Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 68
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The Nusselt n u m b e r for very large m can be found in Piva [9].Hydrodynamically Developing Flow.Solutions for three different flow conditions are pro-vided as follows.Solutions for Very Large Reynolds Number Flows. For very large Reynolds n u m b e r flow,b o u n d a r y layer theory simplifications are involved in the solutions. The numerical solutionfound by H o r n b e c k [10] is the most accurate of the various solutions reviewed by Shah andL o n d o n [1].
The dimensionless axial velocity and pressure drop obtained by H o r n b e c k [10]are presented in Table 5.2.Based on these results, the hydrodynamic entrance length was found to be:L~y = 0.0565K(oo) = 1.28(5.29)It should be noted, however, that the solutions for very large Reynolds n u m b e r s are inaccurate in the duct inlet. For practical computations, the following correlation proposed by ShahTABLE 5.2Axial Velocity and Pressure Drop in the Hydrodynamic Entrance Region of a Circular Duct [10]Axial velocity u/umx÷r/a = 00.10.20.30.40.50.60.70.80.91.0Dimensionlesspressure drop Ap*0.000000.000500.001250.002500.003751.00001.15031.22691.31261.37821.00001.15031.22691.31261.37811.00001.15031.22691.31251.37791.00001.15031.22691.31241.37701.00001.15031.22681.31151.37331.00001.15031.22641.30681.35961.~1.15021.22301.28671.31601.00001.14851.20161.21441.20001.00001.12931.09501.00980.95111.00000.84340.68930.59080.5417000000.00(O0.32200.50340.72040.89600.005000.007500.010000.012500.017501.43321.52391.59771.65951.75551.43311.52321.59601.65621.74881.42341.52041.58931.64481.72691.42991.51201.57271.61881.68311.42141.49021.53581.56751.60731.39591.43951.46231.47511.48741.32921.33491.33081.32451.31251.18141.14761.12181.10231.07570.91070.85850.82610.80400.77560.51020.47200.44960.43460.4159000001.05061.32121.56101.78222.19000.022500.030000.040000.050000.062501.82401.89201.94311.96981.98631.81421.87851.92661.95171.98721.78291.83661.87631.89691.90951.72441.76261.79011.80421.81281.63061.65091.66501.67211.67641.49271.49621.49811.49901.49961.30341.29431.28751.28401.28181.05881.04331.03211.02641.02290.75840.74290.73190.72630.72290.40470.39470.38770.38400.3718000002.56923.10643.78944.45205.26880.38000oo2 .
0 0 0 0 1.9800 1.9200 1.8200 1.6800 1.5000 1.2800 1.0200 0.7200FORCED CONVECTION,INTERNALFLOW IN DUCTS5.9and London [1] can be used to calculate the dimensionless axial pressure drop in the hydrodynamic entrance region:tiP*= 13"74(x+)1"2+1.25 + 64x + - 13.74(x+) 1/21 + 0.00021(x+) -2(5.30)Solutions for the Flow with Re < 400.
It has been found that the effects of axial momentum diffusion and radial pressure variation are significant only in the duct inlet of x + < 0.005.Chen [11] obtained the dimensionless hydrodynamic entrance length L+hyand the fully developed incremental pressure drop number K(oo), which are given by0.60L+hy= 0.056 + Re (1 + 0.035Re)(5.31)38K(oo) = 1.20 + R---~(5.32)For x + > 0.005, the solutions proposed by Hornbeck [10] are quite satisfactory.Solutions for Small Reynolds Number (Re ~ O) Flows.
For small Reynolds number flows,such as creeping flow, in which viscous forces completely overwhelm the inertia forces, thehydrodynamic entrance length L+hyhas been found to approach the value of 0.60 as Re ---) 0with the uniform flow at the inlet of the circular duct [1].The following expression, proposed by Weissberg [12] and verified experimentally byLinehan and Hirsch [13], can be used to compute pressure drop in the entrance region of a circular duct with a very small Reynolds number flow:3n )Ap* = 64 x + + 16Re(5.33)Thermally Developing FlowHeat Transfer on Walls With Uniform Wall Temperature.
