Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 82
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For 60 ° < 2~ < 90 °, the use of circular duct correlations with Dh is probably accurate enough. For 2~ = 90 °, the circular duct correlations inTable 5.8 can be used. For 2~ > 90 °, no definite r e c o m m e n d a t i o n can be made at this moment.Right Triangular Ducts. The fully developed turbulent friction factor for two right-angledtriangular ducts and an equilateral triangular duct are shown in Fig. 5.31 [45]. The data arefrom Nikuradse [178] and Schiller [179]. Also provided in this figure are the correlations forcomputing the friction factor for each triangular duct.Thermally Developing Flow. Altemani and Sparrow [176] have conducted experimentalm e a s u r e m e n t s of the thermally developing flow of air (Pr = 0.7) in an equilateral triangularduct with the ~) boundary condition on two walls and the third wall insulated. The local Nusselt numbers Nux, m and the thermal entrance lengths from their results are given in Figs.
5.32and 5.33, respectively.0.1000.050(a)f = 0.079 Re"°.250.0200.0100.0O5 _ 0 Nikuradse [178]• Schiller [17910.002 -/ / ~(=)t2 X 102 4I6(b)II103!2,, tto)~ ,I6 104Re,4I2,t46 10s2FIGURE 5.31 Fully developed friction factor for turbulent flow in smooth rightangled and equilateral triangular ducts [45].1400 Measurements [176]120AdiabaticwallRe= 59,13010080NUx,HI0040,0--JI6028,57020,1304013,8609,73020 ~100.20..40..606,74080x/DhFIGURE 5.32 Local Nusselt numbers Nux,m for thermally developing andhydrodynamically developing turbulent airflow (Pr = 0.7) in a smooth equilateraltriangular duct [176].5.80FORCED CONVECTION,INTERNALFLOW IN DUCTS50II' II.
. . . . .5.81IO Measurements [176]40Lth, HID,(1,,,,,,30.-20-IIlO....LoI.... I2 x 1(#4 x 1(#,I6 x 1(#ReFIGURE 5.33 Thermal entrance lengths for thermally developing and hydrodynamicallydeveloping turbulent airflow (Pr = 0.7) in a smooth equilateral triangular duct [176].For equilateral triangular ducts having rounded corners with a ratio of the corner radius ofcurvature to the hydraulic diameter of 0.15, Campbell and Perkins [180] have measured thelocal friction factor and heat transfer coefficients with the @ boundary condition on all threewalls over the range 6000 < Re < 4 x 104. The results are reported in terms of the hydrodynamically developed flow friction factor in the thermal entrance region with the local wall(T~) to fluid bulk mean (Tm) temperature ratio in the range 1.1 < Tw/Tm< 2.11, 6000 < Re <4 x 104, and 7.45 < X/Dh < 72.
These data were correlated byfo( Zw l-O'40+(X/Oh)"0"67- \ T,, /(5.248)where f, so denotes the friction factor for isothermal flow, which can be calculated from eitherthe Blasius formula or the PKN formula presented in Table 5.8. In these calculations, kinematic viscosity v entering Re = UmDh/Vmust be evaluated at the duct wall temperature Tw.The following correlation has been obtained by Campbell and Perkins [180] from theirmeasurements for local Nusselt numbers:Nux,H1 = 0.021Re °'8 Pr °'4 (\-~m]Tw / 0.7¢(5.249)For 6 < X/Dh < 50, the correction factor • is given by•=l+\oh]\Tm/ J(5.250)and for X/Dh> 50, (I)= 1.
Equation 5.249 is valid for 6 <X/Dh < 123 and 1.10 < Tw/Tm< 2.11. Thefluid properties v, tx, and k entering Eq. 5.249 must be taken at the fluid bulk mean temperature T,,.5.82CHAPTERFIVEELLIPTICAL DUCTSElliptical ducts can be thought as a family of ducts, including geometries ranging from lenticularto circular ducts. The major and minor axes lengths are represented by 2a and 2b, respectively,in this section. The origin of the coordinate is the intersection of the major and minor axes.LaminarFlowPresented in this section are the friction factors and Nusselt numbers for laminar flow in elliptical ducts.Fully Developed Flow.
The velocity distribution for fully developed laminar flow in ellipticducts with major axis 2a and minor axis 2b is given by Shah and London [1] as follows:uIUmUm-----~1+"~Xo~. 2fRe'2(1=E(m)+ o~ . 2 )(5.252)In the preceding equations, m = 1 + ct.2, and E(m) is the complete elliptic integral of the second kind. The hydraulic diameter of the elliptical duct is Dh = nb/E(m), and the crosssectional area is Ac = nab.The expression for hydrodynamic entrance length L~y developed by Bhatti [181] is as follows:0.5132 ( E ( m ) ) 2Lhy - 1 + O~:~2n(5.253)+The fully developed incremental pressure drop number K(oo) for elliptic ducts has beenfound to be independent of the duct aspect ratio o~* = 2b/2a [187]. The value of K(oo) is recommended to be 1.26 for practical calculations [2].The fully developed Nusselt numbers NuT, Nun1, NUn2 are displayed in Fig.
5.34. The values of the data used in this figure are derived from the results by Tyagi [6], Tao [182], Iqbal etal. [175], and Dunwoody [183].6.0iiI1Ii1iI'1IIIIII['IIm=5.0 ~NUHI~-4.0 -NUT.-iNu~~~ = " ~ =jiium3.0 -,Bu2.0-m,i1.0ma* = 2 b / 2 a0.0-,,,'J'I0.0oaF I G U R E 5.34IFullyI0.2ii0.3IIa0.4developedNusscltI0.sII0.6I .I0.7IIo.sIuI019numbers for elliptical ducts [2].J1.0FORCEDCONVECTION,INTERNALFLOWINDUCTS5.83Hydrodynamically Developing Flow. Bhatti [181] has analyzed hydrodynamically developing flow in elliptic ducts.
