Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 76
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T h e r e f o r e , e c c e n t r i c a n n u l a rd u c t s a r e f r e q u e n t l y e n c o u n t e r e d . T h e v e l o c i t y p r o f i l e f o r fully d e v e l o p e d f l o w in a n e c c e n t r i ca n n u l u s h a s b e e n a n a l y z e d b y P i e r c y et al. [105]. B a s e d o n P i e r c y ' s s o l u t i o n , S h a h a n d L o n d o n [1] h a v e d e r i v e d t h e f r i c t i o n f a c t o r f o r m u l a , as follows:fRe=16(l_r,2)(l_r,)2x(l_r,4+Z_Se,2(l_r,)2S2~ x=n I exp[-n(o~] ) ] ) s[n([~i n h - +00](5.153)FORCEDCONVECTION,INTERNALFLOWIN DUCTSwhere(1- e,2)lr2 [(1+ r,)2S = 1 - r*2e*1 - r*a = sinh -1 S~ = sinh -1- e*2]1/25.49(5.154)(S/r*)Z = 4e.2(1 - r*)2 S 2a-13w h e r e r* = ri/ro a n d e* = e/(ro - ri). The t e r m e d e n o t e s t h e d i s t a n c e b e t w e e n t h e c e n t e r s o f t h et w o c i r c u l a r walls.
E q u a t i o n 5.153 is valid for 0 < e* < 1 a n d 0 < r* < 1.F o r e* = 1 a n d 0 < r* < 1, t h e f o l l o w i n g e q u a t i o n o b t a i n e d by T i e d t [106] c a n be a p p l i e d :16(1 - r'E)(1 - r*) 2f R e = 1 - r . 4 - 4 r ' 2 ~ ' [ 1 / ( 1 - r*)](5.155)w h e r e ~ ' is t h e s o - c a l l e d t r i g a m m a f u n c t i o n with the a r g u m e n t 1/(1 - r*). This is given b y11)2W'( l _ r, ) = ~__o( n + [1/(l _ r,)](5.156)F r o m t h e r e s u l t s o b t a i n e d by B e c k e r [107] f o r 0 < e* < 1 a n d r* ---> 1, T i e d t [106] also d e m o n strated that24f R e = 1 + 1.5e .2(5.157)It s h o u l d b e n o t e d t h a t w h e n e* = 0, t h e e c c e n t r i c a n n u l a r d u c t is r e d u c e d to a c o n c e n t r i ca n n u l a r duct.C h e n g a n d H w a n g [108] a n a l y z e d t h e h e a t t r a n s f e r p r o b l e m in e c c e n t r i c a n n u l a r ducts.T h e N u s s e l t n u m b e r s f o r fully d e v e l o p e d f l o w in e c c e n t r i c a n n u l a r d u c t s w i t h t h e t ~ a n dt h e r m a l b o u n d a r y c o n d i t i o n s a r e given in T a b l e 5.26.
F o r e c c e n t r i c a n n u l a r d u c t s w i t h b o u n d a r y c o n d i t i o n s d i f f e r e n t f r o m t h e f o u r d e s c r i b e d in t h e s e c t i o n e n t i t l e d " F o u r F u n d a m e n t a lTABLE 5.26Nusselt NumberNum and NUll2for Fully Developedr*e* = 00.010.050.10.20.250.500.750.907.8048.1178.2148.2327.8008.1118.2088.226-~-~7.4197.6087.6597.6676.5246.4736.4326.4220.020.050.100.200.300.400.500.600.700.800.900.95mmm~m----~~--mm-~--~--~~~6.3476.7777.1397.4307.4527.2676.8396.0644.7763.2730.8870.2296.2246.5826.8156.7936.3955.6884.6913.4632.1541.0020.2460.0595.8265.9565.8545.1914.2413.1982.2061.3640.7250.2990.0680.016Laminar Flow in Eccentric Annular Ducts [1,108]0.40.60.80.90.950.994.7614.3934.2274.1923.7353.2473.0242.9753.2032.6442.3842.3243.0382.4462.1712.106-~---2.9252.3052.0161.9474.9134.7014.2283.1912.2441.4840.9180.5220.2620.1040.0230.0054.5374.2893.5182.5171.6941.0800.6500.3620.1790.0700.0150.0034.1573.8493.3342.3601.5710.9900.5900.3260.1600.0620.0130.0034.1493.8273.3222.3611.5710.9880.5860.3230.1580.0610.0130.0034.1603.8333.3412.3701.5750.9890.5860.3220.1570.0610.0130.003NumN U l l 2mmmm5.50CHAPTERFIVEThermal Boundary Conditions," caution must be taken in using the superposition technique.The reader is strongly recommended to consult the literature.Turbulent FlowPresented in this section are the friction factor and Nusselt number for turbulent flow andheat transfer in concentric annular ducts.
