Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 77
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— John Wiley & Sons / Page 502 / 2nd Proofs / Heat Transfer Handbook / BejanTURBULENT JETS123456789101112131415161718192021222324252627282930313233343536373839404142434445503the stagnation or impingement zone. Within this region the flow accelerates in thetransverse direction (r or x) and decelerates in the z direction. Farther away inthe transverse direction (in the wall jet region), the flow starts to decelerate due toentrainment of the ambient fluid.Average Nusselt Number for Single Jets Martin (1977) provides an extensive review of heat transfer data for impinging gas jets. For single nozzles, the averageNusselt number is of the form (Martin, 1977; Incropera and DeWitt, 1996)r H(6.188a)Nu = f Re, Pr,,Dh Dhorx HNu = f Re, Pr,,Dh Dh[503], (65)(6.188b)Lines: 2759 to 2822where———Nu =h̄kDh(6.189a)Re =Ve Dhν(6.189b)andwheref1 (Re) = 2Re1/2 (1 + 0.005Re0.55 )1/2(6.191a)and1/2Replacing D/r by 2Ar1 − 1.1(D/r)Dr 1 + 0.1[(H /D) − 6](D/r)(6.191b)yields1/2G = 2A1/2r1 − 2.20Ar1/21 + 0.20[(H /D) − 6]ArBOOKCOMP, Inc.
— John Wiley & Sons / Page 503 / 2nd Proofs / Heat Transfer Handbook / Bejan———Normal Page* PgEnds: Eject[503], (65)where Ve is the uniform exit velocity at the jet nozzle, Dh = D for a round nozzle,and Dh = 2W for a slot nozzle. Figure 6.29 defines the geometric parameters.For a single round nozzle, Martin (1977) recommendsNuD H,f1 (Re)=G(6.190)r DPr0.42G=1.32532pt PgVar(6.192)504123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: EXTERNAL FLOWSThe ranges of validity of eqs. (6.191) are2000 ≤ Re ≤ 400,0002.5 ≤r≤ 7.5Dand2≤H≤ 12D0.004 ≤ At ≤ 0.04For r < 2.5D, the average heat transfer data are provided by Martin (1977) ingraphical form.For a single-slot nozzle, the recommended correlation isNu3.06Rem=(x/W ) + (H /W ) + 2.78Pr0.42(6.193)[504], (66)where x is the distance from the stagnation point, x H 1.33m = 0.695 −+ 3.06+2W2W−1(6.194)———2.63231pt PgVarand the corresponding ranges of validity are3000 ≤ Re ≤ 90,0004≤H≤ 20W4≤———Long Page* PgEnds: Ejectx≤ 50WFor x/W < 4, these results can be used as a first approximation and are within40% of measurements.
More recent measurements by Womac et al. (1993) suggestthat with the Prandtl number effect included in Martin (1977), these correlations arealso applicable as a reasonable approximation for liquid jets.Average Nusselt Number for an Array of Jets Martin (1977) also providescorrelations for arrays of in-line and staggered nozzles, as well as for an array of slotjets.
These configurations are illustrated in Fig. 6.30, and the following correlationsare also provided by Incropera and DeWitt (1996).For an array of round nozzles, NuHH= K Ar ,(6.195)G Ar ,f2 (Re)DDPr0.42wheref2 (Re) = 0.5Re2/36 −0.05HH/DK Ar ,= 1 +1/2D0.6/Arwhere G is the same as for the single nozzle, eq. (6.191b).BOOKCOMP, Inc. — John Wiley & Sons / Page 504 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 2822 to 2885(6.196a)(6.196b)[504], (66)123456789101112131415161718192021222324252627282930313233343536373839404142434445Ar = W/SAr =D 2/ 2公3S 2Ar =D 2/4S 2(c) Slot jet array(b) Staggered circular jet array(a) In-line circular jet arraySDSDBOOKCOMP, Inc. — John Wiley & Sons / Page 505 / 2nd Proofs / Heat Transfer Handbook / BejanSFigure 6.30 Commonly utilized jet array configurations.
(From Incropera and DeWitt, 1996.)SWS505[505], (67)Lines: 2885 to 2894———6.87999pt PgVar———Long Page* PgEnds: PageBreak[505], (67)506123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: EXTERNAL FLOWSThe ranges of validity of eqs. (6.196) are2000 ≤ Re ≤ 100,0002≤H≤ 12D0.004 ≤ Ar ≤ 0.04For an array of slot nozzles,Nu2 3/4= Ar,00.423Pr2ReAr /Ar,0 + Ar,0 /Ar2/3(6.197)whereAr,0H= 60 + 4−22W2 −1/2(6.198)[506], (68)and1500 ≤ Re ≤ 40,0002≤H≤ 80WLines: 2894 to 29400.008 ≤ Ar ≤ 2.5At,0Free Surface Jets The flow regimes associated with a free surface jet are illustrated in Fig. 6.31. As the jet emerges from the nozzle, it tends to achieve a moreuniform profile farther downstream, due to the elimination of wall friction.
