Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 48
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The first isdefined byNu = 1 + ( q i - qio)L(4.72)k(Th - Tc)A iand the second byNu = 1 + [qi+ qr -t- qw - (Clio + qro + qwo)]L(4.73)k( Th - Tc)A iwhere qr is the nonradiative heat transfer rate from the inner area A i of the hot plate, qr is thenet radiant heat transfer from that same area into the cavity, and qw is the heat transfer conducted from the area Aw of the hot plate and into the wall. The heat transfers qi0, q,o, and qwoare the respective values of qi, qr, and qw when the fluid is completely stationary and therefore behaves thermally like a solid--a situation reached in the limit Ra ~ 0. For fluids that areradiantly opaque,qr = qro = 0qio =k A i ( Th - Tc)Lqwo =kwAw( Th - Tc)LIf the fluid is transparent, qro + Clio + qwo must be determined from a combined radiativeconductive analysisnsee, for example, Hollands et al.
[143]. Such an analysis is beyond thescope of this chapter, whose function is to report the additional heat transfer associated withfree convective motion. This motion usually alters the temperature distribution in the wall4.42CHAPTER FOURfrom that which exists when the fluid is stationary. In so doing, it alters not only ql but also qwand qr; the alteration in qr and qw is not incorporated into Nu as defined by Eq.
4.72, but it isincorporated into Nu as defined by Eq. 4.73. Thus the latter (Eq. 4.73) is to be preferred, andit will therefore be the meaning of Nu used in this section.A dimensional analysis reveals that, in the most general case,(H W b kw ~T3mL ew, eh, ec, _T~cc)Nu = Nu Ra, Pr, 0,-~--, -~-, ~-, k ' - - - - - ~ '(4.74)for the rectangular parallelepiped cavity, and that for the circular cylinder cavity,Nu = Nu (Ra, Pr, O, D b kw t~T3 L-~~)L ' L' k ,------~,~-w,e.h,~-c,(4.75)The Rayleigh number is based on dimension L:R a = g ~ ( T h - Tc)L 3vtx(4.76)U n d e r certain (and probably most) conditions, the parameter lists given by Eqs. 4.74 and 4.75can be considerably shortened.
Table 4.6 lists some of the more common conditions and theshortening each permits. Of particular interest are the adiabatic wall and the perfectly conducting wall. The latter imposes a linear temperature rise from the cold plate to the hot plate,regardless of the convective strength.TABLE 4.6Conditions Under Which the List of Parameters in Eq. 4.74 or 4.75 Can Be ShortenedEntryConditionChanges in parameter list permittedName of condition or comment1.Fluid is opaqueCan drop t~T3L/k, ew, eh, e~, and Th/T~Opaque field2.4.b/D ~ 0.055.b/H <<.0.05 andb/W ~ 0.05Can drop oT3mL/k, ~, eh, e~, Th/Tc,k~/k, and b/LCan drop oT3mL/k, e~,,eh, e~, Th/T~,k~/k, and b/Lb/L and kw/k, and b/L can be dropped,but the single group kwb/kD mustbe addedb/L and k~/k, and b/L can be dropped,but the single group kwb/(kH) mustbe addedAdiabatic walls3.Fluid is opaque andkw/k << 1 or b = 0k~/k >> 16.7.Walls behave like fins andkw/(kL) > 25 Nuk~/k < Nu/5 and b/L > 0.75Can drop oTaL/k, ew, Eh, ec, Th/Tc, andkwb/kDCan drop b/LPerfectly conducting walls(obtained with finlike walls)Very thick walls (which behavelike b = oo)8.Th/T~= ICan drop Th/Tc9.H/L ~ 10 andW/L >>-10Can drop kJk, b/L, ~T3L/k, ew, eh, e~,and Th/TcRadiation effects can belinearizedExtensive plates (for whichheat transfer between wallsand fluid is unimportant)D/L ~ 40, or W/L >>-10 andH/L >>-40Can drop kw/k, b/L, oT3mL/k, ew, eh, ec,Th/Tc, and either D/L or W/L andH/L10.Perfectly conducting wallsFinlike walls on a cylinderFinlike walls on a rectangularparallelepipedVery extensive plates (forwhich extent of walls isunimportant)NATURAL CONVECTION4.43The Conduction Layer M o d e lThe concept of surrounding the surfaces by a layer of stationary fluid, called the conductionlayer, is useful for the present enclosure problem as well as for the external and open cavityproblems.
Unless the conduction layer thickness is greater than the cavity dimensions, a central region is produced (Fig. 4.26a and b), which experience has shown takes up a nearly uniform temperature; this region can therefore be modeled as isothermal. Once the thicknessesof the conduction layers have been specified, finding the heat transfer and the temperature Tcrof this central region is a relatively straightforward heat conduction problem.The conduction layer thickness A on each individual surface of the enclosure may be calculated using the equation A = X/Nux, where X is the characteristic dimension used for thatsurface and Nux is the Nusselt number on that surface calculated using the methods previously discussed; that is, in calculating the Nusselt number for a particular surface, one assumesthat the surface is immersed in an infinite fluid of uniform temperature Tcr.
