Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 72
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Its predictions are within+_5% of the available measurements.This correlation is valid for 0.5 < Pr <5000, 0.001 < 0.002 < e../Dh < 0.05, andRe > 2300. Its predictions are within+15 % of the available measurements.Bhatti and Shah [45] listed these eigenvalues and constants. On the other hand, the local Nusselt n u m b e r s for uniform wall t e m p e r a t u r e and uniform wall heat flux NUx,T and NUx,H in turbulent developing flow are nearly identical for Pr > 0.2. Therefore, the subscripts T or H ared r o p p e d in the equations in this section.The m e a n Nusselt n u m b e r Num for thermally developing flow with uniform wall t emperature or uniform wall heat flux conditions can be calculated using A1-Arabi's [95] correlation:NunNu~.C-1 + ~(5.85)X/Dhwhere Nu~. stands for the fully d e v e l o p e d Nusselt n u m b e r NUT or NUll andC=(X/Dh)°l (3000 /prl/60.68 + Re0.81(5.86)However, the thermal entrance lengths for uniform wall t e m p e r a t u r e and uniform wallheat flux are much different.
These are shown in Figs. 5.10 and 5.11, respectively.5.28CHAPTER FIVE35-'m'm I mmlmm ~ i I "1'' ' '"1i30 -Pr=0.010.004o2~"'~~,,,,~d~~~o25 -003~0.04~"'L'~,--" ' "20 -l0.060.10Pr =DA15-I00.10-50.720.03~/Jf0.023.00:l0~0.720.06 /0.04'I ' 0.01.'~3X 10 31040 004"~J3.00II2I I ==m,I46 8 105=2tm m m=,ml46 8 106ReFIGURE 5.10duct [80].Thermal entrance lengths for the turbulent Oraetz problem for a smooth circularI~=350.02~.H3°E0.03250.06-20ohPr~--150.06 /100.72 ~0.030.023.000.01I J J m=Jl03 X 103lO 4I2II, I I tatlI4268 105II4m I Itml3"j 00168 106ReF I G U R E 5.11 Thermal entrance lengths for the turbulent thermal entrance problem for asmooth circular duct with uniform wall heat flux [80].FORCED CONVECTION, INTERNAL FLOW IN DUCTS5.29For liquid metals (Pr < 0.03), when X/Dh > 2 and Pe > 500, the local and mean Nusselt numbers for the uniform wall temperature boundary condition have been proposed by Chen andChiou [96] as follows:2.41NUT - 1 + x/D----~h (X/Dh)2NUx,TNUm,T72.8NUT - 1 + x/Dh + x/Dh In(5.87)( X/Oh I(5.88)10 ]NUT = 4.5 + 0.0156Re °85 Pr °'86(5.89)When the uniform heat flux condition is applied to the duct wall, Eqs.
5.87 and 5.88 are stillused, but with NUx,H, NUm,H, and Null replacing NUx,T, NUm,T, and NUT, respectively, while Nullcan be obtained from:NUll = 5.6 + 0.0165Re °85 Pr °86(5.90)The thermal entrance length Lth, H for liquid metals has been found by Genin et al. [97]:Lth,H0.04PeDh - 1 + 0.002Pe(5.91)Simultaneously Developing Flow.The local Nusselt numbers obtained theoretically byDeissler [92] for simultaneously developing velocity and temperature fields in a smooth circular duct subject to uniform wall temperature and the uniform heat flux for Pr = 0.73 areplotted in Fig. 5.12.
It can be seen from this figure that the Nusselt numbers for two differentthermal boundary conditions are identical for X/Dh > 8.It is worth noting that the duct entrance configuration affects simultaneously developingflow [98, 99]. The local Nusselt number is different for each duct entrance configuration. Forpractice usage, Bhatti and Shah [45] suggest the following formula for the calculation of themean Nusselt number:NumC- 1+~Nu.(x/Dh) n(5.92)where Nuo.
denotes the fully developed Nusselt number Null or NUT. The terms C and n havebeen determined from the NUm,H m e a s u r e m e n t given by Mills [99] for air (Pr = 0.7). Table 5.13lists the resulting C and n of Eq. 5.92 for each configuration. Equation 5.92 may be used in thecase of the uniform wall temperature and uniform wall heat flux boundary conditions.For liquid metals (Pr < 0.03), Chen and Chiou [95] have obtained the correlations forsimultaneously developing flow in a smooth circular duct with a uniform velocity profile atthe inlet. These follow:2.4NuxNu.. - 0.88 + X/Dh1.25(X/Dh)2A51.86(X/Dh)Nux1+x/Dh+X/DhIn10 - BNu..(5.93)(5.94)where for the uniform wall temperature boundary condition,A=40 - X/Dh190'B = 0.09(5.95)5.30CHAPTER FIVE8OO700...,,Nu=, 1".Nuz.
