Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 69
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( 1 )(5.51))1exp(-E132x*)111 1NUm,H = ~ + -2n= CnRn(1)(5.50)exp(-2132x*) )-1~7(5.52)(5.53)where 13~,R,(r/a), and (7, are eigenvalues, eigenfunctions, and constants, respectively.Hsu [26] extended the work conducted by Siegel et al. [25] and reported the first 20 valuesfor 132,R~(1), and C~. These are listed in Table 5.5.
In addition, Hsu [26] presented approximate formulas for higher eigenvalues and constants. The following are of particular interest:13n= 4n + 4/3(5.54)R,(1) = (-1) n0.77475900313; 1/3pt-4/3C , = (-1) 3.099036005[~(5.55)(5.5.6)The local and mean Nusselt numbers for the @ boundary condition are displayed in Fig.5.2. For the inlet of the circular duct, the local and mean Nusselt numbers can be computed byf1.302X *-1/3 - 1NUx,H = ~1.302x *-m - 0.5[4.364 + 8.68(103x , ) -0506• e -41x*I1"953x*-1/3 - 1NU=,H = [4.364 + 0.0722/x*for x* < 0.00005for 0.00005 ___x* ___0.0015for x* _>0.0015for x* < 0.03for x* > 0.03(5.57)(5.58)FORCED CONVECTION, INTERNAL FLOW IN DUCTSTABLE 5.55.13Eigenvalues and Constants for Eqs. 5.50-5.53 [26]n13,2C,R,(1)1234525.67961183.861753174.16674296.53630450.94720-0.492516580.39550848-0.345873670.31404646-0.291251440.40348318-0.175109930.10559168-0.0732823700.055036482678910637.38735855.8495321106.3290351388.8225941703.32785210.27380693-0.259852960.24833186-0.238590240.23019903-0.0434843550.035595085-0.0299084520.025640098-0.02233368511121314152049.8430452438.3668252838.8981423281.4361733755.980271-0.222862800.21637034-0.210565960.20533190-0.200577160.019706916-0.0175764560.015818436-0.0143463690.01309817116171819204262.5299264801.0847475371.6444445974.2088126608.7777270.19623013-0.192233500.18854081-0.185113890.18192104-0.0120282020.011102223-0.0102940710.0095834495-0.0089543767The thermal entrance length for thermally developing flow under the uniform wall heatflux boundary condition is equal to the following:L *th,H = 0.0430(5.59)The effects of fluid axial conduction on the thermal entrance problem with uniform wall heatflux are negligible for Pe > 10 when x* > 0.005 [24].
However, the thermal entrance length,Lth,H obtained by Nguyen [23] is expressed in terms of Pe:f-0.000518 + 0.4686/PeLth,H* = ~0.03263 + 0.3090/Pe1,0.04217 + 0.1309/Pefor 1 < Pe < 5for 5 <_Pe <_20for 20 < Pe < 1000(5.60)The effects of viscous dissipation on the thermal entrance problem with the uniform wallheat flux boundary condition can be found in Brinkman [27], Tyagi [6], Ou and Cheng [28],and Basu and Roy [29]. Other effects, such as inlet temperature, internal heat source, and wallheat flux variation, are reviewed by Shah and London [1] in detail.Heat Transfer on the Walls With Exponential Heat Flux. Heat transfer on walls with exponential wall heat flux is denoted as the (~ boundary condition.
According to the analysis bySiegel et al. [25], the local Nusselt number for a circular duct with exponential variation of thewall heat flux, as represented by q~ = q'o"exp(mx*), can be determined using the following formula:)1-C'R'(1)~2 11 - e x p [ - ( m + 2132)x*]}NUxH5 ='.=m+2132(5.61)The constants C,, R,(1), and 132 in Eq.
