BOOKCOMP, Inc. — John Wiley & Sons / Page 502 / 2nd Proofs / Heat Transfer Handbook / Bejan 502 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 [502], (64) Lines: 2746 to 2759 ——— 0.927pt PgVar ——— Normal Page PgEnds: T E X [502], (64) The impinging jet may be circular (round) or planar (rectangular or slot), based on its cross section. It may be submerged (fluid discharged in the same ambient medium), or free surface (a liquid discharged into ambient gas). The flows in each of these cases may be unconfined or partially confined. Moreover, in the case of multiple jets, interaction effects arise. 6.7.2 Submerged Jets A schematic of a single submerged circular or planar jet is seen in Fig. 6.29. Typically, the jet is turbulent at the nozzle exit and can be characterized by a nearly uniform axial velocity profile. With increasing distance from the nozzle exit, the potential core region within which the uniform velocity profile persists shrinks as the jet interacts with the ambient. Farther downstream, in the free jet region, the velocity profile is nonuniform across the entire jet cross section. The centerline velocity decreases with distance from the nozzle exit in this region. The effect of the impingement surface is not felt in this region. The impingement surface influences the flow in Figure 6.29 Transport regimes in a submerged circular unconfined jet impinging on a surface. BOOKCOMP, Inc. — John Wiley & Sons / Page 503 / 2nd Proofs / Heat Transfer Handbook / Bejan TURBULENT JETS 503 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 [503], (65) Lines: 2759 to 2822 ——— 1.32532pt PgVar ——— Normal Page * PgEnds: Eject [503], (65) the stagnation or impingement zone. Within this region the flow accelerates in the transverse direction (r or x) and decelerates in the z direction. Farther away in the transverse direction (in the wall jet region), the flow starts to decelerate due to entrainment of the ambient fluid. Average Nusselt Number for Single Jets Martin (1977) provides an exten- sive review of heat transfer data for impinging gas jets. For single nozzles, the average Nusselt number is of the form (Martin, 1977; Incropera and DeWitt, 1996) Nu = f Re, Pr, r D h , H D h (6.188a) or Nu = f Re, Pr, x D h , H D h (6.188b) where Nu = ¯ h D h k (6.189a) and Re = V e D h ν (6.189b) where V e is the uniform exit velocity at the jet nozzle, D h = D for a round nozzle, and D h = 2W for a slot nozzle. Figure 6.29 defines the geometric parameters. For a single round nozzle, Martin (1977) recommends Nu Pr 0.42 = G D r , H D f 1 (Re) (6.190) where f 1 (Re) = 2Re 1/2 (1 + 0.005Re 0.55 ) 1/2 (6.191a) and G = D r 1 − 1.1(D/r) 1 + 0.1[(H/D) − 6](D/r) (6.191b) Replacing D/r by 2A 1/2 r yields G = 2A 1/2 r 1 − 2.20A 1/2 r 1 + 0.20[(H/D) − 6]A 1/2 r (6.192) BOOKCOMP, Inc. — John Wiley & Sons / Page 504 / 2nd Proofs / Heat Transfer Handbook / Bejan 504 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 [504], (66) Lines: 2822 to 2885 ——— 2.63231pt PgVar ——— Long Page * PgEnds: Eject [504], (66) The ranges of validity of eqs. (6.191) are 2000 ≤ Re ≤ 400,000 2 ≤ H D ≤ 12 2.5 ≤ r D ≤ 7.5 and 0.004 ≤ A t ≤ 0.04 For r<2.5D, the average heat transfer data are provided by Martin (1977) in graphical form. For a single-slot nozzle, the recommended correlation is Nu Pr 0.42 = 3.06Re m (x/W ) + (H/W ) + 2.78 (6.193) where x is the distance from the stagnation point, m = 0.695 − x 2W + H 2W 1.33 + 3.06 −1 (6.194) and the corresponding ranges of validity are 3000 ≤ Re ≤ 90,000 4 ≤ H W ≤ 20 4 ≤ x W ≤ 50 