Ht600W500Z:3z400zR e = 2 x lO s30010 s2006X 10410010 402468101214161820x/D hFIGURE 5.12 Local Nusselt numbers Nux,r and NUx,Hfor simultaneously developing turbulentflow in a smooth circular duct for Pr = 0.73 [92].and for the uniform wall heat flux b o u n d a r y condition,A =B =0(5.96)The Nux in Eqs. 5.93 and 5.94 denotes the local Nusselt n u m b e r Nux,~ or NUx,H, and Nu~. represents the fully developed NuT or NUll. E q u a t i o n s 5.93 and 5.94 are valid for 2 < X/Dh< 35 andPe > 500.Transition FlowAs seen in the previous section, flow is considered to be laminar when R e < 2300 and turbulent when Re > 104. Transition flow occurs in the range of 2300 < Re < 104.
Few correlations orformulas for computing the friction factor and heat transfer coefficient in transition flow areavailable. In this section, the formula developed by Bhatti and Shah [45] is p r e s e n t e d to compute the friction factor. It follows:Bf = A + Re1/--------~(5.97)E q u a t i o n 5.97 is applicable to the laminar, transition, and turbulent flow regions. For laminar flow (Re < 2100), A = 0, B = 16, and m - 1.
For transition flow, 2100 < Re _< 4000, A 0.0054, B - 2.3 x 10 -8, and m = -2/3. For turbulent flow, (Re > 4000) A - 1.28 x 10 -3, B = 0.1143,and m = 3.2154. Blasius's [49] formula (see Table 5.8) is also applicable for calculating the friction factor in the range of 4000 < R e < 10 s.FORCED CONVECTION,INTERNALFLOWIN DUCTSNusseltNumber Ratios for a Smooth Circular Duct with VariousEntrance Configurations for Pr = 0.7 [2]TABLE 5.13Entrance configurationsLong calming sectionSchematicsAdiabaticsurfaceq~,C0.9756 0.760,..,.,~.,,,,.,,,,,,II IiiI"~'~"''~"''tti"tttSquare entranceq~,2.4254 0.676I~tilliilliili[~ttttitf-'t t f f180° Round bend0.9759 0.700l ii I i i l il~1.0517 0.62990°R°undbend~~l90° Elbow~~ I ~ i~N~ I I i""~tt~ t ttt ti2.0152 0.6145.315.32CHAPTER FIVEHeat transfer results for transition flow are rather uncertain due to the fact that so manyparameters are needed to characterize heat-affected flow.
In the range of 0 < Pr < ~ and2100 _<Re < 10 6, Churchill [100] recommends that the following equations be used to calculatethe Nusselt number for the laminar, transition, and turbulent regimes:Nu 10= Nu~° +f3.657Nut = [4.364{ exp[(2200- Re)/365] N_~2}-sNu 2+for the uniform wall temperature boundary conditionfor the uniform wall heat flux boundary conditionNu, = Nu0 +4.8Nu0 = 6.30.079(f/2) la Re Pr(1 + pr4/5)5/6for the uniform wall temperature boundary conditionfor the uniform wall heat flux boundary condition(5.98)(5.99)(5.100)(5.101)where superscript 10 indicates transition region and Nu0 denotes an overall Nusselt numberassociated with the convection boundary condition.Kaupas et al.
[101] experimentally investigated heat transfer in transition gas flow in a circular duct at high heat flux in the range of 2 x 103 ~-- Re < 3 x 10 4.More research is needed to determine reliable friction factors and Nusselt numbers fortransition flow.CONCENTRIC ANNULAR DUCTSConcentric annular ducts are a common and important geometry for fluid flow and heattransfer devices. The double pipe heat exchanger is a simple example. In this device, one fluidflows through an inside pipe, while the other flows through the concentric annular passages.The friction factor and the heat transfer coefficient are essential for the design of such heattransfer devices.Four Fundamental Thermal Boundary ConditionsAs shown in Fig. 5.13, there are two walls in concentric annular ducts.
Either or both of themcan be involved in heat transfer to a flowing fluid in the annulus. Four fundamental thermalboundary conditions, which follow, can be used to define any other desired boundary condition. Correspondingly, the solutions for these four fundamental boundary conditions can beadopted to obtain the solutions for other boundary conditions using superposition techniques.The four fundamental thermal boundary conditions are as follows:First kind. Uniform temperature (different from the entering fluid temperature) at onewall; the other wall at the uniform entering fluid temperatureSecond kind. Uniform wall heat flux at one wall; the other wall insulated (i.e., adiabaticwith zero heat flux)Third kind.