The effects of eccentricity on the friction factor andNusselt number are also discussed.Critical Reynolds Number.For concentric annular ducts, the critical Reynolds number atwhich turbulent flow occurs varies with the radius ratio. Hanks [109] has determined thelower limit of Recrit for concentric annular ducts from a theoretical perspective for the case ofa uniform flow at the duct inlet. This is shown in Fig. 5.16. The critical Reynolds number iswithin +_3 percent of the selected measurements for air and water [109].Fully D e v e l o p e dFlow.Knudsen and Katz [110] obtained the following velocity distributions for fully developed turbulent flow in a smooth concentric annular duct in terms of wallcoordinates u ÷ and y+:2500Uo÷ = 3.0 + 2.6492 In yo+for rm < r < ro(5.158)uT = 6.2 + 1.9109 In y~for ri < r < rm(5.159)iiiiiiliIII24002300Recnt2200ro21002000I__00.2m0.4II0.60.81.0r*FIGURE 5.16 Lower limits of the critical Reynolds numbers for concentric annular ductswith uniform velocity at the inlet [109].F 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 SwhereUo+ -UuT -Ut, oY +o --andut,o(ro -5.51UUt, ir)yi ÷=VUt,o =ut, i(r -ri)V(5.160)Ut, i =The radius of maximum velocity r* in Eqs.
5.158 and 5.159 can be determined by the formulaobtained by Kays and Leung [111]. It follows:r* = rm = r,0.343(1 +ror *0"657 -r*)(5.161)A critical review of the extensive friction factor data has been made by Jones and Leung[112]. The researchers recommend that the fully developed friction factor formulas forsmooth circular ducts given in Table 5.8 be used for calculating the friction factor for concentric annular ducts by replacing 2a with the laminar equivalent diameter Dt for concentricannular ducts. The term Dt is defined by1 + r .2 + (1 - r 2*)/In r*Dr= Dh(1 - r*) 2(5.162)where r* = ri/ro; Dh = 2 ( r o - ri) and r~ and ro are the radii of the inner and outer tubes, respectively.The fully developed Nusselt numbers Nuo and Nui at the outer and inner walls of a smoothconcentric annular duct can be determined from the following relations for uniform wall heatfluxes qo' and q;' at the outer and inner walls:whereNuo-hoDhk -Nui-hiDhk - 1q~= ho(To - Tm),1--Nuoo,, ,, ,( q i ]qo )0o(5.163)Nuii,, ,, ,(qo/qi )Oi(5.164)q;" = h~(Ti - Tm)(5.165)The terms To and T/denote the duct wall temperatures at the outer and inner walls.
