Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 70
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A great number of experimental investigations have been performed to ascertain the critical Reynolds number atwhich laminar flow transits to turbulent flow. It has been found that the transition from laminar flow to fully developed turbulent flow occurs in the range of 2300 < Re < 104 for circularducts [41]. Correspondingly, flow in this region is termed transitionflow. More conservatively,the lower end of the critical Reynolds number is set at 2100 in most applications.Generally, the duct inlet configuration and surface roughness have significant effects onthe value of Recnt. Other factors, such as noise, vibration, and flow pulsation, affect Re,it aswell.
Caution should be taken to choose Rent for the particular application. On the otherhand, flow and heat transfer characteristics are difficult to predict in transition flow. Thereader is encouraged to consult the literature for the cases not mentioned in this chapter.Fully Developed Flow. In this section, the characteristics of the fully developed turbulentflow and heat transfer are presented for both a smooth and a rough circular duct with a diameter of 2a.FORCED CONVECTION, INTERNAL FLOW IN DUCTS5.19The Power Law Velocity Distribution.
The solution for the power law velocity distribution is introduced in Prandtl [42] in the following form:u{y~u2n 2Umax(n -t- 1)(2n d- 1)TMUmax ~ a )(5.65)where y = a - r represents the radial distance measured from the wall. The exponent n varieswith the Reynolds number. The values of n are listed in Table 5.7; these were obtained fromthe measurements by Nikuradse [43].TABLE 5.7Constants in Eqs. 5.65, 5.71, and 5.72RenmC40002.3 x 1 0 41.1 x 1051.1 x 10 62.0 X 1063.2 X 10666.678.810103.53.844.95.55.50.10640.08800.08040.04900.03630.0366Universal Velocity-Defect Law. Figure 5.7 shows the velocity profile computed from Eq.5.65 together with Nikuradse's [43] measurement data.
It can be observed that the velocityprofile becomes flatter over most of the duct section, and the exponent 1/n of the power lawof the velocity distribution, Eq. 5.65, decreases as Re increases. This observation led to thederivation of another form for velocity distribution, the universal velocity-defect law. The for1.00.90.80.7Re3.2 X 1060.62.0 X 10 6Mmax1.1 X 10 60.51.1 X 10 50.42.3 X 104• N~'s4.0 X 10 30.3~ta Hal0.200.20.40.60.8l.Oy/aPower law distribution for fully developed turbulent flow in a smoothcircular duct [45].F I G U R E 5.75.20CHAPTERFIVEmula, which follows, was discovered by Prandtl [44] and can be used in high Reynolds numberflow, applicable in the turbulent core:Uma x --U_.
2.5In autwhereut"-(5.66)y=Um(5.67)denotes friction velocity.Von K~irm~in [46] derived the following form for the universal velocity-defect law:Umau:U=-2.5[ln ( 1 - ~ / 1 - Y ) + ~/1 - y ](5.68)Wang [47] proposed a third form for the universal velocity-defect law. It follows:.Umax-u = 2.5[ln l + V ' l - y / a _ 2 t a n - a / l _ Y _ o . g 7 2 1 n 2 " 5 3 - y / a + l ' 7 5 X / 1 - y / a]u,1 - ~/1 - y/aa2.53 - y/a - 1.75%/1 - y/a+ 1.43 tan -1 1"75%/1- y/a0.53 + y/a(5.69)Darcy's [48] experimental measurements led to the following formula for the velocitydefect law:Um,~- u _ 5.08 1 -(5.70)UtFigure 5.8 displays the velocity distributions in terms of the velocity defect obtained from Eqs.5.66, 5.68, 5.69, and 5.70. When compared to the experimental data presented by Nikuradse[43], it can be observed that the Eq.
5.69 is in overall best accord with the data; however, itis too complicated to be used. Equation 5.70 agrees well with the data except near the wall,y/a < 0.25.Friction Factor. From the power law velocity distribution of Eq. 5.65, the friction factorcan be expressed as:Cf = Re1/m(5.71)where C is an experimentally determined constant and m is related to the n in Eq. 5.66 as follows:m-n+l2(5.72)The constants n, m, and C have been determined based on Nikuradse's [43] measurements, asis shown in Table 5.7.Several friction factor correlations for fully developed turbulent flow in smooth, circularducts are listed in Table 5.8.
