pipeline friction

8/9/2019 Pipeline Friction

1/17

IMPROVEMENTS IN ESTIMATING PIPELINE FRICTION

Guillermo Octavio Cordero Techint Enineerin and Con!truction

Recent refinements in the prediction of Darcy friction factor1show that the currently used

Colebrook-White formula falls below experimental results up to 2.9 for all Reynolds abo!e

"#$$ in smooth pipe. %urthermore& the Colebrook friction factor for some typical industrial casesof rou'h pipes is shown to be less than the experimental factor for smooth pipes 2.

(his presentation re!iews the impact on li)uid and 'as pipeline desi'n& points out the

differences in the formulas applied in both fields and su''ests the use of a unified treatment for

all cases.

E"#erimental re!ult! $or $riction in !mooth #i#e!%

*s early as 1911& +lasius proposed a formula based on an exponential !elocity profile for

Reynolds , 1$$&$$$

25.0Re

3164.0=f 1/

0n 19##& randtl published the formula deduced from a lo'arithmic !elocity profile& with the

constants deri!ed from ikuradse3s tests for Reynolds up to #&"$$&$$$

( )

==

ff

f Re

51.2log28.0Relog2

12/

0n 1945& the tests performed by the 6 7 +ureau of 8ines #found a different set of constants&

later adopted by the *merican as *ssociation **/

=

=

f

f

f Re

825.2log23.0

2

Relog2

1#/

0n 2$$4& 8c:eon& ;a'arola and 7mits& after refinin' the ori'inal measurements obtained with

the 7uperipe facility at rinceton 6ni!ersity for Reynolds between #1$$ up to #4&4$$&$$$ in

1995& proposed another set of constants1

( ) 537.0Relog930.11 = ff

"/

(hey estimated their experimental error at 1.1. (he formula de!iation from the published test

results is between and ?$.>"& narrowin' to $.4 for Reynolds between "$$&$$$

and 2"&$$$&$$$. (hey !alidated +lasius3s formula and the exponential !elocity profile for

Reynolds below 9>&$$$. 8c:eon3s and +lasius3s formulas intersect at Reynolds e)ual to

55&95". +elow this Reynolds& +lasius de!iates


2/17

%i'ure 1

%i'ure 2

Friction Factor $or Smooth Pi#e

$.$1

$.$14

$.$2

$.$24

$.$#

$.$#4

$.$"

1$$$$ 1$$$$$

Re&nold!

'arc&$riction

$actor

Blasius

Prandtl

McKeonnon

Re = 38383

Re = 66964

Prandtl and (S)M ver!u! Mc*eon

$.$$>

$.$$>4

$.$$9

$.$$94

$.$1

$.$1$4

$.$11

$.$114

$.$12

1$$$$$$ 1$$$$$$$

Re&nold!

'arc&$riction

$actor

Prandtl

!BM

McKeon

Re = 1"563"705


3/17

E"#erimental re!ult! $or $riction in rouh #i#e!%

0n 19##& ikuradse published the results of his tests with pipes artificially rou'hened with sand

'rains of uniform siBe. *t low Reynolds numbers in the turbulent Bone& the friction factor was the

same as in smooth pipe. (hen a transition Bone was reached where the friction factor still

decreased& but not as fast as in smooth pipe& passed throu'h a minimum and started to

increase until attainin' a constant !alue dependin' only on the sand rou'hness kand the inside

diameter D %i'. #/

=+

=

D

k

k

D

f 706.3log274.1

2log2

14/

Colebrook and White ran similar tests usin' sand of different siBe on the same pipe and with

hi'hly corroded pipes. (he 'eneral beha!ior was similar to ikuradse3s test& but 0n some cases

of hi'hly non-uniform rou'hness the minimum disappeared and they proposed a combination of

randtl3s smooth-pipe-law and ikuradse3s rou'h-pipe-law in order to pro!ide a safe estimate of

the friction factor in the transition Bone 19#9/

+=

fDk

f Re51.2

7.3log21 5/

0n 19""& 8oody published his chart of the friction factor !ersus Reynolds& usin' sand rou'hness

as a parameter& based on a'en-oiseuille Re#64=f / for Reynolds below 2$$$ and on

Colebrook-White for Reynolds abo!e "$$$. (his chart was enthusiastically adopted and

remains the standard for calculatin' pipe friction throu'h different computer approximations.