The temperature difference T o - Ti is given byDh[ ( 10* /7 ( 1--~0* )]To - Ti = --if- q'o" Nuoo + Nuii ] - q \ Nuii + Nuoo(5.166)The Nusselt numbers Nuoo and Nuu., as well as the influence coefficients 0* and 0* in Eqs.5.163, 5.164, and 5.166 are provided by Kays and Leung [111]. These are given in Table 5.27for wide ranges of Re and Pr and for r* = 0.1, 0.2, 0.5, and 0.8.For r* = 1, the concentric annular duct is reduced to a parallel plate duct. The applicableresults are given in Table 5.28, the simple Nu being used for the Nusselt number at the heatedwall.It should be noted that for laminar flow (Re < 2300) in parallel plate ducts, Nu is equal to5.385 and 0* is equal to 0.346 for all values of Pr.Dwyer [113] has developed semiempirical equations for liquid metal flow (Pr < 0.03) in aconcentric annular duct (0 < r* < 1) with one wall subjected to uniform heat flux and the other5.52CHAPTER FIVETABLE 5.27 Nusselt Numbers and Influence Coefficients for Fully Developed Turbulent Flow in a ConcentricAnnular Duct with Uniform Heat Flux at One Wall and the Other Wall Insulated [111]r* = 0.10Heating from outer wall with inner wall insulatedRe3= 10 4105× 10 43 × 10510 6PrNuoo0*Nuoo0*Nuoo0*Nuoo0*Nuoo0*00.0010.0030.010.030.50.71.03103010010006.006.006.006.136.4524.829.836.561.599.2143.0205.0378.00.0770.0770.0770.0760.0760.0390.0320.0260.0130.0060.0030.0026.126.126.246.507.9553.466.081.8147.0246.0360.0525.0980.00.0790.0790.0810.0810.0750.0320.0280.0230.0130.0060.0030.0026.326.406.557.8013.71341672123956851030150028500.0810.0820.0830.0770.0650.0280.0240.0210.0120.0060.0030.0026.506.607.3412.128.2320.0409.0520.01000.01780.02720.04030.07600.00.0840.0820.0820.0670.0510.0250.0220.0190.0120.0060.0030.0026.687.2010.826.471.8860.01100.01430.02830.05200.08030.012,10023,0000.0850.0820.0710.0520.0360.0220.0200.0170.0110.0060.0030.002r* = 0.10Heating from inner wall with outer wall insulatedRe = 1043105X 10 43 x 10510 6PrNuii0'~Nuii0'~NuiiO*NuiiO*NuiiO*00.0010.0030.010.030.50.71.031030100100011.511.511.511.812.540.848.558.593.5140.0195.0272.0486.01.4751.4751.4751.4721.4720.6320.5120.4120.2020.0890.0410.0170.00411.511.511.511.814.181.098.0120.0206.0328.0478.0673.01240.01.5021.5021.4751.4421.3300.4860.4070.3380.1750.0810.0390.0150.00311.511.511.713.521.8191.0235.0292.0535.0890.01320.01910.03600.01.5001.4801.4731.3231.0270.3940.3380.2860.1620.0780.0380.0150.00311.511.712.619.442.0443.0550.0700.01300.02300.03470.05030.09600.01.4601.4621.3911.0900.7600.3390.2920.2560.1520.0780.0380.0160.00411.612.317.039.0103.01160.01510.01910.03720.06700.010,300.015,200.028,700.01.4771.4101.1240.7600.5260.2940.2690.2320.1480.0770.0400.0180.004r* = 0.2Heating from outer wall with inner wall insulatedRe3= 10 4105× 10 43 × 10510 