new correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing...

7/23/2019 New Correlations of Single-phase Friction Factor for Turbulent Pipe Flow and Evaluation of Existing Single-phase Fri

1/7

See discussions, stats, and author profiles for this publication at: http://www.researchgate.net/publication/251486588

New correlations of single-phase friction factorfor turbulent pipe flow and evaluation of

existing single-phase friction factor

correlations

ARTICLE in NUCLEAR ENGINEERING AND DESIGN MARCH 2011

Impact Factor: 0.95 DOI: 10.1016/j.nucengdes.2010.12.019

CITATIONS

36

READS

1,467

3 AUTHORS, INCLUDING:

XD Fang

Nanjing University of Aeronautics & Astron

48PUBLICATIONS 272CITATIONS

SEE PROFILE

Yu Xu

Nanjing University of Aeronautics & Astron

7PUBLICATIONS 53CITATIONS

SEE PROFILE

Available from: Yu Xu

Retrieved on: 19 October 2015
http://www.researchgate.net/profile/Xd_Fang?enrichId=rgreq-51307dcf-3c56-49ff-84e4-322011f22398&enrichSource=Y292ZXJQYWdlOzI1MTQ4NjU4ODtBUzo5ODQ1MjI1NjA2NzYwMkAxNDAwNDg0MjU2MTUx&el=1_x_4http://www.researchgate.net/?enrichId=rgreq-51307dcf-3c56-49ff-84e4-322011f22398&enrichSource=Y292ZXJQYWdlOzI1MTQ4NjU4ODtBUzo5ODQ1MjI1NjA2NzYwMkAxNDAwNDg0MjU2MTUx&el=1_x_1http://www.researchgate.net/profile/Yu_Xu5?enrichId=rgreq-51307dcf-3c56-49ff-84e4-322011f22398&enrichSource=Y292ZXJQYWdlOzI1MTQ4NjU4ODtBUzo5ODQ1MjI1NjA2NzYwMkAxNDAwNDg0MjU2MTUx&el=1_x_7http://www.researchgate.net/institution/Nanjing_University_of_Aeronautics_Astronautics?enrichId=rgreq-51307dcf-3c56-49ff-84e4-322011f22398&enrichSource=Y292ZXJQYWdlOzI1MTQ4NjU4ODtBUzo5ODQ1MjI1NjA2NzYwMkAxNDAwNDg0MjU2MTUx&el=1_x_6http://www.researchgate.net/profile/Yu_Xu5?enrichId=rgreq-51307dcf-3c56-49ff-84e4-322011f22398&enrichSource=Y292ZXJQYWdlOzI1MTQ4NjU4ODtBUzo5ODQ1MjI1NjA2NzYwMkAxNDAwNDg0MjU2MTUx&el=1_x_5http://www.researchgate.net/profile/Yu_Xu5?enrichId=rgreq-51307dcf-3c56-49ff-84e4-322011f22398&enrichSource=Y292ZXJQYWdlOzI1MTQ4NjU4ODtBUzo5ODQ1MjI1NjA2NzYwMkAxNDAwNDg0MjU2MTUx&el=1_x_4http://www.researchgate.net/profile/Xd_Fang?enrichId=rgreq-51307dcf-3c56-49ff-84e4-322011f22398&enrichSource=Y292ZXJQYWdlOzI1MTQ4NjU4ODtBUzo5ODQ1MjI1NjA2NzYwMkAxNDAwNDg0MjU2MTUx&el=1_x_7http://www.researchgate.net/institution/Nanjing_University_of_Aeronautics_Astronautics?enrichId=rgreq-51307dcf-3c56-49ff-84e4-322011f22398&enrichSource=Y292ZXJQYWdlOzI1MTQ4NjU4ODtBUzo5ODQ1MjI1NjA2NzYwMkAxNDAwNDg0MjU2MTUx&el=1_x_6http://www.researchgate.net/profile/Xd_Fang?enrichId=rgreq-51307dcf-3c56-49ff-84e4-322011f22398&enrichSource=Y292ZXJQYWdlOzI1MTQ4NjU4ODtBUzo5ODQ1MjI1NjA2NzYwMkAxNDAwNDg0MjU2MTUx&el=1_x_5http://www.researchgate.net/profile/Xd_Fang?enrichId=rgreq-51307dcf-3c56-49ff-84e4-322011f22398&enrichSource=Y292ZXJQYWdlOzI1MTQ4NjU4ODtBUzo5ODQ1MjI1NjA2NzYwMkAxNDAwNDg0MjU2MTUx&el=1_x_4http://www.researchgate.net/?enrichId=rgreq-51307dcf-3c56-49ff-84e4-322011f22398&enrichSource=Y292ZXJQYWdlOzI1MTQ4NjU4ODtBUzo5ODQ1MjI1NjA2NzYwMkAxNDAwNDg0MjU2MTUx&el=1_x_1http://www.researchgate.net/publication/251486588_New_correlations_of_single-phase_friction_factor_for_turbulent_pipe_flow_and_evaluation_of_existing_single-phase_friction_factor_correlations?enrichId=rgreq-51307dcf-3c56-49ff-84e4-322011f22398&enrichSource=Y292ZXJQYWdlOzI1MTQ4NjU4ODtBUzo5ODQ1MjI1NjA2NzYwMkAxNDAwNDg0MjU2MTUx&el=1_x_3http://www.researchgate.net/publication/251486588_New_correlations_of_single-phase_friction_factor_for_turbulent_pipe_flow_and_evaluation_of_existing_single-phase_friction_factor_correlations?enrichId=rgreq-51307dcf-3c56-49ff-84e4-322011f22398&enrichSource=Y292ZXJQYWdlOzI1MTQ4NjU4ODtBUzo5ODQ1MjI1NjA2NzYwMkAxNDAwNDg0MjU2MTUx&el=1_x_2


