Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 50
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4.29 pertaining to W >> H are valid, provided Car is redefined askL(o T3L2e(1-Sk) )-1(4.87)Car---- "~w 1 + {8b[1 - (1 - ¢)Sk]}where, for L/H > 2, Sk is given by [253]:nSk = 1.0102- 1.4388 ~ -(~__)29.4653(~)3+ 31.44(~__)4- 27.515(4.88)This method has been found [253] to predict Rac with reasonable accuracy for L/H at least assmall as 3.4.50CHAPTERFOURHeat Transfer Across Horizontal Cavities Restricted in the Horizontal Direction.
The recommended equation [140] for the heat transfer across horizontally nonextensive cavities isNu = 1 + [1---Ra-ajeac]'[kl+2(Ral/3)l-ln(Ral/3/kz)Jk2(4.89)where Rac is the critical Rayleigh number appropriate to the particular cavity, calculated bymethods outlined in the previous section, and k~ and k2 are as given by Eqs.
4.80 and 4.81. Thislargely empirical equation has been tested against experimental data for gases [29, 141,253],and liquids of various Prandtl numbers [36, 265,266] (but not liquid metals) using circular cylinder cavities (as approximated by hexagons) with 0.2 ~< D/L ~< 5, and rectangular parallelepipedcavities with 1 ~< H/L <<_10 and planforms ranging from square (H/W = 1) to long (W/H >> 1).For circular cylinder, and for rectangular parallelepiped, cavities with W = H, Eq. 4.89 generallyagrees with measurements to within 10 percent, but for rectangular parallelepiped cavities withW >> H, differences of up to 25 percent occur.
The equation from Smart et al. [253]11"})(4.90)fits the data for W >> H better than Eq. 4.89, but in contrast to Eq. 4.89, it does not have theproper asymptote, as Ra ---> ~. Equation 4.90 agrees well with data for Ra < 100 Rac andL / H = 3, 5, and 10. It is not recommended for Ra > 100 Rac.Figure 4.31 shows a plot of Eq. 4.89 for a circular cylinder cavity with perfectly conductingwalls and various values of D/L.
As is clear from the graph, the Nusselt number rises verysteeply with Ra after initiation of convection, and very rapidly approaches the value of Nu forthe horizontally extensive cavity. This behavior is consistent with the conduction layer model:at high Ra, the conduction layers on the walls at the sides are so thin that they have no effecton the heat transfer; at sufficiently low Ra, they are so thick that they overlap (even thoughthose on the horizontal plates do not), so that their presence governs the condition for a stationary fluid.HeatTransferin V e r t i c a lRectangularParallelepipedCavities:9 = 90 °Cavities with HIL > 5 and WIL >~5. In contrast to the horizontal cavity, for which there isflow only when Ra > Rac, the vertical cavity experiences flow for any finite Ra.
At small Ra,70,503020-z7,532!1103104105I 1 !111111061I I I lllllI0 ?L I I tltttlI0 e1 I AtiitlI0 9RoFIGURE 4.31 Relation between Nu and Ra for horizontal cylindrical cavities with perfectlyconducting walls, for various values of D/L, as given by Eq. 4.89 with Pr = 6.0.NATURAL CONVECTION4.51however, the velocities are small and essentially parallel to the plates, so that they contributelittle to the heat transfer, and for all practical purposes, Nu = 1.
These conditions constitutethe conduction regime. The development of the flow as Ra increases beyond this regimedepends on H/L.If H/L >~ 40, the conduction regime becomes unstable at a critical Rayleigh number Ra~,which is plotted as a function of Pr in Fig. 4.32 (from Ref. 162). Increases in Ra past Rac leadthrough a turbulent transition regime and finally into a fully developed turbulent boundarylayer regime characterized by turbulent boundary layers on each plate and a well-mixed corebetween them in which there is a vertical temperature gradient of about 0.36(Th - Tc)/H [170].If H / L <~ 40, the flow enters a laminar boundary layer regime before becoming unstable andentering the turbulent transition regime.
This laminar boundary layer regime [112] is characterized by laminar boundary layers on each plate with an essentially stationary core betweenthem: this core is nearly isothermal in the horizontal direction, but it has a positive gradient inthe vertical direction of approximately 0.5(Th - Tc)/H. The stability analyses [17] of the laminar boundary layer regime predict higher critical Rayleigh numbers than for the conductionregime plotted in Fig. 4.32. In summary, for H / L >~40, the regimes encountered as Ra increasesare first conduction, then turbulent transition, and then turbulent boundary layer; for H / L <~40 they are conduction, then laminar boundary layer, then turbulent boundary layer.104~,fAsymptoteor Pr ~ m,..
10313_n"Pr -- 12.7102II0 -I....IIIII0 iPrI0 2LI0 3FIGURE 4.32 Korpela's [162] plot of Rac governing the stability of theconduction regime in a vertical rectangular parallelepiped cavity withW / L ~>5 and H / L >~ 40. For Pr < 12.7, the instability leads to stationary,horizontal axis rolls; for Pr > 12.7, it leads to unsteady, vertically travelingwaves.The recommended correlation equations for the Nusselt number relation are based onexperimental data.