5.61 can be obtained from Table 5.5 and Eqs. 5.54through 5.55.5.14CHAPTERFIVE1°3 ~......NUxj.iNUmj-Il021011-100f10-6,,,,,,,,I,10-510-410-3X•, ......I........10-2I10-1........100FIGURE 5.2 Local and mean Nusselt numbers NUx.Hand NUm,H for thermal developingflow in a circular duct.Heat Transfer on Walls With External Convection. Figure 5.3 presents the results obtained by Hsu [30] for the thermal entrance problem with the convective duct wall boundarycondition (~ without consideration of viscous dissipation, fluid axial conduction, flow work, orinternal heat sources. As limiting cases of the (~ boundary condition, the curves corresponding to Bi = 0 and Bi = oo are identical to NUx,H and NUx,T, respectively. Significant viscous dissipation effects have been found by Lin et al. [31] for larger Bi values.Heat Transfer on Walls With Radiation. The local Nusselt numbers normalized withrespect to NUx,H have been obtained by Kadaner et al.
[8] for thermally developing flow withthe radioactive duct wall boundary condition @. This is expressed as:NUx,T40.0061- 0.0053 In x + ( _ ~ )NUx,H - 0.94 1 + 0.0242 In x ÷ In(5.62)Equation 5.62 is valid in the ranges 0.001 < x ÷ < 0.2 and 0.2 < Sk < 100 for zero ambient temperature. It should be noted that Nusselt numbers NUx,T4 with Sk = 0 and Sk = oo are identicalto Nux,n and NUx,T, respectively.Simultaneously Developing Flow. Simultaneously developing flow usually occurs when thefluid exhibits a moderate Prandtl number. In such a flow, the velocity and the temperatureprofiles develop simultaneously along the flow direction. Therefore, the heat transfer ratestrongly depends on the Prandtl number of the fluid and the thermal boundary condition.Heat Transfer on Walls With Uniform Temperature.
The local and mean Nusselt numbersNUx,T and NU~T are shown in Figs. 5.4 and 5.5, respectively. The data corresponding to these figures can be found in Jensen [32] and Shah and London [1]. It can be observed that for fluids witha large Prandtl number (Pr --> oo), the solution for simultaneously developing flow correspondsto the solution for thermally developing flow because the velocity profile develops before thetemperature profile begins developing. However, for a fluid with a very small Prandtl number(Pr = 0), the temperature profile develops much more quickly than the velocity profile, while thevelocity remains uniform. This is termed slug flow. The local and mean Nusselt numbers for slugflow (Pr = 0) and for Pr --> oo are shown in Figs.
5.4 and 5.5, respectively.FORCED CONVECTION, INTERNAL FLOW IN DUCTS282420Bi = 0 ( N u ~ )Bi=2Bi = 20Bi=200Bi=® (NUx,T)NUx,T312'L210-424610-324610-224610-1X*FIGURE 5.3Local Nusselt number NUx,T3 for thermally developing flow in a circular duct [30].28Pr=O240.720516NU=,T125.78322E_42 X 10-43.6568 . . . .46810 -a2xt,46810 -22FIGURE 5.4 Local Nusselt number NUx,T for simultaneously developing flow in a circularduct [1].5.155.16CHAPTERFIVE44Pr=O40--------0.7--------2-------- 532P=,,,=,,~,~ oo24Nun,.
T165.7832_..~3.6568 . . . .02 X l O -446810-3246810 -22XsFIGURE 5.5 Mean Nusselt number NU,,,T for simultaneously developing flow in a circularduct [1].The thermal entrance lengths for simultaneously developing flow with the thermal boundary condition of uniform wall temperature provided by Shah and London [1] are as follows:[0.028L *th, T = ]0.037L0.033for Pr = 0for Pr = 0.7for Pr = ,,o(5.63)Heat Transfer on the Walls With Uniform Heat Flux. The solutions for simultaneouslydeveloping flow in circular ducts with uniform wall heat flux O are reviewed by Shah andLondon [1].
