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BOOKCOMP, Inc. — John Wiley & Sons / Page 462 / 2nd Proofs / Heat Transfer Handbook / Bejan 462 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 [462], (24) Lines: 1085 to 1137 ——— 8.63919pt PgVar ——— Long Page PgEnds: T E X [462], (24) C 2 =−C 1 = 2c p (T 0 − T AW ) U 2 The solution for θ AW (η B ) is obtained numerically by solving eq. (6.51). From this θ AW (0) ≡ r c = b(Pr) Pr 1/2 (for gases) where r c is called the recovery factor. Using this, the adiabatic wall temperature is calculated as T AW = T ∞ + r c U 2 2c p For low velocities this can be approximated as T AW = T ∞ The temperature profile for various choices of T 0 is shown in Fig. 6.9 from Gebhart (1971). The heat flux can be written as q 0 =−k ∂T ∂y y=0 =−k U 2 2c p U νx 1/2 [θ AW (0) + C 1 φ (0)] = k(T 0 − T AW ) 0.332 U νx 1/2 · Pr 1/3 (6.54) Figure 6.9 Boundary layer temperature profiles with viscous dissipation. (From Gebhart, 1971.) BOOKCOMP, Inc. — John Wiley & Sons / Page 463 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW 463 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 [463], (25) Lines: 1137 to 1175 ——— 0.47308pt PgVar ——— Long Page * PgEnds: Eject [463], (25) and with the definition q 0 = h x (T 0 − T AW ) the Nusselt number becomes the well-known Pohlhausen (1921) solution: Nu = 0.332Re 1/2 x · Pr 1/3 where the properties can be evaluated at the reference temperature recommended by Eckert: T ∗ = T ∞ + (T W − T ∞ ) + 0.22(T AW − T ∞ ) 6.4.7 Integral Solutions for a Flat Plate Boundary Layer with Unheated Starting Length Consider the configuration illustrated in Fig. 6.10. The solution for this configuration can be used as a building block for an arbitrarily varying surface temperature where a similarity solution does not exist. Assuming steady flow at constant properties and no viscous dissipation, the boundary layer momentum and energy equations can be integrated across the respective boundary layers to yield d dx δ 0 u(U − u) dy = ν ∂u ∂y y=0 (6.55) d dx δ T 0 u(T ∞ − T)dy = α ∂T ∂y y=0 (6.56) To integrate eqs. (6.55) and (6.56), approximate profiles for tangential velocity and temperature across the boundary layer must be defined. For example, a cubic parabola profile of the type T = a +by + cy 2 + dy 3 can be employed, with the conditions UT, ϱ ␦ ␦ T x T 0 x 0 Figure 6.10 Hydrodynamic and thermal boundary layer development along a flat plate with an unheated starting length in a uniform stream. BOOKCOMP, Inc. — John Wiley & Sons / Page 464 / 2nd Proofs / Heat Transfer Handbook / Bejan 464 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 [464], (26) Lines: 1175 to 1233 ——— 1.94324pt PgVar ——— Long Page * PgEnds: Eject [464], (26) y(T = T 0 ) = 0 (6.57a) y(T = T ∞ ) = δ T (6.57b) ∂T ∂y T =T ∞ = 0 (6.57c) By evaluating the energy equation at the surface, an additional condition can be developed: ∂ 2 T ∂y 2 y=0 = 0 Using these boundary conditions, the temperature profile can be determined as T − T 0 T ∞ − T 0 = 3 2 y δ T − 1 2 y δ T 3 (6.58) Polynomials of higher order can be selected, with the additional conditions deter- mined by prescribing additional higher-order derivatives set to zero at the edge of the boundary layer. In a similar manner, a cubic velocity profile can be determined by prescribing u(y = 0) = v(y = 0) = 0 u(y = δ) = u ∞ ∂u ∂y y=δ = 0 and determining ∂ 2 u ∂y 2 y=0 = 0 This results in u U = 3 2 y δ − 1 2 y δ 3 (6.59) These profiles are next substituted into the integral energy equation and the inte- gration carried out. Assuming that Pr > 1, the upper limit needs only to be extended to y = δ T , because beyond this, θ = T 0 − T = T − T ∞ = θ ∞ and the integrand is zero. In addition, defining, r = δ T /δ, it noted that δ T 0 (θ ∞ − θ)u dy = θ ∞ Uδ 3 20 r 2 − 3 280 r 4 (6.60) BOOKCOMP, Inc. — John