0n 1945& the 6. 7. +ureau of 8ines#& testin' strai'ht commercial pipes& found the same beha!ior

as ikuradse& with a minimum in the transition Bone only about 2 below the final rou'h-pipe

friction factor.

0n 1954& the outstandin' work of 6hl and others 4for the *merican as *ssociation **/&

testin' actual 'as pipelines& !alidated the 67+8 formula for the smooth-pipe-law and adopted

ikuradse3s e)ui!alent sand rou'hness for the rou'h-pipe-law. (he tests su''ested that the

transition Bone reduced to a sin'le point& correspondin' to the transition Reynolds number Ret.

(hese results confirmed that Colebrook3s friction factor was too conser!ati!e for commercial

pipes with low rou'hness& reachin' a maximum o!er-desi'n about 12 at the transition

Reynolds.

0n aB de %rance5& the tests on 'as pipelines pro!ided results similar to **& althou'h the

transition Bone not always reduced to a sin'le point. (hey preferred to use a transition exponent

n& between 1 and 1$& to adust the extent of Colebrook transition Bone

+

=

nn

fD

k

nf Re

51.2

7.3log

21=/

(he ER roupe EuropFen de Recherches aBiGres/ adopted this techni)ue to different

formulas for the smooth-pipe-law& includin' ;a'aBola ori'inal formula for the 7uperipe=.

Friction Factor $or Rouh Pi#e

$.$1"

$.$15

$.$1>

$.$2

$.$22

$.$2"

$.$25

$.$2>

$.$#

$.$#2

1$$$$ 1$$$$$ 1$$$$$$ 1$$$$$$$

Re&nold!

'arc&$riction

$actor

$ole%roo&

!BM'i&uradse

()(


4/17

%i'ure #


5/17

Com#ari!on o$ the u!ual $riction $ormula! +ith e"#erimental re!ult!

Cole,roo-.! $ormulafor non-corroded pipes was expected to pro!ide a conser!ati!e estimate

of the friction factor& since it was approximately 12 abo!e experimental results at the transition

Reynolds number and con!er'ed asymptotically to the rou'h-pipe law for increasin' Reynolds.

%or decreasin' Reynolds& howe!er& randtl3s law remainin' always below the experimental

results& Colebrook3s friction factor for rou'h pipe may fall below smooth-pipe test points.

%or example a #4.24-in. 0D li)uid pipeline& with $.$$1>-in. rou'hness& would ha!e a Colebrook

factor 2." below smooth-pipe friction factor at Reynolds 14&$$$& in the +lasius Bone %i'. "/.

(he effect is less noticeable in 'as pipelines& operatin' normally abo!e Reynolds 1&$$$&$$$.

%or instance a #9-in. 0D epoxy-lined 'as pipeline& with "-micron rou'hness& operatin' at

Reynolds 1&2$$&$$$& would ha!e a Colebrook friction factor 1 below 8c:eon3s smooth-pipe

formula. (he obection to use Colebrook for 'as pipelines lies mainly on o!er-estimated friction

for the maority of practical cases. owe!er& both for li)uid and 'as pipelines& Colebrook is

normally used with the strai'ht pipe len'th in the pressure drop formula& disre'ardin' the effect

of minor losses it will be shown that this practice may cause an underestimate of the pressure

drop reachin' !alues about 5 in the lower Reynolds re'ion..

AGA.! $ormula for smooth-pipe intersects with 8c:eon3s at Reynolds 1&45#&=$4. %or lower

Reynolds& ** 'rows abo!e 8c:eon up to $.>2 at Reynolds "$$&$$$. %or hi'her Reynolds&

** falls below 8c:eon up to 1.#5 for Reynolds #4&4$$&$$$. 0n the usual operatin' re'ion&

** would be sli'htly on the unsafe side& but almost within the experimental error in rinceton

tests.

8inor losses are correctly accounted for throu'h the use of a drag factorin the pressure drop

formula.


6/17

%i'ure "

A !inle $ormula $or $riction in #i#eline!

*ccordin' to the conclusions of the rinceton research 'roup1& both +lasius and 8c:eon3s

formulas are re)uired for co!erin' the whole ran'e of turbulent Reynolds numbers for smooth

pipe.

