Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 56
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For any selected rate of increase of temperature with time, an energy balancegives the heat that must be added through the boundaries, and Eq. 4.138 gives the corresponding wall-to-fluid temperature difference.A number of studies [50, 91,247] have discussed the thermal development of the flow priorto the quasi-steady state under uniform flux conditions Quasi-steady results for a uniform fluxapplied to the side wall of a short vertical cylinder are described by Hess and Miller [135].NATURAL CONVECTION WITH INTERNAL GENERA TIONInternal ProblemsBackground. Natural convection driven by internal heat sources is of interest in geophysics,and the heat transfer associated with such motion is important in the design of tanks in whichfermentation or other chemical reactions occur and in the safety analysis of nuclear reactorswhere a core meltdown is postulated.
The last of these applications has led to the intensivestudy of internally generating horizontal fluid layers.The connection between internal generation problems and the quasi-steady transientproblems described in the previous section permits data obtained in one area to be applied inthe other.Horizontal Fluid Layers. A uniform volumetric heat production q'" in a horizontal layerbounded above by an isothermal surface and on the sides and bottom by adiabatic surfacesis depicted in Fig.
4.40. For a stationary fluid, the Nusselt number defined in the figure isNu - 2, and the temperature difference used to construct the Rayleigh number is To- 7'1 =q"L2/2k. As Ra increases from zero, the layer remains stable and heat flow is by conductionuntil a critical Rayleigh number of 1386 is reached [167].
Thereafter convection promotes amonotonic increase in Nu with Ra. For water (2.5 < Pr < 7), and for Ra < 1012, the heat transfer data of Kulacki et al. [166-168] are accurately represented byNu=[2.0, 0.389 Ra°'228]max(4.139)The lateral extent of the layer and the boundary shape appear to play a small role in this problem until the smallest horizontal dimension of the fluid cavity is equal to or smaller than thedepth of the layer [174].Tt, isothermal//lq"."AdiabaticToq'l Lq"= q'"L Nu = (To_TI)-------~~Re= g,BL3 q'"L 2va2k(a)(b)F I G U R E 4.41t Geometry and nomenclature for some natural problems wheremotion is driven by internal heat generation within the fluid.NATURALCONVECTION4.69The transient thermal response of the horizontal layer after turning on or shutting off ofthe internal generation has been studied extensively also by Kulacki et al.
[166-168] and Keylani and Kulacki [159].When both bottom and top surfaces are maintained at constant temperatures and there isinternal generation, there is a superposition of the horizontal layer problem discussed in thesection on natural convection within enclosures and the internal generation problem previously described. These are characterized by the "external" Rayleigh number defined in thesection on natural convection within enclosures and the "internal" Rayleigh number definedin Fig. 4.40a.
The dependence of the layer stability on these parameters has been discussed byNing et al. [208]. The heat transfer at the top and bottom surfaces has been estimated forthese conditions by Baker et al. [13], Suo-Anttila and Catton [276], and Cheung [51].Other Enclosure Geometries. The steady and transient heat transfer in internally generating fluid layers bounded above by an isothermal flat surface, below by an adiabatic sphericalsegment, and on the sides by an adiabatic cylinder, as shown in Fig. 4.40b, has been measuredby Min and Kulacki [192, 193]. Kee et al. [157] present numerical heat transfer predictions fora heat-generating fluid within an isothermal sphere. Murgatroyd and Watson [202] and Watson [279] have examined the corresponding problem for closed vertical cylinders, andBergholz [18] has presented an approximate analysis for a rectangular enclosure. The quasisteady data of Deaver and Eckert [73] for convection in a long cylinder can be converted tothe corresponding internal generation problem, as already described.Convective heat transfer across two immiscible stably stratified layers, bounded above byan isothermal plate and below by an adiabatic plate, and with the bottom layer heated internally, has been measured by Nguyen and Kulacki [206].
A similar problem was earlier studiedby Schramm and Reineke [245].CONVECTION IN POROUS MEDIAA porous medium consists of a packed bed of solid particles in which the fluid in the poresbetween particles is free to move. The superficial fluid velocity fz is defined as the volumetricflow rate of the fluid per unit of cross-sectional area normal to the motion. It is the imbalancebetween the pressure gradient (VP) and the hydrostatic pressure gradient (p/~) that drives thefluid motion.
The relation that includes both viscous and inertial effects is the Forscheimerequation [47]-~P'-- ( ~ P - p~) = ~-f,+pz-~-119 I17"(4.140)where p and la are the fluid properties, K is the permeability of the medium, and )C is theForscheimer coefficient. The coefficients K and )(; are approximated by [90]d2q~3K = 150(1 - 0) 21.75dZ = 150(1 - O)(4.141)where d is the average particle diameter and ~ is the medium porosity, or volume fractionoccupied by the fluid. For Rep/(1 -qb) < 10, where Rep =pd 157I/g is the particle Reynolds number, the Forscheimer term, the term with coefficient ~, in Eq.
4.140 can be dropped, and theresulting linear relation between velocity and pressure gradient is Darcy's equation. The relations in the chapter are restricted to Darcy flow.Properties and Dimensionless GroupsHeat transfer in a porous medium depends on the thermal properties of both the fluid and thesolid. For porous media km is the thermal conductivity of the medium (fluid and solid) in the4.70CHAPTERFOURabsence of no fluid motion. Relations here are from Kaviany [155]. If the heat transferthrough the solid and fluid are assumed to act in parallel,k m = ~ k f a t .
