many circumstances where experiments are the only means for ob- taining useful heat transfer data. In many instances empirical relationships still serve as invaluable tools to estimate heat transfer, particularly for the design of industrial equipment. 6.3 ANALYSIS OF EXTERNAL FLOW HEAT TRANSFER For a general three-dimensional flow (Fig. 6.1), the six unknowns at any instant at a given location are the three velocity components and the pressure, temperature, and density. For constant fluid properties, the flow field is not coupled to the tempera- ture field and can be obtained independently. With a known flow field, the energy equation subsequently provides the temperature variation. The determination of flow and temperature fields and gradients permits computation of the local heat transfer coefficient h: h = −k f (∂T /∂n) s T s − T ref (6.1a) and its nondimensional representation, known as the Nusselt number: Nu = hL k f (6.1b) The surface heat flux can be obtained from q = h(T s − T ref ) (6.1c) BOOKCOMP, Inc. — John Wiley & Sons / Page 446 / 2nd Proofs / Heat Transfer Handbook / Bejan 446 FORCED CONVECTION: EXTERNAL 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 [446], (8) Lines: 232 to 268 ——— 3.70003pt PgVar ——— Normal Page PgEnds: T E X [446], (8) Here T s is the local surface temperature and T ref is the fluid reference temperature. For external flow, this is typically the local ambient environment temperature. Experimental Determination of Convection Transport The experimental tech- nique has been employed extensively. For a given geometrical configuration, it in- volves the simulation of thermal loads and flow conditions and the determination of resulting thermal fields. Experimental techniques are often used for validation of numerical simulations of laminar flows in complex geometries and for transport char- acterization for turbulent flow. Analytical Solutions The key nondimensional flow parameter in convection anal- ysis is the Reynolds number, Re, which is defined later. For Re 1, the nonlinear advection terms in the momentum equations can be neglected, and the resulting lin- earized set of equations can then be solved analytically for a limited set of conditions. Numerical Simulations of Transport With the advent of high-speed digital com- puters, it has become possible to carry out numerical simulations with great accuracy for virtually any geometrical configuration in laminar flow. The focus in these solu- tions is on the determination of generalized transport correlations and on the detailed descriptions of the thermal and flow fields. For turbulent and transitional flow, reliance must be placed on experimental data and correlations. 6.4 HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW 6.4.1 High Reynolds Number Flow over a Wedge In the absence of mass transfer, the governing equations in Cartesian coordinates (x,y,z) for constant property, incompressible three-dimensional flow with the ve- locity V = ui + vj + wk account for continuity (mass conservation), conservation of momentum, and conser- vation of energy. The mass conservation equation is ∇·V = 0 (6.2) For the conservation of momentum, the x, y, and z components are ρ Du Dt = ρ ∂u ∂t +∇·(uV) =− ∂p ∂x + µ ∇ 2 u (6.3a) ρ Dv Dt = ρ ∂v ∂t +∇·(vV) =− ∂p ∂y + µ ∇ 2 v (6.3b) BOOKCOMP, Inc. — John Wiley & Sons / Page 447 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW 447 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 [447], (9) Lines: 268 to 313 ——— 5.68127pt PgVar ——— Normal Page * PgEnds: Eject [447], (9) ρ Dw Dt = ρ ∂w ∂t +∇·(wV) =− ∂p ∂z + µ ∇ 2 w (6.3c) The conservation of energy requires ρc p DT Dt = ρc p ∂T ∂t +∇·(T V) = k ∇ 2 T + µΦ + β Dp Dt (6.4) The last two terms on the right-hand side of the energy equation, eq. (6.4), account for viscous dissipation and pressure stress effects, where, as indicated in Chapters 1 and 5, Φ is given by Φ = 2 ∂u ∂x 2 + ∂v ∂y 2 + ∂w ∂z 2 × ∂v ∂x + ∂u ∂y 2 + ∂w ∂y + ∂v ∂z 2 + ∂u ∂z + ∂w ∂x 2 − 2 3 (∇·V) 2 Viscous dissipation effects are