2/7

Nuclear Engineering and Design 241 (2011) 897902

Contents lists available atScienceDirect

Nuclear Engineering and Design

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / n u c e n g d e s

New correlations of single-phase friction factor for turbulent pipe flow and

evaluation of existing single-phase friction factor correlations

Xiande Fang , Yu Xu, Zhanru Zhou

Institute of Air Conditioning and Refrigeration, Nanjing University of Aeronautics and Astronautics, 29 Yudao St., Nanjing 210016, China

a r t i c l e i n f o

Article history:

Received 13 August 2010

Received in revised form22 November 2010

Accepted 21 December 2010

a b s t r a c t

Thedetermination of single-phase friction factorof pipe flowis essentialto a variety of industrial applica-

tions, such as single-phase flow systems, two-phase flow systems and supercritical flow systems. There

are a number of correlations for the single-phase friction factor. It still remains an issue to examine

similarities and differences between them to avoid misusing. This paper evaluates the correlations for

the single-phase friction factor against the Nikuradseequation and the Colebrook equation, respectively.

These two equations are the base for the turbulent portion of the Moody diagram, and are deemed as

the standard to test the explicit counterparts. The widely used correlations for smooth pipes, the Bla-

sius correlation and the Filonenko correlation, have big errors in some Re ranges. Simpler forms of the

single-phase friction factor covering large ranges are needed. For this reason, two new correlations of

single-phase friction factor for turbulent flow are proposed, one for smooth pipes and the other for

both smooth and rough pipes. Compared with the Nikuradse equation, the new correlation for smooth

pipes has the mean absolute relative error of 0.022%, with the maximum relative error of0.045% in

the Reynolds number (Re) range from 3000 through 108. It is an idea replacement of the correlations

of Blasius and Filonenko. The new correlation for both smooth and rough pipes has the mean absolute

relative error of 0.16% and the maximum relative error of 0.50% compared with the Colebrook equation

inthe range ofRe = 3000108 and Rr= 0.00.05,whichis themost simplest correlation in that error band.

2011 Elsevier B.V. All rights reserved.

1. Introduction

In-pipe (including channel) pressure drop calculations are

important for designing a variety of industrial thermo-fluid

equipment and systems, such as tubes, ducts, heat exchangers,

hydraulic systems, nuclear, chemical and petroleum processes,

various renewable energy systems, and heating, ventilation, air-

conditioning and refrigerating systems, etc.

The single-phase friction factor of pipe flow is not only the base

for determining single-phase friction pressure drop, but also the

foundation for pressure drop calculations of supercritical flow andtwo-phaseflow. Fornuclear industries andsystems with CO2as the

refrigerant or coolant, pressure drop of supercritical flow has been

animportantissuetobeexplored.Theflowpatternundersupercrit-

ical pressures is somewhat similar to the conventional single-phase

flow, which results in the practice to develop supercritical friction

factor correlations based on single-phase friction factor equations

(Petrov and Popov, 1988; Pioro et al., 2004; Yamshita et al., 2003 ).

Corresponding author. Tel.: +86 25 8489 6381; fax: +86 25 8489 6381.

E-mail address:xd [email protected](X. Fang).

Methods for predicting two-phase friction pressure drop in

pipes can be classified as two categories: Homogeneous and sep-

arated flow approaches. The former treats two-phase flow as a

pseudo single-phase flow characterized by suitably averaged prop-

erties of the liquid and vapor phase (Chen et al., 2001; Shannak,

2008). The latter considers a two-phase flow to be artificially sepa-

rated into twostreams,each flowingin itsown pipe (Cavallini et al.,

2009; Chisholm, 1967; Dalkilic et al., 2010; Friedel, 1979; Lockhart

and Martinelli, 1949; Lee and Mudawar, 2005; Sun and Mishima,

2009; Zhang et al., 2010),and then chooses a single-phase friction

factor correlation to calculate the related friction pressure drops.There are a number of correlations for the single-phase fric-

tion factor of pipe flow, whose ranges of validity were described

by the corresponding author(s). The criteria vary, however, from

author(s) to author(s). Therefore, a through evaluation is needed

to provide the guide to the users. Romeo et al. (2002)compared

the available correlations of the single-phase friction factor and

ranked them. However, he did not provide details of the errors

each correlation has.Yldrm (2009)conducted the most compre-

hensive analysis of existing correlations for single-phase friction

factor. He provided the maximum and minimum errors each cor-

relation has in the ranges of 4000Re108 and 106 Rr0.05,

0029-5493/$ see front matter 2011 Elsevier B.V. All rights reserved.

