Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 75
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5.129 through 5.144, the coefficients 0 and • in various combinations of subscripts andsuperscripts can be found in Tables 5.16 through 5.19.Simultaneously Developing Flow. For the four fundamental thermal boundary conditions,the solutions to simultaneously developing velocity and temperature fields in concentricannuli with r* = 0.1, 0.25, 0.50, and 1.0 and Pr = 0.01, 0.7, and 10 have been obtained by Kakaqand Yticel [104]. Presented in Tables 5.21 to 5.23 are the results for Pr = 0.7. The results forPr = 0.01 and Pr = 10 have also been tabulated in Kakaq and Yticel [104].Unlike thermally developing flow, the superposition method cannot be applied directly tothe simultaneously developing flow because of the dependence of the velocity profile on theaxial locations.
However, certain influence coefficients are introduced to determine the localNusselt number for simultaneous developing flow in concentric annuli with thermal boundary conditions that are different from the four fundamental conditions; the influence coefficients 0* through 0*2, determined by Kakaq and Yticel [104] are listed in Tables 5.24 and 5.25.Several examples of the application of the influence coefficients and fundamental solutions are detailed in the following paragraphs.The fundamental solution of the first kind, presented in Table 5.21, is only valid if one ofthe duct walls is at the same temperature as the entering fluid. When the duct walls are maintained at uniform and equal or unequal temperatures Ti and To, the local Nusselt numbersNux,~ and Nux, o at the two walls can be determined from the following [1]:Nux,, 1- [(To- Te)/(T,- Te)]0*Nu(x~],1 - [(To- Te)/(Ti- Te)]0*(5.145)5.45FORCED CONVECTION, INTERNAL FLOW IN DUCTSTABLE 5.21 Fu ndamental Solution of the First Kind for Simultaneously Developing Flow in Concentric A nnul a rDucts for Pr = 0.7 [104]N U x, iiN U x , oiNu(1)x, oo0.000050.00010.00050.00100.00250.010.050.1~61.93039.82019.17014.60010.5107.0855.4655.0304.892-----0.10662.76293.37493.522856.31036.56017.70013.1109.0605.6404.1083.6583.5180.14853.81214.68244.88810.000050.00010.00050.0010.0050.010.50.1oo58.85037.61518.14013.4407.4846.1264.5764.1404.000-----0.11523.13213.83924.000058.85037.61518.14013.4407.4846.1264.5764.1404.0000.11523.13213.83924.0000r*x*Nu °!.x,uN @x, oz) •Nu (1)x, ooNu(~!x, zor*X*0.100.000050.00010.00050.0010.0050.010.050.168.03046.99026.96022.02015.03013.33011.16210.56710.450-----0.06492.34362.93073.097057.45036.86017.48012.9106.6905.3103.8563.3533.095--~--0.24737.60649.712510.46030.500.000050.00010.00050.00100.00250.010.050.1oo63.50041.70021.31016.66012.4608.8707.0996.6266.471-----0.08432.53343.12223.266956.99036.69017.62013.0008.9305.4003.8593.3943.267-----0.17724.93986.15066.47131.000.25(1)(1)Nu(~!x, louTABLE 5.7.2 Fundamental Solution of the Second and Third Kinds for Simultaneously Developing Flowin Concentric A n n u l a r Ducts for Pr = 0.7 [104]r*x*Nu(2!.X, llN u (2)x, o oNu(3.).X, ilN u x,(3)o or*x*Nu(2.).x,uN u x,(2)o oNu(3!.Nu (3)x, ltx, o o0.100.000050.00010.00050.0010.0050.010.050.1o.91.41064.67033.24026.35016.89014.63012.04311.84011.90082.51055.52024.30017.6609.0147.0444.9694.8414.83468.03046.99026.96022.02015.03013.33011.50011.41611.56057.45036.86017.48012.9106.6905.3134.0994.0454.1130.500.000050.00010.00050.00100.00250.100.050.1oo83.34058.64025.90019.24013.3958.5006.3516.1906.18181.37054.87024.49017.86012.0907.2505.1885.0445.03661.93039.82019.17014.60010.5107.0855.7775.7345.73856.31036.56017.70013.1109.0605.6394.4054.3784.4290.250.000050.00010.00050.00100.00250.010.050.1,o87.59060.17027.87021.16015.22010.1907.9317.7597.73582.05055.24024.37017.72011.9407.1005.0464.9154.90463.50041.70021.31016.66012.4608.8707.4127.3577.37056.99036.69017.62013.0008.9305.3974.1424.0844.2321.000.000050.00010.00050.010.0050.010.050.1o.83.62056.22024.88018.2709.6017.6315.5425.3875.38483.62056.22024.88018.2709.6017.6315.5425.3875.38458.85037.61518.14013.4407.4846.1264.8904.8474.86058.85037.61518.14013.4407.4846.1264.8904.8474.860,5.46CHAPTER