Heat transfer in a duct with uniform wall temperature is known as the Graetz or Graetz-Nusselt problem. In this case, a fluidwith a fully developed velocity distribution (Eq. 5.15) and a uniform temperature flows intothe entrance, and the fluid axial conduction, viscous dissipation flow work, and energyresources are negligible. Graetz [14] and Nusselt [15] solved this problem as follows:0 = Tw----~e =C,R, aexp(-2~'2x*)(5.34)n=Om= Tw----~ _8 ,,__ ~Nu,,.-r =G,, exp(-2X 2 x*),,--02 ~" (G,,/~.2 ) exp(-2~ 2 x*))(5.35)(5.36)n=0NUm,T =In 0 m4x*(5.37)where ~,,, R,(r/a), and C, denote the eigenvalues, eigenfunctions, and constants, respectively,and G, =-(C,/2)R~, (1), where R" (1) is the derivative of R,(r/a) evaluated at r/a = 1.5.10CHAPTER FIVEThe e i g e n v a l u e s and c o n s t a n t s in Eqs.
5.34-5.36 are listed in Table 5.3, while t h o s e forgiven in Table 5.4. W h e n n is g r e a t e r t h a n 10, the following c o r r e l a t i o n s are usedto calculate the ;~, and G n [16]:R,(r/a) are~'n "- ~" "F S1 ~--4/3 Jr- S2 ~-8/3 -k- 53~, -10/3 -k- 54 ~-11/3 --I-0(~g3 g4Gn---~C [ 1 + - ~gl5 - + - ~g2+-~+~+'~+o(~,where;~ = 4n + 8/3$1 = 0.159152288andL1 = 0.144335160](5.39)(5.40)$3 = -0.224731440(5.41)C = 1.012787288L2 = 0.115555556L4 = -0.187130142-4)n = 0, 1, 2 . .
. .$2 = 0.0114856354$4 = -0.033772601g5(5.38)-14/3)L3 = -0.21220305(5.42)L5 = 0.0918850832TABLE 5.3 Eigenvalues and Constantsfor Eqs. 5.34-5.36 [17]TABLE 5.4Eigenfunctions,n~.C.G.0123456789102.704366.6790310.6733814.6710818.6698722.6691426.6686630.6683234.6680738.6678842.667731.47644-0.806120.58876-0.475850.40502-0.355760.31917-0.290730.26789-0.249060.233230.748770.543830.462860.415420.382920.358690.339620.324060.311010.299840.29012R,(r/a),for Eq. 5.34 [17,18]R,(r/a)n r/a =0012345678910111213141.01.01.01.01.01.01.01.01.01.01.01.01.01.01.00.10.20.30.40.50.60.70.80.91.00.981840.891810.735450.531080.302290.074880.12642-0.28107-0.37523-0.40326-0.36817-0.28088-0.15836-0.021180.109530.928890.604700.15247-0.23303-0.40260-0.32121-0.076130.177160.299740.239150.04829-0.15310-0.24999-0.19545-0.031820.845470.23386-0.31521-0.359140.000540.289820.20122-0.10751-0.25305-0.085580.166450.19847-0.00845-0.18955-0.130830.73809-0.10959-0.392080.067930.29907-0.04766-0.251680.034520.22174-0.02486-0.200580.017140.18456-0.01074-0.171830.61460-0.342140.142340.31507-0.07973-0.205320.193950.05514-0.205020.081260.13289--0.15931-0.019270.15967-0.085600.48310-0.432180.169680.11417-0.255230.19750-0.01391-0.153680.19303-0.09176-0.064740.16099-0.133930.012580.109270.35101-0.397630.33149-0.196040.036100.10372-0.188830.20290-0.150990.056520.04681-0.125770.15742-0.135390.070690.22426-0.284490.30272-0.292240.25918-0.208930.14716-0.079850.012980.04787-0.097970.13375-0.153110.15549-0.141890.10674-0.141130.16262-0.17762-0.18817-0.195220.19927-0.200680.19967-0.196450.19120-0.184090.17527-0.164910.153190.00.00.00.00.00.00.00.00.00.00.00.00.00.00.0FORCED CONVECTION,INTERNALFLOWIN DUCTS10 ~,I......5.11,,Nu x,TNUm,T102101........1oo10-6I10-5,,,,....