doi:10.1016/j.nucengdes.2010.12.019
http://localhost/var/www/apps/conversion/tmp/scratch_2/dx.doi.org/10.1016/j.nucengdes.2010.12.019http://localhost/var/www/apps/conversion/tmp/scratch_2/dx.doi.org/10.1016/j.nucengdes.2010.12.019http://www.sciencedirect.com/science/journal/00295493http://www.elsevier.com/locate/nucengdeshttp://-/?-http://-/?-http://-/?-mailto:[email protected]://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://localhost/var/www/apps/conversion/tmp/scratch_2/dx.doi.org/10.1016/j.nucengdes.2010.12.019http://localhost/var/www/apps/conversion/tmp/scratch_2/dx.doi.org/10.1016/j.nucengdes.2010.12.019http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-mailto:[email protected]://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://www.elsevier.com/locate/nucengdeshttp://www.sciencedirect.com/science/journal/00295493http://localhost/var/www/apps/conversion/tmp/scratch_2/dx.doi.org/10.1016/j.nucengdes.2010.12.019


3/7

898 X. Fang et al. / Nuclear Engineering and Design241 (2011) 897902

whereReis the Reynolds number and Rris the relative roughness.

The standard data bank Yldrm used was generated by read-

ing a Moody diagram (Moody, 1944)using Techdig2.0 software.

This method can cause remarkable reading errors, which makes

the finding disputable. Our evaluation does not support Yldrms

error estimations because Yldrm greatly overestimated errors

and offered differentaccuracy-basedrank orderfor the correlations

evaluated. For example, Yldrm gave maximum relative errors

(MREs) of3.76% and 3.96 to theChen (1979)and theZigrang

and Sylvester (1982) correlations, respectively. However, our eval-

uation shows that the Chen correlation has a MRE of0.5%, while

the ZigrangSylvester correlation has a MRE of 0.2%, which is

smaller than Chens. Furthermore, both Romeo et al. and Yldrm

did not pay attention to correlations for smooth pipes. It is the

single-phase friction factor correlations for smooth pipes that are

used most widely. Unfortunately, the widely used correlations for

smooth pipes, the Blasius correlation and the Filonenko correla-

tion, do not have good accuracy in the reported Re ranges, which

not only needs to redefine their Re ranges of validity,but also needs

to develop new alternatives.

This work will evaluate the correlations of the single-phase fric-

tion factor for turbulent pipe flow so that the Re and Rrranges

of validity of each correlation are identified to provide a clear

vision for users. The issue of single-phase friction factor correla-tions for smooth pipes is also addressed. New compact accurate

correlations of the single-phase friction factor for smooth pipes

and covering both smooth and rough pipes will be proposed,

respectively.

2. Brief review of single-phase friction factor correlations

for pipe flow

For single-phase fully developed internal laminar

flow (Re2000), the widely used equation is given by

HagenPoiseuilles law, which can be expressed as

f =

64

Re (1)

For fully developed turbulent flow in smooth pipes, Nikuradse

(1933)proposed the following equation:

1f= 2log(Re

f) 0.8 (2)

The Nikuradseequation is the basefor theturbulentsmoothportion

of the Moody diagram (Moody, 1944).However, it is implicit for

f, thus needs iteration that is not convenient. Consequently, the

Blasius equation and the Filonenko equation are widely used for

calculating turbulentflow in smoothpipes(Dang andHihara, 2004;

Huai et al., 2005; Incropera and DeWitt, 2001; Son and Park, 2006;

Yoon et al., 2003).For Re 2104, the Blasius equation is of the

form

f =0.316

Re1/4 (3a)

ForRe2104, the Blasius equation is of the form

f =0.184

Re1/5 (3b)

The Filonenko equation is of the form

f = (0.79lnRe 1.64)2 (4)

Incropera and DeWitt (2001) gave the Filonenko equation appli-

cableRerange of 3000Re5106

.

Danish et al. (2011) proposed the following correlation for

smooth pipes both in laminar and in turbulent regimes:

1

2

f= A

1.73718Aln A

1.73718 +A +

2.62122A(lnA)2

(1.73718+A)3

+3.03568A(lnA)3

(1.73718 +A)4 (5a)

A = 4logRe 0.4 (5b)

Colebrook (19381939) developed the following equation that

combines experimental results of studies of turbulent flow in

smooth and rough pipes:

1f= 2log

Rr

3.7+

2.51

Re

f

(6)

The validity of the Colebrook equation was reported in the range

ofRe = 4000108 andRr= 00.05. It should be mentioned that the

Colebrook equation was developed by Colebrook (19381939), but

the Colebrook and White paper (1937)is often erroneously cited as

the source of the equation.