(he Colebrook-+lasius-8c:eon-ikuradse C+8/2formula with n H 1 is apt to replace the

traditional Colebrook formula for pipes subect to corrosion.

(he ** report4conclusi!ely shows that the extended transition Bone resultin' from

Colebrook3s method is too conser!ati!e for 'as pipelines. (he transition exponent n& howe!er&

already adopted by ER& allows the use of the rinceton formulas and ikuradse3s rou'h-pipe

law to co!er all practical cases C+8n/& reducin' the extension of the transition Bone to match

actual pipeline rou'hness.

*pplyin' C+8n with nH "$ pro!ides a friction factor only $.# abo!e ** rou'h-pipe factor

at the transition Reynolds for typical cases of the rou'h-pipe transmission factorf

Ft 1=

around 1$. *doptin' nH 1$& the difference increases to 1.2. Ialues of nbetween 1 and 1$

ha!e been used to match friction measurements in some European 'as lines.

Cole,roo- /01203in1 I'4 5155673in1 Rouhne!! ver!u! Smooth Pi#e

$.$1

$.$14

$.$2

$.$24

$.$#

$.$#4

$.$"

1$$$$ 1$$$$$

Re&nold!

'arc&$riction

$actor

$BM'

& = 0

$ole%roo&& = 0.0018 in.


7/17

(he C+8n formulas are

%or Re55&95"

+=

nn

D

k

nf 706.310log

21125.1

Re 125.0

>/

%or ReA 55&95"

+

=

nn

D

k

fnf 706.3Re

897747.1log

21965.0

9/

* basic procedure for calculatin' C+8n in a spreadsheet is included in *ppendix 1.


8/17

Additional minor lo!!e!

** tests co!ered 22 cases of partially turbulent flow with four or more points at the left of the

transition Reynolds number. (hey calculated the transmission factorf

Ft 1= from the

pressure drop formula for a strai'ht pipe and plotted it !ersus Reynolds& findin' that the cur!ekept approximately parallel& but always below the 67+8 smooth-pipe law.

(hey identified this displacement as the effect of minor losses owin' mainly to bends and weld

beads at pipe oints and also to deposits& block !al!es and fittin's. (hey kept the 67+8 Ftand

added a Jdra' factorK Ff& less than unity& multiplyin' Ftin the pressure drop formula. %rom the

scarce information a!ailable& they built a 'raph %i'. 4 is a computeriBed !ersion/ for Ffas a

function of the Jbend indexK total chan'es of direction in de'rees per mile/ showin' four cur!es

for bare steel& plastic-lined& pi'-burnished and sandblasted pipe. (he effect of weld beads

decreased in that order and the dra' factor conse)uently increased.

%i'ure 4

%or the #$ cases of totally turbulent flow with at least four points at the ri'ht of the transition

Reynolds number& they calculated Ftin the same way and deri!ed the Jeffecti!e rou'hnessK ke

from ikuradse3s law. (hey acknowled'ed that ke& concentratin' between 9$$-1$$$

microinches 22.9-24." micron/& was hi'her than the absolute rou'hness kobtained by 67+8&

concentratin' between 5$$-=$$ microinches 14.2-1=.> micron/& because of the absence of

bends and weld beads in the 67+8 test pipes.

(he intersection of the smooth-pipe law and the rou'h pipe law pro!ided the formula for the

transition Reynolds number

FfFf

tke

D

ke

D11

7.3log

7.365.5Re

= 11/

(hey pointed out that Retshould depend only on k/Dthe transition is reached when the

'ra Factor

$.9$

1.$$

1 1$ 1$$ 1$$$

)end Inde" 89mile

'ra

FactorF$

%are steel*lastic+lined

*ig+%urnis,ed

sand%lasted


9/17

thickness of the laminar sublayer becomes less than the absolute rou'hness/. 6nfortunately&

they did not follow this clue to find the relationship between k& keand Ff

%or the 67+8 tests FfH 1. 0f the same pipes were welded and laid on a bendin' route& the

transition Reynolds remainin' the same& the transmission factor calculated throu'h the smooth-

