Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 74
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e ,~mo!.-.-x,u+(Zoq,,(la) _x,o-Te)Ox,,o](1)k- O-"-~[(Ti - Te,~O (1)'x,o,+ (To - Te)t~(lx,)o]Nu~.~=( T ~ - T ~e l (- =I ) x. ,+u ( T o(To - T,)(1Nu~,~,-(To(To --- T ~e l O °)x, m o0<l)x.,no,~ -- (Ti - Te)Ox.miT ~e l O °)x, o o + ( T ,Te)~(lx,)oT,)(1 - ..x,,,,o,A(')~ - ( T ~ - Te)Ox.,,(5.129)(5.130)(5.131)(5.132)(5.133)TABLE .5.17 F u n d a m e n t a l Solutions of the Second Kind for T h e r m a l l y D e v e l o p i n g F l o win C o n c e n t r i c A n n u l a r Ducts (compiled from Shah and L o n d o n [1])0 (2!N u (2)0(2?x , tt0.020.000050.00010.00050.0010.0050.010.050.10.51.0~0.01150.01640.018590.024290.0270360.0341100.0383970.0697920.109008~~~-~~0.00(0)020.0015670.0052960.0366570.075872~~86.961.254.0141.8538.09333.12632.72932.70532.70532.7050.0278530.0353760.0622420.0798850.1462170.1930180.4019280.6030712.1720114.132796~0.000050.00010.00050.0010.0050.010.050.10.51.0~0.017250.026020.030340.042270.048370.0648010.0751420.1518320.246620--~~~0.0000050.0038720.0129470.0891100.184348~58.038.633.1724.2021.52118.09117.82717.81117.8110.0275540.0349900.0615200.0789200.1442010.1901250.3939120.5893342.1134444.018208~~17.811~36.54428.89416.77413.3147.9906.5784.9164.7994.7924.7924.7920.000050.00010.00050.0010.0050.010.050.10.51.00.0172540.0211940.0334860.040430.060240.0707380.1006340.1202670.2658110.447629~~~-~~0.0000120.0076420.0247940.1701970.352015~58.01947.26530.02724.9617.1214.90312.12811.91811.90611.90611.9060.0272520.0345960.0607560.0778800.1418920.1867000.3833910.5710902.0520413.843224~36.94029.21216.97613.4698.0836.6524.9614.8414.8344.8344.8340.000050.00010.00050.0010.0050.010.050.10.51.00.0211010.0263270.0435910.0538850.0868190.1058560.1659710.2087920.5289740.928974~~~-~~0.0000300.0164920.0545720.3744480.774448~47.48038.09923.15318.83812.07510.2197.9387.7647.7537.7537.7530.0266760.0338320.0592150.0757390.1368450.1790050.3583240.5235461.8038833.403883~0.000050.00010.00050.0010.0050.010.050.0233340.0293450.0498050.0624740.1054110.1319110.224075----~0.0000520.02426342.96034.23320.35116.35610.1278.4336.3530.0259940.0329190.0573140.0730540.1303270.1689840.3261570.100.250.505.400(,20)0x*0.05O(x2)oi,NU(x2,1ir*x, to0.0000970.0783390.2648081.8328393.7936240.0001080.0774330.2589311.7821953.6869580.0001160.0746210.2479441.7019673.5201500.0001180.0659680.2182901.4977923.0977930.0001050.054896x, oo36.15728.58416.58913.1647.8986.5024.5854.7414.7344.7344.73437.71329.84017.35713.7868.2756.8034.0424.9134.9054.9054.90538.66830.62617.86414.2078.5487.0275.186FORCED CONVECTION, INTERNAL FLOW IN DUCTS5.41TABLE 5.17 Fundamental Solutions of the Second Kind for Thermally Developing Flowin Concentric Annular Ducts (compiled from Shah and London [1])r*1.0x*0(2!.x, l|a(2)V x ,oiNR(2!.x, t!0(2)0(2!NR(2)x, oo0.10.51.0~,0.2948400.8284521.495118~0.0905990.6239101.290576oo6.1926.1816.1816.1810.4648511.5318812.865214~,0.1816051.2478192.581152~5.0465.0375.0375.0370.000050.00010.00050.0010.0050.010.050.10.50.0249400.0314980.0543220.0688210.1201210.1535170.2803070.3853621.185714~mm~~-0.0000800.0411180.1360060.935714~40.25731.95018.75414.9659.0817.4905.5465.3955.3855.3850.0249400.0314980.0543220.0688210.1201210.1535170.2803070.3853621.185714~--~-~0.0000800.0008530.1360660.935714~40.25731.95018.75414.9659.0817.4906.7735.3955.3855.385I 4r, )= 1 + r * x*0 (2)....=4r* ~ ,x, oo(i)(2!=1x,.O~x,2o)i= 0(I) (2)Ox,(2)oi -- 0x,oo = 11 + r.
