JOURNAL OF THE HYDRAULICS DIVISION
EXPLICIT EQUATIONS FOR P PIPE-FLOW PROBLEMS
By Prabhata K. Swamee a!" A#a$a!# K. %a&!
'
The three major pipe-flow problems encountered in hydraulic engineering practice relate to the determination of pipe diameter, discharge, and head loss for a given set of known variables. The problem of obtaining discharge and head loss are classified as problems of analysis, whereas the determination of diameter falls into the category of synthesis. The main hurdle in obtaining obtaining a direct solution solution of these problems problems has been the implicit form of the Colebrook Colebrook-White -White equation, which governs governs the resistance resistance to flow through commercial pipes. n accurate solution for f or the unknown variable using the Colebrook-White equation invo involv lves es many many trial trials. s. !ood !oody" y"ss diag diagra ram, m, whic which h is a grap graphi hical cal repr represe esent ntati ation on of the the Colebrook-White equation, eliminates trials otherwise involved in the evaluation of the friction factor. The direct determination of head loss is thus possible by use of !oody"s diagra diagram m in conjun conjuncti ction on with with the #arcy-W #arcy-Weis eisbac bach h equati equation. on. The soluti solution on of pipe-f pipe-flow low problems involving use of the computer, computer, however, makes it obligatory to use the Colebrook-White equation. $y proper choice of variables in formulating the nondimensional groups and employing a suitable curve fitting method, it has been possible to obtain an e%plicit equation for the pipe diameter, which yields results of high accuracy. &or the determination of discharge, a proper grouping of variables has made it possible to transform the Colebrook-White equation to yield a closed-form solution. n e%plicit equation for the friction factor, which is an accurate mathematical appro%imation of the Colebrook-White equation, has been develop developed. ed. The use of this this equati equation, on, in conjun conjunctio ction n with with that that of the #arcy#arcy-W Weisbach eisbach equation, provides a direction solution for the head loss. The development of these e%plicit equations will not only solve the straightforward pipeflow problems in a single step, but will provide mathematical e%pressions for constraints for optimi'ation optimi'ation studies studies of pipelines and water distribution distribution networks that has attracted the attention of hydraulic engineers. (((((((( )ote. * #iscussion open until +ctober , /. To e%tend the t he closing date one month, a written request must be filed with the 0ditor of Technical 1ublications. 2C0. This paper is part of the copyrighted 3ournal of the 4ydraulics #ivision, 1roceedings of the merican 2ociety of Civil 0ngineers. 5ol. 67, )o. 489. !ay, /. !anuscript was submitted for review for possible publication on 3une :6, 9.
;eader in Civ. 0ngrg.. , ?ndia.
7
2r . ;esearch &ellow =on deputation under @.?.1.>, @.?.1.>, #ept. of Civ. 0ngrg., , =<.1.>, ?ndia, Aovernment Coll. of 0ngrg. and Tech., ;aipur =!.1.>.
PIPE-FLOW PROBLEMS
The variables variables influencing influencing the pipe flow may be listed asB => The discharge, discharge, Q =7> the pipe diameter, D; =:> the equiva equivalen lentt sandgr sandgrain ain roughn roughness ess height height,, =D> =D> the the kine kinema mati ticc viscos viscosity ity of the flowing flowing fluid, fluid, v; and and =9> =9> the the head head loss per unit unit pipe pipe lengt length. h. S f . The interrelationship among these variables is e%pressed by the #arcy-Weisbach equation for the head lossB
h f
fLV 7
=>
7 gD
in which h, = energy head loss in a pipe of length L V E mean velocity g E gravitational acceleration and f E friction factor. The continuity equation for the flow is Q
D
D 7V
=7>
Combining 0qs. and 7 yields S f
F fQ 7
=:>
7
gD 9
&or commercial pipes, the friction factor is described by the well-known Colebrook-White semi-empirical formula =D>B , f
7log
D
,.,D 7log
-.:9 D R
f
=D>
in which R E VD/v. the pipe ;eynolds number. &or a given pipe and known flow and fluid conditions, the determination of friction factor using 0q. D involves a trial procedure. The problem becomes further complicated if the discharge, Q, or the diameter, D, is to be determined. The pipe-flow problems of importance to hydraulic engineers fall into three categoriesB => #etermination of diameter when Q, , S f ,. and v are given =7> determination of discharge when D, , S f ,. and v are known, and =:> determination of head loss when Q, D, , and v are given. These problems can be solved utili'ing 0q. : provided that the friction factor, f , e%pressed by 0q. D is determinable. &or problems of the first category, since the diameter is not known, the friction factor cannot be computed, since it is dependent upon the relative roughness, G D, and the ;eynolds number, R . ?t is also not possible to determine the friction factor for the second category of problems, because in the absence of discharge the ;eynolds number is again an unknown. ?n the third category of problems, however, despite both the relative roughness and the ;eynolds number being known, the difficulty is posed by the inherent implicit nature of 0q. D, which requires many trials for computing the friction factor.