There isa corresponding reduction in jet centerline velocity (or midplane velocity for the slotjet). As for the submerged jet, a stagnation zone occurs. This zone is associated withthe concurrent deceleration of the jet in a direction normal to surface and accelerationparallel to it and is also characterized by a strong favorable pressure gradient parallelto the surface.Within the stagnation zone, hydrodynamic and thermal boundary layers are ofuniform thickness.
Beyond this region, the boundary layers begin to grow in the walljet region, eventually reaching the free surface. The viscous effects extend throughoutthe film thickness t (r), and the surface velocity Vs starts to decrease with increasingradius. The velocity profiles are similar to each other in a region that ends at r = rc ,where the transition to turbulence begins.The flow development associated with the planar jet is less complicated. Followingthe bifurcation at the stagnation line, the two oppositely directed films are of a fixedfilm thickness and the free surface velocity is Vs = Vi .
Boundary layers grow outsidethe stagnation zone. These reach the film thickness, before or after transition toturbulence, depending on the initial conditions.Womac et al. (1993) considered free surface impinging jets of water and FC-77on a square nearly isothermal heater of side 12.7 mm. Nozzle diameters Dn rangedfrom 0.978 to 6.55 mm, and nozzle-to-surface spacings varied from 3.5 to 10. Theycorrelated their average heat transfer coefficient data using an area-weighted averageof standard correlations for the impingement and wall jet regions. The correlation forthe impingement region is of the formBOOKCOMP, Inc. — John Wiley & Sons / Page 506 / 2nd Proofs / Heat Transfer Handbook / Bejan———0.57813pt PgVar———Normal PagePgEnds: TEX[506], (68)TURBULENT JETS123456789101112131415161718192021222324252627282930313233343536373839404142434445507[507], (69)Lines: 2940 to 2959———0.71075pt PgVar———Normal Page* PgEnds: EjectFigure 6.31 Transport regions in a circular, unconfined, free surface jet impinging on asurface.Nu= C1 · RemDiPr0.4(6.199)where m = 0.5 and the Reynolds number is defined in terms of the impingementvelocity Vi diameter Di = Dn (Vn /Vi )1/2 .
The wall jet correlation is of the formNu= C2 · RenL∗Pr0.4(6.200)where the Reynolds number is defined in terms of the impingement velocity vi , andthe average length L∗ of the wall jet region for a square heater is√0.5( 2 Lh − Di ) + 0.5(Lh − Di )∗L =(6.201)2Combining the correlations of eqs. (6.199) and (6.200) in an area-weighted fashiongives (Incropera, 1999)BOOKCOMP, Inc. — John Wiley & Sons / Page 507 / 2nd Proofs / Heat Transfer Handbook / Bejan[507], (69)508123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: EXTERNAL FLOWSNuLhLh= C1 · RemAr + C2 · RenL∗ ∗ (1 − Ar )Di0.4DLPri(6.202)whereAr =πDi24lh2The data were found to be best correlated in the range 1000 < ReDn < 51,000 forC2 = 0.516, C2 = 0.491, and n = 0.532, where the fluid properties are evaluated atthe mean of the surface and ambient fluid temperature.6.8[508], (70)SUMMARY OF HEAT TRANSFER CORRELATIONS• Isothermal flat plate in uniform laminar flow (Sections 6.4.3 through 6.4.5): NearPr = 1,1/3Nux = 0.332Re1/2x · Pr(6.36)For P r 1,1/2Nux = 0.565Re1/2x · Pr(6.44)For P r 1,Lines: 2959 to 3023———3.51332pt PgVar———Normal PagePgEnds: TEX[508], (70)1/3Nux = 0.339Re1/2x · Pr(6.48b)• Isothermal flat plate in uniform laminar flow with appreciable viscous dissipation(Section 6.4.6):1/3Nux = 0.332Re1/2x · PrThe local heat flux is given byq = hx (To − TAW )whereTAW = T∞ + rcU22cpand for gasesrc = b(Pr Pr)1/2BOOKCOMP, Inc.
