Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 86
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5.47. Swirl flow is created in this manner. The0.0II1 IIIIIIl0.0width of the tape is usually the same as the internal diameter0.0 0.20.40.60.8 1.0of the duct. The tape twist ratio X~ is defined as H / d . Whenl/aXL approaches infinity, the circular duct with the twistedFIGURE 5.44 Friction factor and Nusselt number tape becomes two semicircular straight ducts separated byfor fully developed laminar flow in rectangular ductsthe tape.with longitudinal thin fins from opposite walls [1].Manglik and Bergles [272, 273] made an extensive reviewon the study of laminar and turbulent flow in circular ductswith inserted tape.
For laminar flow, the dimensional swirl parameter S w was incorporated inthe correlation of friction factor. This parameter considers the thickness of inserted tape &-I0080/.=±/60//////40/~40'f'".30202o.--"'..I'I'_ . 1. ~ - '- -I2~ =0 °- 24 = 3*/~"'~'\'~"-'"- 24 =6 ° /f.~'"'"\\-1"=/ ' i ' , " / / -/ /,J16I08NUH~d 6 ~4.3644[3- !/.....2~ =6 °24812n1620""FIGURE 5.45 Friction factors for fully developedlaminar flow in a circular duct with longitudinal triangular fins [1].l =0.6///""~ £ = o . , - . . . jL/,""80-"'-~....24•""/I.": .-'li~ .....'V,'"/2~ = 0", 3",6"I0"".
' ~ ~ -/4, , _ ~ ~ . _=u.z- , , ,Ft8121620~I,24FIGURE 5.46 Nusselt numbers for fully developedlaminar flow in a circular duct with longitudinal triangular fins [1].FORCED CONVECTION, INTERNAL FLOW IN DUCTSH~A5.103TAPESECTION AAF I G U R E 5.47A circular tube with a twisted tape inserted [1].the twist ratio Xz,, and the helicoidally twisting flow velocity. The definition of dimensionalswirl parameter Sw is given as:Sw = Resw V ~ L1+n - 45/d(5.330)For most applications, the following equation is recommended for the calculation of friction factor in laminar flow in circular ducts with inserted tape:f Re = 15.767rl:(rl: + 2 - -~)2(rl: - 4 ~-)-3(1 + 2--~t )(1 + 10-6Sw255)1/6(5.331)where f R e is based on the empty tube diameter d.
The above equation can be applied in therange 0 < (8/d) < 0.1 and 300 ___Sw < 1400.The mean Nusselt number for the laminar flow in isothermal circular ducts with insertedtape can be obtained from the following equation suggested by Manglik and Bergles [272]:Num = 4.612{[(1 + 0.0951Gz°-894) z-5 + 6.413 x 10-9(Sw • Pr°391)3853] °2+ 2.132 x10-14(Re~x•Ra)223}°1(\ ~~l'mw // 0"14 (5.332)Where Gz, Re~x, and Ra are the Graetz number, the Reynolds number based on axialvelocity, and the Rayleigh number, respectively. Their definitions are expressed as follows:Gz = m c p / K L(5.333)rnktRe~ = n,d/4 - 8~(5.334)Ra = pgd3~SATw~ta(5.335)For the turbulent flow in circular duct with inserted tape, it was proposed by Manglik andBergles [273] that the friction factor can be calculated by the following equation:0.0791( )1.75(2.752)f = °2---------TRen - 48/d1 d- Sl.29(5.336)It was found that the flow rates with Re > 104 can be considered as fully developed turbulent flow.
Therefore, the above equation is a more generalized correlation that covers a broaddatabase of available empirical data for turbulent flow [273].5.104CHAPTER FIVETABLE 5.51 The Fully Developed (fRe)d and Nud Values for Forced Convection of Laminar Flowin a Semicircular Duct With Internal Fins [274]Fin length (l*)n0.10.20.30.40.51358111743.307-46.83649.207---46.12953.34560.20569.10375.92784.65650.369-84.306105.634--55.71787.943122.503167.660201.532243.45461.674m175.639265.071---0.60.70.80.91.067.462140.762234.651391.297547.204815.77272.148-280.714506.379--75.047170.601303.208568.948910.7261798.8076.175-309.012585.854---76.314173.382309.420587.054953.7621959.2311.15924.81638.58141.32932.08419.04111.765m47.00177.458~11.83925.84344.12779.918124.440211.24311.839-42.66776.329-~11.82125.18042.55876.021118.145226.977(fRe)dNum.a135811176.806-6.8786.904m~7.1967.5317.6687.6277.4677.1767.895-9.3838.954~8.89612.17113.10411.76010.2538.80810.086m21.89219.148m~A g e n e r a l i z e d c o r r e l a t i o n of m e a n Nusselt n u m b e r for t u r b u l e n t h e a t transfer in ani s o t h e r m a l circular duct with i n s e r t e d t a p e was d e v e l o p e d by M a n g l i k and B e r g l e s [273] b a s e don the e x p e r i m e n t a l data.
