Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 49
Текст из файла (страница 49)
Also, for some values of Pr, the narrow range of Ra overwhich Eq. 4.78 has been tested should be noted. The data of Kek and Mtiller's [158] recentexperiments using liquid sodium with Pr = 0.0058 are fit reasonably well by Eqs. 4.78-4.81, butthe fit is improved considerably if kl is set equal to 0.087 and the power on (Ra/5830) ischanged from 1/5 to 1A.TABLE 4.7Values of kl and k2 to Be Used in Eq.
4.78Pr (approximate)klk2Range of Ra testedReference0.020.763410020030000.351.401.441.441.441.441.44>200>400140100-8585-75Ra < 108Ra < 10111 0 3 < Ra < 2 x 105103 < Ra < 1051 0 3 ___Ra < 3 × 1 0 610 3 < Ra < 5 × 1051 0 3 < Ra < 3 x 1 0 4233See 142 for list.See 142 for list; also 11724324323324310 3 <10 3 <Equation 4.78 with kl and k2 having values appropriate to water at moderate temperatures(Pr --- 6) is plotted in Fig. 4.27, together with relevant data for water. Figure 4.28 shows a plotof Eq.
4.78 for various values of Pr, covering only those ranges in Ra at which the equationhas been tested. Also plotted are the predictions of the conduction layer model given by Eq.4.77. This model is seen to be correct only in the limit of small Ra (Ra < 1708) and large RaI00Z60402(• Garon 8, Goldstein41o Chu 8, GoldsteinI• Rossby-103I104I0 5L ....I0 611Ii0 rI0 e109RoFIGURE 4.27 Comparisonof Eq. 4.78 (solid curve) and the data of Garon and Goldstein [104],Chu and Goldstein [53], and Rossby [233] for water (Pr = 6).4.4t5CHAPTERFOUR2o-,o=4-Pr --~ 30~2-~Conduction'~J~11~w~-~foyer model I1103104105106RoI10r10a109FIGURE 4.28 Plot of Eq.
4.78 for various values of Prandtl numbers, describing the heat transfer across a horizontal cavity with D/L > 10 and heating from below; also shown is the heat transfer predicted by the conduction layer model (Eq. 4.77).(Ra ~> 108). The maximum error, which occurs near Ra = 5830, varies from 20 percent forPr = 0.024 to 50 percent for Pr > 0.7.Critical Rayleigh Numbers f o r Horizontal Cavities Restricted in the Horizontal Direction.A critical Rayleigh number Rac governs the initiation of convective motion in the horizontalcavity that is restricted in the horizontal direction, just as in the extensive cavity discussed inthe previous section.
In restricted cavities, Ra~ depends on the geometric parameters describing the cavity and on the thermal properties of the wall, but not on the Prandtl number. For0 = 180 °, Rac = o,,, as for horizontally extensive cavities. For 0 = 0 °, Rac is bounded between twovalues, Ra~p and Raci. Evaluated by Catton [31-33], and Buell and Carton [27], these boundsare tabulated in Table 4.8 for rectangular parallelepiped cavities and in Table 4.9 for circularcylinder cavities. The greater of the two, Racp, applies to the perfectly conducting wall case(Table 4.6, entry 3 or 5); and the lesser, Ra~i, applies to the adiabatic wall case (see Table 4.6,entry 2; note that because of radiation, a wall with kw = 0 is not necessarily adiabatic).
Interpolation and extrapolation in Tables 4.8 and 4.9 may be assisted by knowledge of certainasymptotes: for the circular cylinder,asD-7- --> 0%L,Ra~p ---> 1708andRaci ~ 1708and [216, 289]aso-~- ~ 0,Racp~ 3456andRa~i---->1086.4For the rectangular parallelepiped,asHW-~- and ~ ~ ~,Ra~p ---> 1708andRa~i ---> 1708For the rectangular parallelepiped having W/H = oo [285],as-~- ~ 0,Racp~ 97.4andRaci~ 473.7The asymptotic relations for the cylinder suggest that for the square-section ( H / W = 1)rectangular parallelepiped, Racp (H/L) 4 and Raci (H/L) 4 should approach constant valuesas H/L --->O.
