Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 77
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For the case of the outer wall's being heated, the semiempirical equations areas follows:Nuoo : Ao + Bo(~ Pe) n°0.05Ao = 5.26 + r---g-whereBo = 0.01848 +no = 0.78 -0.0031540.0001333r---------~r, 20.013330.000833r-------T--+r,---------5~1.8213= 1 - Pr (l~m/V)max1.4(5.167)(5.168)(5.169)(5.170)(5.171)where (l~m/V)maxcan be calculated from the relation1: 2" (--V--)max,c(5.172)An expression for (Em/V)max,c applicable to a circular duct (r* = 0) was developed by Bhatti andShah [45].
It is given by(ff~-)max,c-----0.037Re V f(5.173)In Eq. 5.173, the friction factor f can be calculated from the explicit formula given byTecho et al. [56], which is shown in Table 5.8.For a concentric annular duct with the inner wall heated, the semiempirical equationsdeveloped by Dwyer [113] are applicable:Nuii = Ai + ni( ~ Pe) niwhere0.686r-----g--(5.175)0.000043r~(5.176)0.016570.000883r----------Z----r,-------------T----(5.177)Ai = 4.63 +Bi 0.02154 =ni = 0.752 +(5.174)The values of 13for this case can also be calculated from Eqs.
5.171 to 5.173. Both Eqs. 5.166and 5.173 are valid for Pe values above the critical values. For Pr = 0.005, 0.01, 0.02, and 0.03,the critical Pe values are 270, 300, 330, and 345, respectively. For liquid metals, only the heattransfer mode for Pe < Pe~t is molecular conduction.Hydrodynamically Developing Flow.
Hydrodynamically developing turbulent flow in concentric annular ducts has been investigated by Rothfus et al. [114], Olson and Sparrow [115],and Okiishi and Serouy [116]. The measured apparent friction factors at the inner wall of twoconcentric annuli (r* = 0.3367 and r* = 0.5618) with a square entrance are shown in Fig. 5.17(r* = 0.5618), where 3] is the fully developed friction factor at the inner wall. The values of fequal 0.01, 0.008, and 0.0066 for Re = 6000, 1.5 x 104, and 3 x 104, respectively [114].5.56CHAPTERFIVE3.5Re3.0tot104lapp,iXX 10 410 4XX 104X 1042.5 Iri(,2.01.51.00~I~1~2~2~xlD hFIGURE 5.17 Normalized apparent friction factors for turbulent flow in the hydrodynamic entrance region of a smooth concentric annular duct (r* = 0.5168) [114].Having determined fapp, i from Fig. 5.17, the apparent friction factor fapp,o at the outer wallcan be determined fromfapp,ofapp~r*(1 - r .2 )r*2 _ r*Z(5.178)where r* is given byr* = r*°343(1 + r *0"657- r*)Having identified both fapp,o andcalculated as follows:fapp,i, the(5.179)circumferentially averaged friction factor can be£ro + Lpp~rilapp = ~-pp,o(5.180)ro + riThermally Developing Flow.Kays and Leung [111] present experimental results for thermally developing turbulent flow in four concentric annular ducts, r* = 0.192, 0.255, 0.376, and0.500, with the boundary condition of one wall at uniform heat flux and the other insulated,that is, the fundamental solution of the second kind.
In accordance with this solution, the localNusselt numbers Nu~,o and Nux,i at the outer and inner walls are expressed asNu~o= 1'Nux,oo"* . . . .- Vx, oqi I q o '1Nuxi='1 - Ox,i i - Ox,mi(5.181)where q~ and q7 are the uniform heat fluxes at the outer and inner walls. Both q~ and q7 arepositive whenever heat is added to the fluid and negative whenever heat is transferred out of5.57FORCED CONVECTION, INTERNAL FLOW IN DUCTSthe fluid. The Nusselt numbers Nux,oo and Nux, ii and the influence coefficients 0*x,oand 0~i aregiven by:1Nux, oo = 0 .... - 0 .
