Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 78
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. . . )for Pe << 1(5.196)Uniform Heat Flux at Each Wall When the heat fluxes on the two walls of parallel plateducts are equal, q:l = q~2, the temperature distribution of fully developed laminar flow isgiven by:Tw-TH35 [(b) 2 (y)41Tw-Tm, H- 136 5 - 6+ b'Tw-Tm,H17(5.197)q"Dh/k - 140and the Nusselt number is obtained as:NUl = Nu2 = Null = 8.235(5.198)When qwl g: q~2, then140NUl = 2 6 - 9(q'~2/q'~1)'140Nu2 = 2 6 - 9(q'~1/q'~2)(5.199)To consider the effect of internal energy generation and viscous dissipation, the following formula obtained by Tyagi [6] is recommended:140Nun = 17 + 3AS* + 108Br'(5.200)Uniform Temperature at One Wall and Uniform Heat Flux at the Other When the twowalls of a parallel plate duct are subject to a thermal boundary condition such as uniform temperature at one wall and uniform heat flux at the other, the Nusselt numbers for fully developed laminar flow for q'~ = 0 and q" # 0 are determined to be:NUl = 4.8608Nu2 = 0NUl = Nu2 = 4for q'~ = 0for q" ~ 0(5.201)(5.202)The Convective Boundary Condition @.
The fully developed Nusselt number with theconvective boundary condition at both walls can be computed from Hickman's [7] analysis:4620 + 561BiNuT3 = 561 + 74Bi(5.203)The Exponential Wall Heat Flux Boundary Condition ~. When both walls of parallelplate duct are subjected to the exponential heat flux of q" = q~ exp(mx*), the fully developedNusselt number can be obtained as follows [2]:NUlls = 8.24 + 2.1611 x 10-3m - 4.4397 x 10-Sm 2 + 1.2856 x 10-7m 3 - 2.7035 x 10-1°m 4 (5.204)When m = -30.16, then NUlls = NUT, and when m = 0, then Nuas = NUll.Hydrodynamically Developing Flow. For hydrodynamically developing flow in parallel plateducts, Shah and London [1] obtained the apparent friction factor fapp and Chen [11] has obtained Lh.
K(~) as the function of x ÷ and Re, respectively, as shown in the following equations:3.4424 + 0.674/(4x ÷) - 3.44/(x÷) 1/2fapp Re - (x÷)l,2 +1 + 0.000029(x*) -2(5.205)5.62CHAPTERFIVE0.315Lhy _ 0.011 Re +Dh1 + 0.0175 Re(5.206)38K(oo) =0.64 + R---e(5.207)Thermally Developing Flow. The results for thermally developing flow in parallel plateducts are presented for the following practical thermal boundary conditions of interest.Equal and Uniform Temperatures on Both Walls. The local and mean Nusselt numbersfor parallel plate ducts with equal and uniform temperatures on both walls can be computedfrom Nusselt's [131] solution, which is displayed in Fig. 5.21.
The tabulated values for Fig. 5.21are available in Shah and London [1].9.50200 ~ . . \100~~.60Hn40~LlquxJ!"'-.\"~.lqnnt,T"/---~:r201010.6 24 6 10-5 24 6 10-4 24 6 10"3 2 4 6 10"2 24 6 10"1 2X10-1X*FIGURE 5.21 Local and mean Nusselt numbers in the thermal entrance region of a parallel plateduct with the (~ and (8)boundary conditions [1].From these results, the dimensionless thermal entrance length can be determined as follows:L,~T = 0.00797(5.208)It is also suggested that the following set of empirical equations proposed by Shah and London [1] be used for the practical calculation of the local Nusselt number:I1.233x -1/3+ 0.4NUx,T = [7.541 + 6.874(103x*)-°~e -245x"fl.849x *-1/3NUm,T = ]1.849x *-1/3 + 0.61,7.541 + 0.0235/x*for x* < 0.001for x* > 0.001for x* ___0.0005for 0.0005 < x* < 0.006for x* > 0.006(5.209)(5.210)F O R C E D CONVECTION, I N T E R N A L FLOW IN DUCTS5.63Uniform and Equal Heat Flux at Both Walls.
Thermally developing flow in a parallelplate duct with uniform and equal heat flux at both walls has been investigated by Cess andShaffer [132] and Sparrow et al. [133] in terms of a series format for the local and mean Nusselt numbers. The dimensionless thermal entrance length for this problem has been found byShah and London [1] to be as follows:(5.211)Lth, H = 0 . 0 1 1 5 4The local and mean Nusselt numbers are also displayed in Fig. 5.21.The following set of equations are recommended for practical computations without lossof accuracy [1]:1.490x *-1/3NUx,H = ~1.490x *-v3 - 0.4[8.235 + 8.68(103x*)-°5°re-164x"NUm, H -r2.236x *-1/3]2.236x *-1/3 + 0.9[8.235 + 0.0364/x*for x* < 0.0002for 0.0002 < x* < 0.001for x* > 0.001for x* < 0.001for 0.001 < x* < 0.01for x* > 0.01(5.212)(5.213)It has been concluded that except in the neighborhood of the duct inlet (x* < 10-2), the effectof the fluid axial conduction is negligible for Pe > 50 [134, 135].Convective Boundary Condition at Both Walls or One Wall The solutions for the convective boundary condition on both walls or one wall are reviewed in Shah and London [1],where more detailed descriptions are available.Simultaneously Developing Flow.The results for simultaneously developing flow in parallel plate ducts are provided for the following thermal boundary conditions.Equal and Uniform Temperatures at Both Walls.