pipe and the rou'h-pipe law become displaced by the same factor& so that

=

=

Dke

D

k

FffFf t 7.3log27.3log2Re

825.2

log2 12/

Conse)uently

=

D

ke

D

kFf

7.37.31#/

Replacin' in 11/

=

k

D

k

Dt

7.3log

7.365.5Re 1"/

(he ** 'roup also identified the relationship of the dra' factor to the total len'th Lt e)ual to

the sum of strai'ht len'th Lplus total e)ui!alent len'th Leof bends& weld beads& !al!es and

fittin's/ or to the respecti!e total loss coefficient K

=+=

+=

=

D

L

Ff

fK

D

Lf

D

LeLf

D

LtfN

214/

5.0

2 11

1

+==

=

Lf

DK

Lt

LFf

FfD

LfK 15/

(hese formulas can predict the effect on the dra' factor when addin' intermediate scraper

traps& where indi!idual Kcoefficients are known. * more accurate prediction of the dra' factor

for a new pipeline would be possible& if !alues of Kbecame a!ailable for weld beads and for

an'les between #L and 1$L usually present alon' the route. (he )ualitati!e relationship between

the dra' factor and the loss coefficients or e)ui!alent len'th of minor-loss elements has already

been analyBed by 6hl.

(he number of friction !elocity heads N& usually applied to 'as and li)uid pipes in refineries and

industrial plants& could also be applied to li)uid and 'as pipelines for both the partially and

totally turbulent flow re'ions usin' absolute instead of effecti!e rou'hness/.

6sin' function CBMNn(k,D,Re,n)for calculatin' f and Nin the li)uid or 'as pressure drop

formulas pro!ides a unified treatment for all cases of pipin' and pipelines within the precision

afforded by the rinceton tests.

The a##lication! o$ a,!olute rouhne!!

(he absolute e)ui!alent sand rou'hness kfor actual 'as pipelines can be obtained from the

fourteen series of ** tests for bare steel crossin' the transition point. (ests ;*-1 and ;+-1 >-

1$ years without cleanin'/ can be statistically reected as not belon'in' to the same population.

(he remainin' tests 'i!e mean effecti!e rou'hness of 5=$-in. or 1=-micron with "> standard

de!iation. (he same tests pro!ide #15-in. or >-micron mean absolute rou'hness with 1>

standard de!iation almost three times less/.

*n absolute rou'hness of "##-in. or 11-micron mean ? 2 standard de!iations/ can be adopted


10/17

for desi'n with CBMNnor the effecti!e rou'hness can be estimated from e)uation 1#/& as a

function of the dra' factor& to be used in ** or ER formulas.

(he only three tests for pi'-burnished pipe crossin' the transition Reynolds 'i!e 221-in.

absolute rou'hness 4.5-micron/.

(he tests with plastic-lined pipe did not attain the transition Reynolds. *bsolute rou'hness was

below 1##-1=5 in. #."-".4 micron/.

7and-blasted pipes also remained below the transition Reynolds with absolute rou'hness less

than 159-1>> in. ".#-".> micron/.

(he application of the absolute rou'hness to assess potential capacity increase in existin'

pipelines has already been analyBed> %or example& let us consider ** tests * and ;+-1 for

bare steel pipe

(est 0D inches ke microinches %f bend index LMmile

* 12 1522 $.>552 "4$

;+-1 24.#=4 1519 $.92$$ flat country

(o ascertain the possibility of increasin' pipeline capacity& the effecti!e rou'hness alone 'i!es

no practical 'uidance. (he absolute rou'hness calculated from e)uation 1#/ is ##4 in. !ery

near the mean !alue/ for test * and 52# in. for ;+-1. (he dra' factor is within the expected

ran'e for test * with hi'h bend index& and looks low for ;+-1 in flat country

Case * is conse)uently operatin' in the normal ran'e and no cleanin' is necessary.