For Pr -- 0.7 (gases), and H/L > 40, the equation of Shewen et el. [250]is recommended:[I 0.0665Ral/3]211/2Nu = 1 + 1 + (9000/Re) TM(4.91)This equation has been validated for Ra < 106 and 40 < H / L < 110. For 5 < H / L < 40, theequation of E1Sherbiny et el. [84, 85] equates Nu to the maximum of three Nusselt numbersas follows:Nu = [Nuc,, Nut, Nut]max[where{ 0.104Ra°293(4.92)}3] 1/3Nuc,= 1 + 1 + (6310/Ra) 1"36(4.93)4.52CHAPTER FOURNu~ = 0.242Ra L )0.273H(4.94)Nut = 0.0605 Ra 1/3(4.95)Nuct applies to the conduction and the turbulent transition regime, Nut to the laminar boundary layer regime, and Nut to the turbulent boundary layer regime. Equation 4.92 fits the datawith a maximum deviation of about 10 percent and mean deviation of about 4 percent.
Itcompares well with computer simulations [177]. A more accurate but also more complex setof equations is also available in [84, 85]. Equation 4.92 has been validated up to Ra (H/L) 3 1.5 × 101°for H/L - 5, 20, and 40. Equation 4.92 is based on data for perfectly conducting walls,but for H/L > 10 the effect of wall properties is not expected to be important (see Table 4.6,entry 9).For fluids with Pr > 4, the recommended equations are based on the proposals of Sekiet al.
[248]. For Ra (H/L) 3 < 4 × 1012,[Nu = 1, 0.36 Pr °°51()036Ra °25, 0.084 Pr °°511Ra °3(4.96)dmaxand for Ra (H/L) 3 > 4 x 1012,Nu = 0.039 Ra 1/3(4.97)These equations have been tested for values of H/L ranging from 5 to 47.5. The middle termin Eq. 4.96 has been tested for 3 < Pr < 40,000, and the last term for 3 ~ Pr ~< 200. Equation4.97 has been tested only for Pr = 5, and may underpredict measurements by as much as20 percent. For 5 _<H/L < 10, the equations should be most accurate for adiabatic walls. ForHIL >~ 10, the plates are extensive (Table 4.6, entry 9), and the wall thermal properties are notimportant.Vertical Cavities (0 = 90 °) with U H > 2 and WIL >~5.
Except in an end region immediatelyadjacent to the two vertical plates, the flow in a cavity with L >> H is everywhere parallel tothe horizontal walls, with hot fluid in the upper half of the cavity streaming toward the coldplate and cold fluid in the lower half streaming toward the hot plate (only at very highRayleigh numbers, where turbulent eddies of a scale smaller than H are possible, will this simple flow pattern break down). The plates at temperatures Th and Tc deflect the streams intoboundary layers on each vertical surface.
The predictions of Bejan and Tien [16] for adiabaticwalls are correlated to within 8 percent by their equationNu=l+(['y1Ra2(~)8]m+[72Ral/5(-~)E/5]m}l/m(4.98)in which m = -0.386, 71 = 2.756 × 10-6, and 3'2= 0.623. The analysis is for laminar flow and hencethis equation is not recommended for large Ra (H/L) 3. Because of the dominance of the wallsin this problem, departures from the adiabatic wall conditions can be expected to have amarked effect on Nu.Rectangular cavities of practical interest are very often not isolated cells but rather members of a multicellular array, such as that sketched in Fig. 4.33.
When 0 - 0, the central planeof each partition forms an adiabatic plane of symmetry, so that each cell behaves like an isolated cell (of the type defined in the section on geometry and parameters for cavities withoutinterior solids) having wall thickness b equal to one-half the partition thickness. When 0 ~ 0,there is usually heat transfer between cells, the magnitude of which is established by the coupling parameter kwL/kb.Smart et al. [253] carried out an experimental study on multicellular arrays with air as thefluid and 0 = 90 ° and found that for Ra at least as high as 107, Eq.
4.98 fit their data providedNATURAL CONVECTION4.53Plate at TcDetail A,2b/--Plate at T hFIGURE 4.33 Sketch of a multicellular array in which many rectangularparallelepiped cavities such as sketched in Fig. 4.25 may be contained.the values of 71 and 72 were slightly altered. The altered values of 7~ and Y2, tabulated in Table4.10, depended on the conductive and radiative wall properties, as noted in the table. W h e t h e rthe changes in 71 and 72 were attributable to the multicellular array effect, the radiative effect,or both, cannot be resolved from the data.Vertical Cavities with 0.5 < I t / L < 5 a n d W / L >- 5. In this intermediate range of H/L, thelow-to-moderate Rayleigh number flow consists of a two-dimensional roll.
The problem, particularly with L / H = 1, has been the subject of many numerical studies, and indeed for the adiabatic wall case with Pr = 0.7, it has formed the basis of a "benchmark problem" [72] forcomputational fluid dynamic (CFD) codes (even though it is virtually impossible to duplicatethis situation in the real world because real fluids with Pr = 0.7 can never be properly insulated). For both the perfectly conducting and the adiabatic boundary conditions, Table 4.11gives a tabulation of Nu as a function of Ra for values of H / L of 0.5, 1, 2, and 5, as calculatedby Catton et al.
[35] (and reported by Catton [34]) for very large Pr, by Wong and Raithby[284] and Raithby and Wong [230] for Pr = 0.7, and by Le Qu6r6 [176]. The effect of Pr overValues of Y1and Y2to Be Used in Eq. 4.98 for Air-Filled Cavitiesin Multicellular Arrays [253]TABLE 4.10L/HeweheckL/kwb71 × 106"Y235555100.130.130.90.90.90.130.0650.0650.0650.90.90.0650.0650.0650.0650.0650.90.0651001664242423321.2741.3240.9701.5244.763.9520.4150.4740.5940.4300.5110.502See Fig. 4.33 for the meaning of b.4.54CHAPTER FOURthe r a n g e 0.7 < Pr < oo is seen to be quite modest.
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