Uniform temperature (different from the entering fluid temperature) at onewall; the other wall insulatedFourth kind. Uniform wall heat flux at one wall; the other wall maintained at the entering fluid temperature.The previously mentioned boundary conditions can be applied in both laminar and turbulentflow.FORCED CONVECTION, INTERNAL FLOW IN DUCTS5.332520Case|l!t|ttt----.r~,, ro(la)---r~= To(lb)='_LtlroI%%%15%%%%Nu%%%%Nu~lb)I0NUo(lb ) " " ~ " .
. . .)~__Nu~la)Nu(Ia)OL0.00.10.20.30.40.50.60.70.80.91.0r*FIGURE 5.13 Fully developed Nusselt numbers for uniform temperatures atboth walls in concentric annular ducts [1].Laminar FlowIn this section, the characteristics of laminar flow and heat transfer in concentric annularducts are presented, and the effect of eccentricity is discussed.Fully Developed Flow.Velocity distribution, the friction factor, and heat transfer for fullydeveloped laminar flow in concentric annular ducts are described sequentially.Velocity Distribution and the Friction Factor.
For a concentric annular duct with innerradius ri and outer radius ro, the velocity distribution and friction factor for fully developedflow in a concentric annular duct are as follows [1]:u =- ~u= = - ~rE 1 --+ 2r .2 inr2o(1 + r . 2 - 2r .2)urn= 2(1 - r .2 + 2r .2 In r*)u= 1 + r . 2 - 2r .2(5.102)(5.103)(5.104)5.34CHAPTER FIVERe= - - - 1_Oh_- ~ ),Dh = 2(ro- ri)(5.105)ri( -~xdp ) D--2-h( r2°-ror2m)fiRe:-~t(5.106)16(1 - r * ) 2f R e = 1 + r . 2 - 2r .2(5.107)rm in the preceding equations is the radius where the m a x i m u m[(i)u/br) = 0], and r* is its dimensionless form, which is defined asr.
= rm = ( l_r,2 )1/2ro 2 In (l/r*)wherevelocity achieves(5.108)The terms f and fo represent the friction factor at the inner and the outer walls, respectively.The circumferentially averaged friction factor is related to j~ and fo as follows:f= firi + forori + ro(5.109)Natarajan and L a k s h m a n a n [102] present a simple equation for f Re that is easy to use.This equation agrees with the values calculated from Eq. 5.107 within +_2 percent for r* >0.005. It follows:f R e = 24r *°'°35(5.110)Heat Transfer F u n d a m e n t a l solutions for b o u n d a r y conditions of the first, second, andthird kinds for fully developed flow in concentric annular ducts are given in Table 5.14. Thenomenclature used in describing the corresponding solutions can best be explained with reference to the specific heat transfer p a r a m e t e r s AV ~k)lj and a(k)" ' m j ' which are the dimensionless ductwall and fluid bulk mean temperature, respectively.
The superscript k denotes the type of thefundamental solution according to the four types of b o u n d a r y conditions described in the section entitled "Four F u n d a m e n t a l Thermal B o u n d a r y Conditions." Thus, k = 1, 2, 3, or 4. Thesubscript l in ..qA~k)refers to the particular wall at which the t e m p e r a t u r e is evaluated; l = i or owhen the t e m p e r a t u r e is evaluated at the inner or the outer wall. The subscript j in --qA~k)refersTABLE 5.14 Fundamental Solutions of the First, Second, and Third Kinds of Boundary Conditions for Fully DevelopedFlow in Concentric Annular Ducts [1]r*00.010.020.040.050.080.100.150.250.400.500.801.00¢~/1)~42.9951525.0509814.9120412.684719.106287.817305.973974.328093.274072.885392.240712.00000Nu~)~50.4539630.1794218.6138716.0584311.9425110.458708.341636.471395.305114.888964.230354.00000Nu¢olo)012)_ 0(,.2o)Nu~2)Nu(2o)Nu~))NU(o3o)2.666672.908342.948362.999283.018873.067513.095283.157083.267003.420773.520353.811344.00000-0.145833-0.130725-0.127945-0.124122-0.122568-0.118559-0.116214-0.110999-0.102207-0.091495-0.085513-0.071409-0.064286~54.0166932.7051220.5092517.8112813.4680611.905789.687037.753476.583306.181025.578495.384624.363644.692344.734244.778034.790984.802704.834214.860264.904754.979175.036535.236545.38462~-32.337m17.460~11.560~7.3708~5.7382m4.86083.65683.99344.05654.1135m4.23214.42934.8608F O R C E D C O N V E C T I O N , I N T E R N A L F L O W IN D U C T S5.35to the wall at which T ~ Te or q~ ¢ 0 (i.e., the active wall that participates in the heat transferwith flowing fluid).
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