6PrNuoo0o*Nuoo0o*Nuoo0o*Nuoo0o*Nuoo0o*00.0010.0030.010.030.50.71.03103010010005.835.835.835.956.2222.529.435.560.098.0142.0205.0380.00.1400.1400.1400.1400.1400.0710.0630.0510.0260.0130.0040.0030.0015.925.926.006.207.5551.564.380.0145.0243.0360.0520.0980.00.1450.1440.1460.1460.1400.0640.0550.0460.0260.0130.0060.0030.0016.106.106.227.4012.7130.0165.0206.0390.9680.01030.01500.02830.00.1510.1510.1500.1440.1250.0550.0490.0420.0240.0120.0060.0030.0016.166.306.9011.426.3310.0397.0504.0980.01750.02700.04000.07500.00.1520.1540.1500.1310.0980.0490.0440.0390.0240.0120.0060.0030.0016.356.9210.224.680.0823.01070.01390.02760.04980.07850.012,000.022,500.00.1570.1530.1360.1020.0740.0440.0400.0350.0230.0120.0060.0030.001FORCED CONVECTION, INTERNAL FLOW IN DUCTS5.53T,~BI.I: 5.27 Nusselt Numbers and Influence Coefficients for Fully Developed Turbulent Flow in a ConcentricAnnular Duct with Uniform Heat Flux at One Wall and the Other Wall Insulated [111] (Continued)Heating from inner wall with outer wall insulatedr* = 0.2Re3= 10 4105× 10 43 × 10510 6PrNuiiOffNuiiO*NuiiO~Nuii0'~NuiiO~00.0010.0030.010.030.50.71.03.010.030.0100.01000.08.408.408.408.509.0031.238.646.877.4120.0172.0243.0448.01.0091.0091.0091.0001.0120.5200.4120.3390.1720.1200.0360.0140.0048.308.408.408.6010.164.079.899.0175.0290.0428.0617.01400.01.0281.0401.0271.0180.9430.3980.3380.2840.1510.0740.0340.0140.0028.308.308.509.7015.8157.0196.0247.0465.0800.01210.01760.03280.01.0201.0201.0250.9440.7710.3330.2860.2480.1430.0720.0350.0150.0028.308.409.0514.031.7370.0473.0600.01150.02050.03150.04630.08800.01.0381.0140.9800.7960.6000.2950.2600.2290.1370.0730.0360.0160.0048.308.9012.533.681.0980.01270.01640.03250.06000.09300.013,800.026,000.01.0200.9760.8340.7480.3740.2620.2350.2090.1350.0770.0380.0160.003r* = 0.5Heating from outer wall with inner wall insulatedRe3 x 104= 10 41053 x 10510 6PrNuoo0*Nuoo0*Nuoo0*Nuoo0*Nuoo0o*00.0010.0030.010.030.50.71.03.010.030.0100.01000.05.665.665.665.736.032.628.334.860.5100.0143.0207.0387.00.2810.2810.2810.2810.2790.1620.1370.1110.0590.0280.0130.0060.0015.785.785.785.887.0549.862.078.0144.0246.0365.0530.0990.00.2940.2940.2940.2890.2840.1420.1190.1010.0580.0280.0130.0060.0015.805.805.856.8011.6125.0158.0200.0384.0680.01030.01500.02830.00.2960.2960.2940.2890.2580.1230.1070.0920.0550.0280.0140.0060.0015.835.926.4510.324.4298.0380.0490.0960.01750.02700.04000.07600.00.3020.3020.3010.2640.2140.1110.0970.0850.0540.0280.0140.0060.0015.956.409.0022.664.0795.01040.01340.02730.05030.08000.012,000.023,000.00.3100.3040.2780.2170.1630.0980.0900.0780.0520.0280.0150.0060.001r* = 0.5Heating from inner wall with outer wall insulatedRe3 x 104= 10 41053 x 105106PrNuiiO*NuiiO*NuiiO*NuiiO*NuiiO*00.0010.0030.010.030.50.71.03.010.030.0100.01000.06.286.286.286.376.7524.630.938.266.8106.0153.0220.0408.00.6200.6200.6200.6220.6270.3430.3000.2470.1290.0590.0280.0060.0026.306.306.306.457.5352.066.083.5152.0260.0386.0558.01040.00.6320.6320.6320.6360.5980.2920.2580.2180.1210.0590.0270.0060.0026.306.306.407.3012.0130.0166.0212.0402.0715.01080.01600.03000.00.6510.6510.6560.6230.5330.2530.2250.2080.1150.0590.0280.0060.0026.306.406.8510.824.8310.0400.0520.01010.01850.02850.04250.08000.00.6590.6590.6370.5400.4300.2990.2060.1830.1140.0590.0310.0070.0026.306.759.4023.235.5835.01080.01420.02870.05400.08400.012,600.024,000.00.6540.6440.5850.4270.3330.2080.1850.1700.1110.0610.0320.0070.002TABLE 