Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 45
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Examples are shown in Table4.3a. For these bodies, the correlation equation has the formBNu r = GCe Ra TMNue = [(NUcoND)n +(4.49a)(NuT)n]TM(4.49b)Nu, = Ct Ra 1/3(4.49d)Nu = [Nu~n + Nu~n]1/m(4.49c)The length scales on which Nu and Ra are based, and values of the constants, are provided inTable 4.3a. The basis for these relations was discussed in the section on heat transfer correlation method.In the rightmost column in Table 4.3a, NA is an abbreviation for none available, meaningthat the correlation is based entirely on the approximate method [227].
Values of n - 1.07, reco m m e n d e d by Hassani and Hollands [126, 127], and m - 10 are used for these cases. Whendata are available, the values of G have been adjusted to provide a best fit of the data. Exceptfor the thin oblate spheroid, C/D - 0.1, the value G was never adjusted by more than a fewpercent from the value predicted by the approximate method. C, was never adjusted, mainlybecause so few of the available data fall in the turbulent regime. Table 4.3b provides references to all data used, and the RMS error (ERMs) and maximum error (EMAx) from Eq. 4.49fit using the constants in Table 4.3a. The range of Ra and Pr covered by the data is also noted.The Nusselt number in the Ra --> 0 limit is the conduction Nusselt number NUCOND.
Thiswas calculated in all cases using the recommendations of Yovanovich et al. [291-294].Table 4.3b shows that Eq. 4.49 fits the data very closely. The range of data is, however,limited.For body shapes not treated in Table 4.3, the general correlation of Hassani and Hollands[126, 127] is recommended.Correlations for Spheres in a Thermally Stratified Medium. Consider the case of anisothermal sphere in a thermally stratified medium with constant vertical temperature gradient dT~/dz and with a temperature difference at the mid-height of the sphere of AT.
A Nusselt number Nuiso is first calculated for an isothermal sphere in an isothermal environment4.26¢00<v<-d0d~a.=.mQul#oO0~c~ZIIC~oR.~5c5oOa~§0a=~©e~~Dc~eqe~IIII'-a0aZZZ~#II~II~II~l ~z"00I.-~IIIIIIII1"-'4II~ZZ;,II! ~l ~ ~"G;>.00~0N0r~II~II. .
o ~ o ozIIIINOII~ ~IIII+IIIIm.IIIIe~L~0IIIIII4.274.28~20C0.<<0D0ILl.Jt'4o.8Ze~0E,.=e~m.o.t~m.•<.<<,~c-~IIzII~II-~c-IIIzzIIMMMmMd~m.IImz.=."C0,.Cr~.=0mZC¢3(1,)e~1¢3o.C¢3"2.i/3oo~"~0JHo.ooo.0IIII.,,c)II..DoObr~oOc~ooc~ooIIII0ZZII-~II8~z~II~z0ZZ~<<~ZZ<<8oo o-,~IIIIq•.~" ~"~ t"-~q ~IIIaa0IIZV'~IIIII4.294.30O.---tt-¢3b.~,<N<UN©I~8ovo.oIIo.x,-IIo.vxo.I8IIqo.oTdo~o.8008TZ8]lT~ILflIf;>IfNOONe~O,.J8IfTABLE 4.3b Measurements for the Shapes in Table 4.3a. ERMSand EMAx Are the RMS and Maximum Differencesof Eq. 4.49 From the Data,,ShapeSphereProlate spheroidC/L = 0.52Oblate spheroidC/L = 0.5C/L = 0.1Short vertical cylinderL/D = 0.1L D = 1.0Short horizontal cylinderL/D = 0.1L/D = 1.00.069 < L/D < 0.155Short inclined cylinderL/D = 1.0, 0 = 45 °Vertical cylinder with spherical end capsL/D = 2.0Horizontal cylinder with spherical end capsL/D = 2.0BisphereShort vertical square cylinderL / D =0.1Cube---corner upCube--edge upCube--face upShort square cylinder----edge upL/D = 0.1L/D = 1.0Short square cylindermface upL/D = 0.1L/D = 1.0Horizontal cones3.5 < ~ < 11.5L cornerV cornerReference38, 39242301280124, 126, 127229Ra range101-108109-1012108-101°107-1011101-108103-107PrE ~ s (%)EMAX(%)0.71-6200020000.710.75.29.74.118.59.518.90.710.711.86.04.79.33.37.23.36.411.97.4124, 126, 127229124, 126, 127101-107103-106101-1080.710.710.71124, 126, 127255101-108101-107104-1050.710.710.713.11.45.010.25.15.9124, 126, 127124, 126, 12730110-10710-107102-1050.710.710.711.31.0-3.03.02.9-10.0124, 126, 12710-1070.712.49.9124, 126, 1271-1070.712.04.9124, 126, 1271-1070.71124, 126, 1272801-107108-1090.7120001.9?5.632%124, 126, 12710-1080.712.74.938, 3926226228038, 3926226228038, 392622622862801-107105-107105-107108-10111-1071.50.92.94.31.85.6106-1090.710.716.020000.710.71-6020000.710.716.0200020002.11.92.57.90.90.95.37.88.47.33.23.7192.61.59.715.114.2124, 126, 127see Cube--edge up102-1070.711.74.2124, 126, 127see Cubemface up10-1080.711.63.7212280234234105-106106-1011106-109106-1090.7120006.06.03.110.14.23.410.633.9103-106105-107108-101°103-107103-106105-107106-101°4.0NA, none available.4.314.32CHAPTER FOUR(use Eq.
