Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 84
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The influence of the pitch of coil onthe friction factor has been found to be negligible [226, 227]. Friction factors for curved rectangular ducts are provided by Cheng et al. [222], as follows:f~ fsCo De *°5 (1.0 + C1 De -lr2 + C2 De -1 + C3 De -3/2 + C4 De -2)(5.285)Equation 5.284 is valid for De < 700. Co, C1, C2, C3, and C 4 in Eq. 5.284 are constants givenin Table 5.45 and De is defined as Re (Dh/R) ~/2.The following correlation, obtained by Cheng et al. [228], is recommended for curvedsquare ducts:Nuts2 = NuT = 0.152 + 0.627(1.414De) °5 Pr °'25(5.286)Equation 5.286 is valid for 0.7 < Pr < 5 and 20 _<De < 705.
This correlation agrees quite wellwith the experimental data for air for the ~ boundary condition [224]. It also represents theNusselt number for the ~ boundary condition [229] quite well.TABLE 5.45Constants for Eq. 5.285 [222]~*CoC~C2C3C40.51.02.05.00.09740.12780.27360.08054.366-0.257-24.79-5.217-13.560.699325.2104.4131.8187.7-1591.0-202.8-182.6-512.22728.00.05.92CHAPTER FIVEIt should be noted again that the effect of the wall thermal boundary condition on the Nusselt number for coils is not significant for the fluids with Pr > 0.7. Equation 5.286 can also beused for the 0), ~ , and @ thermal boundary conditions. Furthermore, the appropriate correlation for circular cross section coiled tubes can be adopted with the substitution of the appropriate hydraulic diameter for 2a to calculate the Nusselt number when the parameters are outof the application range as is the case in Eq.
5.286.Fully Developed Turbulent Flow in Curved Rectangular and Square DuctsFor curved rectangular ducts as well as square ducts, when Re* < 8000, the fully developedfriction factors can be computed from the following correlation obtained by Butuzov et al.[230] and Kadambi [231]:fc _ 0.435 x 10_3 Re,0.96 [ R__R_/°22fs\ d* ](5.287)where d* is the short-side length of the rectangular duct and Re* is defined as umd*/v. Theterm f~ refers to the friction factor in a straight duct with the same aspect ratio as that ofcurved coil.
For Re* > 8000, Eq. 5.276 or Eq. 5.277 for circular ducts can be used with areplaced by 0.5Dh, where D h is the hydraulic diameter of the rectangular duct.The Nusselt numbers for turbulent flow in curved rectangular ducts have been studied byButuzov et al. [230] and Kadambi [231]. The correlation suggested by Butuzov et al.
[230] isas follows:Nuc = 0.117 × 10_2 Re,O.93 [ R_R~°24Nus\ d* /(5.288)This correlation is valid for 450 < Re* (R/d*) °5 < 7500 and 25 < R/d* < 164. The term Nus inEq. 5.288 is fully developed Nusselt number for a straight duct.Laminar Flow in Coiled Annular DuctsXin et al. [232] experimentally investigated the laminar flow and turbulent flow in coiledannular ducts. The pressure drop was measured for air and water flows. Based on these experimental measurements, the friction factor data can be correlated for laminar and turbulentflow as follows:f = 0.02985 + 75.89[0.5 - a tan ( D e77.56- 39"88)/~'](d°Ddi)l45(5.289)where D is coil diameter. This equation is valid in the region of De = 35-20,000, do/di =1.61-1.67, and D/(do - d i ) = 21-32.For the heat transfer in laminar flow in coiled annular ducts, Garimella et al. [233] experimentally obtained the following correlation to calculate the heat transfer coefficient:000e094 r069(5.290)This equation indeed shows that the Dean number represents the heat transfer in laminarflow; the coil ratio (do- di)/D is another factor to affect the heat transfer.Laminar Flow in Curved Ducts With Elliptic Cross SectionsDong and Ebadian [234] numerically obtained the friction factor for laminar flow in curvedelliptic ducts.
The friction factor ratio fc/fs is represented by the following expression:5.93F O R C E D C O N V E C T I O N , I N T E R N A L F L O W IN D U C T S= 1 + 0.0031~X.3 De 1"°7(5.291)f,where f~ is the friction factor for straight elliptic ducts and a* is the ratio of the minor axis tothe major axis of the elliptic duct.In subsequent research [235], thermally developing flow in curved elliptic ducts is analyzedfor different 0~* and Prandtl numbers. The local Nusselt numbers along the flow direction areshown in graph form, and the asymptotic values of the Nusselt numbers have been obtained,as is shown in Table 5.46.