It is e x p r e s s e d as:Nu =O.O23(l + O.769/XL) Re°8pr°4(rt)°8( ~ + 2 + 28/d ) °2n - 48/dn - 45/d~(~*'l'minor(rm) mwhere~ = \-g--~-/(5.337)(5.338)n - 0.18 for liquid heating and 0.30 for liquid cooling; m = 0.45 for gas h e a t i n g and 0.15 for gascooling.Semicircular Ducts With Internal FinsD o n g and E b a d i a n [274] have used a very fine grid to p e r f o r m a n u m e r i c a l analysis of fullyd e v e l o p e d l a m i n a r flow in a semicircular duct with internal l o n g i t u d i n a l fins. T h e fins are considered to h a v e z e r o thickness, and the n u m b e r of fins n and relative fin length l* = / / a areyt a k e n into account.
The (~ t h e r m a l b o u n d a r y c o n d i t i o n isapplied. Their results are given in Table 5.51.Elliptical Ducts With Internal Longitudinal Fins1txa-- XFIGURE 5.48 An elliptical duct with internal fins.A n elliptical duct with f o u r internal l o n g i t u d i n a l finsm o u n t e d on the m a j o r and m i n o r axes, as s h o w n in Fig. 5.48,has b e e n a n a l y z e d by D o n g and E b a d i a n [275] for fullyd e v e l o p e d l a m i n a r flow and h e a t transfer.
In this analysis,the fins are c o n s i d e r e d to have z e r o thickness. T h e ~ t h e r m a lb o u n d a r y c o n d i t i o n is a p p l i e d to the duct wall, and l* isdefined as a ratio of Ha/a = Hb/b. The friction factors andNusselt n u m b e r s for fully d e v e l o p e d l a m i n a r flow are givenin Table 5.52.FORCED CONVECTION, INTERNAL FLOW IN DUCTS5.105TABLE 5.52 Friction Factors and Nusselt Numbers for FullyDeveloped Flow in an Elliptical Duct With Internal Fins [275]o~*1"(f Re)dNU.,.d(f Re)Num0.00.50.80.91.072.20108.61270.63301.73313.623.783.7812.5116.0445.8572.2045.3669.3471.5367.263.781.583.203.803.400.50.00.50.91.067.26133.39297.78309.203.754.9716.3116.1167.2650.0268.7861.713.751.863.763.210.80.00.50.91.064.33135.53391.06303.383.675.3616.1915.9664.3350.7267.3458.913.672.003.743.101.00.00.50.91.063.99140.57294.18301.823.675.6016.1415.9663.9951.4665.8958.403.672.053.613.090.25OTHER SINGLY CONNECTED DUCTSThe fluid flow and heat transfer characteristics for 14 typesof singly connected ducts are described in this section.YSine DuctsI..-.2,-,,.1A sine duct with associated coordinates is shown in Fig.
5.49.The characteristics of fully d e v e l o p e d laminar flow and heattransfer in such a duct are given in Table 5.53. These resultsare based on the analysis by Shah [172].r..-2FIGURE 5.49 A sine duct.TABLE 5.53 Fully Developed Fluid Flow and Heat Transfer Characteristicsof Sine Ducts [172]b/aK(oo)Lh+yf ReNuxNumNUll2oo23/21V/3/23.2181.8841.8061.7441.7390.17010.04030.03940.04000.040815.30314.55314.02213.02312.6300.739m2.602.452.5213.3113.2673.1023.01400.951.371.5551.473A½¼1/801.7441.8102.0132.1732.2710.04190.04640.05530.06120.064812.23411.20710.1239.7469.6002.332.121.80m1.1782.9162.6172.2132.0171.9201.340.900.330.09505.106CHAPTER FIVETrapezoidal DuctsA trapezoidal duct is displayed in the inset of Fig.
