Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 57
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4.42).The rectangular cavity of width wand height H in Fig. 4.42b has side boundaries held at different temperatures, 7"1 and T2, andis insulated on the top and bottom. The cavity is considered to be extensive in the thirddimension. Bejan [15] compiled heat transfer calculations for aspect ratios 0.1 < A < 30, andthis is reproduced in Fig. 4.43.10• Bankvall(i974)ZkHome (1975)200 ~ ~uejan and Tien [16]( ~~ ~..;-NuW~lkerand['Ho m s y T / /~ ~ 5 0l~!978)~/ J l ~1l"r~i-'~'.~.za~---cc-q" f ,0.1~ " , ~~ ~ - - ~ ' ~Bejan [15] ~, -,-, . . i1,A,,~~ , , .
,i10~,30FIGURE 4.43 Summary of heat transfer calculation for a cavity filled with afluid-saturated porous medium, from Bejan [15]. (Reprinted by permission ofPergamon Press.)NATURAL CONVECTION4.73MIXED CONVECTIONMixed convection occurs when both natural convection and forced convection play significantroles in the transfer of heat. In applications it is important to first establish whether satisfactory predictions will result by ignoring either one, or if the combined effects must be considered. Guidelines for delineating the forced, natural, and mixed convection regimes arereported for external and internal flows in the sections that follow.
Some design equationsand graphs for heat transfer in the mixed convection regime are also given. Attention isfocused on laminar flows, in which mixed convection effects are most frequently important.External FlowIntroduction. For the problem depicted in Fig. 4.44, the heat transfer by "pure" forced convection would increase monotonically with Reynolds number along the curve shown.
Theheat transfer by pure natural convection from the same surface for various Ra is denoted bythe horizontal lines in the figure. If Re is slowly increased from zero in the real problem, themeasured values of Nu would at first follow the natural convection curve, since the superimposed forced convection velocities are too feeble to affect the heat transfer. If the forced convection "assists" the natural convection, the Nu curve in Fig.
4.44 will break upward alongpath A at larger Re and approach the pure forced convection curve from above. If the flowsare "opposed," Nu passes through a minimum along path B in Fig. 4.44 and approaches theforced convection curve from below. Mixed convection occurs when the heat transfer is significantly different from that for either pure natural convection or pure forced convection.From the Nusselt number for pure natural convection, NuN, and that for pure forced convection, NUF, a rough estimate of the actual Nusselt number for a given problem isNu = [NUN, NUF]max(4.153)That is, the maximum of the two Nusselt numbers is used.
The error in this equation is oftenless than 25 percent, with the maximum deviation near the intersection point of the curves,denoted by the dot in Fig. 4.44.convectionT.>,.' ' ' ' v "eTA Natural,~Ra/ //(A)~~AssistingI / 7/ ~O/~.Forced..-pposedV"Assisting. . . .mixedconvect ionTw>Too~-'--/Opposed"mixedconvect ionconvectionRe = VL/vq"LLVg,~ (Tw-Tm)~,o L3 Gr = _RoNu: (Tw-Too)k ' R e = T ' Re='Prq" xxvgB (Tw-Too) x3ltGNux = (Tw_Too)k , Rex = -:~-, Rax =FIGURE 4.44 Definitionof terms relating to mixed convection in opposedand assisting flows.4.74CHAPTERFOURThe intersection points of the pure natural convection and pure forced convection equationalso provide valuable information on the conditions for which forced and natural convectionare equally important.
For example, for laminar flow along the heated isothermal vertical platein Fig. 4.6 if Eq. 4.33a for NUNis equated to the forced convection Nusselt number given byNUF 0.664Re 1/2 Pr 1/3(4.154)=This yields the following relation between Gr; and Re2~•Gri = ( 0-664PrU12 ) 4CtRe2(4.155)where i refers to the intersection point. For a 0.30-m (1-ft) vertical plate at 60°C (140°F)immersed in air at 20°C (68°F), the superimposed forced convection velocity that satisfies Eq.4.155 is 0.25 m/s (0.82 ft/s); under the same conditions but in water, this velocity is 0.084 m/s(0.28 ft/s).The intersection relation (see, for example, Eq.
4.155) also permits one to roughly estimatewhen mixed convection should be considered. If, for a given Reynolds number, the value ofRayleigh number Ra for a laminar problem greatly exceeds the intersection Rayleigh numberRai, forced convection effects can be ignored. If Ra is much less than Rai, natural convectioneffects may be ignored.
The change from the natural convection regime to the forced convectionregime occurs over a smaller range of Ra, for a given Re, for turbulent flow than for laminar.Vertical Plates. If Eq. 4.33a (Eq. 4.33b should be used if Ra < 104 is of interest) and Eq.4.33c are used for laminar and turbulent natural convection, respectively, and if Eq. 4.154 andNUF = (0.037Re °8 - 871.3) Pr 1/3 are used, respectively, for laminar and turbulent forced convection, then Eq. 4.155 gives the laminar-laminar (i.e., laminar forced convection and laminarnatural convection) intersection, while the laminar-turbulent intersection is given byGr;0.664=cY(4.156)R e 3/2and the turbulent-turbulent intersection byOr,:[13~-fRe~ 4(4.157)These are plotted in Fig.