important in very viscous fluids or in the presence of large velocity gradients. Two other orthogonal coordinate systems of frequent interest are the cylindrical and spherical coordinate systems. Governing equations for continuity, the force– momentum balance and the conservation of energy are provided in Chapter 1. Equations (6.2)–(6.4) can be normalized using the variables (x ∗ ,y ∗ ,z ∗ ) = (x,y,z) L (6.5a) (u ∗ ,v ∗ ,w ∗ ) = (u, v, w) U (6.5b) t ∗ = t L/U (6.5c) p ∗ = p ρU 2 (6.5d) where L and U are appropriate length and velocity scales, respectively. The normal- ized form of eq. (6.3a) is then ∂u ∗ ∂t ∗ + u ∗ ∂u ∗ ∂x ∗ + v ∗ ∂u ∗ ∂y ∗ + w ∗ ∂u ∗ ∂z ∗ =− ∂p ∗ ∂x ∗ + µ ULρ ∂ 2 u ∗ ∂x ∗2 + ∂ 2 u ∗ ∂y ∗2 + ∂ 2 u ∗ ∂z ∗2 (6.6) BOOKCOMP, Inc. — John Wiley & Sons / Page 448 / 2nd Proofs / Heat Transfer Handbook / Bejan 448 FORCED CONVECTION: EXTERNAL 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 [448], (10) Lines: 313 to 378 ——— 1.50526pt PgVar ——— Long Page * PgEnds: Eject [448], (10) The normalized forms of eqs. (6.3b) and (6.3c) are similar, and for brevity are not shown. Moreover, when all the variables in eq. (6.2) are replaced by their normalized versions, the Reynolds number emerges as the key nondimensional solution parame- ter, relating the inertia forces to the viscous forces: Re = ρU 2 L 2 µ(U/L)L 2 = ρUL µ (6.7) To nondimensionalize the energy equation, a normalized temperature is defined as T ∗ = T − T ∞ T s − T ∞ (6.8) where T ∞ and T s are the local ambient and surface temperatures, respectively. Both temperatures could, in general, vary with location and/or time. The resulting normal- ized energy equation is DT ∗ Dt ∗ = 1 Re · Pr ∇ 2 T ∗ + 2Ec · Pr · Φ + 2βT · Re · Pr · Ec Dp ∗ Dt ∗ (6.9) where β is the coefficient of volumetric thermal expansion and Pr is the Prandtl number: Pr = ν α = c p u k (6.10a) a measure of the ratio of diffusivity of momentum to diffusivity of heat. The Eckert number Ec = V 2 2c p (T s − T ∞ ) (6.10b) which expresses the magnitude of the kinetic energy of the flow relative to the en- thalpy difference. In eq. (6.9), βT = 1 for ideal gases, and typically, βT 1 for liquids. The pressure stress term is thus negligible in forced convection whenever the viscous dissipation is small. The wall values of shear stress and heat flux are, respectively, τ s = µ ∂u ∂y s = µU L ∂u ∗ ∂y ∗ y ∗ =0 (6.11a) and q s =−k ∂T ∂y s = k(T s − T ∞ ) L − ∂T ∗ ∂y ∗ y ∗ =0 (6.11b) From a practical perspective, the friction drag and the heat transfer rate are the most important quantities. These are determined from the friction coefficient, C f : BOOKCOMP, Inc. — John Wiley & Sons / Page 449 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW 449 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 [449], (11) Lines: 378 to 418 ——— 2.23122pt PgVar ——— Long Page PgEnds: T E X [449], (11) C f = 2τ s ρU 2 = 2 Re ∂u ∗ ∂y ∗ y ∗ =0 (6.12a) and the Nusselt number Nu, defined as Nu = q s L k(T s − T ∞ ) =− ∂T ∗ ∂y ∗ y ∗ =0 (6.12b) The boundary layer development for two-dimensional flow in the x- and y-coor- dinate directions over the wedge shown in Fig. 6.6 is based on the governing equations representing continuity, the x and y components of momentum, and the conservation of energy, ∂u ∂x + ∂v ∂y = 0 (6.13) ∂u ∂t + u ∂u ∂x + v ∂u ∂y = X ρ − 1 ρ ∂p ∂x + ν ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 (6.14a) ∂v ∂t + u ∂v ∂x + v ∂v ∂y = Y ρ − 1 ρ ∂p ∂y + ν ∂ 2 v ∂x 2 + ∂ 2 v ∂y 2 (6.14b) ∂T ∂t + u ∂T ∂x + v ∂T ∂y = α ∂ 2 T ∂x 2 + ∂ 2 T ∂y 2 + µ ρc p Φ + βT ρc p Dp Dt + q ρc p (6.15) In eqs. (6.14), X and Y are the body forces in the x and y directions. The body force and the static pressure gradient terms on the right-hand sides of eqs. (6.14) can be combined to yield − ∂p m ∂x = X ρ − 1 ρ ∂p ∂x and − ∂p m ∂y = Y ρ − 1 ρ ∂p ∂y where the motion pressure p m is the difference between the local static and local hydrostatic pressures. On each face of the wedge shown in Fig. 6.6, the x direction is measured along the surface from the point of contact (the