Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 51
Текст из файла (страница 51)
But for Pr < 0.7, the effect of Pr has b e e nf o u n d to be stronger: the effect of low P r a n d t l n u m b e r on cavities with adiabatic walls andH / L = 1 was investigated, using a C F D code, by L a g e and B e j a n [173]. T h e y f o u n d that atR a = 1 x 105, N u w e n t f r o m 4.9 at Pr = 1, to 3.35 at Pr = 0.1, and to 2.77 at Pr = 0.01.
T h e s ew o r k e r s also give a criterion to establish w h e t h e r the flow is l a m i n a r or t u r b u l e n t . E x t e n s i o nto Table 4.11 to higher R a a p p e a r s only to have b e e n m a d e for the " b e n c h m a r k " configuration (i.e., adiabatic walls, Pr = 0.7). Thus K u y p e r et al. [171] c o r r e l a t e d their C F D results bythe e q u a t i o n sN u = 0.171 R a °'282 for104 < R a < 108(4.99a)N u = 0.050 R a °341 for108 < R a < 1012(4.99b)E q u a t i o n 4.99b fits predictions o b t a i n e d using a t u r b u l e n c e model. H s i e h and W a n g [146] corr e l a t e d their high R a y l e i g h n u m b e r e x p e r i m e n t a l results on a d i a b a t i c - w a l l e d cavities by theequationsN u = 0.321andR a 0"241( H / L ) -°'°95 Pr °'°53 forN u = 0.133 R a °3°1 ( H / L ) -°°95 Pr °°53 for106 < R a < 1.4 x 107(4.100)R a > 1.4 x 107 ~ R a ~ 2 x 109(4.101)This e q u a t i o n pair was derived f r o m d a t a covering the range 0.7 < Pr < 464 and 3 < H / L < 5.H e a t Transfer in Vertical Cavities With W l L ~ 5: A n Overview.
Figure 4.34 p r e s e n t s ac o m p i l a t i o n of s o m e of the d a t a of the previous t h r e e sections in terms of N u versus H/L, withR a as a p a r a m e t e r , for adiabatic walls with ew = 0 and Pr = 0.7. T h e figure shows a p e a k thatm o v e s to lower values of H / L as R a is increased.TABLE 4.11 Tabulation of Numerically Computed Nusselt Numberfor Vertical (0 = 90 °) Rectangular Parallelepiped CavitiesHaving W/L ~> 5 and 0.5 < H/L < 5.H/L25Prandtl number0.5oo0.71oo0.70.7Reference35176, 230352302301.05~1.772.503.66~1.111.421.972.613.53~1.051.281.812.453.301.12-2.243.164.52~~~1.191.642.343.124.26----1.091.392.002.723.68Perfectly conducting wallsRa = 1 0 33 x 10 31043 x 10 41053 x 1051.001.011.071.482.513.641.051.251.752.413.404.47Adiabatic wallsRa = 1033 x 1031043 x 10 41051.001.051.302.183.8210 6--107108~--See Fig. 4.25 for meaning of symbols.1.121.502.243.144.518.8316.5230.22NATURAL CONVECTIONII -III I I I 1 --1II1 I I 1I I'!4.55I 1 1 l__lcI0.Ip_L-a.-r-r'Y-1.0~J--'I"-r---~ JiI0lLIn4 0 ooH/LFIGURE4.34 N u as a f u n c t i o n o f H / L f o r various values o f R a f o r a v e r t i c a l r e c t a n g u l a rp a r a l l e l e p i p e d cavity w i t h W/L >~ 5, Pr = 0.7, and ~w = 0 (no r a d i a t i o n effects).
F o r H / L < 0.5,the p l o t is based on Eq. 4.98; f o r 0.5 _<H / L <_.5, it is based on Table 4.11; f o r H / L > 5, it is basedon Eq. 4.92.For L / H = 4, and for cavities withadiabatic walls, Arnold et al. [9] showed experimentally that Nu is unaffected when W / H isdecreased from essentially infinity to 2.
On the other hand, Edwards et al. [80] demonstratedthat when W / H is further reduced to unity, a very substantial reduction in Nu results. For cavities with 2 <__L / H < 10 within a multicellular array (Fig. 4.33), the experimental data of Caneet al. [29] with W / H = 1 show a constant reduction factor in Nu - 1 of about 2.2 when compared to the W / H ~ oo data of Smart et al. [253]. That is, for all other factors (including Ra)constant,E f f e c t o f W I L on the H e a t Transfer in Vertical Cavities.(Nu - 1)win= 1= 0.45 + 0.05(Nu - 1)win_, o~(4.102)The reduction relation given by Eq.
