Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 83
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Mori and Nakayama [190] have obtained the solution for De >100 for a coil with R > a. Their results are in agreement with the experimental data reportedby Mori and Nakayama [190] and Adler [191] and numerical simulations [192].The friction factors for fully developed flow in helicoidal pipe proposed by Srinivasan et al.[193] in the range of 7 << 104 follow:R/afc [1-- =/0.419De °275fs [0.1125De °5for D e < 3 0for 30 < De < 300for De > 300(5.265)However, after reviewing the available experimental data and theoretical predications,Manlapaz and Churchill [194] recommended the following correlations, which contain boththe Dean number De and radius ratio ofR/a:fc_ I(1 .
0 -f, -018[1 + (35/De)2] °5+ 1.0 + ~88.33(5.266)where m = 2 for De < 20; m = 1 for 20 < De < 40; and m = 0 for De > 40. It can be observed thatEq. 5.265 does not include the parameter R/a and will not be used for all the range of R/a.After a comparison of Eq. 5.266 with experimental data, Shah and Joshi [195] suggested thatEq.
5.265 be used for the coils with R/a < 3 and that either Eq. 5.265 or Eq. 5.266 be used forcoils with R/a > 3.5.86CHAPTER FIVEThe friction factors for spiral coils can be obtained using the following correlation [193],which is valid for 500 < Re (b/a) °5 < 20,000 and 7.3 < b/a < 15.5.fc =0"62(n°7 - n°7)2Re0.6(b/a)O.3(5.267)where nl and n2 represent the numbers of turns from the origin to the start and the end of aspiral.The critical Reynolds number, which is used to identify the transition from laminar to turbulent flow in curved or helicoidal pipes, has been recommended for design purposes bySrinivasan et al.
[193]:Recrit = 210011+ 12(R) -°5](5.268)However, for spiral pipes, no single Recrit exists because of varying curvature. The minimumvalue of Recrit can be obtained using Rmax to replace R in Eq. 5.268, and the maximum value ofRecnt can be determined using Rmin tO replace the R in Eq. 5.268.The fully developed Nusselt numbers for laminar flow in helicoidal pipes subjected to theuniform wall temperature have been obtained theoretically and experimentally by Mori andNakayama [196], Tarbell and Samuels [197], Dravid et al.
[198], Akiyama and Cheng [199],and Kalb and Seader [200]. A comparison of these results has been made using the ManlapazChurchill [201] correlation.In Fig. 5.36, experimental and theoretical results [196-200] are compared with the following Manlapaz-Churchill [201] correlation based on a regression analysis of the available data:NuT =where[(4"343)3.657 +3xl957 )2,x3 = 1.0 + De 2 Pr( De/3/211/3+ 1.158 ~\x2/ Jx2 = 1.0 +0.477Pr(5.269)(5.270)It can be seen that the prediction calculated from Eq.
5.269 agrees well with the experimental data in most cases, except for Pr = 0.01 and 0.1 at intermediate He values.The fully developed Nusselt numbers for spiral coils with uniform wall temperature aresuggested by Kubair and Kuldor [202, 203], as follows:NuT- 1.98 + 1.8which is valid in the ranges of 9 <_Gz _<1000, 80 < Re < 6000, and 20 < Pr < 100. Although thisequation can be used to obtain the Nusselt number in the thermal entrance region, the fullydeveloped Nusselt number may be calculated by substituting Gz - 20.For helicoidal pipes with the (~ thermal boundary condition, the Nusselt number has beendeveloped by Manlapaz and Churchill [201]:NUll1 = I( 4.364 + 4"636) 3 + 1.816(~4e)3/211/3X3where1342x3 = 1.0 + De 2 Pr)2X4 =1.151.0 + p----~(5.272)(5.273)Figure 5.37 compares Eq.
5.272 with some of the available theoretical predictions [204, 205]and experimental Nusselt number data [190,198]. The figure indicates a fairly good agreementbetween the correlation and most of the available data.J0|0i"oi"o!IIII;I1-÷NO£•o4Q•I~o~~ a O o Ozdgoo\Co0 '~,\!to~o.,=,-C"0&~"o"0 ;~5.875.88o,.JxII.t-4e l mo~zI> ,", 0¢, 0DDrt ,"-,oz""~®~'~FORCED CONVECTION, INTERNAL FLOW IN DUCTSTABLE 5.445.89Numerically Calculated NUll2 for Helical Coils of Circular Cross Sections [201]NUp,H2R/ab/R5.00.00.51.05.00.00.51.010.00.00.51.05.00.00.51.010.00.00.51.05.00.00.51.0Re9.1969.1979.19446.7047.7246.79DeHe4.1134.1134.1124.1134.1004.061Pr = 0.10.31621.010.04.6424.4624.4624.6394.6394.6404.6334.6334.6344.6204.6204.6214.9364.9344.9298.4478.4388.41420.8820.8920.9320.8820.8320.674.7694.7684.7654.7594.7584.755392.6393.0394.4124.14124.29124.72124.14123.90123.175.6045.6025.5967.5417.5357.518402.5403.1404.9180.01180.28181.07180.01179.71178.826.0586.0786.0719.3129.3079.292100810091013318.8319.1320.5318.8318.1316.57.1207.1147.103104310451051466.6467.4469.8466.6465.9464.09.6809.6009.58814.3014.2714.23The fully developed Nusselt numbers NUll2 for helicoidal pipes have been obtained numerically by Manlapaz and Churchill [201].
