Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 79
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5.221 should be used to obtainthe friction factor for fully developed turbulent flow in a parallel plate duct. However, use ofthe hydraulic diameter to substitute for the circular duct diameter in the Blasius equation isreasonable for the prediction of the fraction factor [45].Kays and Leung [111] analyzed turbulent heat transfer in a parallel flat plate duct for arbitrarily prescribed heat flux on the two duct walls. The fully developed Nusselt number Nullcan be obtained from the following expression:NuNull - 1 - ~,e*(5.222)where ), is the ratio of the prescribed heat fluxes on the two duct walls. The Nusselt numberNu and the influence coefficient 0* in Eq.
5.222 are given in Table 5.28 for different Re andPr numbers. It should be noted that ~,= 0 signifies that one wall is heated and the other is insulated; ? = 1 indicates that uniform heat fluxes of equal magnitudes are applied to both walls;5.66CHAPTER FIVEand 7 = -1 refers to heat transfer into one wall and out of the other wall, while the absolutevalues of the heat fluxes at both walls are the same.Bhatti and Shah [45] and Sparrow and Lin [133] have performed a comparison of Nusseltnumbers predicted using Eq. 5.222 or other equations for parallel plate ducts and the Nusseltnumber calculated using the equation for circular ducts replacing 2a with the hydraulic diameter of the parallel plate duct. It was concluded that the Nusselt number for parallel plateducts can be determined using the circular duct correlations.Analogous to circular ducts, the fully developed turbulent Nusselt numbers for uniformwall temperature and uniform wall heat flux boundary conditions in parallel plate ducts arenearly identical for Pr > 0.7 and Re > 105.
This is also true for the Nusselt number of turbulentthermally developing flow in a parallel plate duct [147].For liquid metal, when one wall of the parallel plate duct is heated and the other is adiabatic, the following empirical equation is recommended for Pr < 0.03 by Duchatelle andVautrey [148]:(5.223)NUll = 5.14 + 0.0127 Pe °'8Fully developed fluid flow and heat transfer results for rough parallel plate ducts can bepredicted using the results for rough circular ducts with the use of hydraulic diameter [45].Hydrodynamically Developing Flow. Hydrodynamically developing flow in smooth parallel plate ducts with uniform velocity at the duct inlet has been analyzed by Deissler [92] bymeans of an integral method.
The apparent friction factors fapp in the hydrodynamic entranceare presented in Fig. 5.24.0.020ttt0.015ll\f.~=1o,I•I . ~~ ~ xII3×104I5o.o~o0.0050.00004812162024~DaFIGURE 5.24 Turbulent flow apparent friction factors in the hydrodynamic entrance regionof a parallel plate duct with uniform inlet velocity [45].FORCED CONVECTION,INTERNALFLOW IN DUCTS5.67Thermally Developing and Simultaneously Developing Flow. Thermally developing turbulent flow in a parallel plate duct with uniform and equal temperatures at both walls hasbeen solved by Sakakibara and Endo [149] and by Shibani and Ozisik [150]. A discussion ofthe solution can be found in Bhatti and Shah [45]. Hatton and Quarmby [151] and Sakakibaraand Endo [149] have obtained the solution for a thermally developing turbulent flow problemin a flat duct with one wall at uniform temperature and the outer wall insulated (i.e., the fundamental solution of the third kind).
Hatton and Quarmby [151], Hatton et al. [152], andSakakibara [153] have analyzed thermally developing turbulent flow in a flat duct with uniform heat flux at one wall and the other insulated (i.e., the fundamental solution of the second kind). Ozisik et al. [154] have obtained the solution of the thermal entry region heattransfer of turbulent developing flow in a parallel plate duct with uniform wall temperature.Detailed discussions of these solutions can be found in the previously mentioned references.Few investigations have been conducted for simultaneously developing flow in parallelplate ducts. Therefore, no correlations are provided for practical usage.RECTANGULAR DUCTSRectangular ducts are also often used in the design of heat transfer devices such as compactheat exchangers.
Unlike circular and parallel plate ducts, two-dimensional analysis is requiredto obtain the friction factors and Nusselt numbers for rectangular ducts.Laminar FlowIn this section, the friction factors and Nusselt numbers for fully developed, hydrodynamically developing, thermally developing, and simultaneously developing laminar flows in rectangular ducts are presented.Fully Developed FlowVelocity Distribution and the Friction Factor. Marco and Han [155] have obtained thefully developed velocity distribution in a rectangular duct with cross-sectional dimensions 2aand 2b.
It follows:16(dp)a2u = - --~ ~where the pressure gradient~l3 (--1)(n-1)/2(cosh(n~y/2a))~ ......n31 - cosh(n~z I(n~b/2a) cos \ - ~ a ](5.224)dp/dx is related to Umas follows:Urn=---~ ~--~ [1----~--/ ......~- t a n h \[ nrcb2a l]]](5.225)The origin of the Cartesian coordinate is at the center of the rectangular duct. To avoid computational complexity, the following simple approximations have been suggested [156]:uUmax[1 y " (z)mlUmaxum(mX)(nl)m nNatarajan and Lakshmanan [157] have provided the relation for the values of rn andm = 1.7 + 0.50[*-1"45226,5227,n:(5.228)5.68CHAPTERFIVEn =where ~* =[~+ 0.3(~* - V~)for a* < 1,6for ~* > 1/3(5.229)b/a is the aspect ratio.