According to Bhatti and Shah [45], these formulas were derivedfrom highly accurate experimental data for a certain Reynolds number range.The Prandtl-K~irm~in-Nikuradse (PKN) correlation is based on the universal velocitydefect law with the coefficients slightly modified to fit the highly accurate experimental datareported by Nikuradse [43], which is known to be the most accurate. This correlation is alsoreferred to as the universal law of friction. However, since the PKN formula gives f valuesFORCED CONVECTION, INTERNAL FLOW IN DUCTS5.2110~ , ~ l~Yilnl,lH( ~~ .
5.67Eq. 5.68F:~1. 5.69~ . 5.70MIdt•[44][46][47][48]NiKuradse's data [43]Re = 3.24 X 10601-.0.00.10.20.30.40.50.60.70.80.91.0y/aFIGURE 5.8 Universalvelocity-defect law distribution for fully developed turbulentflow in a smooth circular duct [45].implicitly, the explicit formulas by Colebrook [54] and Techo et al.
[56], which are closeapproximations to the PKN formula, may be used.Velocity Distribution a n d the Friction Factor f o r R o u g h Circular Ducts. Fully developedvelocity distribution in a completely rough circular duct has been expressed by Schlichting[57] as follows:yu+= 2.5 In ~whereu+ - u _Utu;V'xwlp++ 8.5y+_ yu, _ ~Y~/~w/Pvv(5.73)(5.74)and Re~ is the roughness Reynolds number, defined as Re~ = eut/v. T h e term e denotes thesurface-roughness element height.
The value Re~ < 5 corresponds to the hydraulically smoothregime; 5 _<Re~ > 70 corresponds to the transition from the hydraulically smooth to the completely rough regime; and Re > 70 corresponds to the completely rough regime. Furthermore,y+ < 5 corresponds to the laminar sublayer region, whereas y+ > 70 is the fully turbulent regionand 5 < y+ < 70 is the transition region.The friction factor correlations for fully developed turbulent flow in a rough circular ductare summarized in Table 5.9. The friction factor for turbulent flow in an artificially roughedcircular duct can be found in Rao [59].Moody's [58] plot, shown in Fig.
5.9, gives the friction factor for laminar and turbulent flowin both smooth and rough circular ducts. Relative roughness E/Dhis used as a parameter for5.22CHAPTER FIVETABLE 5.8Fully Developed Turbulent Flow Friction Factor in Smooth, Circular Ducts [45]InvestigatorsCorrelationRecommended Re rangeBlasius [49]f = 0.0791 Re -°254×10 3 t o 10 5McAdams [50]f = 0.46 Re -°2f = 0.036 Re -°'1818f = 0.0366 Re -°18183×4×4×10 4 t o 10 6f=f=f=f=105 to 1074 × 103 to 5 X 1064 × 10 3 to 1074 × 10 3 to 10 7Bhatti and Shah[45]Nikuradse [43]Drew et al.[51]Bhatti and Shah[45]Prandtl [5210.0008 + 0.0553 Re -°:370.00128 + 0.1143 R e -°'3110.0014 + 0.125 R e -°'320.00128 + 0.1143 R e -°311110 4 to 10 710 4 t o 10 7RemarksWithin +2.6 and-1.3% ofPKN (see the following)Within +2.6 and -0.4% ofPKNWithin +2.4 a n d - 3 % ofPKNWithin -2% of PKNWithin +3% of PKNWithin +1.2 and -2% ofPKNN/~ - 1.7272 In (Re V/f) - 0.39464 × 103 to 107Classical correlation, herecalled PKN, has atheoretical basis and isvalid for arbitrarily largeRe.