leading edge) and the y direction is measured normal to the surface. The free stream velocity, which is designated by U(x) at a Figure 6.6 Boundary layer development for high-Reynolds-number flow over a wedge. BOOKCOMP, Inc. — John Wiley & Sons / Page 450 / 2nd Proofs / Heat Transfer Handbook / Bejan 450 FORCED CONVECTION: EXTERNAL 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 [450], (12) Lines: 418 to 454 ——— 10.15335pt PgVar ——— Normal Page * PgEnds: Eject [450], (12) given x, remains unchanged for large values of y. The velocity varies from zero at the surface [u(y = 0) = 0] to the free stream velocity within a fluid layer, the thickness of which increases with x. The region close to the surface where the velocity approaches a value close to the local free stream level defines the hydrodynamic boundary layer, which has the thickness δ(x). When heat transfer occurs, the temperature at the surface, T(y = 0) = T s , is not equal to the free stream temperature T ∞ , and a thermal boundary layer also exists. In the thermal boundary, the temperature adjusts from the wall value to near the free stream level. The thermal boundary layer thickness δ T (x) characterizes the thermal boundary layer. When R 1, thin hydrodynamic and thermal boundary layers of thickness δ(x) and δ T (x), respectively, develop along an object in general and on the surfaces of the wedge in Fig. 6.6 in particular. Under this condition, δ/L 1 and δ T /L 1 and the following scaled variables can be defined: t ∗ = t t 0 (6.16a) x ∗ = x L (6.16b) y ∗ = y δ (6.16c) u ∗ = u U(x) (6.16d) v ∗ = v V s (6.16e) p ∗ m = p m − p ∞ ρU 2 (6.16f) T ∗ = T − T ∞ T s − T ∞ (6.16g) y ∗ T = y δ T (6.16h) where V s is the fluid velocity normal to the surface imposed by induced or forced fluid motion, U(x)the velocity field in the potential flow region outside the boundary layers, t 0 a time reference, and T ∞ the temperature of the environment. The corresponding continuity equation is ∂u ∗ ∂x ∗ + V s L Uδ ∂v ∗ ∂y ∗ = 0 (6.17) and in order for the two terms to be of the same order of magnitude, V s must be of the order of δU/L: BOOKCOMP, Inc. — John Wiley & Sons / Page 451 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW 451 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 [451], (13) Lines: 454 to 509 ——— 6.25131pt PgVar ——— Normal Page * PgEnds: Eject [451], (13) V s = O δU L Then the normalized momentum equations become L Ut 0 ∂u ∗ ∂t ∗ + u ∂u ∗ ∂x ∗ + v ∂u ∗ ∂y ∗ =− ∂p ∗ m ∂x ∗ + νL Uδ 2 δ L 2 ∂ 2 u ∗ ∂x ∗2 + ∂ 2 u ∗ ∂y ∗2 (6.18) and as Re 1, νL Uδ 2 = O(1) so that δ L = O 1 Re 1/2 L 1 and therefore, δ L 2 L Ut o ∂v ∗ ∂t ∗ + u ∗ ∂v ∗ ∂x ∗ + v ∗ ∂v ∗ ∂y ∗ =− ∂p ∗ m ∂y ∗ + 1 Re δ L 2 ∂ 2 v ∗ ∂x ∗2 + ∂ 2 v ∗ ∂y ∗2 (6.19) As Re →∞, eq. (6.19) becomes 0 =− ∂p ∗ m ∂y ∗ + 0 which implies that the pressure inside the boundary layer is a function of x and can be evaluated outside the boundary layer in the potential flow via the solution to the Euler equation − 1 ρ dp m dx = U dU dx The energy equation is L Ut 0 ∂T ∗ ∂t ∗ + u ∗ ∂T ∗ ∂x ∗ + δ δ T v ∗ ∂T ∗ ∂y ∗ T = k ρc p UL ∂ 2 T ∗ ∂x ∗2 + L δ T 2 ∂ 2 T ∗ ∂y ∗2 T + µUL ρc p ∆T δ 2 Φ ∗ + βTUL c p ∆Tt 0 Dp ∗ Dt ∗ + q L ρc p U ∆T (6.20) Here the transient term is important if L/t 0 U = O(1) and steady conditions are approached for L/t 0 U 1. In addition, the pressure stress term vp y up x and the normalized equation resulting from letting Re = O(L 2 /δ 2 ) is . because heat conduction in the substrate is coupled with surface heat transfer on the object as well as the substrate. This coupled process is called conjugate heat transfer. In the case of heat. electronics. BOOKCOMP, Inc. — John Wiley & Sons / Page 443 / 2nd Proofs / Heat Transfer Handbook / Bejan MORPHOLOGY OF EXTERNAL FLOW HEAT TRANSFER 443 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 [443],. substrates. BOOKCOMP, Inc. — John Wiley & Sons / Page 445 / 2nd Proofs / Heat Transfer Handbook / Bejan ANALYSIS OF EXTERNAL FLOW HEAT TRANSFER 445 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 [445],