Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 88
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The f R e and Num for fully developed laminar flow in rectangularducts are also included for the purpose of comparison.Moon-Shaped DuctsA moon-shaped duct is depicted in Fig. 5.58. Shah and London [1] have determined the fully developed f Re and thevelocity profile for moon-shaped ducts. These follow:s S-"- - - - -Zcl(2acos0)u = --~ (r a - b z) 1 - ~ rFIGURE 5.58¢1D2fRe - - ~2u.,A moon-shaped duct.whereDh = 2a[ (2 - (x*2)~J +( o~*)~+ 2sin 2~Cla2 (1~13~'4+ 2CX. 2 - 1 ) ¢ - 8/'30~.3 sin ¢ + (0~'2 - ~ ) sin 2 ¢ - 1/12 sin 4~U m --4(2 - (x'z), + sin 2 ,(5.342)(5.343)(5.344)(5.345)In the preceding equations, o~* - b/a and cl = gdp/dx.Corrugated DuctsThree corrugated ducts are schematically shown in the insets of Fig.
5.59. Hu and Chang [265]have analyzed the f Re for fully developed laminar flow in circumferentially corrugated circular ducts with n sinusoidal corrugations over the circumference as shown in Fig. 5.59, inseta, for e* = ~ a = 0.06. The perimeter and hydraulic diameter of these ducts must be evaluatednumerically. However, their free flow area Ac is given by Ac = no~2(1 + 0.5e2).The f R e , Null1, and Null2 values determined by Hu and Chang [265] for circumferentiallycorrugated circular ducts with sinusoidal corrugations are presented in Table 5.63 as functions5.114CHAPTER FIVE16121-1/c10e* = ~afRe= 0.06E'r2 !0sin(a)I20I4060'(b)I60I80(c)I100I120I140i160i18024,, dellFIGURE 5.59 Fully developed friction factors for circumferentially corrugated circular ducts [2].of n and e*, which are defined in Fig.
5.59. A n g l e 2¢ in Fig. 5.59 is related to n simply as2~ = 360°/n.Schenkel [279] has d e t e r m i n e d the fully d e v e l o p e d friction factors for circular ducts withsemicircular corrugations, as that shown in Fig. 5.59, inset b. For this kind of duct,Ac = r~a2 sin ¢]sin ¢ + cos ¢~ ,sin0P = n2a ~TABLE 5.63 Fully Developed Friction Factors and NusseltNumbers for Circumferentially Corrugated Circular DuctsWith Sinusoidal Corrugations [1]e*fReNumNumDh/2a0.020.040.060.080.100.1215.99015.96215.91515.85015.76515.6784.3564.3344.2974.2444.1764.0904.3574.3354.2994.2464.1774.0890.99860.99440.98740.97760.96500.9501120.020.040.060.080.1015.95215.80615.55915.20014.7114.3404.2674.1423.9623.7234.3404.2674.1403.9563.7090.99660.98630.96890.94390.9107160.020.040.060.0815.88715.54214.94314.0514.3164.1683.9123.5404.3164.1673.9063.5270.99380.97470.94180.8934240.020.040.0615.67914.67112.8724.2453.8753.2314.2453.8703.2190.98560.94020.8583n(5.346)FORCED CONVECTION,INTERNALFLOW IN DUCTS5.115The radius of the semicircular corrugation is a sin ~.
The f Re values for ducts with semicircular corrugations can be determined using the following expression given by Schenkel [279]:f R e = 6.4537 + 0.8350(I) - 3.6909 x 10-2~2 + 8.6674 x 10--4t~3 - 1.0588 x 10-st~4+ 6.2094 × 10-8(I)5 - 1.3261 x 10-4(~6 (5.347)where ~ is in degrees. Equation 5.347 is valid for 0 < 2~ < 180 °. When 2~ = 180, this geometryreduces to a circular duct. The prediction of f Re = 16 was obtained from Eq.
5.347 for circular ducts.Schenkel [279] has also determined the fully developed friction factors for laminar flow incircular ducts having triangular corrugations with an angle of 60 ° , as shown in Fig. 5.59, insetc. For this type of duct, the cross section of the fluid flow area Ac and wetted perimeter P canbe calculated as follows:Ac = rra 2 cos ~ + Vr3 sin ~P = 4rra sin___¢_¢(5.348)The f Re values for ducts with triangular corrugations can be obtained with the followingexpression [279]:f R e = 3.8952 + 0.3692~ - 3.2483 x 10-3(~2 - 3.3187 x 10-st~3 + 4.5962 x 10-Tt~4(5.349)where ~ is in degrees.
Equation 5.349 is valid for 0 < 2¢ < 120 °.A comparison of f Re for these three types of corrugated ducts with e* = 0.06 is displayedin Fig. 5.59.Parallel Plate Ducts W i t h S p a n w i s e Periodic C o r r u g a t i o n s at O n e WallTwo types of corrugations (triangular and rectangular) in parallel plate ducts are displayed inthe insets of Figs. 5.60 and 5.61, respectively.