. . .Ox~,o1O x ' m i - Ox'm°= 0 . . . . -- 0 . . . .Ox'm° -- Ox'i°O x, ii __ O x, m i(5.183)4r*(x/Dh)Ox,mi = Re Pr (1 + r*)(5.184)Ox~,,i =4(x/Dh)and(5.182)N u x , ii -~ Ox, ii _ Ox, m i0 . . . . = Re Pr (1 + r*)The nondimensional temperatures Ox,oo, Ox,,, Ox, oi, and Ox,io for r* = 0.192 and 0.5 are presented in Fig. 5.18 as an example. Additional graphical results for r* = 0.192, 0.255, and 0.376are available in Kays and Leung [111].0.050.050.040.04Oz,°° 0.03#x,// 0.030.020.02= 11,0100.010Re= I0,80~70_~ -..i ro ~ . ~O=,io 0 .
0 1O030:420..-~\~ ~'2040x/D h6010J O=.ai 0.0180f0(~Re= 11,01015 240221410--,,\\3 0 , 841,II20,40rri/~I60'•'i.j'80x/D hFIGURE 5.18 0.... Ox,io, Ox,ii, Ox.oifor use with Eqs 5.182 and 5.183 for thermally developing flow in asmooth concentric annular duct with r* = 0.5 and Pr = 0.7 [111].The preceding solution is restricted to a fluid with Pr = 0.7, 104 < Re < 1.61 x 105, and 0.192 <r* < 0.5. Cross plotting and interpolation can be employed to increase the application rangeof the results in terms of Re and r*. For Pr = 0.01 and Pr = 1000, an eigenvalue solution to thefundamental problem of the second kind for four concentric annular ducts (r* = 0.02, 0.1067,0.1778, and 0.3422) can be found in Q u a r m b y and A n a n d [117].Developing Flow.Little information is available on simultaneously developing turbulent flow in concentric annular ducts.
However, the theoretical and experimentalstudies by Roberts and Barrow [118] indicate that the Nusselt numbers for simultaneouslydeveloping flow are not significantly different from those for thermally developing flow.Simultaneously5.58CHAPTERFIVEEffects of Eccentricity.Jonsson and Sparrow [119] have conducted a careful experimentalinvestigation of fully developed turbulent flow in smooth, eccentric annular ducts. Theresearchers have provided the velocity measurements graphically in terms of the wall coordinate u ÷ as well as the velocity-defect representation.
From their results, the circumferentiallyaveraged fully developed friction factor is correlated by a power-law relationship of the following type:Cf = Re n(5.185)where C is a strong function of e*, a relatively weak function of r*, and independent of theReynolds number, which is given in Fig. 5.19. A single value, n = 0.18, has been suggested byJonsson and Sparrow [119] for all r*, e*, and Re.
More details regarding the friction factors j~and fo for each of the two surfaces are also available [120]. Other investigations of fully developed turbulent flow in eccentric annular ducts have been conducted by Lee and Barrow [121],Deissler and Tayler [122], Yu and Dwyer [123], and Ricker et al. [124].0.20r*=0.2810.160.140.12C0.10O.080.06m0.02r•~-00I0.1I0.210.3I0.410.5I0.6_J0.7I0.8I0.91.0e*FIGURE 5.19 Empiricalconstant C in Eq. 5.185 [119].The effects of the eccentricity on turbulent heat transfer in eccentric annular ducts havebeen investigated by Judd and Wade [125], Leung et al.