The exact expression for the fully developed friction factor is241 +1~1~,50~, . . .1,3,..n5However, for ease in practical calculations, the following empirical equation suggested byShah and L o n d o n [1] is used to a p p r o x i m a t e Eq.
5.230:f R e = 24(1 - 1.3553~* + 1.9467~ .2 - 1.7012~ .3 + 0.9564~ .4 - 0.2537~ .5)(5.231)Heat Transfer The fully d e v e l o p e d Nusselt n u m b e r s NUT for the case of the uniform temp e r a t u r e at four walls are a p p r o x i m a t e d by the following formula [1]:NUT = 7.541(1 - 2.610~* + 4.970~ .2 - 5.119~ .3 + 2.702~ .4 + 0.548C~.5)(5.232)For rectangular ducts with uniform t e m p e r a t u r e at one or m o r e walls, the Nusselt numbers, available in Shah and L o n d o n [1], are displayed in Table 5.30.For rectangular ducts with a uniform wall heat flux at four walls u n d e r the ~ b o u n d a r ycondition, the fully d e v e l o p e d Nusselt n u m b e r s N u m can be c o m p u t e d with the following formula [1]:NUll1 = 8.235(1 - 2.0421tx* + 3.0853tx .2 - 2.4765~t .3 + 1.0578t~ .4 - 0.18610~ .5)(5.233)For rectangular ducts with one or m o r e walls subjected to the t~ b o u n d a r y condition withthe other wall insulated, the fully d e v e l oped Nusselt numbers NUll1 are displayed in Table 5.30.TABLE 5.30 Nusselt Number for Fully Developed Laminar Flow in Rectangular Ducts With One Wallor More Walls Heating_~I,~--2a---*l1//////0~*NUTNUll1NUTNRH1NUTNumNUTNUll1NUTNUll10.00.10.20.30.40.50.60.70.80.91.02.02.55.010.0o,7.5415.8584.8034.1143.6703.3833.1983.0833.0142.9802.9703.3833.6704.8035.8587.5418.2356.7855.7384.9904.4724.1233.8953.7503.6643.6203.6084.1234.4725.7386.7858.2357.5416.0955.1954.5794.1533.842.3.408..3.0182.6022.6032.9823.5904.8618.2356.9396.0725.3934.8854.5057.5416.3995.7035.2244.8844.619.4.192..3.7032.6572.3331.4670.84308.2357.2486.5615.9975.5555.203.4.662..4.0942.9472.5981.6640.975000.4570.8331.1481.4161.64700.5380.9641.3121.6041.8542.0232.2632.4373.1853.3903.9094.2704.8612.7123.5393.7774.4114.8515.3854.8613.8233.3302.9962.7682.6132.5092.4422.4012.3812.3752.6132.7683.3303.8234.8615.3854.4103.9143.5383.2793.1042.9872.9112.8662.8432.8362.9113.2793.9144.4105.385..3.991....3.5563.1463.1693.6364.2525.385...iFORCED CONVECTION, INTERNAL FLOW IN DUCTS5.69TABLE 5.31 Fully Developed fRe, NUT,NUll1,and NUll2for Laminar Flow in Rectangular Ducts With AllFour Walls Transferring Heat [1]ix*fReNUTNUll1NUll2ct*fReNUTNum1.0000.9001/1.20.8000.7501/1.40.700~60.6000.5000.4001,60.30014.22714.26114.32814.37814.47614.56514.60514.70114.98015.54816.36817.09017.5122.9702.980~3.014~3.0773.0833.1173.1983.3833.6703.9564.1143.607953.620453.645313.663823.700523.734193.749613.790333.894564.123304.471854.794804.989893.0910.2500.20018.23319.07119.70220.19320.58520.90421.16921.58322.01922.47723.36324.0004.4394.8035.137-5.597-5.858----7.5415.331065.737696.049466.294046.490336.651076.784956.995077.216837.450837.905898.235293.022.971//70.1251/'91/101/121/151/201/500NUH22.942.932.932.942.942.942.958.235For rectangular ducts with four walls h e a t e d u n d e r the @ thermal b o u n d a r y condition, thefully d e v e l o p e d Null2 can be d e t e r m i n e d from Table 5.31 [1].
The f R e , NUT, and NUH] are alsogiven in Table 5.31 for convenience of usage.Hydrodynamically Developing Flow.Shah and L o n d o n [1] have reviewed and c o m p a r e dseveral analytical and e x p e r i m e n t a l investigations of hydrodynamically developing flow inrectangular ducts. They concluded that the numerical results r e p o r t e d by Curr et al. [158] andthe analytical results r e p o r t e d by Tachibana and I e m o t o [159] best fit the experi ment al data.The a p p a r e n t friction factors obtained by Curr et al. [158] are shown in Fig.
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