Extrapolating the values in Table 4.8 gives the following estimates for theseasymptotes:NATURAL CONVECTION4.47Critical Rayleigh Numbers Ra~p and Ra~; for Horizontal Rectangular Parallelepiped Cavities Having aPerfectly Conducting Wall (Ra~p) or an Adiabatic Wall (Ra~i) [31-33]TABLE 4.8H/L0.125W/L0.1250.250.501.001.502.002.503.003.504.004.505.005.506.006.5012.000.25RacpRaci3,011,718333,01370,04037,68939,79836,26237,05835,87536,20935,66435,79435,48635,55635,38035,45135,193Rac~9,802,9601,554,480606,001469,377444,995444,363457,007473,725494,7410.5Racp203,16328,45211,96212,54011,02011,25110,75710,85810,63510,66610,54410,57110,49910,51810,426638,754115,59664,27153,52950,81650,13650,08850,4101.00RacpRa~17,3075,2626,3414,5244,5674,3304,3554,2454,2614,1864,1964,1584,1654,11848,17814,61511,3749,8319,3129,0998,9802.00Rac~Racp3446327027892754262226092552254525022498248024472453697451383906363434463558Ra~Racp2276222221212098205720442009200119891984196737742754253123602286H/L3.00Ra~i3.003.504.004.505.005.506.006.5012.004.00Ra~p200419781941192718971888187918711855Raci5.00RacpRa~6.00Ra~p12.00Rac~Ra~p1797178917681992RacpRaci25572337189418781852184218331826180821742101227020822111203718101803178320081741TABLE 4.9 Critical Rayleigh Numbers Race and Rac; at Different Values of D/L for Horizontal Circulary CavitiesHaving a Perfectly Conducting Wall (Ra~p) or an Adiabatic Wall (Ra~) [27]iRaceRaci0.40.50.71.01.4234151,20051,80066,60023,80021,3008,4208010377043502650254022602010190018801830asH-~--~ 0,Ra~p~ 2350andRa~i17081708~ 710For the limit H/L equal to infinity, Daniels and Ong [70] have obtained the following valuesfor Racp: 8955 at W/L = 0.5; 2944 at W/L = 1; 1870 at W/L = 2; and 1719 at W/L = 4.For opaque fluids contained in rectangular parallelepiped cavities with finlike walls (seeTable 4.6, entry 5), Catton [32, 33] has calculated Rac as a function of L/H, L/W, and the wallthermal admittance Car, defined by4.40CHAPTER FOUR107-L-~ : 8i0 r -4~o4oiooo-~ : 44~,oIOooCor106ILi0 7L-~-:6o~° ~o~co,4oI06"Eo,-,~orr"CorI0 5,0%105 -io 5i2[416I8i004_I2618I00.0n"Cor1041i6JB141I68/__~..~,o4 I -~:'105o4,105I04040uoor",eel///1030,oo®Car1044L/WI2106 -4o'°4 ~-:2~-:3" lOsI0L/W1062jO 4L/WI060II0-I4L/WCar10424L/W68103l0l2l4L/W16_l8FIGURE 4.29 Catton's [32, 33] plots of Rac as a function of L/H, L/W, and Ca, for horizontal rectangular parallelepiped cavities with finlike walls of arbitrary conductivity.kLCar -kwb(4.82)The results are plotted in Fig.
4.29. Note that Rac ~ Raci as Ca,. ~ 0% and Rac ~ Racp as Ca,0. (Values of Rac for cases when opposing walls have different admittances are also available[32, 33].) Figure 4.30 gives similar plots taken from the data of Buell and Catton [27] for thecircular cylinder with a finlike wall (Table 4.6, entry 4), for which the wall thermal admittanceC,c is defined byCa~ -kD2k*wtb(4.83)where (for the finlike wall and an opaque fluid) k*wt= kw.No exact solutions for Rac are available when the walls are not finlike and L is finite. Butfor circular cylinder cavities with thick walls and for L/D ~ 0% Ostroumov [216] showed thatFig.
4.30 is valid, provided k*wtin Eq. 4.83 is equated to kwt, where kwt (the equivalent finlikewall conductivity of a very thick wall materialmsee Table 4.6, entry 7) is defined bykwD (D + 2b) z - D 2kw,- 2b (D + 2b) 2 + D 2(4.84)NATURALCONVECTION' ....14-For D/L< 2read curves tothe left-handoI ='°°~dl~l~1~Io , f4.492000I- I Io'900'\\k,~~.\~,_,~.read curves to lthe right-hand |1 7 0 0 6OILFIGURE 4.30 Buell's [26] plots of Rac as a function of DIL and C.c for horizontal cylindrical cavities with finlike walls of arbitrary conductivity.The effect of the radiant exchange in a cavity containing a transparent fluid is similar to thatof an increase in wall conductivity. For circular cylinders in which Th/Tc -- 1, Edwards and Sun[83] showed that for large L/D the results of Fig.
4.30 apply to radiantly transparent fluids ifk*, in Eq. 4.83 is given byk*, : kw, +8oT3 Dew4 -(4.85)~wwhere Tm = (Th + Tc)/2. Thus the effect of radiation is to raise the apparent conductivity of thewalls and thereby raise Ra~, Experiments by Hollands [141] and Cane et al. [29] confirmedthat, although technically correct only in the limit L/D ~ oo, Eq. 4.85 can be applied with little error for L/D as small as unity. They also showed that Eq. 4.85 can be applied to squarecelled (H/W = 1) rectangular parallelepiped cavities with finlike walls if D is set equal to H;that is, be redefining Car for transparent fluids contained in square cavities askLCar: b[kw '1- 8t~T3mHew/(4 - ew)](4.86)the points on Fig.
4.29 pertaining to H/W = 1 are made valid for both transparent and opaquefluids. For rectangular parallelepiped gas-filled cavities in which W >> H and Th/Tc = 1,Edwards and Sun [82] and Sun [265] showed that for large L/H the results of Fig.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
