Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 58
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For a heated horizontal cylinder inperpendicular cross flow, the angle of the approachingstream, ~ in Fig. 4.47, greatly affects the heat flow in theDVmixed convection regime. For ~ = 0 the forced flow assists theRe= Tnatural convection and the dependence of the average NusTgB (T.-T®) D~selt number on Re resembles path A in Fig. 4.44. For ~)= 90 °Ro viithere is a sharper transition from natural to forced convecRation than when ~ = 0, while for opposed flow ((~ = 180 °) thereGr Pris a minimum as shown by path B in Fig.
4.44. For a cooledgcylinder the same description applies except that ~) is measured from the vertical axis extending upward from theTw > T~~ . .cylinder.Equating the Nusselt numbers for pure natural convecFIGURE 4.47 Perpendicular flow across a horizon- tion and pure forced convection provides a good estimate ofthe Ra-Re curve along which mixed convection effects aretal circular cylinder in mixed convection.most important, as already discussed. After a careful studyof available data, Morgan [198] proposed the following equation for forced convection heattransfer from a cylinder for cross flow in a low-turbulence airstream:Nu =q"D(Tw-T~) kNUF = a Re"(4.163)where a and n are given in Table 4.14.
This equation can also be used as a first approximationfor other fluids if the right side is multiplied by the factor (Pr/0.71) 1/3. If Eq. 4.163 is equatedto Eq. 4.45 for NuN, the Re/versus Gri relation indicated by the solid curve in Fig. 4.48 isobtained, which denotes the approximate center of the mixed convection regime. The approximate bounds of this regime, based on a 5 percent deviation in heat transfer from pure forcedconvection and from pure natural convection, respectively, have been estimated by Morgan[198] to lie in the shaded bands.NATURAL CONVECTION4.77Constants for Forced Convection Over a Circular Cylinder (Eq.
4.163)TABLE 4.14Re rangean10-4 to4 x 1 0 -34 × 1 0 -3 to9 x 10-29 X 1 0 -2to 1.01.0 to 3535 to5 × 10 35 X 1 0 3 to5 × 10 45 x 104 to2 x 1050.4370.08950.5650.1360.8000.2800.7950.3840.5830.4710.1480.6330.02080.814A procedure for calculating the heat transfer in the mixed convection regime for the problem has also been proposed by Morgan [198] on the basis of work of B6rner [22] and Hattonet al. [131].
For a given Ra and Re, the value of NuN is computed from Eq. 4.45. For the givenRe, the constants a and n are chosen from Table 4.14. The value of Re/is then found from Eq.4.163 with NUF= NuN; that is,Re~=a(Pr/0.71)l/3(4.164)An effective Reynolds number Re~, is then calculated fromNee. = [(Re/+ Re cos ,)2 + (Re sin (i))211/2(4.165)and Nu is computed by insertion of Re~.
into Eq. 4.163 to obtain( Pr /1/3Nu = a RebUff\ 0.71 ]It will be seen that if Rei >> Re, the natural convection result is recovered, while if Rei << Re,Nu - NUF. Morgan showed that this calculation procedure gave good agreement with experiments for air and water for ~ = 90 °. For assisting flow (~ = 0 °) the agreement was poorer, andit was still poorer for opposed flow (~ = 180°).Gebhart et al.
[109, 111] have provided accurate heat transfer measurements spanning themixed convection zone for extremely fine wires for Pr = 0.7, 6.3, and 63, and have also provided equations for the bounds of the mixed convection zone.10 4rr....... .......Nu N = Nu F~l•""-IO- 3I0 -II10 2-~u-"Nutv106lO 4108I0 I01012GrFIGURE 4.48 Regimes of forced, mixed, and natural convection for assistingflow over a horizontal circular cylinder.4.78CHAPTER FOUROther Shapes. Churchill [55] proposed that Eq.
(4.159) with m = 3 be used also for assistingmixed convection flow around other surface shapes, such as spheres and cylinders. The appropriate expressions for NUF and NuN for the body shape of interest must be used.Internal FlowsHorizontal TubesUniform Heat Flux.For laminar flow in a horizontal tube where uniform heat flux isapplied at the outer boundary of the tube, the bulk temperature Tb, increases linearly in theaxial direction. To maintain the heat flow to the fluid, the wall temperature must remainhigher than the fluid temperature, and under these conditions a fully developed natural convection motion becomes established in which velocity and temperature gradients becomeindependent of the axial location.
Because the fully developed Nusselt number for laminarpure forced convection is small (NUF--> 4.36), the buoyancy-induced mixing motion cangreatly enhance the heat transfer.Marcos and Bergles [189] correlated their data for fully developed heat transfer for waterand ethylene glycol in glass and metal tubes by the equationNUfd = [4.362 + [0.055( Ra Pr°35 )°412}1/2elw/4(4.166)where the nomenclature is defined in Fig. 4.49. The Pw term accounts for the redistributionby circumferential conduction of the uniform heat flux on the outside surface of the tubebefore entering the fluid. Fluid properties are to be evaluated at the mean film temperature,0.5(Twi + Tb).