FIVETABLE 5.23 Fundamental Solution of the Fourth Kind for Simultaneously Developing Flow in Concentric AnnularDucts for Pr = 0.7 [104]r*x*Nu(4!.N u (4) .N u (4)0.100.000050.00010.0050.0010.00250.010.050.191.41064.67033.24026.35016.89014.62611.78010.99710.450~~D~~--2.0910-2.7783-3.96082.51055.52034.30017.6609.0147.0444.8544.3523.1110.000050.0010.0050.00100.00250.010.050.187.59060.17027.87021.16015.22010.1907.7037.0346.471D~~~-~2.1972.90063.26710.25x,ux, otx, oo87.590060.17024.37017.72011.9407.1004.8844.3213.267Nu(4!r*x*Su~x,4~0.500.000050.00010.0050.00100.00250.010.050.186.34058.64025.90019.24013.3958.4976.1365.5024.8900.000050.00010.00050.0010.00250.010.050.1oo83.62056.22024.88018.2709.6017.6315.3394.7194.000x, to5.67658.206410.45921.000.09393.79745.31396.4714Nux, o1 -[(Vi-Te)/(Vo-N U(4)x,oiS u (4)....81.37054.87024.49017.860Nu(4!x,,om12.0900.07522.31503.09953.52117.2495.0004.3993.5180.07523.00984.12454.89120.05032.54533.45714.000083.62056.22024.88018.2709.6017.6315.3394.7194.0000.05032.54533.45714.0000Te)]0~'Nu(x~o)o - 1 - [(Ti - T e ) / ( T o - Te)]0*(5.146)w h e r e N u x,,,(17.and r.T~ u ....(1) are available in Table 5.21 and 0", 0~, 0~', and 0* are listed in Table 5.24.The f u n d a m e n t a l solution of the s e c o n d kind p r e s e n t e d in Table 5.22 is valid in the case ofo n e duct wall's b e i n g adiabatic, that is, either q~" = 0 or q" = 0.
H o w e v e r , for b o t h duct wallss u b j e c t e d to u n i f o r m and e q u a l or u n e q u a l wall fluxes q;' and qo'; the local N u s s e l t n u m b e r sN u x , i and N u x , o at the two walls can be d e t e r m i n e d f r o m the following e x p r e s s i o n [1]:Nux,~Nu(2).....-1(5.147)1 - (qo'/qi" " )05*Nux, o _1N u ....(2) - 1 - (qi"'/qo" )06*(5.148),,T (1)w h e r e N u (2!x,,and l~Ux, oo are t a k e n f r o m Table 5.22 and 0", 0~, 0~, and 0* are listed in Table 5.24.T h e third e x a m p l e is for the case of u n i f o r m h e a t flux at the o u t e r wall and u n i f o r m t e m p e r a t u r e on the i n n e r wall, t h a t is, q~ = qo" at r = ro and Tw = Ti at r = ri. T h e local Nusselt n u m bers at the two walls are d e t e r m i n e d f r o m the f u n d a m e n t a l solutions of the third and f o u r t hkinds f r o m Tables 5.22 and 5.23 and the influence coefficients f r o m Table 5.25, which aregiven asNux, i-_-NUx,(3)ii1 -[(q'~Dh/k)/(T~-- 1 - [(T i - - T e ) / ( T ONux, o _(4)N u ....--Te)]0~-T e ) ] 0 ff(5.149)11 --[(T i - Tel/(qo'Dh/k)lOT2(5.150)If Tw = To at r = ro and q~ = q;' at r = ri, the local Nusselt n u m b e r s Nux,/and N u x , o at the twowalls are given byFORCED CONVECTION, INTERNAL FLOW IN DUCTS5.47TABLE 5.24Influence Coefficients from F u n d a m e n t a l Solutions of the Third and F o u r t h Kindsfor S i m u l t a n e o u s l y D e v e l o p i n g F l o w in C o n c e n t r i c A n n u l a r D u c t s for Pr = 0.7 [104]r*x*0*0*0]0*0]0~0.100.000050.00010.00050.0010.0050.010.050.1oo~~~-~0.00540.48850.84421.00000.01910.02730.05950.08910.19960.29300.71670.91841.0000m~~~-0.00110.26980.69081.00000.00220.00330.00860.01360.04600.08490.44340.78921.00000.01600.2280.05870.09440.30600.52891.28551.37051.38350.00150.00200.00440.00640.01650.02560.05380.05650.05620.220.000050.00010.00050.00100.00250.010.050.1oo~~m~~0.00560.49960.85961.00000.01690.02410.05300.07620.12460.27440.71820.92621.0000-~~~-0.00200.35880.79091.00000.00440.00660.01590.02420.04400.12630.54740.86041.00000.01360.01870.04330.06670.12090.32420.74430.78970.79320.00310.00430.00960.01400.02380.05650.11890.12490.12500.500.000050.00010.00050.00100.00250.010.050.1oom-~~~0.00510.49220.86241.00000.01420.02020.04490.06520.10810.24810.70580.92651.0000~m~~~0.00300.49220.82801.00000.0720.01060.02450.03650.06370.16740.70580.89711.00000.01130.01530.03360.05060.08870.22520.49790.52700.52880.00510.00700.01600.02350.04000.09600.20370.21470.21601.000.000050.00010.00050.0010.0050.010.050.1~,---~~0.00410.46350.85121.00000.00870.01380.03390.05030.13210.21010.47450.91651.0000NN-~~0.00410.46350.85121.00000.00870.01380.03390.05030.13210.21010.47450.91651.00000.00640.00950.02340.03540.09510.15120.32570.34270.34600.00640.00950.02340.03540.09510.15120.32570.32470.3460Nux, i-Nu(x~li - 1Nux, oN,., ....•(3)---[(To-T.1e)/(qi'Dh/k)]Oll"*1 - [ ( q ; ' D f l k ) / ( T o - Te)]0~'l1-[(q~'Ddk)/(To-Te)]O*o(5.151)(5.152)where N,.,x,.
( 3 )oo a n d Nux,. ( 4 )ii a r e f o u n d in T a b l e s 5.22 a n d 5.23. T h e t e r m s 0 " , 0"0, a n d 0"1 a r e l i s t e din T a b l e 5.25.I t s h o u l d b e n o t e d t h a t t h e f u n d a m e n t a l s o l u t i o n o f t h e t h i r d k i n d , w h i c h is p r e s e n t e d inT a b l e 5.22, is r e s t r i c t e d in t h e c a s e o f u n i f o r m t e m p e r a t u r e ( d i f f e r e n t f r o m T~) a t o n e w a l l w i t h5.48CHAPTER FIVETABLE 5.25Influence Coefficients from Fundamental Solutions of the Third and Fourth Kindsfor Simultaneously Developing Flow in Concentric Annular Ducts for Pr - 0.7 [104]r*x*0~'0*0"0"0"0~20.100.000050.00010.00050.0010.0050.010.050.1.0.00020.00040.00180.00370.01880.03870.22330.5080---~~0.00010.02170.0944-0.00010.00020.00040.00230.00510.04250.1374.---~~0.00030.11020.6352.1.74161.75781.96082.21463.24523.98063.06301.37220.00000.17540.17820.19560.21980.33480.43180.54600.40860.0000~~~m-0.00010.04480.18600.00010.00010.00040.00090.00230.01060.08490.2612.~~m-~0.00030.11050.3732.1.47461.44241.45361.57701.82282.50661.92580.90590.00000.35720.35620.36680.39660.46300.67000.72560.48820.0000~~~-~0.00020.08130.26120.00010.00020.00140.00270.00700.01700.14560.4486.-~~~~0.00030.10360.3620.1.21921.17441.13541.21061.36501.81521.44460.72760.~0.57520.57000.57460.61040.69200.94520.90360.55180.0000~~-m-0.00030.08810.32430.00010.00020.00100.00210.01120.02410.16830.4548.~~~-~0.00030.08810.3243.0.72240.76700.81520.87501.12021.31461.12740.62080.00000.72240.76700.81520.87501.12021.31461.12740.62080.00000.250.501.000.000050.00010.00050.00100.00250.010.050.1.0.000050.00010.00050.00100.00250.010.050.1.0.000050.00010.00050.0010.0050.010.050.1...0.00020.00030.00160.00330.00830.03530.21590.5168..0.00020.00030.00140.00270.00700.03070.19910.5034..0.00010.00020.00100.00210.01120.02410.16830.4548..t h e o t h e r a d i a b a t i c .
T h e f u n d a m e n t a l s o l u t i o n o f t h e f o u r t h k i n d ( T a b l e 5.23) is v a l i d w h e no n e wall is at u n i f o r m t e m p e r a t u r e (4 Te) a n d t h e o t h e r wall h a s a u n i f o r m t e m p e r a t u r e e q u a lto Te. T h e r e a d e r s h o u l d t h e r e f o r e e x e r c i s e c a u t i o n w h e n u s i n g T a b l e s 5.21-5.25.Effects of Eccentricity.In practice, a perfect concentric annular duct c a n n o t be achievedb e c a u s e o f m a n u f a c t u r e r t o l e r a n c e s , i n s t a l l a t i o n , a n d so f o r t h .
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