The Colebrook (or ColebrookWhite) equation contributes theroughportionof theMoody diagram.Due to its demonstratedappli-

cability andMoodys work, the Colebrook equation has becomethe

acceptable standard for testing single-phase friction factor corre-

lations in turbulent regimes. It is not convenient to use, however,

because its implicit expression in frequires iteration. For this rea-

son, a number of approximate explicit counterparts have been

proposed (Table 1).

3. Evaluation of the existing correlations

Based on theabovereview, we evaluate theexplicitcorrelations

against the Colebrook equation in the range ofRe = 4000108 and

Rr= 00.05. Therank,whichis based on accuracy,is given in Table 1.

The detail accuracies are listed in Tables 27,where the relativeerror (RE) is defined as

RE =f(i)pred f(i)st

f(i)st(7)

where f(i)pred is the single-phase friction factor predicted by the

individualapproximate correlation, andf(i)stis thestandardsingle-

phase friction factor value calculated with the Colebrook equation

for the turbulent rough region and the Nikuradse equation for the

turbulent smooth portion.

To reduce the complexity ofTables 27, the following measures

are taken:

(1) Thecorrelations ofChurchill (1973) and SwameeJain are omit-

ted because they are very close to the Jain correlation bothin form and in prediction, with the Jain correlation perform-

ing a little better, so that the Jain correlation is chosen as the

representative.

(2) If any of Haaland, Moody, Wood and Round correlations does

not have considerableRe range in the specified RE, it will not

appear in that table.

(3) If any of Serghides, ZigrangSylvester, Romeo et al., Chen, Barr

and SonnadGoudar correlations has higher degree accuracy

than the RE specified in a table, it will not appear in that table.

3.1. Summary of evaluation of the existing correlations covering

rough pipes

FromTables 27, it follows:
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


4/7

X. Fang et al. / Nuclear Engineering and Design241 (2011) 897902 899

Table 1

Single-phase friction factor correlations covering roughness: rank by accuracy in the range ofRe =4000108 andRr=00.05.

Rank Model Correlation Range of validity

reported in the

original paper

1 Serghides (1984) 1/

f =A (B A)2/(C 2B A), Re > 2100

A =2log(12/Re + Rr/3.7), 0Rr0.05

B =2log(2.51A/Re + Rr/3.7)

C=2log(2.51/Re + Rr/3.7)

2 Zigrang and Sylvester (1982) 1/

f = 2log{Rr/3.7 5.02/Re log[Rr/3.7 5.02/Re log(Rr/3.7 + 13/Re)]} a

3 Romeo et al. (2002) 1/

f = 2log(Rr/3.7065 5.0272/Re A) 3000Re1.5108

A =log{Rr/3.8274.567/Re log[(Rr/7.7918)0.9924 + (5.3326/208.815+ Re)0.9345]} 0Rr0.05

4 Chen (1979) 1/

f = 2log[Rr/3.7065 5.0452/Re log(Rr1.1098 /2.8257 + 5.8506/Re0.8981 )] 4000Re4108

107 Rr0.05

5 Barr (1981) 1/

f = 2log[Rr/3.7+ 4.518 log(Re/7)/Re(1+ Re0.52Rr0.7/29] a

6 Sonnad and Goudar (2006) 1/

f = 0.8686 ln[0.4587Re/SS/(S+1)] 4000Re108

S= 0.124RrRe + ln(0.4587Re) 106 Rr0.05

7 Manadilli (1997) 1/

f = 2log(Rr/3.7 + 95/Re0.983 96.82/Re) 5200Re108

0Rr0.05

8 Haaland (1983) 1/

f = 1.8log[(Rr/3.7)1.11

+ 6.9/Re] 4000Re108

106 Rr0.05

Jain (1976) 1/f = 2log(Rr/3.715+ 5.72/Re0.9) 5000Re107

4105 Rr0.05

Swamee and Jain (1976) 1/

f = 2log(Rr/3.7 + 5/74/Re0.9) 5000Re108

106 Rr0.05

Churchill (1973) 1/

f = 2log[Rr/3.7+ (7/Re0.9)] a

Churchill (1977) f= 8[(8/Re)12+A3/2]

1/12AnyRe > 0

A = (37530/Re)16 [2.457 ln((7/Re)

0.9+ 0.27Rr)]

160Rr0.05

9 Moody (1947) f= 0.0055

1 + (2 104Rr+ 106/Re)1/3

4000Re5108

0Rr0.01

10 Wood (1966) f= 0.53Rr+ 0.094Rr0.225 + 88Rr0.44Re1.62Rr0.134

4000Re5107

105 Rr0.04

11 Round (1980) 1/

f = 1.8log(0.135Rr+ 6.5/Re) 4000Re4108

0Rr0.05

a Not explicitly specified.

(1) The correlations can be ranked as inTable 1.(2) In the range of Re = 4000108 and Rr= 00.05, the Serghides

correlation has accuracy of0.1%, the ZigrangSylvester corre-

lationand theRomeoet al.correlationhave accuracies of0.2%,

the Chen correlation has accuracy of0.5%, the Barr correlation

and the GoudarSonnadcorrelation have accuraciesof1%,the

Haaland correlation has the accuracy of2%,and allothers have

errors exceed 2%.