The data are in the range 3 x 104 < Ra < 106, 4 < Pr < 175, 2 < Pw < 66.The heat transfer immediately downstream of the location where heating begins will bedominated by forced convection and will depend on the velocity profile.
For a parabolic inletprofile, the forced convection Nusselt number can be approximated by [249]:NUF= 1"30(Rex/DPr)1/3~ReX/Dpr<0.01(4.167)For Ra <~ 5 x 105 the transition from forced (Eq. 4.167) to natural convection (Eq. 4.166) isvery sharp [249] but it becomes more rounded with increasing Ra. To find the total heat transTiTw'/ - - q " or T/specifiedfkw-~v////// ~///////////////////////////A8rz//.,////////#///~W~////%T/b///////////z.//////////L////////////;Di////j~TOUniform wallheat fluxq"Diu= (~wi-Tb) kRog/3 Di 3 (Twi-Tb)~aq D,Uniform walltemperatureNu = (Twi-T--b)kRag/3 Oi3 (Tw-%)vaDiVRe =q"Di 2Pw = (Twi_Tb) kw 8%:,-~ (Ti + To)Gz = wc'-'-EpkLFIGURE 4.49 Geometryand definition of terms for mixed convection insidea horizontal tube.N A T U R A L CONVECTION4.79fer over some length of tube, numerically integrate from the inlet of the tube using, at eachlocation, the maximum value of local heat transfer given by Eqs.
4.166 and 4.167, and usingthe energy balance at each step to update the bulk temperature Tb.As a result of natural convection, the fully developed condition is reached much fartherupstream than for pure forced convection. For example, for Ra --- 10 6, Nusa is reached atroughly x/D = 0.001Re Pr as opposed to x/D --- 0.06Re Pr for pure forced convection (i.e., 1diameter as opposed to 60 diameters for Re Pr = 1000).When circular tubes subjected to a uniform heat flux are tilted upward, the Nusselt number has been shown to monotonically decrease with increasing angle for air [236].Isothermal WalL Natural convection also affects the laminar thermal development in atube with an isothermal wall.
In this case the temperature differences in the fluid near thetube inlet initiate a natural convection motion, but as the fluid temperature approaches thewall temperature far downstream, the motion slows and the fully developed Nusselt number(NUF = 3.66) is approached.On the basis of data available up to 1964, Metais and Eckert [190] established the forcedconvection boundary of the mixed convection regime, and their results are presented in Fig.4.50.
The line was drawn where natural convection was thought to alter the heat transfer fromthat for pure forced convection by 10 percent. Figure 4.49 defines the nomenclature for thisproblem.,o5[i.|a,rr...I Transition,laminar t o - ~[turbulent.y ~IV'/'//////.////.//////./~,/-/'///~I~:,okI/...I~~-o~o"Mixed convection,tur bulentoForced flowl laming r'inzL,v I04.Forced convection turbulent1- ~~/1 --~--/--~//~(~/'/~/I /105Ilaminar7/////////,~/.III106107 3 IOeRe 9/9 (Twi;-Tb)D/1109F I G U R E 4.50 Regimes of forced and mixed convection for flowthrough horizontal tubes with uniform wall temperature, for 10 -2 <(Pr D/L) < 1.
From Metais and Eckert [190].Heat transfer relations in the mixed convection regime have been proposed by severalinvestigators, but the equation proposed by Depew and August [74] appears to be mostsuccessful. This relation agrees with their measurements for water (Pr = 6.5), ethyl alcohol(Pr = 15), and a glycerol-water mixture (Pr = 375), as well as with the data from other authors,to about +40 percent:Nu = 1.75{/ lab \}I/4[Gz+ 0.12(Gz R a 1/3 Pr°°3)°88] 1/3(4.168)\~tw Jwhere b and w refer to average bulk and wall conditions, Nu, Gz, and Ra are defined in Fig.4.49, and Tb is obtained by arithmetically averaging the inlet and outlet temperature.
Allphysical properties are to be evaluated at Tb. The data on which this equation was derived liein the following ranges: 28 < L / D < 193, 10 < Gz < 456, and 30 < Gr < 2 x 10 7.Vertical Tubes. The flow regime chart for vertical tube flow shown in Fig. 4.51 was prepared by Metais and Eckert [190] for either a uniform-heat-flux or uniform-wall-temperatureboundary condition. The two boundaries of the mixed convection are defined in such a way4.80CHAPTER FOUR406I1Forced convectionturbulentIiOs104Transition, laminarturbulent.,N\\\\\\ \'x\\\\\\\..\\\'~.J Free convection.turbulent103 Forced convectionlaminar flowF102"/I0IO 2.;J10 3fI0 4Ro---.D =L\F r e e convection laminarI0 5106107\I0 a\109g/3 ( T .- Tb) D4veLF I G U R E 4.51 Regimes of natural, forced, and mixed convection for flowthrough vertical tubes with uniform wall temperature of heat flux, for10 -2 < (Pr D/L) < 1.
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