Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 53
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In the case of vertical cylinders, the height H of the cylinders also plays a role, and so do the two flat end faces, which are assumed to be adiabatic surfaces. Several studies have been done on this problem; Kumar and Kalam [169] have provideda summary of much of that work. For laminar flow and outside the conduction regime, theyrecommend the equationNut =0.09 Ra °278 (Oo/Oi)°34+°'329D'/D°(Z/n)°'122(Oi]L)In(Do/Di)(4.123)Once Nut has been calculated from this equation, Eq.
4.121 should then be applied. The rangeof validity of this approach should extend at least over the following ranges: Ra < 10 6, 2 <(Do/Di) < 15, 1 < H/L < 10.Region Between Concentric and Eccentric Spheres.The geometry, dimensions, andRayleigh number definition are as sketched in Fig. 4.36a, the centers of the (isothermal)4.60CHAPTER FOURspheres being distance E apart.
The flow regimes and heat transfer relations closely parallelthose for the circular cylinder. The Nusselt number is defined byNu =qLrtDiDo(Ti- To)k(4.124)which makes the NUCOND (i.e., the Nu for a stationary fluid) equal to unity when E = 0. NUCONDfor 0 < E < L is found by conduction analysis to be [M.
M. Yovanovich, personal communication]Do- D,(4.125)NUCOND = d p ( l l i ) D ° _ dp(11o)D i[(1)(11) = 1 + ~11i c o s h -1s i n h 11rl=l4DiE(4.126)( 2_~o D, ;D2- D2- 4E2)/ D 2 _ D 2 _ 4E 2V]-1sinh (n + 1)11'11° = c°sh-1+ ~o4DiE(4.127)When 0.3 ___11 < 1.2, Eq. 4.126 can be approximated to within 1 percent by (1)(11) = 0.6591"1°42;when 11 > 1.2, it can be similarly approximated by (I)(11)= (2 cosh 11- 1)/(2 cosh 11). As was thecase for cylinders, Nu is expected to be independent of E when Ra is large enough to makeA i --I- mo < L - E.
Using a modified conduction layer method, Raithby and Hollands [223]obtained an explicit relation for the Nusselt number Nu: namely Eq. 4.121 with NUCONDgivenby Eqs. 4.125-4.127 (or given by unity if E = 0) and Nut given by- - [ L ~1/4Nu/1.16Cik-~/-//""Ral/4[(Di/Do)315 _{_(Do[Di)4/5] 5/4(4.128)This equation was shown to closely fit the E =0 data of Scanlan et al. [241], which covered theranges 1.3 x 103 < Ra < 6 × 108, 5 <__Pr < 4000, and 1.25 < Do/Di < 2.5. The measurements ofWeber et al. [281] for eccentric spheres, where the displacement from the concentric positionis vertically up or down, showed that for downward displacement E had little effect on Nu for0 < E/L < 0.75, but for upward displacement with 0.25 < E/L <_0.75, the Nusselt number wasobserved to be about 10 percent higher than that given by Eq.
4.128. The same reservationsas discussed for the cylinder when NUcoND = Nut apply here.Other 3D Enclosures With Interior Solids. Warrington and Powe [278] showed that so faras the heat transfer is concerned, cubes and stubby cylinders behave similarly to equivalentspheres of the same volume. This appears to be the case for both the inner and outer bodyshape. So Eqs. 4.121, 4.124, and 4.128 appear to be applicable to other inner and outer bodyshapes as well, it being understood that Do = ( 6 V o / ~ , ) 1/3 and Di = (6Vi/x) 1/3,where Vo and Vi arethe inner and outer body volumes, respectively. Sparrow and Charmichi [258], using stubbycylinders for the inner and outer body shapes, confirmed the conduction layer model prediction that the heat transfer is independent of eccentricity E when Ra (based on inner cylinderdiameter) is greater than about 1500.Partitioned EnclosuresClassification of Partitions.Partitions are relatively thin, solid walls mounted inside theenclosure, as for example in Fig.
4.33. The partitions are "passive" in the sense that neithertheir temperature nor heat flux is prescribed. Depending on their extent and orientation, suchpartitions can have profound effect on the heat transfer. Partitions can be classified in variousways. Complete partitions run continuously from one side to another; partial partitions haveNATURALCONVECTION4.61breaks, or windows. Referring to Fig. 4.25, parallel partitions run parallel to the hot and coldplates; perpendicular partitions run perpendicular to these plates, or parallel to the side wallsas shown in Fig. 4.33. Partitions can also be either single (one partition only) or multiple (morethan one partition).Complete, Parallel Partitions.