4.49) with the constants in Table 4.3a and with a temperature difference of AT. Thecorresponding total heat flow is qiso. The actual heat flow q is corrected to account for thestratification as follows:(q _ Nu qiso1+NuisoS_<2(4.50a)2(4.50b)S >= c a TMFor gases: a = 1, b = 0.36, and c = 1.25. For water (Pr = 6): a = 2.0, b = 0.47, and c = 1.17.Equation 4.50 was obtained by fitting the analytical results of Chen and Eichhorn [42].They also obtained measurements for 0 < S < 3.5, Pr -= 6, and 10 6 < Ra < 108, and Eq.
4.50agrees with these data to within about 10 percent. Extrapolation beyond the range of theexperimental data is not recommended.OPEN CAVITY PROBLEMSIn open cavity problems, buoyancy generated by heat exchange with the enclosure wallsdrives flow through the cavity (Fig. 4.20a). Either the wall temperature or the heat flux can bespecified on the cavity walls, and cavities may take a variety of forms (Fig. 4.20). The fluidtemperature far from the cavity is assumed constant at Too.The cooling of electronic equipment and the augmentation of heat transfer using finned surfaces are two important areaswhere open cavity problems arise.III111I II II II,I,,i,~I I ll,/(a)FIGURE 4.20(b)I[)(c)(d)Various configurations in which "open cavity" natural convection occurs.In the equations presented in this section, the characteristic temperature differenceappearing in the Nusselt and Rayleigh numbers is (Tw - Too),where Tw is the average wall temperature of the cavity. Properties are to be evaluated at the film temperature TI = 0.5(Tw + Too)unless otherwise specified.Cooling ChannelsCooling channels of the type depicted in Fig.
4.20b and c will first be discussed. For channelsthat are very long relative to the spacing of the vertical surfaces, the flow and heat transferbecome fully developed (i.e., velocity and temperature profiles become invariant with distance along the channel) and are described by simple equations. For short channels or widelyspaced vertical surfaces, a boundary layer regime is observed in which the boundary layers onthe vertical walls remain well separated. In the latter case the heat transfer relation is similarNATURAL CONVECTION4.33in form to that for a vertical plate but the heat transfer is usually found to be slightly higher.This augmentation results from the flow induced through the cavity by the chimney effect.Parallel Isothermal Plates.
For parallel isothermal plates of either equal or different temperatures (see Fig. 4.21), Aung [11] has shown that the Nusselt number in the fully developedregime is given by4 T .2 + 7 T* + 4Nula=90(1Ra-+ T*) 2Ra24Ra < 10(4.51)Both plate temperatures are assumed in the analysis to be equal to or greater than T=, and qin the definition of Nu in Fig. 4.21 is the total heat delivered to the fluid from both plates; theremaining symbols are defined in the figure.Speeified woll temperoturesSpecified well fluxes"-'G/Nu =//qSNu =2HW (T.-Too)kRe = gB(~w-Too)$3 SuaIIHq S(Tw½-Too) kRe'- g/~q''$4 SuakH' (q;' + q"2)Re: ~ SluIH1_t£._"/~~, : ~-. (T,+Tz)(~T*q~, ="-n'"q2 q l ->q z .
O -< q~ <- IqlT2- TOOT*: m,_moo T, >_m20<- <1-,isF I G U R E 4.21 Geometry and nomenclature for natural convection heat transfer from a wide (W >> S) rectangular cooling slot with heat flux or temperaturespecified conditions on the walls.For Ra > 10, the observed values of Nu depart from the value of Nura given in Eq. 4.51 andbeyond Ra --- 10 3 follow a relation of the formNu = cCe Ra TM(4.52)where Ce is given by Eq. 4.13 and Table 4.1. This is called the laminar boundary layer regime.For air, measurements of Elenbaas [88, 89] and Aung et al. [12] indicate that c = 1.20, whichgives 20 percent higher heat transfer than a vertical isolated plate.
The analysis of Aung et al.[12] and Bodoia and Osterle [21] yielded a value of c = 1.32. The measurements of Novotny[210] for Pr = 6 and the analysis of Miyatake and Fujii [196] for Pr = 10 suggest that capproaches 1.0 at higher Pr.For all Ra up to Ra = 105, the following heat transfer equation is recommended:NU = [(NUfd) m + ( e f tRal/n)m]1/mm =-1.9( 4.53)While property values are normally evaluated at 0.5(Tw + T=), for large temperature differences and small Ra better agreement between Eq. 4.53 and measurements is obtained (e.g.,Ref. 178) by evaluating the properties at Tw.The heat transfer per unit surface area, and the heat transfer coefficient, are relativelyinsensitive to the plate spacing until the spacing is reduced to the point at which the thermalboundary layers begin to interfere. If the objective is to transfer the maximum heat from agiven volume of height H, adding more channels proportionately increases the heat transferby adding surface area, until the boundary layers begin to interfere.
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