Wiley & Sons / Page 465 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW 465 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 [465], (27) Lines: 1233 to 1299 ——— -0.20381pt PgVar ——— Long Page * PgEnds: Eject [465], (27) Because r<1, the second term may be neglected in comparison to the first. Now r is a function of x and the integral of eq. (6.60) may be put into the integral energy equation of eq. (6.56), yielding 2r 2 δ 2 dr dx + r 3 δ dδ dx = 10α U (6.61) From the integral momentum equation, δ(x) can be determined as δ(x) = 4.64 νx U 1/2 This yields r 3 + 4r 2 x dr dx = 13 14Pr which can be solved to yield r = 1 1.026Pr 1/3 1 − x 0 x 3/4 1/3 where x 0 is the unheated starting length. The heat transfer coefficient and the Nusselt number are then h = −k(∂T/∂y)| y=0 T 0 − T ∞ = k θ ∞ ∂θ ∂y y=0 = 3 2 k δ T = 3 2 k rδ or h = 0.332k Pr 1/3 [1 − (x 0 /x) 3/4 ] 1/3 U νx 1/2 (6.62) and Nu = 0.332Pr 1/2 · Re 1/2 x [1 − (x 0 /x) 3/4 ] 1/3 (6.63) Arbitrarily Varying Surface Temperature The foregoing results can easily be generalized for any surface temperature variation of the type T 0 = T ∞ + A + ∞ n=1 B n x n (6.64) For a single step, T 0 − T T 0 − T ∞ = θ(x 0 ,x,y) BOOKCOMP, Inc. — John Wiley & Sons / Page 466 / 2nd Proofs / Heat Transfer Handbook / Bejan 466 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 [466], (28) Lines: 1299 to 1339 ——— -0.54802pt PgVar ——— Normal Page * PgEnds: Eject [466], (28) and for an arbitrary variation, T 0 − T ∞ = x 0 [1 − θ(x 0 ,x,y)] ∂T ∂x 0 dx 0 + k i=1 [1 − θ(x 0 i ,x,y)] ∆T 0,i (6.65) q 0 = k x 0 ∂θ(x 0 ,x,0) ∂y dT 0 dx 0 dx 0 + k i=1 ∂θ(x 0 i ,x,0) ∂y ∆T 0,i (6.66) This procedure using the integral momentum equation can also be generalized to a turbulent boundary layer with an arbitrary surface temperature variation. 6.4.8 Two-Dimensional Nonsimilar Flows When similarity conditions do not apply, as in the case of an unheated starting length plate, two classes of approaches exist for solution of the governing equations. The integral method results in an ordinary differential equation with the downstream coordinate x as the independent variable and parameters associated with the body profile shape and the various boundary layer thicknesses as the dependent variable. Generally, such solutions result in correlation relationships that have a limited range of applicability. These are typically much faster to compute. Differential methods solve the partial differential equations describing numerically the conservation of mass, force momentum balance, and conservation of energy. These equations are discretized over a number of control volumes, resulting in a set of algebraic equations that are solved simultaneously using numerical techniques. These solutions provide the detailed velocity, pressure, temperature, and densityfields for compressible flows. The heat transfer rates from various surfaces can also be determined from these results. In the 1980s and 1990s, the computational hardware capabilities expanded dramatically and the differential methods have become the most commonly used methods for various complex and realistic geometries. 6.4.9 Smith–Spalding Integral Method The heat transfer in a constant-property laminar boundary layer with variable veloc- ity U(x) but uniform surface temperature can be obtained via the Smith–Spalding integral method (1958), as described by Cebeci and Bradshaw (1984). A conduction thickness is defined as δ c = k(T 0 − T ∞ ) q 0 =− T 0 − T ∞ (∂T /∂y) y=0 (6.67) This is expressed in nondimensional form as U(x) ν dδ 2 c dx = f δ 2 c ν dU dx , Pr = A − B δ 2 c ν dU dx (6.68) BOOKCOMP, Inc. — John Wiley & Sons / Page 467 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW 467 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 [467], (29) Lines: 1339 to 1418 ——— 1.08134pt PgVar ——— Normal Page PgEnds: T E X [467], (29) where A and B are Prandtl number–dependent constants. For similar laminar flows, the Nusselt number is given as Nu = q x k(T 0 − T ∞ ) =− x(∂T/∂y) 0 T 0 − T ∞ = f(Pr,m,0)Re 1/2 x (6.69) and using the definition of the conduction thickness yields δ 2 c = νx C 2 U (6.70) where C = f(Pr,m,0). The parameters in eq. (6.68) can be expressed as U(x) ν dδ 2 c dx = 1 − m C 2 (6.71a) δ 2 c ν dU dx = m C 2 (6.71b) Equation (6.69) is a first-order ordinary differential equation that can be integrated as δ 2 c = νA x 0 U B−1 dx U B + δ 2 c U B i U B where the subscript i denotes initial conditions. The normalized heat transfer coeffi- cient in the form of a local Stanton number (St = Nu/Re x · Pr) is St = q ρc p U(T 0 − T ∞ ) = k ρc p Uδ = c 1 (U ∗ ) c 2 x ∗ 0 (U ∗ ) c 3 dx ∗ 1/2 1 Re L 1/2 (6.72) Here c 1 = Pr −1 · A −1/2 ,c 2 = B 2 ,c 3 = B − 1 and provided in Table 6.1 and U ∗ = U(x) U ∞ x ∗ = x L Re L = U ∞ L ν where U ∞ is a reference velocity, typically the uniform upstream velocity, and L is a length scale characteristic of the object. As an example of the use of the Smith–Spalding approach, consider the heat transfer in cross flow past a cylinder of radius r 0 heated at a uniform temperature. The velocity distribution outside the boundary layer is given by BOOKCOMP, Inc. — John Wiley & Sons / Page 468 / 2nd Proofs / Heat Transfer Handbook / Bejan 468 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 [468], (30) Lines: 1418 to 1436 ——— 1.97408pt PgVar ——— Short Page PgEnds: T E X [468], (30) TABLE 6.1 Parameters c 1 ,c 2 ,andc 3 as Functions of the Prandtl Number Pr c 1 c 2 c 3 0.70 0.418 0.435 1.87 1.00 0.332 0.475 1.95 5.00 0.117 0.595 2.19 10.0 0.073 0.685 2.37 Source: Cebeci and Bradshaw (1984). U(x) = 2U ∞ sin θ = 2U ∞ sin x r 0 (6.73) where x is measured around the circumference, beginning with the front stagnation point. For small values of θ, sin θ ≈ θ, and the free stream velocity approaches that for a stagnation point where similarity exists. The computed results for Nu x /Re 1/2 x are shown in Fig. 6.11 for three values of Pr. 0 0.2 0.4 0.6 0.8 1 1.2 Pr = 0.7 Pr=5 Pr=10 0 20406080100120 180 (deg) x r 0 Nu Re x x ͌ Figure 6.11 Local heat transfer results for crossflow past a heated circular cylinder atuniform temperature for various Pr values, using the integral method. (From Cebeci and Bradshaw, 1984.) BOOKCOMP, Inc. — John Wiley & Sons / Page 469 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW 469 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 [469], (31) Lines: 1436 to 1486 ——— 1.81322pt PgVar ——— Short Page PgEnds: T E X [469], (31) 6.4.10 Axisymmetric Nonsimilar Flows The Smith–Spalding method (1958) can be extended to axisymmetric flows by using the Mangler transformation. If the two-dimensional variables are denoted by the sub- script 2 and the axisymmetric variables by the subscript 3 and neglecting transverse curvature, then dx 2 = r 0 L 2K dx 3 θ 2 = r 0 L K θ 3 (6.74) where K is the flow index used in the Mangler transformation to relate the axisym- metric coordinates x and θ to two-dimensional coordinates and r 2 0 (δ c ) 2 3 = νA x 3 0 U B−1 dx 3 U B (6.75) where r 0 is the distance from the axis to the surface (see Fig. 6.13). The Stanton number is given in nondimensional form by St = c 1 (r ∗ 0 ) K (U ∗ ) c 2 x ∗ 3 0 (U ∗ ) c 3 (r ∗ 0 ) 2K dx ∗ 3 1/2 1 Re L 1/2 (6.76) where U ∗ = U(x)/U ∞ ,r ∗ 0 = r 0 /L, and x ∗ 3 = x 3 /L. Here the constants c 1 , c 2 , and c 3 are given in Table 6.1. As an example, heat transfer from a heated sphere of radius a at a uniform tem- perature placed in an undisturbed flow characterized by U ∞ can be calculated by this method. The velocity distribution in the inviscid flow is U = 3 2 U ∞ sin φ = 3 2 U ∞ sin x a (6.77) The variation of Nu x (U ∞ a/ν) −1/2 downstream from the front stagnation point is seen in Fig. 6.12. The result of the similarity solution for the case of axisymmetric stagnation flow is also shown. As described by Cebeci and Bradshaw (1984), this similarity solution requires the transformation of the axisymmetric equations into a nearly two-dimensional form through use of the Mangler transformation. 