Its predictions agreewith extensive experimental measurementswithin +2 %.Colebrook [54]X/~ - 1.5635 In4 × 103 toMathematical approximation to PKN, yieldingnumerical values within_+1% of PKN.Filonenko [55]1X/~ - 1.58 In R e - 3.28104 to 107Within +1.8% of PKN104 toExplicit form of PKN;von Kfirmfin [53]Nikuradse [43]"Techo et al. [56]10 7d1Re_/- - 1.7372 In1.964 In ReVf3.821510 7turbulent curves. The b r o k e n line demarcating fully turbulent flow and transition flow,obtained by M o o d y [58], is as follows:V~=100Re (elDh)(5.75)It should be n o t e d that the horizontal portions of the curves to the right of the b r o k e n line arer e p r e s e n t e d by Nikuradse's [60] correlation, which is presented in Table 5.9.
The downwardsloping line for the smooth turbulent flow is r e p r e s e n t e d by the P K N correlation shown inTable 5.8. The downward-sloping line for laminar flow is r e p r e s e n t e d by Eq. 5.17. Relativeroughness e can be obtained from Table 5.10 for a variety of commercial pipes.Heat Transfer in Smooth Circular Ducts. For gases and liquids (Pr > 0.5), very little difference exists b e t w e e n the Nusselt n u m b e r for uniform wall t e m p e r a t u r e and the Nusseltn u m b e r for uniform wall heat flux in smooth circular ducts.
However, for Pr < 0.1, there is adifference b e t w e e n NuT and NUll. Table 5.11 presents the fully developed t u r b u l e n t flow Nusselt n u m b e r in a s m o o t h circular duct for Pr > 0.5. The correlation p r o p o s e d by Gnielinski [69]is r e c o m m e n d e d for Pr > 0.5, as are those suggested by Bhatti and Shah [45]. In this table, thef in the equation is calculated using the Prandtl [52]-von K~irm~in [53]-Nikuradse [43]; Coleb r o o k [54]; F i l o n e n k o [55]; or Techo et al. [56] correlations shown in Table 5.8.FORCED0.10--~"0.09~0.08-~-o.o7.~:~, ,_ulii1,11INTERNALFLOW'IIIIIIIIIIIIIIII-J ~J .: ., ., .- -IIIIIIIIIIIIIIIII~ ' ~ ~ .
~ ~" ~ . .~. .~--~.5.23IIIllllli -~-0.O6IN DUCTSIIIIIII]~ ii;=u~iii~'n eCONVECTION,-=~_"" -~--,I I I I l l l l liiiiiiiio.o5"'"'":0.03' ' " " l ' 0""..mumwmmmm'0.04-_-i:'~i"0.0250.~o.ol~----t .....~~iiii i iI i~I II~- "--..........-..!-"..''..___ ' L.__~i~. .J ....i i Ji,i iii~-~""0.0o8 _'_"o.o,,.~! ~~" ~ ~~I ! i iii.~"~iiIL !103 2 ( 1 # ) 3 4 S6J i Jilidi2(1~)3I1#2(1#))4 $6--",,,,,,,,'"'"'- ----~~~ _ ~..., ~ " -- :IIIIIIII0 0 ~0.006nnlmlgl7,;;;;;;;~,,-~,,mnnmm-ml l l l l l l l0.002O.®t0.0008::::::::0.0006mnuummmu 0.0o04|nuumnnnllllllll~~IlUlUl-It l Il nIl lImIl nI I I0.00010.000,05-~ ~::"r~--''~,nummmu~.mn,,,,,,,,o ooo.=mllNnmllllt:"----:-- mUllllmnnnnnnmI1#~2(1~)3ReEDhIlllillm"i:4 s6.,,n,,,,,nuummunnnum 0.01::t:::t::.
~~- :~ ,-,. ..,.,;,,.,. ...'__1-~!-~!o.oo9_~~ ....----~- -I:-~_, : : ::. . . .~~).o00.0,4 S6 l l O ? ' ~ : t n u ./t = 0.000,001~,t~= 0.000,005FIGURE 5.9 Moody's [58] friction factor diagram for fully developed flow in a rough circularduct [45].For liquid metal (Pr < 0.1), the most accurate correlations for NUT and NUll a r e those putforth by Notter and Sleicher [80]:NUT = 4.8 + 0.0156Re °'85 P r °'93(5.76)Null = 6.3 + 0.0167Re °85 P r °'93(5.77)These equations are valid for 0.004 < Pr < 0.1 and 10 4 < Re < 10 6.Heat Transfer in Rough Circular Ducts.
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