(3) The correlations of Haaland, Jain and Churchill (1977) are

closely tied in accuracies. The Haaland correlation has theaccu-

racy of2%,while it does not have considerableRe range underaccuracy of0.1%.

(4) The Churchill (1977)correlation is theonlyone covering allflow

regimes. However, its accuracy is compromised.

(5) The Manadilli correlation has the simplest form in the error of

1% andRr0.0005.

(6) The correlation which has high accuracy usually has compli-

cated form. The compact and accurate correlation covering

large ranges is still wanted.

Table 2

Rerange for given relative roughness under RE of0.1%.

Model Rr

0.000001 0.000005 0.00001 0.00005 0.0001 0.0005 0.001 0.005 0.01 0.05

Serghides 4E31E8a

Zigr ang Sylves te r 4E3 9E3 4E 39E3 4 E39E 3 4E 31E4 4E31E 4 4E31E8

3E61E8 2E61E8 8E51E8 3E51E8 2E51E8

Romeo et al. 3E41E8 2E41E8 2E41E8 2E41E8 2E41E8 9E31E8 5E31E8 4E31E8 4E34E5 b

Chen 1E44E7 1E42E6 1E43E5 9E34E4 6E61E8 1E61E8 5E51E8 6E41E8 2E42E6

6E71E8 2E71E8

Barr 2E41E8 2E41E8 2E41E8 3E41E8 2E51E8 6E41E8 4E41E8 8E31E5 2E47E4 4E51E8

2E61E8 2E61E8

SonnadGoudar 7E71E8 2E71E8 2E71E8 3E61E8 2E61E8 4E51E8 2E51E8 5E41E8 3E41E8 7E31E8

Manadilli 4E32E6 4E37E5 4E33E5 4E36E4 4E32E4 2E71E8 7E61E8 2E61E8 1E61E8 3E51E8

6E71E8

Jain 4E71E8 7E61E8 4E61E8 7E54E6 4E51E6

Churchill (1977) 5E71E8 1E71E8 5E61E8 2E61E8 6E51E8 2E53E6

a Bold value covers theRrrange in the whole box it occupies. This also applies to Tables 37.b

Symbol denotes not applicable. This also applies toTables 37.
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


5/7


Table 3


Model Rr

0.000001 0.000005 0.00001 0.00005 0.0001 0.0005 0.001 0.005 0.01 0.05

ZigrangSylvester 4E31E8

Romeo et al. 4E31E8

Chen 7E31E8 7E31E8 7E36E6 7E33E5 6E31E5 5E32E4 5E31E4 4E31E8 4E31E8 4E31E8

2E71E8 5E61E8 3E61E8 5E51E8 2E51E8

Barr 7E31E8 7E31E8 7E31E8 9E31E8 2E41E8 3E41E8 2E41E8 6E31E8 4E31E8 6E31E8


Manadilli 4E38E6 4E32E6 4E31E6 4E31E5 4E35E4 6E61E8 4E61E8 9E51E8 5E51E8 2E51E8

4E71E8 3E71E8

Haaland 5E61E8 3E71E8 3E71E8 9E61E8 5E61E8 2E61E8 6E51E8 8E41E8 2E41E8 4E41E8

Jain 4E71E8 2E71E8 4E61E8 2E61E8 5E51E8 3E51E8 7E41E6

Churchill (1977) 5E71E8 3E71E8 5E61E8 3E61E8 6E51E8 4E51E8 1E51E8

Table 4


Model Rr

0.000001 0.000005 0.00001 0.00005 0.0001 0.0005 0.001 0.005 0.01 0.05

Chen 4E31E8

Barr 4E31E8 5E31E8 5E31E8 4E31E8


Manadilli 4E31E8 4E38E5 4 E32E5 4E32E4 9 E51E8 3E51E8 2 E51E8 5 E41E88E61E8 5E61E8 2E61E8

Haaland 2E61E8 6E61E8 9E61E8 4E61E8 3E61E8 6E51E8 3E51E8 3E41E8 4E31E8 7E31E8

Jain 7E51E7 6E57E6 5E55E6 2E52E6 8E31E8 1E42E5 5E51E8 2E51E8 1E51E8 4E41E8

7E61E8 7E51E8

Churchill (1977) 5E58E6 4E56E6 3E54E6 1E41E6 1E41E6 2E42E5 6E51E8 2E51E8 2E51E8 5E41E8

7E71E8 1E71E8 5E61E8 1E61E8

Moody 2E41E5

Wood 2E41E5

Table 5

Rerange for given relative roughness under RE of1%.

