Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 71
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The Nusselt number for a complete, rough flowregime in a circular duct is given in Table 5.12. The term f i n this table denotes the friction factor for fully rough flow. It is given by the Nikuradse [60] correlation shown in Table 5.9. Therecommended equations for practical calculations are those correlations by Bhatti and Shah[45] shown in Table 5.12.Artificially roughed circular ducts are also often used to enhance heat transfer.
The Nusselt numbers for artificially roughed ducts have been reviewed by Rao [59].Hydrodynamically Developing Flow.An analytical, close-form solution for hydrodynamically developing flow in rough circular ducts has been obtained by Zhiqing [87]. The velocitydistribution in the hydrodynamic entrance region is given asu~(y/8) inUmax - - [ 1Umax"~-for O < y < 5for ;5 < y < a(5.78)+(5.79)where ;5 is the hydrodynamic boundary layer thickness, which varies with axial coordinate x inaccordance with the following relation:5.24CHAPTER FIVETABLE 5.9 Fully Developed Turbulent Flow Friction Factor Correlations for a Rough Circular Duct [48](a = tube radius)InvestigatorsCorrelations1von Kfirm~in [46]MfNikuradse [60]~1vfRemarksE3.36- 1.763 In aThis explicit theoretical formula is applicablefor Re~ > 70.e- 3.48-1.737 I n aThis experimentally derived formula rendersvery nearly the same results as the von Kfirm~in [46] formula.1Colebrook [54]- ~ = 3.48-1.7372 InMoody [58]f = 1.375Wood [61]f = 0.08(~)°'225 + 0.265(ae--) + 66.69(ae--)°4 Re -"[x 10-3+ ReV~)1 + 21.544(~.This implicit formula is applicable for 5 < Re~ <70, spanning the transition, hydraulicallysmooth, and completely rough flow regimes.100'~lc3]+ ~]]Shows a maximum derivation of-15.78% fromthe Colebrook-White equation for 4000 <Re < 108 and 2 x 10-8 < e/a < 0.1./ e ~0.1Mwhere n = 1.778~a )Swamee and Jain [62]Churchill [63]1-~=3.47691.7372 In [ea-_+42.48]ReO.9Shows a maximum deviation of 3.19% fromthe Colebrook-White equation for 4000 <Re < 108 and 2 x 10-8 < rda < 0.1.2[( 8 ~12÷111/12f= L\-Re ](A1 + B1)3r2Unlike other equations in the table, this equation applies to all three flow regimesmlaminar, transition, and turbulent.
Its predictionsfor laminar flow are in agreement with f =16/Re. The predictions for transition floware subject to some uncertainty. However,the predictions for turbulent flow are comparable with those rendered by the preceding equation.whereA1 = {2.2088 + 2.457 In [ e-a + 42"68316R09e ]}ReChen [64]Round [65]Zigrang and Sylvester1- ~ = 3.48-1.7372 InE; - 16R-------e---"6 In A21where(e/a) 11°98 (7.149) °-~1A 2 = 6.0983 +Re1[96.2963], - - 4.2146 1.5635 In eaRevf1- ~ = 3.4769 - 1.7372 In-R----~ In A3[661where,43 = 7-~ - 2.1802 InZigrang and Sylvester-~=3.4769 - 1.7372 InApplicable only for rda > 2 x 10-5; shows a maximum deviation of 6.16% from theColebrook-White equation for 4000 < Re <10a