For simultaneously developing flow in aparallel plate duct with fluids of 0.1 < Pr < 1000, the following equations are recommended forthe computation of the local and mean Nusselt numbers [2, 136, 137]:NUx,T = 7.55 +0.024x*-114[0.0179Pr °'17x *-°'64 - 0.14][1 + 0.0358Pr °17 x*-°64]z(5.214)0 . 0 2 4 X *-l J4NUm,T =7.55 + 1 +0 . 0 3 5 8 P r °'17 x *-°'64(5.215)The thermal entrance length L*th,T has been found to be 0.0064 for 0.01 < Pr < 10,000 [138,139].
A detailed description can be found in Shah and London [1] for the solutions for Pr = ooand Pr = 0.When one duct wall is insulated and the other is at a uniform temperature, the local andmean Nusselt numbers for simultaneously developing flow have been obtained for fluids of0.1 < Pr < 10. These follow [1]:NUx,T = 4.86 +0.0606x*-12[0.0455Pr °ATx *-°'7 - 0.2][1 + 0.0909Pr °17 x*-°7] 2(5.216)0.0606x *-1.2NUm,T = 4.86 + 1 +0 .
0 9 0 9 P r °'17 x *-°'7(5.217)Uniform and Equal Heat Flux at Both Walls. The local Nusselt number for heat transferof laminar flow in a parallel plate duct with uniform and equal heat flux at both walls is displayed in Fig. 5.22 for different Prandtl numbers, Pr = 0 [34] and Pr = 0.01, 0.7, 1, 10, and oo[136].
The thermal entrance lengths obtained from the results presented in this figure are0.016, 0.030, 0.017, 0.014, 0.012, and 0.0115, for Pr = 0, 0.01, 0.7, 1, 10, and 0% respectively.5.64CHAPTER FIVE25Pr=0,]202bNu=,H1512108.2353SLIsxlo10-32468 10-225 X 10-2X*FIGURE 5.22 Local and mean Nusselt numbers for simultaneously developing flow in a parallelplate duct with the @ boundary condition [34, 136].The local Nusselt n u m b e r is displayed in Fig. 5.23 for Pr = 0.0, 0.01, 0.7, 10, and o~ when onewall of the parallel plate duct is insulated and the other wall is subjected to uniform heat fluxheating [140]. Included in Fig. 5.23 are the results for Pr = 0% obtained from the concentricannular duct corresponding to r* = 1.
The local and m e a n Nusselt n u m b e r s for Pr = 0 wereobtained by Bhatti [34].In addition, N g u y e n [141] has obtained the apparent friction factor and Nusselt n u m b e r sfor low Reynolds n u m b e r simultaneously developing flow in a parallel plate duct with a con3530Pr-O250.012O2b"""ffff/'/" T0.7Nux, Httttttt10156.00105.395110-410-310-210-t100X*FIGURE 5.23 Local and mean Nusselt numbers for simultaneously developing flowin a fiat duct with uniform heat flux at one wall and the other wall insulated [34, 140].FORCED CONVECTION, INTERNAL FLOW IN DUCTS5.65stant temperature and constant wall heat flux thermal boundary conditions. Results are presented for Pr = 0.7 and Reynolds numbers between 1 and 20. The results for Pr = 0.2, 0.7, 2, 7,10, and 100 as well as for the Reynolds numbers between 40 and 200 have been numericallydetermined by Nguyen and Maclaine-Cross [142].Turbulent FlowThe characteristics of turbulent flow and heat transfer in parallel plate ducts are discussed inthis section.
The friction factor for transition flow is also addressed.Transition Flow. The lower limit of the critical Reynolds number Rec,it for a parallel plateduct is reported to be between 2200 and 3400, depending on the entrance configurations anddisturbance sources [143]. The following friction factor formula developed by Hrycak andAndrushkiw [144] is recommended for transition flow in the range of 2200 < Re < 4000:f = - 2 . 5 6 x 10-3+4.085 x 10-6 Re - 5.5 x 10-10 Re 2(5.218)The mean Nusselt number in the thermal entrance region of a parallel plate duct with uniform wall temperature at both walls in the range of 2300 < Re < 6000 is given by Hausen [20]as follows:[(NUm,T= 0.116(Re a3- 160t Pr '/3 1 + \D-7]J(5.219)Fully Developed Flow.Beavers et al.
[145] obtained the following friction factor for fullydeveloped turbulent flow in a parallel plate duct for 5000 < Re < 1.2 x 10 6 from very accurateexperimental data:f=0.1268Re0.3(5.220)For 1.2 x 104 < Re < 1.2 x 10 6, Dean [146] has developed the following equation based on acomprehensive survey of the available data:0.0868f=Re1/4(5.221)Comparisons of precision using Eqs. 5.220 and 5.221 and Blasius's formula (Table 5.8) inwhich the diameter of circular duct 2a is replaced by hydraulic diameter 4b, b being the halfspace between two plates, have been conducted by Bhatti and Shah [45]. In the range of5000 < Re < 3 x 10 4, Eq. 5.220 is recommended; otherwise, Eq.
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