Model Rr

0.000001 0.000005 0.00001 0.00005 0.0001 0.0005 0.001 0.005 0.01 0.05

Barr 4E31E8

SonnadGoudar 4E31E8Manadilli 4E31E8 4E31E4 9E41E8 6E41E8 3E41E8

2E51E8

Haaland 5E31E8 5E31E8 5E36E4 5E33E4 5E32E4 4E31E4 4E31E4 4E31E8

4E51E8 2E61E8 8E51E8 3E51E8 1E51E8

Jain 6E34E7 6E31E8 6E31E8 6E31E8 6E31E8 6E31E8 8E31E8 5E41E8 4E41E8 2E41E8

Churchill (1977) 7E33E7 7E31E8 7E31E8 7E31E8 7E31E8 8E31E8 1E41E8 6E41E8 5E41E8 2E41E8

Moody 4E32E4 4E32E4 4E32E4 4E32E4 4E33E4 4E31E5 4E33E4 5E32E4 6E41E8

Wood 7E64E7 2E42E5 2E41E5 6E34E4

5E71E8

Round 2E62E7

3.2. Summary of evaluation of the existing correlations for

smooth pipes

Both the correlations special for smooth pipes and the correla-tions listedin Table 1 are evaluated against the Nikuradse equation

in the range ofRe = 4000108. The results for the RE range from

0.05% to 2% are listed in Table 7. The details are summarized

below:

(1) The Blasius equation does not have good accuracy. The calcu-lation shows that its error increases from 2.6% atRe = 2106

to 22.2% atRe = 108. Therefore, its usage should be limited to

Table 6

Rerange for given relative roughness under RE of2%.

Model Rr

0.000001 0.000005 0.00001 0.00005 0.0001 0.0005 0.001 0.005 0.01 0.05

Manadilli 4E31E8 1E41E8 7E31E8

Haaland 4E31E8

Jain 4E31E8 7 E31E8 1 E41E8 7 E31E8

Churchill (1977) 4E31E8 9 E31E8 2 E41E8 8 E31E8

Moody 4E32E4 4E32E4 4E32E4 4E32E4 4E33E4 4E31E5 4E33E4 5E32E4 6E41E8

Wood 7E64E7 2E42E5 5E71E8 2E41E5 6E34E4

Round 2E62E7
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


6/7

X. Fang et al. / Nuclear Engineering and Design241 (2011) 897902 901

Table 7

Rerange for smooth portion under given RE.

Model RE%

2 1 0.5 0.2 0.1 0.05

Danish et al. Laminar

region and

4E31E8

Filonenko 1E41E8 2E41E8 4E42E7 1E67E6

Blasius 4E38E3

Serghides 4E31E8

ZigrangSylvester 4E31E8 2E71E8

Romeo et al. 4E31E8 3E61E8

Chen 4E31E8 7E31E8 2E47E7 2E49E6

Barr 4E31E8 7E31E8 2E42E7 2E45E6

SonnadGoudar 4E31E8 7E41E8

Manadilli 4E31E8 4E38E7 4E37E6 9E32E6

Haaland 3E31E8 5E34E7 2E61E7

Jain 4E31E8 6E34E7

Churchill (1977) 4E31E8 5E33E7 5E59E6

Moody 4E34E4 4E32E4

Round 3E51E8 4E61E8 2E78E7

Re = 2106. It is suggested to rewrite the Blasius equation as

the following:

f = 0.316Re1/4

(Re 2 104) (8a)

f =0.184

Re1/5 (2 104 Re 2 106) (8b)

(2) For the givenReranges above, both Eqs.(8a)and(8b)have the

maximum RE of2.62%.

(3) The Filonenko equation has the maximum RE of 2% for

Re = 104108. Therefore, it is recommended to rewrite the Filo-

nenko equation as the following:

f = (0.79lnRe 1.64)2 (104 Re 108) (9)

(4) The correlations of Danish et al. and Serghides have the highest

accuracy, but they also are the most complicated ones, which

impedes their applications.

(5) The new compact and accurate correlation of the single-phase

fraction factor for the turbulent smooth portion is needed.

4. New correlations of the single-phase friction factor for

turbulent pipe flow

New correlations of the single-phase friction factor for turbulent

pipe flow are developed based on computer analysis. A date bank

ofRe (i)Rr(j)= 4 424 = 1056 data points covering the regime of

Re = 3000108 and Rr= 0.00.05 is generated with the Colebrook

equation and the Nikuradse equation.

Based on regression and optimization with software, two corre-

lations are proposed, one is for smooth pipes, and the other coversboth smooth and rough regions in the range ofRr= 0.00.05. The

former is developed considering that the single-phase friction fac-

tor for smooth pipes has more widely applications than those for

rough pipes have, and that the commonly used equations have big

errors and can not cover Rerange of 4000108.

4.1. New correlation of the single-phase friction factor for

turbulent flow in smooth pipes

For turbulent flow in smooth pipes, the following correlation is

proposed:

f= 0.25log 150.39Re

0.98865

152.66

Re 2

(10)

In the range ofRe = 3000108, the new correlation has the mean

absolute relative error (MARE) of 0.022% and the maximum RE of

0.045% compared with the Nikuradse equation. Therefore, it hasequivalentaccuracyto but much simpler formthan the correlations

of Danish et al. and Serghides have, just one term more than the

Filonenko equation. The MARE is defined as

MARE =1

N

Ni=1

f(i)pred f(i)stf(i)st

(11)

4.2. New correlation of the single-phase friction factor for

turbulent flow in both smooth and rough pipes

For turbulent flow in smooth pipes, the following correlation is

proposed:

f = 1.613

ln

0.234Rr1.1007 60.525Re1.1105

+ 56.291Re1.0712

2 (13)

Inthe range ofRe = 3000108 and Rr= 0.00.05, the new correlation

has the MARE of 0.16% and the maximum RE of 0.50% compared

with the Colebrook equation. Compared with all existing correla-

tions, the new correlation is the simplest one with the maximum

RE of0.50% in the range ofRe = 3000108 andRr= 0.00.05.