For x/W < 4, these results can be used as a first approximation and are within 40% of measurements. More recent measurements by Womac et al. (1993) suggest that with the Prandtl number effect included in Martin (1977), these correlations are also applicable as a reasonable approximation for liquid jets. Average Nusselt Number for an Array of Jets Martin (1977) also provides correlations for arrays of in-line and staggered nozzles, as well as for an array of slot jets. These configurations are illustrated in Fig. 6.30, and the following correlations are also provided by Incropera and DeWitt (1996). For an array of round nozzles, Nu Pr 0.42 = K A r , H D G A r , H D f 2 (Re) (6.195) where f 2 (Re) = 0.5Re 2/3 (6.196a) K A r , H D = 1 + H/D 0.6/A 1/2 r 6 −0.05 (6.196b) where G is the same as for the single nozzle, eq. (6.191b). BOOKCOMP, Inc. — John Wiley & Sons / Page 505 / 2nd Proofs / Heat Transfer Handbook / Bejan 505 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 [505], (67) Lines: 2885 to 2894 ——— 6.87999pt PgVar ——— Long Page * PgEnds: PageBreak [505], (67) S S D D S S S W ( ) Slot jet arrayc( ) Staggered circular jet arrayb( ) In-line circular jet arraya A= D/S r 22 4 A=W/S r A= D/ S r ͌ 22 23 Figure 6.30 Commonly utilized jet array configurations. (From Incropera and DeWitt, 1996.) BOOKCOMP, Inc. — John Wiley & Sons / Page 506 / 2nd Proofs / Heat Transfer Handbook / Bejan 506 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 [506], (68) Lines: 2894 to 2940 ——— 0.57813pt PgVar ——— Normal Page PgEnds: T E X [506], (68) The ranges of validity of eqs. (6.196) are 2000 ≤ Re ≤ 100,000 2 ≤ H D ≤ 12 0.004 ≤ A r ≤ 0.04 For an array of slot nozzles, Nu Pr 0.42 = 2 3 A 3/4 r,0 2Re A r /A r,0 + A r,0 /A r 2/3 (6.197) where A r,0 = 60 + 4 H 2W − 2 2 −1/2 (6.198) and 1500 ≤ Re ≤ 40,000 2 ≤ H W ≤ 80 0.008 ≤ A r ≤ 2.5A t,0 Free Surface Jets The flow regimes associated with a free surface jet are illus- trated in Fig. 6.31. As the jet emerges from the nozzle, it tends to achieve a more uniform profile farther downstream, due to the elimination of wall friction. There is a corresponding reduction in jet centerline velocity (or midplane velocity for the slot jet). As for the submerged jet, a stagnation zone occurs. This zone is associated with the concurrent deceleration of the jet in a direction normal to surface and acceleration parallel to it and is also characterized by a strong favorable pressure gradient parallel to the surface. Within the stagnation zone, hydrodynamic and thermal boundary layers are of uniform thickness. Beyond this region, the boundary layers begin to grow in the wall jet region, eventually reaching the free surface. The viscous effects extend throughout the film thickness t(r), and the surface velocity V s starts to decrease with increasing radius. The velocity profiles are similar to each other in a region that ends at r = r c , where the transition to turbulence begins. The flow development associated with the planar jet is less complicated. Following the bifurcation at the stagnation line, the two oppositely directed films are of a fixed film thickness and the free surface velocity is V s = V i . Boundary layers grow outside the stagnation zone. These reach the film thickness, before or after transition to turbulence, depending on the initial conditions. Womac et al. (1993) considered free surface impinging jets of water and FC-77 on a square nearly isothermal heater of side 12.7 mm. Nozzle diameters D n ranged from 0.978 to 6.55 mm, and nozzle-to-surface