We first consider a single parallel partition. Such a partitionwill divide the enclosure into two subenclosures. Usually it is a fair approximation to treateach subenclosure as a regular enclosure of the type discussed in the previous parts of this section. Such a strategy requires the determination of the partition's temperature, and this cangenerally be found by trial and error through an energy balance on the partition itself. (If thefluid is a gas it is very important to include radiation in the heat balancemfor a sample of suchcalculations, see Hollands and Wright [145].) The problem with such a strategy is that it inherently assumes that the partition is isothermal.
Under certain conditions, for example at veryhigh Rayleigh numbers or in the triple-paned vertical windows [287], the assumption wouldappear to introduce little or essentially zero error. On the other hand, it is known to introducesubstantial error in at least one case: the horizontal layer at near-critical conditions.Thus consider the rectangular parallelepiped of Fig. 4.25a with 0 = 0 and with a thin partition running parallel to the plates and extending the whole H by W distance between thewalls. It is further assumed that the cavity is extensive in the horizontal direction, i.e., W >> Land H >> L, and (for the moment) that the fluid is opaque to thermal radiation. Catton andLienhard [37] have treated the problem of predicting the critical Rayleigh number in such asituation, for arbitrary thickness LB and conductivity kB of the partition and for arbitraryspacings L~ and L2 between the lower plate and the partition and between the upper plate andthe partition, respectively.
(Note that L = L~ + L2 + LB.) For L1 = L2 they found that the critical Rayleigh number for each layer varied with k/kB, from 1708 at k/kB = 0 to very close to1296 at large values of k/kB. (The isothermal partition model described in the previous paragraph would predict a critical Rayleigh number of 1708.) Keeping to the restriction that L1 =L2, and for thin partitions (implying in this case that LflL < 0.2), their predicted criticalRayleigh number Rac for each layer was found to vary with the group k(L1 + L2)/(kBLB) as follows: for k(L1 + L2)/(kBLB) = 1, Ra~--- 1580; for k(L1 + L2)/(kBLB) = 3, Ra~--- 1480; for k(L1 +LE)/(kBLB) = 10, Ra~ = 1375; and for k(L1 + L2)/(kBLB) = 100, Rac--- 1305. When L1 ¢ L2, theyfound that the critical Rayleigh number for the thicker fluid layer is always bounded between1708 and 1296. Lienhard [182] extended these results to layers of fluids (like gases) that aretransparent to thermal radiation, where the partition is opaque.
In the case where L1 - L2, theeffect of radiation is to raise the upper-layer Rac, applying for the case of the limit of largek/kB, from the value of 1296 that applied for no radiation to a value intermediate between1296 and 1708, given by([[ Tc~°5331[ kR a c = 1 5 0 2 - 2 0 6 tanh In 1.636~-~-h)~46T4L 1)0.46511])(4.129)Once convection starts in either of the two layers, it drives a fluid motion in the other. Usinga certain model (the Landau model), Lienhard and Catton [299] predicted the heat transferin the Rayleigh number range slightly greater than critical. Use of Eq. 4.78 for both layers,with the Rac the one relevant to thicker layer, is also tentatively recommended.Like the single-partition enclosure, the multipartitioned enclosure may be treated by analyzing each subenclosure separately, assuming isothermal partitions, and then making energybalances on the individual partitions to determine the partition temperatures (e.g., see Hollands and Wright [145]).
But, as for the single-partition case, substantial errors can arisearound the critical condition at 0 - 0. Finding the critical Rayleigh numbers for each layer hasbeen treated by Hieber [139] and Lienhard [181] for some special cases. For the case wherethe spacing between partitions is constant, the Rac at the limit of diminishingly small kB isfound to be only 720 for the inner layers and 1296 for the layers that are next to the hot andcold plates. These should be compared to an expected value of 1708 on the basis of theisothermal partition. For further details the interested reader is referred to these papers.4.62CHAPTERFOURComplete Perpendicular Partitions. For this case, where the partition(s) are perpendicularto the plates (as illustrated in Fig.
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