5.50. Fully developed laminar flow and theheat transfer characteristics of trapezoidal ducts have been analyzed by Shah [172]. The fullydeveloped f Re, Nulls, and NUll2 are given in Figs. 5.50 and 5.51. Farhanieh and Sunden [276]numerically investigated the laminar flow and heat transfer in the entrance region of trapezoidal ducts. The fully developed values of f Re, K(oo), and Nu were in accordance with theresults from Shah [172].24.0I_l_J..iI/'i¢=IIlIIIIi,~ " -- , ,I1III /- - - - - - - - fRe....K(==)2.22.01.8- , , ,,22.06(7'%%%75*\ '," ~ "",%%20.0fRe1.61.4 K(oo)/1B.01.216.01.014.00.812o_3°.11.00.0_ _• * = 2a/2b ~0.20.40.60.8a* = 2b/2a1.00.80.60.40.60.00.21/$FIGURE 5.50Fully developed f Re and K(o~) for laminar flow in a trapezoidal duct [172].9.03.285"8.0NUH]....7.0Nufl2/60*'U "l6.02.875"2.4--"!p,~2.01.6 NUH~NUll1 5.0.........4.01.2~ .
_ _ r3.02.°V1.00.0FIGURE30~" S0.80.4=* = 2=/2b ~I5.51=0.2l10.4iI0.6iI0.8I=* = 2b/2=I1.0allII0.8II0.6II0.4II0.2l0.00.0F u l l y d e v e l o p e d Nusselt n u m b e r s for l a m i n a r f l o w in a t r a p e z o i d a l pipe [172].FORCED CONVECTION, INTERNAL FLOW IN DUCTS5.107Chiranjivi and Rao [277] experimentally obtained a correlation for laminar and turbulentflow in trapezoidal ducts with one side heated, which is expressed as:Nu = a Reb pr°52(-~-)(5.339)where a = 6.27 and b = 0.14 for laminar flow, and a - 0.79 and b - 0.4 for turbulent flow ofReynolds n u m b e r from 3000 to 15,000.Rhombic DuctsA rhombic duct is depicted in Fig. 5.52.
The fully developedflow and heat transfer characteristics of rhombic ductsobtained by Shah [172] are shown in Table 5.54._j i -iii__FIGURE 5.52 A rhombic duct.Quadrilateral DuctsA quadrilateral duct is shown schematically in Fig. 5.53.N a k a m u r a et al. [278] analyzed fully developed laminar flowand heat transfer in arbitrary polygonal ducts. Their resultsare presented in Table 5.55.Regular Polygonal DuctsFully developed flow and heat transfer in a regular polygonal duct with n equal sides, each subtending an angle of360°/n at the duct center, have been reviewed by Shah andL o n d o n [1].
The f Re and Nu are given in Table 5.56.FIGURE 5.53 A schematic drawing of a quadrilateral duct.TABLE 5.54Fully Developed Laminar Flow and Heat Transfer Characteristicsof Rhombic Ducts [172]01020304045K(oo)Lh+y2.9712.6932.3842.1201.9251.8500.10480.07320.05700.04770.04190.0397f ReNUll1 NUll2 ¢K(oo)Lh+y12.00012.07312.41612.80313.19313.3812.0592.2162.4572.7222.9693.0801.7781.6731.6031.5641.5510.03800.03530.03360.03270.032400.0700.2790.6241.091.345060708090f ReNUll1 NUll213.54213.83014.04614.18114.2273.1883.3673.5003.5813.6081.622.162.642.973.09TABLE 5.55 Fully Developed Friction Factors, Incremental Pressure Drop Numbers,and Nusselt Numbers for Some Quadrilateral Ducts [278](Ih (deg)(I)2(deg)(I)3(deg)(1)4(deg)fReK(~)NUll1NUll260506060706030304530456032.2321.6771.5779.1114.1614.3614.6914.011.6541.6121.5221.7073.453.553.723.352.802.903.052.685.108CHAPTERFIVEFully Developed Laminar FlowCharacteristics of Regular Polygonal Ducts [1]TABLE 5.56nfReNUHINUll2NUT3456713.33314.22714.73715.05415.313.1113.6083.8594.0024.1021.8923.0913.6053.8624.0092.472.97689102015.41215.5215.6015.8816.0004.1534.1964.2274.3294.3644.1004.1594.2014.3284.364oom3.657,,,For practical calculations, Schenkel [279] proposed the following formula to compute thef Re in regular polygonal ducts:(n2)4f Re = 16 0.44 + n 2(5.340)The values of the predictions from Eq.
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