Sparrow and Charmchi [290] have obtained thesolutions for fully developed laminar flow in these ducts. The flow in the duct is considered tobe perpendicular to the plane of the paper. Both ducts are assumed to be infinite in the span-22.54.5------20.0fRe4.0NUHI3.015.0Null1fRe2.020*~I0.01.o5.0~o00.10.20.30.40.5a/bFIGURE 5.60 Fully developed friction factors and Nusselt numbers for flat ductswith spanwise-periodic triangular corrugations at one wall [290].5.116CHAPTERFIVE40c/d35-'-----....301131122/325b/d ffi" 124f Re 2015b/d= 1/5b/d = 510000.20.40.60.81.0a/bFIGURE 5.61 Fully developed friction factors and Nusselt numbers for flat ductswith spanwise-periodicrectangular corrugations at one wall [291].wise direction; therefore, the end effects due to the short bounding walls are neglected. Thecorrugated wall is subjected to the ~ thermal boundary condition, while the flat wall is considered to be adiabatic.The cross-sectional area and perimeter of a flat duct with spanwise triangular corrugationcan be found by:A c - n ( b 2 - a 2) tan ¢,sin ¢P = 2 n ( b - a) 11 ++ cos(5.350)where n represents the number of triangular corrugations and 2¢ is the angle of the top vertex of the triangle.The fully developed f Re and Null1 values obtained by Sparrow and Charmchi [290] areshown in Fig.
5.60, which is taken from Shah and Bhatti [2]. If a/b = 0, the duct with triangular corrugations reduces to an array of isosceles triangles. The f R e and Null1 values from Fig.5.60 agree well with the values obtained from the corresponding figures in the section concerning triangular ducts.Fully developed laminar flow and heat transfer in a parallel plate duct with spanwiseperiodic rectangular corrugations at one wall have been investigated by Sparrow andChukaev [291]. The end effect is also ignored in their analysis.
The fully developed f Re isshown in Fig. 5.61, which is based on the results reported by Sparrow and Chukaev [291] andthe extension by Shah and Bhatti [2]. The heat transfer characteristics for the three pairs ofgeometric parameters can be found in Sparrow and Chukaev [291].C u s p e d DuctsA c u s p e d duct, also referred to as a s t a r - s h a p e d duct, such as the one shown in Fig. 5.62, ismade up of concave circular arcs.
The fully developed f R e , Null1, and Nut, in laminar flow aregiven in Table 5.64, in which n is the number of the concave circular arcs in the cusped ducts.The values f o r f R e , Nut, and Null1 are taken from Shah and London [1], Dong et al. [292], andFORCED CONVECTION, INTERNAL FLOW IN DUCTS5.117TABLE 5.64 Fully Developed fRe, Nux,and Num for Laminar Flow in Cusped DuctsnfReNuxNUll1345686.5036.6066.6346.6396.6290.921.091.23----1.352----Dong and Ebadian [293]. An analysis of thermally developing laminar flow in cusped ducts can be found in Dong et al.[292].Cardioid DuctsFIGURE 5.62 A cusped duct with four concavewalls.Y.A cardioid duct is shown in Fig.
5.63. Fully developed laminar flow and heat transfer under the (~ boundary conditionhave been analyzed by Tyagi [294]. The f R e and NUn1 valuesderived from this analysis are 5.675 and 4.208, respectively.The Nusselt number for the @ thermal boundary conditionwas found to be 4.097 [1].Unusual Singly Connected Ducts~zFor the fully developed friction factors for laminar flow inunusual singly connected ducts, interested readers are encouraged to consult Shah and Bhatti [2].Z r=2a(l+cosS)FIGURE 5.63 A cardioid duct.OTHER DOUBLY CONNECTED DUCTS,*2b o 2hi[,f26o2a or*-- 2hi2boFully developed laminar flow and heat transfer in severaldoubly connected ducts are discussed in the following sections.Confocal Elliptical DuctsFIGURE 5.64 A confocal elliptical duct.A confocal elliptical duct is shown in Fig.
5.64. According tothe analysis by Topakoglu and Arnas [295], the friction factor for fully developed laminar flow in confocal ellipticalducts can be computed by256A 3f R e = rClooPZ(ao+ bo)4where1(mS)_I o o = - ~ ( 1 - ~ 4) 1+--~- - 2 m 4 1 (o211 + 0.)2 + 4 In co(5.351)((1 - (o2)2 1 -m4 2~J(5.352)5.118CHAPTER FIVEZc(m4)(ao + bo) 2 = 4 (1 - 032) 1 + ~PE(5.353)(m2 ]- 2 (1 + m2)E1 + 1 + --~ 03Eo~(5.354)03 = ( ai + biob = °t*r* + [ 1 - +°t*2(1(x*(5.355)ao + bo1 -(x* / 1/2m=1+o~*](x* -'boao'r* -bibo(5.356)E1 and Eo, are the complete elliptical integrals of the second kind.
These are evaluated usingthe a r g u m e n t s l - bo/ao2 2 and 1 - bi2]ai,2 respectively. In addition, b~/ag is related to 03 and m bym e a n s of the following:bi _ 1 - (m2/032)ai(5.357)1 + (m2/032)The fully developed Nusselt n u m b e r s Null1 d e t e r m i n e d from the analysis of Topakogluand A r n a s [295], together with the f R e calculated from Eq. 5.351, are displayed in Table 5.65.Regular Polygonal Ducts With Centered Circular CoresThe product of fully developed friction factor and Reynolds n u m b e r in laminar flow f R eobtained by R a t k o w s k y and Epstein [296] for polygonal ducts with centered circular cores(see the inset in Fig. 5.65) are shown in Fig. 5.65. The fully developed Num obtained by C h e n gand Jamil [297] are given in Fig. 5.66.
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