Since Tcr dependson the conduction layer thickness, the method will, in fact, require some iteration to find T,;an initial guess for Tcris required. For the side walls (as opposed to the plates), we take for thesurface temperature the average of Th and T~ Once calculated, the appropriate conductionlayers are applied to all the surfaces, where they are all treated as solids of conductivity equalto the fluid conductivity. The remaining core fluid is treated as material of infinite conductiv-r~Coldplate///0'o'layer/interiorexteriorsolidsolid(a)(b)Z~tc[--T cl\\\/,\\\\\\YTcr-II[ ' x , , \ \ ' \ \ \"x,~ \ \ \ ~ \ \ \ \ " / , X , \ \ \ \ \ \ \ ~[-I-Athk--ThD_1-I(c)FIGURE 4.26 The conduction layer model: (a) conduction layer growth onplates and wall for a cavity without interior solids; (b) similar growth for a cavitywith an interior solid; (c) conduction layer model applied to a horizontal cavityhaving 0 = 0 and D >> L.4.44CHAPTERFOURity, and from a simple conduction solution, the resultant heat transfer on each plate can be calculated, as required.
(Also calculated from the conduction analysis is a new value of T~r, to becompared to the previously assumed value in the iterative scheme.) If the conduction layersoverlap to the degree that the central region disappears, the heat transfer may simply beequated to that for pure conduction across the fluid.These ideas can be illustrated by considering a cylindrical cavity with D >> L and 0 = 0 (Fig.4.26c).
The conduction layer thicknesses on the plates are found to be Ah = [VOt,[{g~(Th- Tcr)}]l/3/CUon the hot plate and A~ = [vtz/{gl3(Tcr- T~)}]I/3/CVt on the cold plate. By symmetry, Tcr= (Th +Tc)/2, SO that each conduction layer is in fact of equal thickness A = 21/3[Vff./{g~(Th - Tc)}]x/3/CVt.Since the central core offers no thermal resistance and since D >> L, the combined unit areathermal resistance of the fluid is 2A/k, so that q"= k(Th - Tc)/2A and Nu is found to be given byNu = CUt2-4/3 Ra 1/3.
As Ra is decreased, the conduction layers thicken until at some particularvalue of Ra they touch in the middle of the layer (that is, until 2A = L) and Nu becomes unity.According to the model, Nu remains at unity for smaller values of Ra. Tnus the conductionlayer model prediction for this problem isNu = [1, C,v2 -4,3 Ral/3]max(4.77)This prediction will be compared with measurement in the next section.As yet, the conduction layer approach has only been tested quantitatively on those problems in which the influence of the side walls is unimportant.
Even for these problems themodel has met with only mixed success in closely predicting the heat transfer. However, itdoes predict the correct trends and the correct asymptotes, it is useful in correlating experimental data, and does afford a simple physical understanding to problems that, when viewedfrom a different perspective, often appear very complex. The practitioner may find it usefulfor problems for which there is insufficient information from other sources.Horizontal Rectangular Parallelepiped and Circular Cylinder CavitiesCavities Extensive in the H o r i z o n t a l Direction (H >> L a n d W >> L, or D >> L). This section deals with situations covered by entry 10 in Table 4.6, with the additional proviso thateither 0 = 0 or 0 = 180 °.When 0 = 180 °, the hot, light fluid lies above the cold, heavy fluid, so the stationary fluidlayer (in which there is no fluid motion) is inherently stable, and Nu = 1 for all Ra.
(In termsof the conduction layer model, for 0 = 180 ° both the conduction layers are infinite, so the conduction layers always overlap, and Nu = 1.)In the 0 = 0 ° orientation, hot, light fluid lies below the cold, heavy fluid, so the stationaryfluid layer is inherently unstable. Despite this inherent instability, the fluid remains stationaryprovided Ra is less than a "critical Rayleigh number" denoted by Rat. The value of Rat forthis particular geometry is 1708. For Ra > Rac, the instability leads to a steady-state convective motion, the form and strength of which depends on both Ra and Pr. For Ra only slightlygreater than Rao it consists of steady rolls of order L in size, but as Ra is further increased,more complex flow patterns are observed, and eventually the flow becomes unsteady.
At veryhigh Ra it becomes fully turbulent. The heat transfer characteristics reflect the existence ofthese various flow regimes: for Ra < Rac the fluid is stationary, so Nu is unity; the cellularmotion initiated at Rac produces a sharp rise in Nu with Ra, which ultimately becomes asymptotic to the relation Nu ~= Ra 1/3 at very large Ra.For 0 = 0 °, the recommended equation [140] for Nu is:N u = 1 + 1 - 1708 "Ra 1/3 1-1n(Ral/3/k2)Rak~+2k2+E J[ ( )Ra 1/3_ 158031[( ) ](4.78)where square brackets with dots indicate that only positive values of the argument are to betaken, i.e.,NATURAL CONVECTION4.45[X]" = (IX1 +X)2(4.79)X being any quantity. Values of the parameters kl and k2, both functions of Pr, are tabulatedin Table 4.7 for several values of Pr.
This table also cites the experiments from which the values were inferred, and gives the range in Ra over which Eq. 4.78 has been tested for each Pr.The following equations fit the dependence of kl and k2 on Pr exhibited in Table 4.7.1.44kl = 1 + 0.018/Pr + 0.00136/Pr 2(4.80)k2 = 75 exp(1.5 Pr -'/2)(4.81)The form of Eq. 4.80 resulted from an approximate analysis [118], but the constants in thedenominator have been chosen to fit the values given in Table 4.7 for Pr = 0.7 and Pr = 0.024,and they are therefore based on limited data. Caution is advised in using this equation whenPr < 0.7, except when Pr = 0.024.
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