spacings varied from 3.5 to 10. They correlated their average heat transfer coefficient data using an area-weighted average of standard correlations for the impingement and wall jet regions. The correlation for the impingement region is of the form BOOKCOMP, Inc. — John Wiley & Sons / Page 507 / 2nd Proofs / Heat Transfer Handbook / Bejan TURBULENT JETS 507 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 [507], (69) Lines: 2940 to 2959 ——— 0.71075pt PgVar ——— Normal Page * PgEnds: Eject [507], (69) Figure 6.31 Transport regions in a circular, unconfined, free surface jet impinging on a surface. Nu Pr 0.4 = C 1 · Re m D i (6.199) where m = 0.5 and the Reynolds number is defined in terms of the impingement velocity V i diameter D i = D n (V n /V i ) 1/2 . The wall jet correlation is of the form Nu Pr 0.4 = C 2 · Re n L ∗ (6.200) where the Reynolds number is defined in terms of the impingement velocity v i , and the average length L ∗ of the wall jet region for a square heater is L ∗ = 0.5( √ 2 L h − D i ) + 0.5(L h − D i ) 2 (6.201) Combining the correlations of eqs. (6.199) and (6.200) in an area-weighted fashion gives (Incropera, 1999) BOOKCOMP, Inc. — John Wiley & Sons / Page 508 / 2nd Proofs / Heat Transfer Handbook / Bejan 508 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 [508], (70) Lines: 2959 to 3023 ——— 3.51332pt PgVar ——— Normal Page PgEnds: T E X [508], (70) Nu Pr 0.4 = C 1 · Re m D i L h D i A r + C 2 · Re n L ∗ L h L ∗ (1 − A r ) (6.202) where A r = πD 2 i 4l 2 h The data were found to be best correlated in the range 1000 < Re D n < 51,000 for C 2 = 0.516, C 2 = 0.491, and n = 0.532, where the fluid properties are evaluated at the mean of the surface and ambient fluid temperature. 6.8 SUMMARY OF HEAT TRANSFER CORRELATIONS • Isothermal flat plate in uniform laminar flow (Sections 6.4.3 through 6.4.5): Near Pr = 1, Nu x = 0.332Re 1/2 x · Pr 1/3 (6.36) For Pr 1, Nu x = 0.565Re 1/2 x · Pr 1/2 (6.44) For Pr 1, Nu x = 0.339Re 1/2 x · Pr 1/3 (6.48b) • Isothermal flat plate in uniform laminar flow with appreciable viscous dissipation (Section 6.4.6): Nu x = 0.332Re 1/2 x · Pr 1/3 (6.36) The local heat flux is given by q = h x (T o − T AW ) where T AW = T ∞ + r c U 2 2c p and for gases r c = b(Pr Pr) 1/2 BOOKCOMP, Inc. — John Wiley & Sons / Page 509 / 2nd Proofs / Heat Transfer Handbook / Bejan SUMMARY OF HEAT TRANSFER CORRELATIONS 509 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 [509], (71) Lines: 3023 to 3075 ——— 12.48723pt PgVar ——— Normal Page * PgEnds: Eject [509], (71) and where the fluid properties are evaluated at T ∗ = T ∞ + (T W − T ∞ ) + 0.22(T AW − T ∞ ) The surface-averaged heat transfer coefficient in each of the foregoing cases for an isothermal flat plate is determined from its local value at x = L(h L ) as ¯ h = 2h L (6.37) • Flat plate in uniform laminar flow with an unheated starting length (x 0 ) and heated to a uniform temperature beyond Pr near 1 (Section 6.4.7): Nu = 0.332Pr 1/2 · Re 1/2 x [1 − (x 0 /x) 3/4 ] 1/3 (6.63) • Wedge at uniform temperature with an included angle of βπ in a uniform laminar flow in the range 0.7 < Pr < 10 (Section 6.4.4): Nu x Re 1/2 x = 0.56A (2 − β) 1/2 (6.42) where β = 2m m + 1 and A = (β + 0.2) 0.11 Pr 0.333+0.067β−0.026β 2 • Cylinder at uniform surface temperature in a laminar cross flow (Section 6.4.16): Nu D ≡ ¯ hD k = 0.30 + 0.62Re 1/2 D · Pr 1/3 [1 + (0.40/Pr) 2/3 ] 1/4 1 + Re D 282,000 5/8 4/5 (6.155) • General two-dimensional object at uniform surface temperature in a uniform laminar flow (Section 6.4.9): 