)xx, toW h e n T~ = To, the p r o b l e m is d e s i g n a t e d as lb. The solutions for this p r o b l e m can be o b t a i n e df r o m Eqs. 5.129 t h r o u g h 5.133 with To = Ti. T h e c i r c u m f e r e n t i a l l y a v e r a g e d Nusselt n u m b e rfor this p r o b l e m can be d e t e r m i n e d with the following expression:Nutx~b) + r* Nu~xl,b)1 + r*NUx,T =(5.134)U n i f o r m H e a t Fluxes at B o t h Walls. A s is m e n t i o n e d in the section " F u l l y D e v e l o p e dFlow," w h e n q~'¢ qo'; the p r o b l e m is d e s i g n a t e d as 2a. T h e solution to this p r o b l e m is e x p r e s s e din t e r m s of the following e q u a t i o n s [1]:Dh7"(x~'i) =#T e -~" T[qi(2)Ox,,, +...#ta(2) "Iq o Ox, ioJ(5.135)Dh,, (2) ariA(2) 1~x~, ) -- Te Jr"T[qi Ox,oi Jr"I"1o v x , o o jin.irA(2 ) . + n#A(2)DhT(x'2'm) ~- T e "Jr" Tt~l .
. . . . ."1 . . . . . .(5.136)](5.137)Jq~N u x,t(~) = --(t[A(2)~lt LVx, oo-- 0 (2) .1 -- q ~ [ O (2)Nu(2~)- q~x,o ra(2)_Lv X,OO--x, mlJL--x, m o-- 0 (2) ]q~a(2) ] v x, moJq~,r0(2)I x, ml(5.138)X,IOJ2)0~x,oi](5.139)5.42.CHAPTER FIVETABLE 5.18 Fundamental Solutions of the Third Kind for Thermally Developing Flowin Concentric Annular Ducts (Shah and London [1])r*x *(/)(3!.0.020.010.050.10.51.035.39428.20724.6668.94132.515700.000120.057290.166990.697930.9150110.03310.134070.237400.723500.92220136.77532.57432.34532.33732.33732.3373.88101.52410.688940.001310.0000000.050.010.050.10.51.019.40514.60512.2733.24430.615100.000200.078010.216770.792900.9607410.045620.169800.297220.814190.96477120.33217.59217.46417.46017.46017.4600.100.010.050.10.51.012.9209.16017.36551.37050.167500.000320.101770.269180.863980.9833710.0612210.213620.362950.881440.9855110.250.010.050.10.51.08.23825.24883.87630.36640.019200.000580.144930.361060.939590.9968310.500.010.050.10.51.06.34043.64922.46750.11560.002501.00.010.050.10.5oo5.24242.70281.64680.03370¢T.~(3)'~llx, io -~0 (3)N u (3)0.001180.342760.697220.999421.0000010.256160.621910.827510.999671.0000015.2174.0313.9943.9933.9933.9933.95171.57050.717660.001480.0(K~00.001400.343100.694440.999371.0(0)0010.252540.616310.823110.999631.0000015.2874.0934.0574.0574.0574.05713.76211.64811.56211.56011.56011.5604.04421.64270.769670.001940.0(0)0000.001530.336880.684000.999201.00(O10.245350.604160.812920.999531.0(0g~15.3594.1504.1144.1144.1144.1140.092290.293460.474170.950280.9973919.0767.4297.3727.3717.3717.3714.27731.84510.928040.004120.(gg)0000.001590.312960.648880.998441.0000010.224800.567850.780740.999031.0(0)0015.5184.2694.2334.2324.2324.2320.000870.188080.444050.973940.9994310.12500.36920.57000.97980.999617.2465.7855.7395.7385.7385.7384.62142.15351.18140.010480.0000300.001470.279670.598870.996440.9999910.197890.518030.733300.997630.9999915.7624.4684.4304.4294.4294.4290.001190.235610.527800.9903310.162540.448670.661240.9930616.2604.9024.8614.8614.86115.24212.70281.64680.033700.001190.235610.527800.990331--,(3)~q U x, oi --" 00.162540.448670.661240.9930616.2604.9024.8614.8614.861(3)x,uO x, oi0q~,3)oi= 0(3)-O x, mi(3)~q U x, ii0(3!.x, tt - -<3)0 x, oo=1~(3) _q'Px ......0(3!