I.......10-41,,,,,,,,I10-3X*,10-2,,,,,,,I. . . . . . . .10-1100FIGURE 5.1 Local and mean Nusselt numbers NUx,Tand NUm,Tfor thermal developingflow in a circular duct.The local Nusselt number and mean Nusselt number computed from Eqs. 5.36 and 5.37 areshown in Fig. 5.1. The data corresponding to this figure can be found in Shah and L o n d o n [1].The thermal entrance length for thermally developing flow in circular ducts can beobtained using the following expression:Lth,T = 0.0335(5.43)It should be noted that the solutions derived from Eqs.
5.34 through 5.36 are inaccuratewhen x* < 10 -4. Therefore, L6v6que's [19] asymptotic solution is introduced. The local andmean Nusselt numbers can be computed from the following formulas [1]:NUx,T="077x*-1/3-0"7 3 , -0488 572x3.657+6.874(10X ) " e- • *fl.615x *-1/3 - 0.7NUm,T = ]1.615X *-1/3 - 0.2[3.657 + (0.0499/X*)for x* _<0.01for x* > 0.01for x* < 0.005for 0.005 < x* < 0.03for x* > 0.03(5.44)(5.45)Shah and Bhatti [2] and Hausen [20] have obtained the following correlations for the localand mean Nusselt numbers for the entire x +.0.0018Nuxx = 3.66 + x,1/3(0.04 + x,2/3) 2(5.46)0.0668NUm,T = 3.66 + X,1/3(0.04 + X,2/3)(5.47)The effects of fluid axial conduction on the Graetz solution have been reviewed extensively by Shah and London [1]. Furthermore, Laohakul et al.
[21], Ebadian and Zhang [22],5.12CHAPTER FIVEand Nguyen [23] have investigated this extended Graetz problem with axial heat conduction.For Pe < 50, Nu,,,T can be calculated with the following expression [4]"I(1.227 + ...'~Nu,,,~ = 3.6508 1 + p e 2]for Pe > 5(5.48)for Pe < 1.5[4.1807(1 - 0.0439Pe + ...)It has been confirmed that the effect of fluid axial conduction can be neglected for Pe > 50[24]. However, the thermal entrance length L'fh,T varies with the P6clet number. Nguyen [23]expresses L~h,Tas follows:--0.003079 + 0.4663/Pefor 1 < Pe < 5for 5 _<Pe _<20for 20 < Pe < 1000Lib,T* = ~0.02020 + 0.3550/Pe[0.03258 + 0.1295/Pe(5.49)Other extended Graetz problems in which the effect of viscous dissipation, inlet velocity,and temperature profiles are considered are reviewed in detail by Shah and London [1].Heat Transfer on Walls With Uniform Heat Flux.
The temperature profile and the localand mean Nusselt numbers for thermally developing flow in a circular duct with uniform wallheat flux are provided by Siegel et al. [25] as follows:O=1 ~ 1 Cngn( r ) exp(-E~Ex *)T - Te = 4x* + l ( r ) 2 - 1 ( a ) 4 - 7q~(Dh/k------~2-ff- ~ + -2 =aO,,, = ~NUx, H =(111~T=-T~q"(Dh/k)= 4x*+ "~',,= C ~ R .
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