Case ;+-1& ha!in' the same effecti!e rou'hness& shows unusually hi'h absolute rou'hness

indicatin' corrosion. i' burnishin' would pro!ide a si'nificant capacity impro!ement factor

087.1

375.25-7.3

000623.0log

375.25-7.3000221.0log

=

(he dra' factor should increase at least 2 weld beads reduced and deposits eliminated/

pro!idin' a total capacity increase around 11 for the same initial and final pressures. *s a

matter of fact& test ;+-5 after pi'-burnishin'& 'a!e the followin' results

(est 0D inches ke microinches %f bend index LMmile

;+-5 24.#=4 ""= $.9#>5 flat country

*bsolute rou'hness is 2$$.4 in. better than a!era'e/ and total capacity increase factor is

117.1

375.25-7.3

000623.0log-9200.0

375.25-7.3

0002005.0log-9386.0

=

%i'ure 5 shows the contribution of hi'her Ffand lower absolute rou'hness to obtain 11.=

capacity increase. (he pressure drop formula in *ppendix 2 pro!es that capacity is directly

proportional to Ff Ft. 7ince Reynolds is directly proportional to capacity& it is easy to represent

the operatin' cur!e when keepin' the same initial and final pressures.


11/17

%i'ure 5

67+8 tests showed that for perfectly dry 'as the sand rou'hness obtained from pressure drop

measurements approximately coincided with the mean peak-to-!alley hei'ht measured on the

wall rou'hness. Recent research9has impro!ed the correlation between different techni)ues of

direct rou'hness measurement and ikuradse3s sand rou'hness. (he presence of small

)uantities of li)uid in two 67+8 tests considerably reduced the effecti!e sand rou'hness

obtained from pressure drop. (his effect is certainly present in the actual 'as pipelines tested by

**& but does not completely co!er the wall rou'hness effect test ;*-1 pro!ided k H =49 in

19.# m/. *fter pi'-burnishin' test ;*-5/& k H 2"9 in 5.# m/. *fter sandblastin' test ;*-9/&

k , 159 in ".# m& transition Reynolds was not reached/. 0n the two first cases& at least& wall

rou'hness had an effect on pressure drop. 0n the third case& wall rou'hness may ha!e been

co!ered by li)uid presence. (his effect is ob!iously not present in li)uid pipelines& where

modern wall-rou'hness measurement techni)ues can be fully applied.

& = 623 icroinc,es

/ = 0.9200

& = 200.5 icroinc,es/ = 0.9386

*erating line" sae initial

and inal *ressures

!traig,t soot, *i*e

& = 0" / = 1


12/17

The u!e o$ the dra $actor

Colebrook formula is traditionally applied to pipelines usin' its strai'ht len'th& i.e. adoptin' FfH

1. (he conse)uence is that the o!erestimate in the pressure drop lies about 5 near the

transition Reynolds& but an underestimate also about 5 appears at Reynolds 1&$$$&$$$ for

typical 'as pipelines. ** test %-1 is shown on %i'ure = illustratin' this effect. (he C+8n

cur!e based on rinceton3s results indicates that ** would be only $.=5 on the unsafe side

near the transition Reynolds.

%i'ure =

()( / = 0.9483&e = 642 icroinc,es

$ole%roo& / = 1

&e = 642 icroinc,es$BM'n / = 0.9483

& = 340.6 icroinc,es


13/17

%or li)uid pipelines& there is a firm tradition to use Colebrook formula with an effecti!e

rou'hness of 1>$$ microinches "5 microns/ and the strai'ht len'th. owe!er& the theoretical

concepts of the thickness of the laminar sub-layer compared to absolute wall rou'hness to

define the transition from smooth pipe beha!ior to rou'h pipe beha!ior are the same for li)uid

and 'ases. 8inor-loss effects should be masked inside the effecti!e rou'hness in the rou'h-

pipe Bone& but should cause an underestimate of the pressure drop in the smooth> $.#1 $.1"

24$$$ $.$2"5= $.$2"51 $.$2"42 $.52 $.2=

4$$$$ $.$2111 $.$21$2 $.$2$>9 1.$= $."5

1$$$$$ $.$1>#2 $.$1>1> $.$1=99 1.># $.=9

24$$$$ $.$144# $.$1429 $.$1"9= #.=# 1.45

4$$$$$ $.$1#99 $.$1#5" $.$1#15 5.#1 2.4"