Their results are listed in Table 5.44.In Table 5.44, it can be seen that the pitch of the helicoidal coil has almost no influence onthe Nusselt number. However, the studies by Yang et al. [206, 207] have shown a positiveeffect of the pitch on the Nusselt number when Pr > 1. In addition, the experiments conductedby Austen and Soliman [208] indicated that the Nusselt number for the laminar flow of water(3 < Pr < 6) in the uniformly heated helicoidal pipe is in good agreement with the predictionfrom Manlapaz and Churchill [201].To consider the effect of variable viscosity, the viscosity ratio (~.l,m/~.[w) 0"14 is applied.
The useof Eqs. 5.269 and 5.272 with their right sides multiplied by (gm/gw) °14 is recommended. Thedensity change of fluids leads to natural convection; consequently, heat transfer is normallyenhanced.An experimental correlation has been obtained by Abul-Hamayel and Bell [209] toaccount for the density and viscosity variations in helicoidal pipe.
Experiments with water,ethylene glycol, and n-butyl alcohol in a helicoidal pipe with the @ boundary condition wereconducted. The following equation was derived from the measurement data:Num=/ Gr ~3.94]f/ / G r \z78 ][4 . 3 6 + 2 . 8 4 ~ ~ e 2 ) ] [ 1 + 0.9348~~e2 ) x s ] l + [ 0.0276 De °75 Pr °'97 (Urn/°"'4]\Uw/J(5.274)wherexs=exp -1.33 G r ' )De:(5.275)This correlation is valid for 92 < Re < 5500, 2.2 < Pr < 101 and 760 < Gr' < 106.
It reduces to thestraight circular duct forced convection Nusselt number value of 4.36 upon neglecting the coil5.90CHAPTER FIVEeffect (De ---) 0). Equation 5.274 is recommended for those fluids whose densities are stronglydependent on temperature.Developing Laminar FlowHydrodynamically developing laminar flow, thermally developing laminar flow, and simultaneously developing laminar flow in helical coils are still under investigation [210-212]. Accurate formulas for engineering applications are limited.
However, the entrance region of ahelical coil is approximately 20 to 50 percent shorter than that of a straight tube. For mostengineering applications, when De > 200, the design can be based on fully developed valueswithout significant errors [195].Turbulent Flow in Coils With Circular Cross SectionsThe research conducted by Hogg [213] has indicated that turbulent flow entrance length incoils with circular cross sections is much shorter than that for laminar flow. Turbulent flow canbecome fully developed within the first half-turn of the coil. Therefore, most of the turbulentflow and heat transfer analyses concentrate on the fully developed region.Ito [214] has proposed the following correlation to calculate the friction factor for turbulent flow in helicoidal coils:f~a)= 0.00725 + 0.076 Refor 0.034 < Re< 300(5.276)However, Srinivasan et al.
[193] has obtained another formula for the turbulent friction factor, as follows:f~|-2-~= 0.084 Refor Re< 700and7 < - - < 104a(5.277)Either Eq. 5.276 or Eq. 5.277 can be used for design purposes since they are very similarand agree quite well (within +10 percent) with the experimental data for air [215] and water[216] and with the numerical predictions by Patankar et al. [217].The friction factor for spiral coils can be obtained using the following experimental correlation [193]:0.0074(nfc =0.9 _ nO.9) 1.5[Re (b/a)°5] °2(5.278)Equation 5.278 is valid for 40,000 < Re (b/a) °5 < 150,000 and 7.3 < b/a < 15.5.Since the turbulent Nusselt numbers are independent of the thermal boundary conditionfor Pr > 0.7, the Nusselt numbers that appear in this section will not be specified with thermalboundary conditions.The following correlation, developed by Schmidt [218] to calculate the turbulent Nusseltnumber, is suggested for 2 x 104 < Re < 1.5 x 105 and 5 < R/a < 84:10 36[Equation 5.279 was obtained using air and water flow in coils.
For low Reynolds numbers,Pratt's [219] correlation is recommended:NU t = l + 3 . 4 -~aNusfor 1.5x103 < Re < 2 x 104This correlation was obtained from experiments using water and isopropyl alcohol.(5.280)FORCED CONVECTION, INTERNAL FLOW IN DUCTS5.91When the variable thermal properties of the fluid are considered, Orlov and Tselishchev[220] recommend the following correlation:Nuc (al(Prml °'°25Nu~ - 1 + 3.54 R-/\-~r~ /Rfor --a > 6(5.281)The pitch effect in helicoidal circular pipe has been considered in the investigation conducted by Yang and Ebadian [221]. These researchers have concluded that the effect of pitchis minimum on heat transfer.Fully Developed Laminar Flow in Curved, Square, and Rectangular DuctsThe following formulas are suggested by Shah and Joshi [195] to compute the friction factorfor fully developed laminar flow in curved square ducts:(fRe)c- 0.1520De °5 (1.0 - 0.216De °5 + 0.473De -1 + l l l . 6 D e -~-5- 256.1De -2)(fRe)sfor De < 100(5.282)(fRe)c- 0.2576De °'39(fRe)sfor 100 < De < 1500(5.283)(fRe)c- 0.1115De °5(fRe)sfor De > 1500(5.284)The preceding three equations were obtained through the comparison of theoretical investigations [222-224] and experimental measurement [225].
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