4.45 as a solid line. The dashed curves are drawn at Ra - 10 Rai andRa = 0.1 Rai as estimates of the bounds on the mixed convection regime.l1I1IIITurbulentIIPr = 0.71,o~L......J.....•~!~..'@"..'-..,~..r:.~..-.:.~:- T ransmon''105i i"/11//.,,I104~V Sparrow and~"~/111//1/f/I/..<.,,,,.,'// ~.-0~'~~It G co,,t" " P " ---~". -I" "loZl / II0 4I05I0 6n=-~-- Laminar - - 4 ~ i ; t - - - -II0 7I E:]~.:~';.ib]II0 8Tur b u lent ----~-I09IIII0 IOI0 ~li0 IzGr= RaPrFIGURE 4.45 Regimesof forced, mixed, and natural convection for flow alonga vertical plate.N A T U R A L CONVECTION4.75Sparrow and Gregg [260] established by a perturbation analysis that the forced convectionNusselt number was altered by less than 5 percent by either an assisting or opposing naturalconvection ifIGrl < 0.225Re 2(4.158)for 0.01 < Pr < 10. The curve representing this equation in Fig.
4.45 agrees closely with the estimate given by the dashed lines of the forced convection boundary of the mixed convectionregime.The mean heat transfer in the mixed convection regime for assisting flows has been correlated by Churchill [55] using the equationN u = [(NUF) m +(NUN)m] 1/m(4.159)m = 3 was found to best fit the results of laminar boundary layer analyses. The same equationwas found to apply to both isothermal and uniform heat flux plates.
Oosthuizen and Bassey[214] also correlated their data with Eq. 4.159, but with m = 4; the Reynolds and Grashofnumbers in their experiments were below the ranges where boundary layer theory holds, andthey also provide measurements (but no correlation) for opposed mixed convection. Churchill [55] reviews other available data.The authors of most experimental studies and analyses have focused their attention on thelocal values defined in Fig.
4.44. Churchill [55] fitted the local Nux values for assisting flowusing Eq. 4.159 with m = 3 and with each average value on the right side of the equationreplaced by its local value counterpart. Mucoglu and Chen [201] have solved the inclined flatplate problem for uniform wall temperature and heat flux and have presented local heattransfer results for mixed convection.Horizontal Flow.For laminar flow over the upper surface of a horizontal heated plate (orover the bottom surface of a cooled plate), the center of the mixed convection regime canagain be estimated by equating the forced convection Nusselt number from Eq. 4.154 to thatfor natural convection from Eq.
4.39c (for detached turbulent convection). This results in(0.664) 3Gr; = ~, CtvR~/2= 107Re 3/2(4.160)where the terms are defined in Fig. 4.46. As a rough approximation, one would expect thatfor Gr ~< 11Re 3/2 forced convection dominates, and for Gr ~> 1100Re 3/2 natural convectiondominates.The details of the flow in the mixed convection regime have been clarified by Gilpin et al.[113].
After an initial development of the laminar forced convection boundary layer, rollswith axes aligned with the flow appear at the location marked Onset in Fig. 46. These persistuntil the end of the transition regime, marked Breakup, after which the motion appears asfully detached turbulent natural convection flow.The experiments for water, 7 < Pr < 10, revealed that the rolls first become visible at the xlocation at whichGrVOnset=KRe : LV/~,BreakupIGr - g/~ (Tw-Too)L3/v 2ikj,xF I G U R E 4.46(4.161)R e 3/2LJ-IRe, = x V / v ; Gr, = g/9 (Tw-Too)x3/v zNu =q(T.-Too)L-kNuxMixed convection on a heated horizontal plate.q('l-.-Too)xk4.76CHAPTERFOURwhere K = 100.
The stability analysis for longitudinal rolls by Wu and Cheng [288] is in goodagreement with these observations and also predicts a Prandtl number dependence of K, butChen and Mucoglu [44] have questioned the validity of these predictions. Despite the uncertainties that still exist, there is general agreement that buoyancy effects can greatly decreasethe value of Rex at which the first instability appears.To obtain an average heat transfer equation, the flow is modeled as purely forced convection up to xc and purely natural convection for x > xc; from the experiments [148] for water, Xcis given roughly by Eq. 4.161 with K = 155. Integrating the local heat transfer relations resultsin the following expression for average heat transfer:Nu = V~0.664Re lr2 P r=[29Re 1Gr2/3 , 11/3 + ( 1 - ~ ) C U R a 1/3Ra = Gr Pr(4.162a)(4.162b)minwhere the minimum of the two quantities in brackets is to be used in Eq.
4.162b. The coefficient 29 in Eq. 4.162b will likely be somewhat dependent on Prandtl number, but data are notavailable to resolve this dependence.For laminar flow above a cooled surface or below a heated surface, the presence of buoyancy forces stabilizes the flow (inhibits transition) and tends to diminish the heat transfer. Theanalysis of Chen et al. [43] predicts that natural convection will alter the local convective heattransfer by less than 5 percent if IGrl(x/L)3rZ/Re 5r2 < 0.03 for Pr = 0.7. Robertson et al. [232]show that for IGr I/Re 5r2 > 0.8 and Pr = 0.7, buoyancy may inhibit the flow so strongly that aseparation bubble may form over the surface.In turbulent flow, stable stratification significantly damps turbulence and reduces heattransfer in the vertical direction.H o r i z o n t a l Cylinders.
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