5. Conclusions and suggestions

Fifteen correlations for the single-phase friction factor of pipe

flow are reviewed and evaluated. According to accuracy, the corre-

lations covering rough portion are ranked inTable 1.In the range

ofRe = 4000108

andRr= 00.05, the accuracies are, respectively,0.1%for the Serghides correlation,0.2%for the ZigrangSylvester

correlation and Romeo et al. correlation, 0.5% for the Chen cor-

relation, 1% for the Barr correlation and the GoudarSonnad

correlation, 2% for the Haaland correlation, and greater than 2%

for all other correlations listed in Table 1. The Churchill (1977)

correlation is the only one covering all flow regimes. Its accuracy,

however, is compromised.

The new applicableRe ranges for the Blasius equation and the

Filonenko equation are recommended, as shown in Eqs. (8) and

(9).Within the recommended Re range, the Blasius equation has

the maximum RE of2.62%, and the Filonenko equation has the

maximum RE of 2%.

Two new correlations of single-phase friction factor for turbu-

lent flow are proposed. One is for smooth pipes, and the other is for
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


7/7


both smooth and rough pipes. The former is the form of

f = 0.25

log

150.39

Re0.98865

152.66

Re

2with the MARE of 0.022% and the maximum RE of0.045% in the

range ofRe = 3000108. The latter is the form of

f = 1.613ln0.234Rr1.1007 60.525

Re1.1105+

56.291

Re1.07122

with the MARE of 0.16% and the maximum RE of 0.50% in the range

ofRe = 3000108 andRr= 0.00.05.

Acknowledgment

This work was funded by AVIC Chengdu Aircraft Design &

Research institute, China.

References

Barr, D.I.H., 1981. Solutions of the ColebrookWhite functions for resistance to uni-form turbulent flows. Proc. Inst. Civil Eng. 2, 71.

Cavallini, A., Col, D., Matkovic, M., Rossetto, L., 2009. Pressure drop during two-phaseflowof R134aand R32 ina singleminichannel.ASMEJ. Heat Transfer131,033107-1033107-8.

Chen, I.Y., Yang, K.S., Chang, Y.J., Wang, C.C., 2001. Two-phase pressure drop ofairwater and R-410a in small horizontal tubes. Int. J. Multiphase Flow 27,12931299.

Chen, N.H., 1979. An explicit equation for friction factor in pipe. Ind. Eng. Chem.Fundam. 18 (3), 296297.

Chisholm, D., 1967. A theoretical basis for the LockhartMartinelli correlation fortwo-phase flow. Int. J. Heat Mass Transfer 10, 17671778.

Churchill, S.W., 1973. Empirical expressions for the shear stressing turbulent flowin commercial pipe. AIChE J. 19 (2), 375376.

Churchill, S.W., 1977. Friction-factor equation spans all fluid-flow regimes. Chem.Eng. 7, 9192.

Colebrook, C.F., 19381939.Turbulent flowin pipes, withparticular reference to thetransition region between the smooth and rough pipe laws. J. Inst. Civil Eng. 11,133.

Colebrook, C.F., White, C.M., 1937. Experiments withfluid friction roughened pipes.In: Proc. Roy. Soc. Lond. Ser. A. Math. Phys. Sci. , pp. 367381, 161 (906).

Dalkilic,A.S.,Agra,O., Teke, I.,Wongwises,S., 2010. Comparison offrictional pressuredrop models during annular flow condensation of R600a in a horizontal tube atlow mass flux and of R134a in a vertical tube at high mass flux. Int. J. Heat Mass

Transfer 53, 20522064.Dang, C., Hihara, E., 2004. In-tube cooling heattransfer of supercriticalcarbon diox-

ide. Part 1. Experimental measurement. Int. J. Refrig. 27, 736747.Danish,M., Kumar,S., Kumar,S., 2011. Approximateexplicit analytical expressionsof

frictionfactor forflow of Binghamfluids in smooth pipes using Adomiandecom-position method. Commun. Nonlinear Sci. Numer. Simulat. 16 (1), 239251.

Friedel, L., 1979. Improved friction pressure drop correlation for horizontal andvertical two-phase pipe flow. Eur. Two-phase Flow Group Meet. Pap. E2 (18),485492.

Haaland,S.E., 1983. Simple and explicit formulasfor friction factor in turbulent pipeflow. Trans. ASME, J. Fluids Eng. 105, 89.

Huai, X.L., Koyama,S., Zhao, T.S., 2005. Anexperimentalstudyof flowand heattrans-fer of supercritical carbon dioxide in multi-port mini channels under coolingconditions. Chem. Eng. Sci. 60, 33373345.