1$$$$$$ $.$12>5 $.$12#= $.$1154 1$.#9 #.91

24$$$$$ $.$11>9 $.$112$ $.$1$$1 1>.># 5.1#

4$$$$$$ $.$11"= $.$1$54 $.$$>9> 2=.=# =.="

friction factor safety mar'in

Water pipelines pro!ide a 'ood example of operation abo!e Reynolds 1&$$$&$$$ where the

dra' factor should be similar to 'as pipelines/& well known li)uid properties and uniform

rou'hness when usin' epoxy-lined steel pipe. Direct measurement of wall rou'hness has

shown an initial !alue of 14$ in.11

0n this field& the aBen-Williams formula has been widely used since the be'innin' of the 2$ th

century

87"4

852"1

67"10DL

CQHf =

17

%riction head Hfin m& flow rate Qin m#Ms& len'th Lin m& inside diameter Din m/


14/17

(he *merican Water Works *ssociation *WW*/ recommends a coefficient CH1"$?$.1=dfor

new pipe with smooth linin'12& where dis the diameter in inches.

(able 2 compares the results of aBen-Williams with CBMNnH"$ for uniform rou'hness/&

adustin' the dra' factor to obtain the best match. * !ery 'ood coincidence is obtained with FfH

$.9"#& which is the a!era'e dra' factor in ** tests. *t lower Reynolds& the comparison

confirms that the dra' factor should be increased.

Colebrook& with the traditional 1>$$-in. rou'hness& shows the same beha!ior as in 'as

pipelines about 5 underestimate for the lower Reynolds and about 5 o!erestimate for the

hi'her Reynolds.

(able 2

/0120 in1 I' ;ater Pi#eline 94#4$ $.$$$=52 $.$$$==# $.$$$=#1 -1."9 -4.4

1.4 1#"#$24 $.$$151" $.$$152= $.$$14=# -$.== -#.#

2 1=9$=$$ $.$$2=4$ $.$$2=51 $.$$2=21 -$.#> -1."2.4 22#>#=4 $.$$"14= $.$$"15# $.$$"1=" -$.1" $.#

# 25>5$4$ $.$$4>2= $.$$4>25 $.$$49#1 $.$1 1.>

#.4 #1##=24 $.$$==42 $.$$=="" $.$$=99$ $.11 #.2

" #4>1"$$ $.$$992= $.$$991$ $.$1$#42 $.1= ".4

".4 "$29$=4 $.$12#"5 $.$12#2$ $.$1#$14 $.21 4.5

4 ""=5=4$ $.$14$$5 $.$1"9=$ $.$149>1 $.2" 5.>

friction loss mMm de!iation

%urther experimental research on the dra' factor or minor-loss Kcoefficients/& includin' low

Reynolds numbers& and absolute wall rou'hness measurements would certainly impro!e the

desi'n and maintenance of li)uid and 'as pipelines.

APPEN'I> 6

)a!ic #rocedure $or C)MNn $unction%

/unction $BM'n&" " Re" n

arc riction actor co%ining $ole%roo&+Blasius+McKeon+'i&uradse orulas

& = a%solute roug,ness" = inside diaeter" Re = Renolds" n = transition actor

& and in t,e sae units

i Rau" /to" /t" /&

Rau = auiliar Renolds" /t = transission actor = arc+0.5

/to = *re:ious guess or /t" /& = relati:e roug,ness actor = .7-

; Re Renolds


15/17

or tur%ulent lo> Re A= 3080

Rau = Re

@nd ;

$BM'n = 0.3164 # Rau 0.25

; & A 0 ,en

$ole%roo&+,ite to account or roug,ness contri%ution

Cog = natural logarit," Cog10 = coon logarit, = Cog#2.302585093

$BM'n = +2 # n - Cog& # 3.7 - n D 10 +n # 2 - !Er$BM'n # 2.302585093

+2

@nd ;

; Re < 3080 ,en

linear inter*olation or transition lo> 2000


16/17

( ) ( )2122

2

2

1

5.0

2

1 2ln11

zzgTRZ

Mpp

A

NQ M

b

b +

+=

21/

(he lo'arithmic term accounts for the increase in the kinetic ener'y as the 'as expands alon'

the line. %or natural 'as pipelines& where the !elocity is kept below 2$ mMs& this term remains

under $.$$# and is usually ne'lected.