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) where c 1 through c 3 are as given in Table 6.1. • Laminar flow over a sphere at a uniform surface temperature (Section 6.4.16): Nu D = 2 + 0.4Re 1/2 D + 6Re 2/3 D Pr 0.4 µ µ s 1/4 (6.156) BOOKCOMP, Inc. — John Wiley & Sons / Page 510 / 2nd Proofs / Heat Transfer Handbook / Bejan 510 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 [510], (72) Lines: 3075 to 3129 ——— 0.73122pt PgVar ——— Normal Page * PgEnds: Eject [510], (72) • General axisymmetric object at uniform surface temperature in a uniform lami- nar flow (Section 6.4.10): 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 c 1 through c 3 are as given in Table 6.1. • Isothermal flat plate with a turbulent boundary layer from the leading edge for Pr and Pr T near 1 (Section 6.4.14): Nu x = 0.0296Re 0.80 x (6.108) • Isothermal flat plate with turbulent boundary layer transition from laminar to turbulent for Pr and Pr T near 1 (Section 6.4.14): Nu L = 0.664Re 0.5 T + 0.36 Re 0.8 L − Re 0.8 T (6.110) • Isothermal flat plate with turbulent boundary Pr T near 1 (Section 6.4.14): Nu x = 0.029Re 0.8 x G (6.130) where G = Pr (0.029/Re x ) 1/2 {5Pr + 5ln[(1 + 5Pr)/6] − 5}+1 (6.131) • Flat plate with a turbulent boundary layer from the leading edge and with an unheated starting length followed by uniform surface temperature for Pr and Pr T near 1 (Section 6.4.14): St · Pr 0.8 = 0.287Re −0.2 x 1 − x 0 x 9/10 1/9 (6.140) • Uniform flux plate with a turbulent boundary layer from the leading edge for Pr and Pr T near 1 (Section 6.4.14): St · Pr 0.4 = 0.03Re −0.2 x (6.141) • Isothermal rough flat plate with a turbulent boundary layer from the leading for Pr and Pr T near 1 (Section 6.4.15): St = C f /2 Pr T + (C f /2) 1/2 /St k (6.154) BOOKCOMP, Inc. — John Wiley & Sons / Page 511 / 2nd Proofs / Heat Transfer Handbook / Bejan SUMMARY OF HEAT TRANSFER CORRELATIONS 511 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 [511], (73) Lines: 3129 to 3190 ——— 4.71225pt PgVar ——— Normal Page PgEnds: T E X [511], (73) where St k 0.8Re −0.2 k · Pr −0.4 • Cross flow across a bank of cylinders at uniform surface temperature (Section 6.5.1): Nu D = C · Re m D,max · Pr 0.36 Pr Pr s 1/4 (6.157) for N L ≥ 20 0.7 ≤ Pr < 500 1 < Re D,max < 2 × 10 6 where C and m are given in Table 6.2. • Plate stack (Section 6.5.2): The optimum number of plates in a given cross- sectional area, L × H , n opt 0.26(H/L)Pr 1/4 · Re 1/2 L 1 + 0.26(t/L)Pr 1/4 · Re 1/2 L (6.161) for Pr ≥ 0.7 and n 1. • Offset strips (Section 6.5.2): In the laminar range (Re ≤ Re ∗ ), f = 8.12Re −0.74 d h −0.15 α −0.02 (6.166) j = 0.53Re −0.50 d h −0.15 α −0.14 (6.167) In the turbulent range (Re ≤ Re ∗ + 1000), f = 1.12Re −0.36 d h −0.65 t d h 0.17 (6.168) j = 0.21Re −0.40 d h −0.24 t d h 0.02 (6.169) The transition Reynolds number Re ∗ is obtained from the set of equations Re ∗ = Re ∗ b d h b (6.170) Re ∗ b = 257 s 1.23 t 0.58 (6.171) . (C f /2) 1/2 /St k (6.154) BOOKCOMP, Inc. — John Wiley & Sons / Page 511 / 2nd Proofs / Heat Transfer Handbook / Bejan SUMMARY OF HEAT TRANSFER CORRELATIONS 511 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 [511],. impinging on a surface. BOOKCOMP, Inc. — John Wiley & Sons / Page 503 / 2nd Proofs / Heat Transfer Handbook / Bejan TURBULENT JETS 503 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 [503],. 0.20[(H/D) − 6]A 1/2 r (6.192) BOOKCOMP, Inc. — John Wiley & Sons / Page 504 / 2nd Proofs / Heat Transfer Handbook / Bejan 504 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 [504],