4.102 has been tested for Ra <_10 6 with L / H = 2, 3, 4, and5, and for Ra < 3 x 106 for L / H = 10.Very little information exists on the effect of W on Nu at moderate to high values of H/L.Edwards et al. [80] indicate that when H / L > 0.5, provided W / L > 2, Nu is insensitive to W. Asomewhat similar insensitivity to W / L was found in the CFD study of Fusegi et al. [298] foradiabatic-walled cavities; they found that changing W / L from oo to 1 decreased Nu by only 8.3percent at most.
At higher Ra an even smaller decrease in Nu would be expected. More workis required to sort out the effect of W/L.Heat Transfer in Inclined Rectangular CavitiesDepending on the inclination 0, flow in an inclined cavity with W / H > 8can resemble that in the corresponding horizontal cavity or that in the corresponding verticalcavity; it rarely combines the characteristics of both. Consequently, with few exceptions, theNusselt number in the inclined cavity can be determined, to a reasonable approximation,from either the vertical or the horizontal Nusselt number relation, by means of simple angular scaling laws.In this section Null (Ra) will refer to the Nusselt number-Rayleigh number relation for ahorizontal cavity (as determined by methods given in the section on natural convection inthese cavities) having the same values for all the other relevant dimensionless groups as theinclined cavity at hand. Similarly, Nuv (Ra) will refer to the Nu (Ra) relation for the corresponding cavity at 0 = 90 ° (as determined by methods given in the section on heat transfer invertical rectangular parallelepiped cavities), while Nu0 (Ra) will be the sought Nu (Ra) relation at the angle 0.A n g u l a r Scaling.4.56CHAPTER FOURThe scaling laws are found to be slightly dependent on the Prandtl number; the laws willfirst be reported for Pr ~> 4 (nonmetallic liquids), then for Pr = 0.7 (gases).Cavities with Pr >~4 and WItt >_ 8.
For 90 ° < 0 < 180 °, i.e., cavities heated from above, thescaling law suggested by Arnold et al. [7],Nue (Ra) = 1 + (Nuv (Ra) - 1) sin 0(4.103)has been experimentally validated by Arnold et al. [8] for cavities with H/L = 1, 3, 6, and 12.For 0 < O < 90 ° (heating from below), two scaling laws are particularly useful: the horizontalscaling law of Clever [65],Nue (Ra) = Null (Ra cos O)(4.104)Nu0 (Ra) = Nuv (Ra sin 0)(4.105)and the vertical scaling law,for H/L _>6, the maximum of the values of Nu0 given by each is recommended, i.e.,Nu0 (Ra) = [Null (Ra cos 0), Nuv (Ra sin8O6o2OI1I246II8 I0H/L----III121416F I G U R E 4.35 Plot of the crossover angle 0c governing the transition from horizontal-like flow to verticallike flow in an inclined cavity containing nonmetallicliquids. Adapted from Arnold et al.
[8]. 0c is in degrees.0)]ma x(4.106)Equation 4.106 yields a single value of the crossover angle Oodefined so that for 0 < 0c the horizontal scaling law appliesand for 0 > 0c the vertical scaling law applies. Angle 0c isobtained by equating Nun (Ra cos 0) to Nuv (Ra sin 0) andsolving for 0. This angle (which also locates a minimum inNu0 when Nu0 is plotted against 0 with Ra held constant) isplotted as a function of H/L in Fig. 4.35, as given by Arnoldet al. [8].
Equation 4.106, with a different but very similarvertical scaling law, was validated by Arnold et al. [8] forH/L = 6 and 12 and for 104 _<Ra _<106.For 1 < H/L _<6, the variation of Nu0 over the range 0 <0 < 90 ° is found to be quite modest, and the horizontal scaling law breaks down. For this regime in H/L the followingrelations, similar to those proposed by Catton [34], are recommended:Nu0 (Ra) = Nuv (Ra sin 0)Nuo (Ra) = NuH (Ra)[ Nuv (Ra sin Oc) ] °/°~Null (Ra)0c < 0 < 90 °(4.107)0 < 0 < 0c(4.108)where 0~ is obtained from Fig. 4.35. Equations 4.107 and 4.108 are in agreement with the dataof Arnold et al.
[8] for H/L = 1 and 3. Agreement with the data and theory of Ozoe et al. [217]is not so satisfactory, but the differences are mostly within 10 percent. For 1 < H/L < 6, and for104 < Ra < 5 x 105, the Nuv (Ra) relation can be approximated by a V4-power law so that forthis range Eq. 4.107 can be approximated by [35]Nu0 (Ra) = Nuv (Ra) sin TM 0(4.109)For H/L <_ 0.25, Eq.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