[126], Lee and Barrow [121], and Yuand Dwyer [123] for the boundary condition of a uniform wall heat flux on the inner or outersurfaces while the other wall is insulated. The results were obtained under specific conditions.Further details can be found in the previously mentioned references.Few investigations have been conducted on hydrodynamically developing flow in eccentric annular ducts. Jonsson [120] has obtained experimental information on the pressure gradient in hydrodynamically developing flow and provided the hydrodynamic lengths Lhy/Dhfor 1.8 x 104 < Re < 1.8 × 105. These are presented in Table 5.29.FORCED CONVECTION, INTERNAL FLOW IN DUCTS5.59TABLE 5.29 Turbulent Flow HydrodynamicEntrance Lengths for Smooth, EccentricAnnular Ducts [120]Lhy/Dhr*e* = 00.50.91.00.2810.5610.750292628323850385969387891Few results that can be used in practice are available for thermally developing flow andsimultaneously developing flow in eccentric annuli.
According to the discussion in Bhatti andShah [45], the Nusselt numbers may be estimated from the corresponding results for concentric annuli (e* = 0).PARALLEL PLATE DUCTSParallel plate ducts, also referred to as flat ducts or parallel plates, possess the simplest ductgeometry. This is also the limiting geometry for the family of rectangular ducts and concentricannular ducts. For most cases, the friction factor and Nusselt number for parallel plate ductsare the maximum values for the friction factor and the Nusselt number for rectangular ductsand concentric annular ducts.Laminar FlowLaminar flow and heat transfer in parallel plate ducts are described in this section.
The friction factor and Nusselt number are given for practical calculations.Fully Developed Flow.For a parallel plate duct with hydraulic diameter D h = 4b (b beingthe half-distance between the plates) and the origin at the duct axis, the velocity distributionand friction factor are given by the following expression:u _ 3 1-UmUm = -(5.186)2l( x)-~b 2,fRe= 24(5.187)Similar to the four fundamental thermal boundary conditions for concentric annuli, thefour kinds of fundamental conditions for parallel plate ducts are shown in Fig. 5.20. The fullydeveloped Nusselt numbers for the four boundary conditions follow [1]:First kind:NUl = Nu2 = 4(5.188)Second kind:Nul = 0Nu2 = 5.385(5.189)Third kind:Nul = 0Nu2 = 4.861(5.190)Fourth kind:NUl = Nu2 = 4(5.191)5.60CHAPTER FIVEz=Oz=OIIIII=zaIIf"IIISeoond kindPlrst kind==0z=O:Wall 1II--0I'I\,IIfT.IIIIIIIThird kindF I G U R E 5.20F o u r t h kindFour fundamental boundary conditions for a parallel plate duct [2].Examples of the application of these fundamental solutions to obtain the fully developedNusselt number for a duct with three different boundary conditions follow.
The Nusselt numbers are defined asq"wjOh(5.192)Nuj = k ( T j - Tm)where ] denotes wall I or 2, and Tj is the temperature of the jth wall.Uniform Temperatureat Each Wall. When the temperatures on two walls are equal, Twl =/'wE, then NUl = Nu2 = NUT. The value of NUT is given by Shah and London [1] as follows:NUT = 7.541(5.193)When the temperatures on two walls are different, Twl ~: Tw2 , then NUl = Nu2 = 4, as shown inEq.
5.188.When the effect of viscous dissipation is considered, the following formulas developed byCheng and Wu [127] for the case T~I > Tw2are used to compute the Nusselt numbers:4(1 - 6ar)Nul - 1 - 48/35Br4(1 + 6ar)(5.194)Nu2 = 1 + 48/35BrWhen the effects of viscous dissipation and flow work are considered together for the caseof Twl = Tw2, Ou and Cheng [128] have shown that for fully developed flow, NuT = 0 and thedimensionless temperature distribution is as follows:T~-T9[(y~2]~T.---~e = -8 Br 1 - \ b ] J 'Tw-Tm27T~----~ - 35 Br(5.195/FORCED CONVECTION, INTERNAL FLOW IN DUCTS5.61Taking the fluid axial condition into account, Pahor and Strand [129] and Grosjean et al.[130] have obtained the following asymptotic formulas for the Nusselt number in the case ofTwl = Tw2:NuT =7.540(1 + 3.79/Pe 2 +.-.)for Pe >> 18.11742(1 -0.030859Pe + 0.0069436Pe 2 .
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