-.NUx,-(3)iox, m o0x, ooFORCED CONVECTION, INTERNAL FLOW IN DUCTSTABLE 5.19 Fundamental Solutions of the Fourth Kind for Thermally DevelopingFlow in Concentric Annular Ducts (Shah and London [1])r*x*0(,4)m~NU(4!.x,uNu(4).x,o,O x, mo(4)Nu(4)x, ooN U x, io0.020.010.050.10.51.0,~0.00078370.00346890.00526650.00678010.00678300.006783438.09332.68931.26030.18130.17930.1790.0301.9762.6482.9482.9482.9480.0392150.1960780.372641.19111.52571.65686.8094.9384.5293.3753.0572.948-15.12822.76129.33829.99530.1790.050.010.050.10.51.0.~0.00190500.00841540.0127960.0165550.0165630.01656321.52217.77616.79816.06016.05916.0580.0372.0382.7133.0183.0193.0190.0375080.185440.347711.00811.20051.24556.5524.8074.3793.2893.0673.0198.35512.44215.72816.00616.0580.100.010.050.10.51.00.00363610.0160930.0246250.0322880.0323070.03230714.90211.86411.07210.46010.45910.4590.0422.0932.7783.0953.0953.0950.0362540.175990.323770.838450.941760.956146.6464.8364.3503.2713.1163.0950.0345.7398.30910.30410.44210.4590.250.010.050.10.51.0,,~0.00799890.0357290.0557800.0764310.0765230.07652310.2257.7107.0406.4746.4716.4710.0502.1902.9083.2663.2673.2670.0319850.152530.271330.585320.616390.618106.8024.8804.3213.3443.2713.2670.0663.8005.3226.4216.6496.4710.500.010.050.10.51.0oo0.0133320.0603430.0967470.141630.142030.142038.4336.1375.5034.8944.8894.8890.0552.3173.0953.5183.5203.5200.0266640.125080.215300.400000.408890.409087.0264.9984.4003.5543.5213.5200.0643.0074.1254.8704.8894.8891.00.010.050.10.51.0.o0.020090.092110.152850.248010.249980.250007.4955.3414.7234.0134.0004.0000.2542.5593.4533.9934.0004.0000.020090.092110.152850.248010.249980.250007.4955.3414.7234.0134.0004.0000.2542.5593.4533.9934.0004.000(I)(4)x,o,= --0(x4,)miNu(4)'x,o,(I)(4-) -- - - 0 (4) Nu~ ~ox,~ox, mo,(1)(4")'x,.= 1(I)(,4L= 10(x,4o)i= 00 (4.) -- 0x,~o(4)5.435.44C H A P T E R FIVETABLE 5.20 Thermal Entrance Lengths for Thermally Developing Flows in Concentric Annular Ducts(Shah and London [1])r*0.020.050.100.250.501.00L*(1)~,i0.058400.064880.069530.076210.082370.09023L*(1)~,oL*(2)~,i0.16500.14580.13110.11260.10030.090230.026990.030430.033340.037260.039750.04101L,(2)th,oL,(3)~,i0.039010.038860.039110.040060.040900.04101L,(3)~,o0.022520.024290.025580.027200.028290.029130.030010.029700.029600.029640.029560.02913I*(4)~ ~,i0.079620.094930.13090.17210.17210.2201/,(4)~,o0.042410.66380.52840.28750.28750.2201Uniform Temperature at One Wall and Uniform Heat Flux at the Other In this case, thesubscripts 1 and 2 refer to either the inside or the outside wall.
The thermal entrance lengthsolution to this problem is expressible in terms of the following equations [1]:Tx,2 = Te +T M~(3)Dh(T1 - • elUx,21 + TTx'm -- T e ~" ( T 1 -.kqw,1-- --~h (7"1 --OhZe)O(x~)ml + TT ~dh(3 )* e ] "*'x, l lt/2 Ux'22.., tt t~(4)(5.140)q~'0(a)m2(5.141),., Hdh(4 )+ tt 2 "~'x,12(7"1- T• e/Ux,,a(3)l l (q({Dh/k)dP(x4~2Nu/1' - -- (7.1-- Te)(1 - Ox,(3),ml)(q2 "O h/k)O(x4)m2l~lUx'2 =/ ' 0 (4) -- 121(4) ~ "t" [ ( T 1 ~, x,22"~'x,rn2]1Te)k/q2"Dh][O(3) 0 (3) ~J~, x,21 x, ml](5.142)(5.143)(5.144)In Eqs.
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