and 2 x 10-8 < e/a < 0.1.This explicit equation is consistently in goodagreement with the Colebrook-White equation for 4000 < Re < 10a and 2 x 10-8 _<e./a <0.1, the maximum deviation being -0.39%.Comparable with Moody's equation.]Shows a maximum deviation of +0.96% fromthe Colebrook-White equation for 4000 _<Re < 10s and 2 x 10-8 < rda < 0.1- 16.1332 In ,43- 16.1332 In A4[661wheree/a[cda13]A, = ~.4 - 2.1802 In - ~ +Predictions not substantially different fromthose of the preceding equation.5.25FORCED CONVECTION, INTERNAL FLOW IN DUCTSTABLE 5.9Fully Developed Turbulent Flow Friction Factor Correlations for a Rough Circular Duct [48] (Continued)InvestigatorsCorrelationsHaaland [67]1- ~ = 3.4735 - 1.5635 InSerghides [68]1(As- B2)2V~ - As - As - 2B + C1where(e./aAs = -0.8686 In \ 7.4 +Remarks6,6 ,0]+ ReShows a maximum deviation of +1.21% fromthe Colebrook-White equation for 4000 <Re < 108 and 2 x 10-8 < rda < 0.1.Shows a maximum deviation of +0.14% fromthe Colebrook-White equation for 4000 <Re < 108 and 2 x 10-8 < e./a< 0.1.12 )(rda) 2.5A5B2 = -0.86868 In - ~ + Re( e.laC1 = -0.8686 In \ ~Serghides [68]1X/f4.781+2.51B2)Re(As -4.781) 2(4.781 - 2A5- B2)Shows a maximum deviation of -0.45 % fromthe Colebrook-White equation for 4000 <Re < 108 and 2 x 10-8 < rda < 0.1.,X/DhReTM _1.4039(~-)5/4[1 + 0.1577(~-) - 0.1793(~-) 2 - 0.0168(~-) 3 + 0.0064(~-) 4](5.80)The axial pressure drop Ap*, the incremental pressure drop n u m b e r K(x), and the apparent Fanning friction factor fapp are given as follows [87]"Ap* = ( Umax]2-1(5.81)\Um]K(X) lapp Rel/4 =X/DhAp* - 0.316 Re1/----zAp*4x/(Dh Re °25)(5.82)(5.83)TABLE 5.10 Average Roughnessof Commercial PipesMaterial (new)Roughness e (mm)Riveted steelReinforced concreteWoodCast ironGalvanized steelAsphalted cast ironBitumen-coated steelStructural and forged steelDrawn tubingGlass0.9-90.3-30.18--0.90.260.150.120.120.0450.0015Smooth5.26CHAPTER FIVE]'ABLE 5.11 Fully Developed Turbulent Flow Nusselt Numbers in a Smooth, Circular Ductfor Gases and Liquids (Pr > 0.5) [48]InvestigatorsDittus and Boelter [70]Correlations~0.024 Re °8 Pr °4Nu = [0.026 Re °8 Pr °'3Colburn [71]Nu = (f/2) Re Pr 1/3Nu = 0.023 Re °8 Pr ~/3von K~irm~in [72]Nu =Application rangefor heatingfor cooling0.7 < Pr < 120 and 2500 < Re <1.24 x 105, L/d > 600.5 < Pr < 3 and 1 0 4 < Re < 105(f/2) Re Prl+5(fi2)a~[Pr-l+ln(0.5 < Pr < 10 and5Pr+1610 4 <Re < 5x 10 6/1Prandtl [52]Nu =(f/2) Re Pr1 + 8.7(f/2)~r2(pr- 1)Drexel and McAdams [73]Nu = 0.021 Re °8 Pr °'4Pr < 0.7 and 104 < Re < 5 xFriend and Metzner [74]Nu =(f/2) Re Pr1.2 + 11.8(f/2)1:E(pr - 1) Pr -1/350 < Pr < 600 and 5 x 104 < Re <5 X 106Petukhov, Kirillov, andPopov [75]Nu =(f12) Re PrC + 12.7(flZ)I/E(Pr 2~ - 1)0.5 < Pr0.5 < Pr < 5 and1 0 4 _< Re< 1 0 6 and<5x10 610 64000 < Re < 5 x 106whereC = 1.07 + 9 0 0 / R e - [0.63/(1 + 10 Pr)]Nu = 0.037(Re °75- 180) Pr °42 [1 + (x/D) -2:3]Hausen [76]Webb [77]Gnielinski [69]Sieder and Tate [78]Sandall et al.