In a related study, the effects of buoyancy on laminar flow in curvedelliptic ducts are discussed by Dong and Ebadian [236].The Asymptotic Values of the Nusselt Numberfor Curved Elliptic Ducts [235]TABLE 5.46Pr0~*R/DhReDe0.10.75500.2410100849.16105.151977.6424.633.3197.89.703.816.3119.224.1811.5526.657.4816.5152.7911.6837.790.54101001271.31058.01514.1635.7334.6151.48.926.735.0723.2315.209.3835.9323.1013.8375.5154.5832.840.8410100881.71336.4118.6440.8422.611.96.576.333.6818.0716.763.7528.9927.584.6264.9161.629.46LONGITUDINAL FLOW BETWEEN CYLINDERSLongitudinal flow between cylinders is encountered in the fuel elements of nuclear powerreactors, shell-and-tube heat exchangers, boilers, and condensers, among other applications.A cylinder is considered to be a long circular pipe or rod.
The flow and heat transfer characteristics between the cylinders are dependent on their arrangement (e.g., triangular array,square array, etc.) as well as the Reynolds number. In this section, the fully developed frictionfactor and Nusselt number for longitudinal flow between the cylinders in triangular andsquare arrays are introduced. For longitudinal flow in other channels formed by the cylindersand the walls, the reader is encouraged to refer to Shah and London [1] and R e h m e [237].Laminar FlowThe friction factor and Nusselt number for longitudinal laminar flow between a triangulararray and a square array are discussed in this section.Triangular Array.
A triangular array is shown in Fig. 5.38. The fluid longitudinally flowsin the virtual channel formed by the triangular array. The friction factor for fully developedlaminar flow in this configuration has been proposed by R e h m e [237] as follows:5.1777(P/D- 1) o.40436.713(P/D- 1) 0.24f R e = 36.947(P/D - 1) 0•372I/16(r, 2 - 1 )37~_2- - - - - ~-~_4-SL--/_21,4r, In r, - 3r, + 4r, - 1for 1.02 < P/D < 1.12for 1.12 < P/D < 1.6for 1.6 < P/D < 2.0(5.292)(5.293)(5.294)for P/D > 2.1(5.295)5.94CHAPTER FIVEP/DforfRe161.01.41.82.22.63.01470J#L1260"NUT10NuNull1"d'/[fRe~,,soo'"5040 fRe3020Null2& Rarnachandra[238],Nut100 i._1.01.21.41.61.82.0P/D for NuH!, NUH2,NUTF I G U R E 5.38 Fully developed f Re and Nusselt numbers for longitudinal laminar flowbetween cylinders in a triangular array [237].whereP x/2X/~Pr, = ~- - n = 1.05 -D(5.296)Equations 5.292 through 5.295 were obtained as a result of comparison with numerousinvestigations such as those by Rosenberg [239], Sparrow and Loeffler [240], Axford [241,242], Shih [243], Rehme [244, 245], Johannsen [246], Malfik et al.
[247], Ramachandra [238],Mikhailov [248], Subbotin et al. [249], Dwyer and Berry [250], Rehme [251], and Cheng andTodreas [252]. The f Re calculated from Eqs. 5.292 through 5.294 is shown in Fig. 5.38.The fully developed Nusselt number for longitudinal flow in a triangular array with uniform cylinder temperature has been analyzed by Ramachandra [238] and is shown in Fig. 5.38.The fully developed Nusselt numbers for the @ and @ boundary conditions have been studiedby Sparrow et al. [253], Dwyer and Berry [250], Hsu [254], and Ramachandra [238].
The differences for NuHi and NUll2 reported by these investigators are small (1 percent). The fullydeveloped Null1 and NUll2 are shown in Fig. 5.38.Miyatake and Iwashita [255] conducted a numerical analysis to determine the characteristics of developing laminar flow between a triangular array of cylinders with a uniform walltemperature and various ratios of pitch to diameter (P/D). The relationships between thelocal Nusselt number NUx,Tand local Graetz number Gz/and between the logarithmic meanNusselt number NUtm,Tand Graetz number Gz were obtained as follows:for P/D = 1.0-1.1"NUx,T = 9.26(1 + 0.0022Gzxl46) TMNUtm,T = 9.26(1 + 0.0179Gz146) TMfor P/D = 1.1-4.0NUx, T = (a 2 +b 2 Gz2/3) 1/2NUtm,T = [a 2 + (3b/2) z Gz2/3] 1/2(5.297)(5.298)(5.299)(5.300)FORCED CONVECTION, INTERNAL FLOW IN DUCTSwherea=b=8.9211 + 2 .
8 2 ( P / D - 1)]1 + 6.86(P/D - 1)5/35.95(5.301)2.3411 + 2 4 ( P / D - 1)][1 + 36.5(P/D - 1)5/4][2V~(P/D - 1) 2 - re]1/3(5.302)Gzx = rftcp/kx(5.303)G z = mcp/kL(5.304)NUx.T = hxD/k(5.305)NUtm.T= hmD/k = rftcp( Th - To)x=LItLA Ttm(5.306)(V~- To)(T~- V.)~:~(5.307)ATom = In [(Tw- To)/(T~- Tb)x=L]In Eqs. 5.306 and 5.307, To, Tw, and Th are the inlet, wall, and fluid bulk temperatures, respectively; L is the length of the cylinder.
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