The apparent friction factors and incremental pressure drop numbers can be expressed as:lapp R e -Ap*4x+,Ap* =K(x) =2(1 - 11)(1 + 311) - (1 + 1"1)2 In 1"133(1 + rl) 2(31]3 + 9rl 2 + 211] + 7)(1 - rl)6(1 + 11)2(5.254)(5.255)where 11 is a boundary layer parameter that refers to the fraction of the duct cross section carrying inviscid flow. The term rl is determined from the following equation implicitly as a function of x+:16(1 + a , 2 )E(m)(5.256)x+ = r12 - 1 - In 112Thermally Developing Flow.
The local Nusselt number for thermally developing flow inelliptic ducts with uniform wall temperature Nux,v was obtained by Dunwoody [183] in termsof a double infinite series. These results are considered the most accurate as x* > 0.005. Dunwoody's formula for calculating the mean Nusselt number is as follows:NU.,,T = ~- 1 + - ~(5.257)The ;~ and C values for Eq. 5.257 are given in Table 5.43.The Constants ~, and Cfor Eq. 5.257 [183]TABLE 5.43a*kC0.06250.1250.250.50.814.5914.9015.1714.9714.670.05780.03880.02390.01580.0138For x* < 0.005, the following formula obtained by Richardson [184] is recommended:3((e-(x*)2+(l-(x*)NUm.T= F(a/3)(9x+)~,3 1 +3)36(5.258)For elliptic ducts subjected to the (~ thermal boundary condition, Someswara et al.
[185]have solved thermally developing flow. The mean Nusselt number Num,m can be computedusing the following expression:2.61FNum.nl- x,1/3(5.259)where factor F is a function of 0~* and can be calculated by the following [2]:F = 1.2089- 0.795(x*- 4.3011(x .2 + 23.8465(x .
3 - 44.7053(x .4 + 37.0874(x . 5 - 11.4809(x .6(5.260)Equation 5.260 is accurate within +3 percent error to the original data given by Someswara etal. [185].5.84CHAPTERFIVETurbulent FlowThe friction factors for fully developed turbulent flow have been measured by Barrow andRoberts [186] in elliptical ducts with co* = 0.316 and 0.415 in the range of 103 < Re < 3.105 andby Cain and Duffy [187] in elliptical ducts with ct* = 0.5 and 0.667 in the range 2 x 104 < Re <1.3 x 105.
Based on the data presented by Barrow and Roberts [186] and Cain and Duffy [ 187],Bhatti and Shah [45] have derived the following correlation to calculate the friction factor:f = 0.4443 + 2.2168o~* - 2.0431ct .2 + 0.3821c~.3(5.261)fcwhere fc is the friction factor for a smooth circular duct (c~* = 1), which can be obtained fromthe Blasius equation in Table 5.8. It should be noted that Eq. 5.261 is valid for 0.136 < c~* < 1.Heat transfer in fully developed turbulent flow in elliptical ducts has been determined inseveral investigations.
A comparison of the different results has been presented by Bhattiand Shah [45]. It was concluded that the Gnielinski correlation for circular ducts can confidently be used to calculate the fully developed Nusselt number for elliptical ducts for fluids ofPr ___0.5.For liquid metals, the fully developed Nusselt numbers can be determined using Eq. 5.236for elliptical ducts with the @ boundary condition. The values of Nus~ugrequired in Eq. 5.236are given in Fig. 5.35.10.09.5Nudug 9.08.58.000.20.40.60.81.0(X*FIGURE 5.35 Slug flow Nusselt numbers for elliptical ducts with the ~ boundarycondition [169].CURVED DUCTS AND HELICOIDAL PIPESThe most prominent characteristic of flow in curved ducts and helicoidal pipes is the secondary flow induced by centrifugal force due to the curvature of the pipe. Consequently, thefriction factor is higher in curved pipes than that in straight pipes for the same Reynolds number.
The pitch of the helicoidal pipe also has an effect on the flow. As a result, the heat trans-FORCED CONVECTION, INTERNAL FLOW IN DUCTS5.85fer rate is higher in curved or helicoidal pipes than in straight pipes. Therefore, curved or helicoidal pipes are widely used in engineering applications.Spiral coils are curved ducts with varying curvature. The friction factor and heat transferrate for spiral coils are also included in this section. In addition to the dimensionless parameters used in straight pipes, the following parameters are particularly useful in the case ofcurved ducts or helicoidal pipes: the Dean number De; the helical number He, and the effective radius of curvature Re. These are defined as follows:De = Re(5.262)[ { b t2]m]He = Re(-~c) ~/2= De 1 + \ - ~ jRc-Rll(5.263)+ ( 2n--~R-)](5.264)where a denotes the radius of a circular pipe, b represents the coil pitch, and R is the radius ofthe coil.In this section, emphasis will be given to the correlations used for calculating the frictionfactors and Nusselt numbers for laminar and turbulent flows in curved ducts, helicoidalpipes, and spiral ducts.
These will be presented as the ratio of the friction factor in curvedducts to the friction factor in straight ductsand the ratio of the Nusselt number in curvedducts to the Nusselt number in straight ducts Nuc/Nu, in most cases. The subscript c represents curved ducts or helicoidal pipes, while the subscript s denotes straight pipes of the sameshape.fc/fsFully Developed Laminar FlowDean [188, 189] first studied the velocity profile of flow in helicoidal pipes using perturbationanalysis. His result is valid only for De < 20, where the velocity distribution is almost identicalto that found in straight ducts.
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