— John Wiley & Sons / Page 508 / 2nd Proofs / Heat Transfer Handbook / Bejan(6.36)SUMMARY OF HEAT TRANSFER CORRELATIONS123456789101112131415161718192021222324252627282930313233343536373839404142434445509and where the fluid properties are evaluated atT ∗ = T∞ + (TW − T∞ ) + 0.22(TAW − T∞ )The surface-averaged heat transfer coefficient in each of the foregoing cases foran isothermal flat plate is determined from its local value at x = L(hL ) ash̄ = 2hL(6.37)• Flat plate in uniform laminar flow with an unheated starting length (x0 ) andheated to a uniform temperature beyond Pr near 1 (Section 6.4.7):Nu =0.332Pr1/2 · Re1/2x[1 − (x0 /x)3/4 ]1/3(6.63)• Wedge at uniform temperature with an included angle of βπ in a uniform laminarflow in the range 0.7 < Pr < 10 (Section 6.4.4):0.56ANux=1/2(2 − β)1/2Rex(6.42)whereβ=2mm+1andA = (β + 0.2)0.11 Pr0.333+0.067β−0.026β25/8 4/51/2ReD0.62ReD · Pr1/3h̄D1+= 0.30 +NuD ≡(6.155)k[1 + (0.40/Pr)2/3 ]1/4282,000• General two-dimensional object at uniform surface temperature in a uniformlaminar flow (Section 6.4.9):c1 (U ∗ )c2x∗∗ c3(U ) dx∗1/21ReL1/2(6.72)0where c1 through c3 are as given in Table 6.1.• Laminar flow over a sphere at a uniform surface temperature (Section 6.4.16): 1/4µ1/22/3NuD = 2 + 0.4ReD + 6ReD Pr0.4µsBOOKCOMP, Inc.
— John Wiley & Sons / Page 509 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 3023 to 3075———12.48723pt PgVar———Normal Page* PgEnds: Eject[509], (71)• Cylinder at uniform surface temperature in a laminar cross flow (Section 6.4.16):kq == St =ρcp U (T0 − T∞ )ρcp U δ[509], (71)(6.156)510123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: EXTERNAL FLOWS• General axisymmetric object at uniform surface temperature in a uniform laminar flow (Section 6.4.10):c1 (r0∗ )K (U ∗ )c2St = x3∗0(U ∗ )c3 (r0∗ )2K dx3∗1/21ReL1/2(6.76)where c1 through c3 are as given in Table 6.1.• Isothermal flat plate with a turbulent boundary layer from the leading edge forPr and PrT near 1 (Section 6.4.14):Nux = 0.0296Re0.80x(6.108)• Isothermal flat plate with turbulent boundary layer transition from laminar toturbulent for Pr and PrT near 1 (Section 6.4.14): 0.80.8NuL = 0.664Re0.5T + 0.36 ReL − ReTLines: 3075 to 3129(6.110)• Isothermal flat plate with turbulent boundary PrT near 1 (Section 6.4.14):Nux = 0.029Re0.8x G[510], (72)———0.73122pt PgVar———Normal Page(6.130)* PgEnds: EjectwhereG=Pr1/2(0.029/Rex ) {5Pr + 5 ln[(1 + 5Pr)/6] − 5} + 1[510], (72)(6.131)• Flat plate with a turbulent boundary layer from the leading edge and with anunheated starting length followed by uniform surface temperature for Pr and PrTnear 1 (Section 6.4.14): x 9/10 1/90St · Pr0.8 = 0.287Re−0.21−xx(6.140)• Uniform flux plate with a turbulent boundary layer from the leading edge for Prand PrT near 1 (Section 6.4.14):St · Pr0.4 = 0.03Re−0.2x(6.141)• Isothermal rough flat plate with a turbulent boundary layer from the leading forPr and PrT near 1 (Section 6.4.15):St =Cf /2PrT + (Cf /2)1/2 /StkBOOKCOMP, Inc.
— John Wiley & Sons / Page 510 / 2nd Proofs / Heat Transfer Handbook / Bejan(6.154)SUMMARY OF HEAT TRANSFER CORRELATIONS123456789101112131415161718192021222324252627282930313233343536373839404142434445511whereStk 0.8Re−0.2· Pr−0.4k• Cross flow across a bank of cylinders at uniform surface temperature (Section6.5.1):NuD = C ·RemD,max· Pr0.36PrPrs1/4(6.157)forNL ≥ 200.7 ≤ Pr < 5001 < ReD,max < 2 × 106[511], (73)where C and m are given in Table 6.2.• Plate stack (Section 6.5.2): The optimum number of plates in a given crosssectional area, L × H ,Lines: 3129 to 3190———1/2nopt 0.26(H /L)Pr1/4 · ReL1/21 + 0.26(t/L)Pr1/4 · ReL(6.161)for Pr ≥ 0.7 and n 1.• Offset strips (Section 6.5.2): In the laminar range (Re ≤ Re∗ ),f = 8.12Re−0.74j = 0.53Re−0.50dhdh−0.15−0.15f = 1.12Rej = 0.21Re−0.40dhdh−0.65 −0.24 (6.166)α−0.14(6.167)tdhtdh0.17(6.168)0.02(6.169)The transition Reynolds number Re∗ is obtained from the set of equationsdhb 1.23 0.58tRe∗b = 257sRe∗ = Re∗bBOOKCOMP, Inc.
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