Recently, a new integral or boundary layer solution has been obtained by A1-AIiand Selim [33] for the same problem. However, the most accurate results for the local Nusseltnumbers [1] are presented in Table 5.6.The thermal entrance lengths for simultaneously developing flow with the thermal boundary condition of uniform wall heat flux [1, 34] can be obtained by[0.042L *th,T = ~0.0531,0.043for Pr = 0for Pr = 0.7for Pr = oo(5.64)The axial diffusions of heat and m o m e n t u m were considered by Pagliarini [35] for simultaneously developing flow, whereas the viscous dissipation effect has been taken into accountby Barletta [36].Heat Transfer With the Convective Boundary Condition. The solution for simultaneouslydeveloping flow with the convective boundary condition @ has been obtained by Javeri [37].The results are listed in Table 5.6.
It should be noted that when Bi = o% Nux,v3 is identical toNux,T. When Bi = 0, NUx,T3 is the same as NUx,H.Conjugate Problem. To this point, uniform wall thickness has been assumed, and no heatconduction in the wall has been involved, meaning that the wall has infinite heat conductivity.If heat conduction in the wall is considered, forced convection and conduction in the wallmust be analyzed simultaneously.
The solution for this combined problem, referred to as aconjugate problem, entails several additional parameters. An extensive review has been per-FORCED CONVECTION, INTERNAL FLOW IN DUCTSTABLE 5.65.17Local Nusselt Number Nux,r3 for Simultaneously Developing Flow in a Circular Duct [1]Nux,a-3Bix*Pr = ~Pr = 10P r = 1.0Pr = 0.7P r = 0.1Pr = 02000.000250.000500.001250.00250.00500.01250.0250.0500.1250.2515.8712.989.6217.5586.0614.6724.0263.7223.6693.66618.2714.049.8377.5015.9774.6774.0273.7263.6693.66624.7817.0511.648.5976.5144.7764.0373.7333.6693.66726.2417.8112.048.8746.6914.8724.0903.7403.6693.66734.6422.9314.5410,647.9145.6494.6664.1763.8923.75043.3830.1718.7813.7110.237.3856.2595.8695.8325.832200.000250.000500.001250.00250.00500.01250.0250.0500.1250.2518.1214.3110.408.1546.4694.9244.1933.8383.7633.76322.5216.4111.028.2766.4714.9274.1943.8383.7633.76329.6920.7513.439.7387.2175.1264.2 443.8533.7643.76331.2621.7213.9710.107.4415.2474.3073.8663.7643.76340.5928.1417.4912.479.0436.2044.9744.3484.0083.85550.3537.0023.7017.0012.368.5186.9426.3106.2246.224100.000250.000500.001250.00250.00500.01250.0250.0500.1250.2518.6814.7510.778.4686.7055.1104.3293.9353.8453.84422.9916.8111.398.5906.7155.1134.3313.9373.8463.84430.5621.5814.0910.267.6085.3684.4073.9653.8463.84432.4022.8914.7610.707.8725.5044.4793.9743.8463.84443.4530.1118.7213.349.6516.5765.2234.4954.1053.94153.1938.8925.3218.3513.419.2557.4736.6846.5476.5470.000250.000500.001250.00250.00500.01250.0250.0500.1250.2520.5515.8211.438.9387.1125.3894.5744.1204.0014.00026.0918.6912.369.2877.0215.3914.5784.1294.0034.00034.9923.8715.2211.028.1595.7604.6844.1654.0034.00034.9624.6215.7411.438.4295.8844.7564.1714.0034.00049.7233.2620.1314.3210.397.1185.6214.7594.2854.10554.4139.3225.6819.0214.2610.098.1817.2897.0887.0875.18C H A P T E R FIVE3028Pr=O24200.716Nuz.
H12Pt=~_..'L22 X 10"-44.363646810-3246810 -22X*FIGURE 5.6 Local Nusselt numberduct [1].NUx,Hfor simultaneously developing flow in a circularformed by Barozzi and Pagliarini [38] for fully developed flow. The reader can also refer toKuo and Lin [39] and Pagliarini [40] for developing flow.Turbulent FlowIn this section, turbulent flow and heat transfer in a circular duct with a diameter of 2a is discussed for fully developed and developing flow.Critical Reynolds Number. The Reynolds number, defined as umDh/v, is widely adopted toidentify flow status such as laminar, turbulent, and transition flows.
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