6.4.11 Heat Transfer in a Turbulent Boundary Layer The two-dimensional boundary layer form of the time-averaged governing equa- tions is ∂ ¯u ∂x + ∂ ¯v ∂y = 0 (6.78) BOOKCOMP, Inc. — John Wiley & Sons / Page 470 / 2nd Proofs / Heat Transfer Handbook / Bejan 470 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 [470], (32) Lines: 1486 to 1512 ——— 2.48503pt PgVar ——— Normal Page PgEnds: T E X [470], (32) ρ ¯u ∂ ¯u ∂x +¯v ∂ ¯v ∂y =− d ¯p dx + µ ∂ 2 ¯u ∂y 2 − ∂ ∂y (ρ u v ) =− d ¯p dx + ∂τ m ∂y − ∂ ∂y (ρ u v ) (6.79) ρc p ¯u ∂ ¯ T ∂x +¯v ∂ ¯ T ∂y = k ∂ 2 ¯ T ∂y 2 − ∂ ∂y (ρc p v T ) =− ∂q m ∂y − ∂ ∂y (ρc p v T ) (6.80) where τ m = µ(∂ ¯u/∂y) and q m =−k(∂ ¯ T/∂y)are the mean shear stress and heat flux, respectively. Axisymmetric Flows Equations (6.78) through (6.80) are derived for two- dimensional flow, with the velocity component as well as all derivatives of time- averaged quantities neglected in the z direction. For axisymmetric flow, such as in a circular jet or within the boundary layer on a body of circular cross section, also called a body of revolution (Fig. 6.13), Cebeci and Bradshaw (1984) show that the governing equations can be generalized: 2 1.6 1.2 0.8 0.4 0 Nu x Ua ϱ Ϫ v 1/2 ( ( Similarity Similarity Pr=1 Pr=10 0 20406080100120 180 (deg) x r 0 Figure 6.12 Local heat transfer behavior for flow past a heated sphere at uniform surface temperature for various Pr values, using the integral method. (From Cebeci and Bradshaw, 1984.) BOOKCOMP, Inc. — John Wiley & Sons / Page 471 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW 471 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 [471], (33) Lines: 1512 to 1539 ——— 1.56715pt PgVar ——— Normal Page * PgEnds: Eject [471], (33) Mainstream Fluid u ϱ x=0 Axis of Symmetry z 90 y ␦ r r o Boundary Layer Figure 6.13 Boundary layer development on a body of revolution. ∂r N ¯u ∂x + ∂r N ¯v ∂y = 0 (6.81) ρ ¯u ∂ ¯u ∂x +¯v ∂ ¯v ∂y =− dp dx + 1 r N ∂ ∂y r N µ ∂ ¯u ∂y − ρ u v (6.82) ρc p ¯u ∂ ¯ T ∂x +¯v ∂ ¯ T ∂y = 1 r N ∂ ∂y k ∂ ¯ T ∂y − ρc p v T (6.83) where N = 1 in axisymmetric flow and N = 0 in two-dimensional flow. The turbulent components of the shear stress and the heat flux are defined as τ T =−ρu v = ρ ∂ ¯u ∂y and q T =−ρc p v T =−ρc p H ∂ ¯ T ∂y where and H are, respectively, the turbulent or eddy diffusivities of momentum and heat. The governing two-dimensional equations with these incorporated become ¯u ∂ ¯u ∂x +¯v ∂ ¯v ∂y =− 1 ρ d ¯p dx + ∂ ∂y (ν + ) ∂ ¯u ∂y (6.84) and ¯u ∂ ¯ T ∂x +¯v ∂ ¯ T ∂y = ∂ ∂y (α + H ) ∂ ¯ T ∂y = ν ∂ ∂y 1 Pr + ν 1 Pr T ∂ ¯ T ∂y (6.85) where Pr T = / H is the turbulent Prandtl number. Solution of (6.84) and (6.85) requires modeling of the turbulent shear stress and heat flux. Analogy Solutions The total shear stress and heat flux are written as a combina- tion of the mean and turbulent components: . Gebhart, 1971.) BOOKCOMP, Inc. — John Wiley & Sons / Page 463 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW 463 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 [463],. θ ∞ Uδ 3 20 r 2 − 3 280 r 4 (6.60) BOOKCOMP, Inc. — John Wiley & Sons / Page 465 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW 465 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 [465],. B δ 2 c ν dU dx (6.68) BOOKCOMP, Inc. — John Wiley & Sons / Page 467 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW 467 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 [467],
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