5.27 Nusselt Numbers and Influence Coefficients for Fully Developed Turbulent Flow in a ConcentricAnnular Duct with Uniform Heat Flux at One Wall and the Other Wall Insulated [111] (Continued)r* = 0.8Heating from outer wall with inner wall insulatedRe = 1043×10510 41063 x 105PrNuoo0o*Nuoo0o*Nuoo0o*Nuoo0o*Nuoo0o*00.0010.0030.010.030.50.71.03.010.030.0100.01000.05.655.655.655.756.1022.428.034.861.3100.0146.0209.0385.00.3790.3790.3790.3810.3880.2250.1920.1590.0830.0390.0190.0080.0025.705.705.705.856.9048.061.076.5142.0243.0365.0533.01000.00.3860.3860.3860.3860.3800.1910.1660.1410.0790.0390.0190.0080.0025.755.755.846.7211.1121.0156.0197.0382.0670.01040.01500.02870.00.3980.3980.3970.3900.3390.1690.1500.1290.0780.0390.0200.0090.0025.805.886.359.9523.2292.0378.0483.0960.01740.02720.04000.07720.00.4070.4060.4070.3610.2900.1530.1360.1200.0760.0400.0210.0090.0025.856.258.8021.062.0790.01020.01330.02730.05050.08000.012,000.023,000.00.4090.4070.3740.2860.2160.1360.1220.1110.0730.0400.0220.0100.002r* = 0.8Heating from inner wall with outer wall insulatedRe = 1043 x 1041051063 x 105PrNu,0*Nuii0*Nu,0*Nuii0*Nui~0*00.0010.0030.010.030.50.71.03.010.030.0100.01000.05.875.875.875.956.2022.928.535.563.0102.0147.0215.0393.00.4890.4890.4890.4850.4780.2680.2440.2000.1080.0510.0270.0100.0025.905.905.906.077.0549.562.378.3145.0248.0370.0540.01000.00.5050.5050.5050.5060.4850.2500.2120.1810.1020.0510.0270.0100.0025.925.926.036.8011.4123.0157.0202.0386.0693.01050.01540.02890.00.5150.5150.4850.4930.4450.2140.1860.1660.0970.0520.0270.0100.0025.956.006.4010.023.0296.0384.0492.0973.01790.02750.04050.07700.00.5250.5180.5040.4520.3570.1930.1720.1540.0960.0510.0290.0110.0025.976.338.8021.761.0800.01050.01350.02750.05150.08100.012,100.023,000.00.5280.5160.4680.3820.2760.1740.1600.1400.0930.0510.0300.0120.002TABLE 5.28 Nusselt Numbers and Influence Coefficients for Fully Developed Turbulent Flowin a Smooth Concentric Annular Duct With r* - 1 (Parallel Plates Duct With Uniform Heat Fluxat One Wall and the Other Wall Insulated* [111]Re = 1043 × 1041051063 × 105PrNu0"Nu0"Nu0*Nu0"Nu0*0.00.0010.0030.010.030.50.71.03.010.030.0100.01000.05.705.705.705.806.1022.527.835.060.8101.0147.0210.0390.00.4280.4280.4280.4280.4280.2560.2200.1820.0950.0450.0210.0090.0025.785.785.805.926.9047.861.276.8142.0214.0367.0214.0997.00.4450.4450.4450.4550.4280.2220.1920.1620.0920.0450.0220.0090.0025.805.805.906.7011.0120.0155.0197.0380.0680.01030.01520.02880.00.4560.4560.4500.4400.3900.1930.1700.1480.0890.0450.0220.0100.0025.805.886.329.8023.0290.0378.0486.0966.01760.02720.04030.07650.00.4600.4600.4500.4070.3300.1740.1560.1380.0870.0450.0230.0100.0025.806.238.6221.561.2780.01030.01340.02700.05080.08000.012,000.023,000.00.4680.4600.4220.3330.2550.1570.1420.1280.0840.0460.0240.0110.0026.54FORCED CONVECTION,INTERNALFLOW IN DUCTS5.55wall insulated.
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