Incropera, F.P., DeWitt, D.P., 2001. Fundamentals of Heat and Mass Transfer, 5th ed.John Wiley & Sons, New York.

Jain, A.K., 1976. Accurate explicit equations for friction factor. J. Hydraul. Div. ASCE102 (5), 674677.

Lee,J., Mudawar,I., 2005. Two-phase flowin high-heat-fluxmicro-channel heatsinkfor refrigeration cooling applications. Part I. Pressure drop characteristics. Int. J.Heat Mass Transfer 48, 928940.

Lockhart, R.W., Martinelli, R.C., 1949. Proposed correlation of data for isother-mal two-phase, two-component flow in pipes. Chem. Eng. Prog. 45 (1),3948.

Manadilli, G.,1997. Replaceimplicitequations withsigmoidal functions. Chem. Eng.J. 104 (8), 129132.

Moody, L.F., 1947. An approximate formula for pipefrictionfactors. Trans. ASME 69,

1005.Moody, L.F., 1944. Friction factors for pipe flow. Trans. ASME, 671684.Nikuradse, J., 1933. Stroemungsgesetze in rauhen Rohren. Ver. Dtsch. Ing. Forsch.

(361), 122.Petrov,N.E., Popov, V.N., 1988. Heattransfer andhydraulic resistance withturbulent

flow in a tube of water under supercritical parameters of state. Thermal Eng. 35(10), 577580.

Pioro, I.L., Duffey, R.B., Dumouchel, T.J., 2004. Hydraulic resistance of fluids flowingin channels at supercritical pressures (survey). Nucl. Eng. Des. 231, 187197.

Romeo, E., Royo, C., Monzon, A., 2002. Improvedexplicit equationsfor estimation ofthe friction factor in rough and smooth pipes. Chem. Eng. J. 86, 369374.

Round, G.F., 1980. Anexplicitapproximationfor thefrictionfactor-Reynoldsnumberrelation for rough and smooth pipes. Can. J. Chem. Eng. 58, 122123.

Serghides, T.K., 1984. Estimate friction factor accurately. Chem. Eng. 91, 6364.Shannak, B.A., 2008. Frictional pressure drop of gas liquid two-phase flow in pipes.

Nucl. Eng. Des. 238, 32773284.Son,C.H., Park, S.-J., 2006. An experimental study on heattransfer andpressure drop

characteristics of carbon dioxide during gascoolingprocessin a horizontal tube.Int. J. Refrig. 29, 539546.

Sonnad, J.R., Goudar, C.T., 2006. Turbulent flow friction factor calculation using amathematically exact alternative to the ColebrookWhite equation. J. Hydraul.Eng. ASCE 132 (8), 863867.

Sun, L., Mishima, K., 2009. Evaluation analysis of prediction methods for two-phaseflow pressure drop in mini-channels. Int. J. Multiphase Flow 35, 4754.

Swamee, P.K., Jain, A.K., 1976. Explicit equation for pipe flow problems. J. Hydraul.Div. ASCE 102 (5), 657664.

Wood, D.J., 1966. An explicit friction factor relationship. Civil Eng. 60 (12), 6061.Yamshita, T., Mori, H., Yoshida, S., Ohno, M., 2003. Heat transfer and pressure drop

of a supercritical pressure fluid flowing in a tube of small diameter. Mem. Fac.Eng., Kyushu Univ. 63 (4), 227243.

Yldrm, G., 2009. Computer-based analysis of explicit approximations to theimplicit ColebrookWhite equation in turbulent flowfriction factor calculation.Adv. Eng. Softw. 40, 11831190.

Yoon, S.H., Kim, J.H., Hwang, Y.W., Kim, M.S., Min, K., Kim, Y., 2003. Heat transferand pressure drop characteristics during the in-tube cooling process of carbondioxide in the supercritical region. Int. J. Refrig. 26, 857864.

Zhang, W.,Hibiki,T., Mishima,K., 2010. Correlationsof two-phasefrictional pressure

drop and void fraction in mini-channel. Int. J. Heat Mass Transfer 53, 453465.Zigrang, D.J., Sylvester, N.D., 1982. Explicit approximations to the Colebrooks fric-tion factor. AICHE J. 28 (3), 514515.

Xiande Fangis a professor atthe Instituteof AirConditioningand Refrigeration, Nan-jing University of Aeronauticsand Astronautics(NUAA), China. Ph.D. in EngineeringThermophysics from University of Science and Technology of China. M.Sci.in Ther-mal Engineering from Tsinghua University, China. B. Eng. in Environmental ControlEngineering from NUAA. His research areas are air conditioning and refrigeration,thermo-fluid engineering, and environmental control engineering.

Yu Xu is a graduate student under the supervision of Prof. Xiande Fang. He receivedhis B. Eng. in Environmental Control Engineering from NUAA.

Zhanru Zhou is a graduate student under the supervision of Prof. Xiande Fang. Shereceived her B. Eng. in Environmental Control Engineering from NUAA.

new correlations of single-phase friction factor for turbulent pipe flow and evaluation of existing...

Documents