[79]Nu =(f/2) Re Pr1.07 + 9(f/2)l~(Pr- 1) Pr TMNu =( f / 2 ) ( R e - 1000) Pr1 + 12.7(f/2)v2(pr ~3- 1)0.7 _<Pr < 3 and1 0 4 _< Re0.5 < Pr < 100 and 104 < Re < 50.5 < Pr < 1.5 andNu = 0.027 Re 4/5 P r 1/30.7 < Pr < 16,700 and Re1 0 4 _< Re<51.5 < Pr < 500 and 3 x 103 < Re([-[/0"14\-~-fw]V ~ Re Pr12.48 Pr 2:3- 7.853 Pr 1/3+ 3.613 In Pr + 5.8 + CwhereC = 2.78 In ( V ~x 10 60.5 < Pr <_2000 and 2300 _<Re <_5 x 106,Nu = 0.0214(Re °'8- 100) Pr °'4Nu = 0.012(Re °-87- 280) Pr °4Nu =< 105x 10 6< 10 6> 10 40.5 < Pr < 2000 and 104 < Re < 5 x 106Re/a5)T h e h y d r o d y n a m i c e n t r a n c e l e n g t h Lhr/Dh can be c a l c u l a t e d by t h e f o l l o w i n g e q u a t i o n[87]"Lhy _ 1.3590 R eDhTM(5.84)T h e results f r o m E q . 5.84 a g r e e fairly well with e x p e r i m e n t a l d a t a [88].Thermally Developing Flow.N u m e r o u s i n v e s t i g a t o r s [80, 89-94] h a v e c a r r i e d o u t t h ei n v e s t i g a t i o n of t u r b u l e n t t h e r m a l l y d e v e l o p i n g flow in a s m o o t h circular d u c t with u n i f o r mwall t e m p e r a t u r e a n d u n i f o r m wall h e a t flux b o u n d a r y c o n d i t i o n s .
It has b e e n f o u n d t h a t thed i m e n s i o n l e s s t e m p e r a t u r e a n d t h e N u s s e l t n u m b e r for t h e r m a l l y d e v e l o p i n g t u r b u l e n t flowh a v e the s a m e f o r m a t s as t h o s e for l a m i n a r t h e r m a l l y d e v e l o p i n g f l o w (i.e., Eqs. 5.34-5.37 a n dEqs.
5.50-5.53). T h e o n l y d i f f e r e n c e s are the e i g e n v a l u e s a n d c o n s t a n t s in t h e e q u a t i o n s .F 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 S5.27TABLE 5.12 Nusselt Numbers for Fully Developed Turbulent Flow in the Fully Rough Flow Regimeof a Circular Duct [45]InvestigatorsMartinelli [81]CorrelationsRemarksRe Pr V ~Nu =5[Pr + In (1 + 5 Pr) + 0.5 In (Re V ~ / 6 0 ) ]Nunner [82]Nu =Re Pr (f/2)1 + 1.5 Re -1~8Pr -'/6 [Pr (f/f~) - 1]Dipprey and Sabersky [83]NH -Re Pr (f/2)1 + V ~ [ 5 .
1 9 Re °2 Pr °44- 8.48]Gowen and Smith [84]Nu =Re Pr4.5 + [0.155(Re V ~ ) 0"54"~"Wr~-f]~Kawase and Ulbrecht [85]N u - 0.0523 Re ~ rKawase and De [86]Nu = 0.0471 Re vrP--rrV f(1.11 + 0.44 Pr -1/3- 0.7 Pr -1/6)Bhatti and Shah [45]Nu =(Re Pr (f/2))1 + V ~ ( 4 . 5 Re °2 Pr °'5- 8.48)Bhatti and Shah [45]Nu =( R e - 1000) Pr (f/2)1 + V~[(17.42 - 13.77 Pr °8) Re °-5- 8.48]This equation differs from that derivedby Martinelli [81] for a smooth ductby the omission of the temperatureratio ( T,,, - T~)/( T,,, - Tin).This correlation is valid for Pr = 0.7; itdoes not give satisfactory results forPr> 1.This correlation is valid for 0.0024 <¢JDh <-0.049, 1.2 < Pr < 5.94, and 1.4 x104 < Re < 5 x 105.This correlation is valid for 0.0021 <E[Dh < 0.095, 0.7 < Pr < 14.3, and 104 <R e < 5 x 104.The predictions of this correlation aresomewhat lower than those of the following correlation.The predictions of this correlation are inreasonable agreement with the experimental data for 0.0024 < elDh < 0.165,5.1 < Pr < 390, and 5000 < Re < 5 x 105.V~This correlation is valid for 0.5 < Pr <10, 0.002 < 0.002 < rdD, < 0.05, andRe > 104.
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