0f the friction factor is calculated with the absolute rou'hness& so that the dra' factor applies forthe partially and totally turbulent re'ions& the pre!ious e)uations become

( )

=

=+=

+=

=

D

LFtFf

D

L

Ff

fK

D

Lf

D

LeLf

D

LtfN

2

222/

( )

+

=

M

TRZzzg

A

Q

D

L

Ff

fpp M

bb12

2

2

22

2

2

2

2

1 2

2#/

( ) ( )2122

2

2

1 2 zzgTRZ

Mpp

A

L

DFtFfQ M

b

b +

=

2"/

NOMENCLAT(RE

A trans!ersal area of pipe

D inside diameter

f Darcy friction factor

Ff dra' factor

Ft transmission factor

g acceleration of 'ra!ity

K sum of loss coefficients for all flow-disturbin' elements in the line

k absolute rou'hness of pipe wall e)ui!alent sand rou'hness obtained from pressure

drop at totally turbulent flow in a strai'ht pipeP it can be related to direct measurement

of wall rou'hness/

ke effecti!e rou'hness of pipe wall e)ui!alent sand rou'hness obtained from pressure

drop at totally turbulent flow in the actual line& includin' the effect of weld beads&

chan'es of direction& deposits& !al!es and fittin's/

L strai'ht len'th of pipe

Le e)ui!alent len'th of weld beads& chan'es of direction& deposits& !al!es and fittin's

Lt total len'th sum of strai'ht len'th plus e)ui!alent len'th/

M 'as molecular wei'ht

N number of friction !elocity heads

n transition exponent applied to reduce the extension of Colebrook transition Bone/

p absolute pressure

Q flow rate

R uni!ersal 'as constant

Re Reynolds number

T absolute temperature of 'as

Z mean !alue of compressibility factor

altitude of pipe centerline


17/17

fluid density

Su,indice!

! at initial pipe section

" at final pipe section

# measured at base temperature and pressure

M a!era'e line !alue

t at the transition from partially turbulent to totally turbulent flow

REFERENCES

1. 8c:eon& +. Q.& 8. I. ;a'arola& and *. Q. 7mits& J* new friction factor relationship for fully

de!eloped pipe flow&K $% F&'d Mec%, Iol.4#>& pp. "29-""#& 2$$4.

2. Cordero& . @.& J*n impro!ed experimental correlation for Darcy friction factor&K

H*drocar#on +roceng& Quly 2$$>.

#. 7mith& R. I.& et al.& J%low of atural as (hou'h Experimental ie Oines and (ransmission

Oines&K 6. 7. +ureau of 8ines 8ono'raph 9& ew ork& *merican as *ssociation& 1945.

". 8c:eon& +. Q.& C. Q. 7wanson& 8. I. ;a'arola& R. Q. Donnelly and *. Q. 7mits& J%riction

factors for smooth pipe flow&K $% F&'d Mec%&Iol.411& pp. "1-""& 2$$".

4. *. E. 6hl et al.& 7teady %low in as ipelines& 0nstitute of as (echnolo'y (echnical Report

o. 1$& *merican as *ssociation& ew ork& 1954.

5. Dewerdt& %.& JOa dFtermination des pertes de char'e dans les canalisations&K aB

d3*uourd3hui& Iol. 1$5 19>"/& pp- >9-9".

=. ersten& :. et al.& Jew transmission-factor formula proposed for 'as pipelines&K Te -&

and .a $o'rna&& %eb. 1"& 2$$$.

>. Cordero& . @.& J* critical analysis of ** and Colebrook methods for calculatin' friction in

'as pipelines&K %irst Oatin *merican as Con'ress& 24-#$ o!ember 19>"& ro!incia del

eu)uFn& *r'entina 0*& 7panish text/.

9. 7letferdin'& E.& Q. 7. udmunsson& J%riction %actor in i'h ressure atural as ipelines

from Rou'hness 8easurements&K 0nternational as Research Conference& o!ember 4->&

2$$1& *msterdam

1$. ooper& W.+. &K(he two-: method predicts head losses in pipe fittin's&K Ce% 0ng%*u'.

2"& 19>1

11. Worthin'am& R. . et al.& JCost study ustifies internal coatin' on ">-in. 'as line&K -& 1 .a

$o'rna&& 8ay #$& 199"

12. *WW* 8anual 811& J7teel ipe < * uide for Desi'n and 0nstallation&K *merican Water

Works *ssociation& 19>=

pipeline friction

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