(1 -- ~ ) k s(4.142)whereas in series1kmg)-kf(1 -g))+~(4.143)ksThe true value of km will lie somewhere above that given by Eq. 4.143 and below that from Eq.4.142. Provided k r is not very much greater than k,, km can be approximated bykm = kl-ek~i(4.144)Properties (zm and ~ are the thermal diffusivity and heat capacity ratio of the mediumdefined bykm0i'm __~ ( p G ) r + (1 - ~)(PG),(~ = ~ ( p G ) / + (1 - ~)(PG),(PG)/(4.145)where subscripts f and s refer to the fluid and solid, respectively. Typical values of porosity aregiven in Table 4.13.TABLE 4.13Values of Porosity for Porous MediaFrom Scheidegger (Data selected from Table 2.1 [155]and Table 15.1.1 [302]FiberglassSilica grainsBlack slate powderLeatherCatalyst (Fisher-Tropsch, granules only)Silica powderSpherical beadsSimple cubic packingBody-centered packingFace-centered packingWell shakenCigarette filtersBrickHot compacted copper powderConcreteCoalGranular crushed rockSoilSandSedimentary rockSandstoneLimestoneChalkChetConglomerateDolomiteShaleSiltstone0.88--0.930.650.57-0.660.56--0.650.450.37-0.490.4760.320.260.36-0.430.17-0.490.12-0.340.09-0.340.02-0.070.02-0.120.44-0.450.43-0.540.37-0.500.1-0.30.06--0.20.290.0380.170.04-0.280.05-0.210.097NATURAL CONVECTION4.71The Darcy-modified Rayleigh number, based on characteristic dimension L, is defined asfollowsRa=Ra x Da = pg~ATL3 Kpg~ATKLxlatXmL2~l,(xm(4.146)where Da = K/L 2is the Darcy number.
The heat transfer rate can be recovered from the Nusselt number, defined in Figs. 4.41 and 4.42 for the problems considered in this section.External Heat Transfer CorrelationsVertical Flat Plate (Fig. 4.41a).Based on a similarity solution [49] for Darcian flow, for anisothermal plate,Nu = 0.89 Ra 1/2(4.147)This relation fits the data of Kaviany and Mittal [156] over the range of their data, 1 < Ra <102, for polyurethane foams with ¢~= 0.98 and 0.4 to 4 pores per millimeter.Horizontal Upward-Facing Plate (Fig. 4.41b).Based on the similarity solution of Chengand Chang [48], the average heat transfer from an isothermal plate in Darcian flow is given byNu = 1.26 f~a 1~2(4.148)There appears to be no experimental validation.Horizontal Cylinder (Fig.
4.410.The boundary layer solution of Cheng, as reported byFand et al. [94], yieldsNu = 0.565 Ra 1/2(4.149)This is found to correlate their data to within about 30 percent for 0.5 < Ra < 10 2. Departuresfrom Eq. 4.149 in the low Ra region occur for oils (high Pr) because the boundary layerapproximations become invalid, and in the high Ra region because the flow is no longer in theDarcy regime. Fand et al. [94] give correlations that fit their data more precisely, but the generality of these equations is unknown.Spheres (Fig. 4.410.Cheng [46] reports the similarity solution for heat transfer from anisothermal sphere to beNu = 0.362 Ra~ 2(4.150)There appears to be no experimental validation of this relation.__IIIIIIIIii1111111IIIIIIIIIIIIIIIIIq"LNuATkr.q"LNuATk,nq"DNu=ATkmFla = pgl3ATKL~o~Ra = pgl3ATKL!~o~Ra = pgI~ATKD~(~Fla = pg[3ATKL!~o~(c)(d)(a)(b)FIGURE 4.41 Definition sketch for vertical plate (a), long horizontal strip (b), long horizontal circular cylinder (c), and short vertical cylinder with insulated ends (d), all in an isothermal porousmedium.4.72CHAPTERFOUR.,._|_dj 0/q"HT2III/~/111111111111111111////Nu.,Ra = pg~ATKH"-INu-///////~//~///~///////; Z J[ t A=H/dTI> T2...........................q"dATkmT1T2 H___~,/////////////////////(a)ha=pgl3ATKd[t(~(b)FIGURE 4.42 Definition sketch for heat transfer between extensive horizontal plates (a) and in long cavitywith isothermal side walls and insulated top and bottom.
The medium is porous.Vertical Cylinder With Adiabatic Ends (Fig. 4.41d). The heat transfer qc from the curvedboundary of the vertical cylinder in Fig. 4.41d is related to the heat transfer from a verticalplate (qp) of the same height and surface area by [194]qcqp- 1 + 0.26~L~L =ro2Lhal/'------~(4.151)qc/q, increases as Ra falls, similar to that for a continuous fluid medium. There is apparentlyno experimental validation of Eq. 4.151.Internal Heat Transfer CorrelationsHorizontal Layer, Heated from Below (Fig. 4.42a).For an extensive horizontal layerheated from below, the Nusselt number is approximately given by [87]Nu = [1, -~-lmaxfta(4.152)There is a very large scatter of experimental data from this relation [46] that has beenattributed to the effect of Prandtl number and the ratio of bead diameter to the depth of theporous layer being too large for the analysis to hold.Rectangular Cavity, Heated From the Side (Fig.
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