AVAILABLE METHODS The pipe-flow problems described earlier can be solved by a method of successive trials that make use of 0qs. : and D. &or problems requiring determination of the diameter, a trial value of the diameter has to be assumed using which G D and R are determined. 0q. D can then be used to compute f by a trial procedure. The !oody"s diagram, however, eliminates the use of 0q. D and, consequently, the trials involved in computing f .
are graphical methods proposed by various researchers, which provide direct solutions of the problems. 3ohnson =9> and ;ouse =F> used similar nondimensional groupsB
V / gDS f
,
D gDS f /
and DG to plot their diagrams. ;ouse, however, superimposed a fourth parameter of ;eynolds number R , as lines of constant R . 1owell =/> adopted QG=v > ,
gDS f /
7
, and DG as nondimensional parameters and proposed a log-log plot for
graphical solution of the three categories of engineering problems. arrived independently at the same basic choice of parameters as 1owell and proposed a resistance diagram that is a log-log plot of V / , g S f / , and DG as the third parameter. lso superimposed on the diagram were lines of equal Q/ =v >. Thiruvengadam => proposed a diagram for a direct solution of the diameter and head loss 7 using Q / g S f , QG= >, and DG . $arr and 2mith =:> in their new diagram proposed a 7 log-log plot of Q / g S f ,
g S f / ,
and DG with lines of constant QG= >
superimposed. The direct solutions for the three categories of pipe-flow problems are : 7 possible with the aid of this diagram. sthana =7> has given a log-log plot of gS f / , QG = >, and G D for a direct solution of pipe-flow problems. ;anga;aju and Aarde => proposed equations for the direct solution of pipe-flow problems, based on the Colebrook-White equation. The equations are Q D
gS f
for
: gS f :.:- 7
:
for
and
7./::
7
,6
Q D
7
7./::
:
gS f
7
,6
6.9D
Q and D
7./::
6.7F7
: gS f 7.F9 7
7
Q and D
=9>
6.967
7./::
6.7F7
=/>
&rom the preceding, it is evident that in all of the approaches, has been selected as a repeating variable for forming nondimensional groups. 2ince it is not possible to measure with a fair degree of accuracy, and since these parameters involve high powers of ( the so-formed parameters will be subjected to a higher degree of error. Therefore, the nondimensional parameter of a dependent variable obtained by the previously described methods will be substantially at variance from the true value.
The e%plicit equations evolved for the solution of pipe-flow problems are based on the Colebrook-White equation =0q. D>, which can be alternatively written as 7.9: 7log :.. D R f f
,
=>
The e%plicit equations for the pipe diameter, the discharge, and the head loss obtained as a result of the present investigation are described in the following paragraphs. )eterm&!at&*! *+ P&,e )&ameter *?n pipe-flow problems requiring the determination of pipe diameter, the known variables are Q, S f , , and v. Considering the variables governing the pipe flow resistance, the following nondimensional groups can be formed gS f non-dimensional diameter D* D 7 Q
nondimensional roughness nondimensional viscosity
*
*
gS f 7 Q
6.7
6.7
, gS Q : f
6.7
=F>
&rom 0q. :, the friction factor can be written as f
7
gS f D 9
=>
FQ 7
and using the relationships defined in 0q. B gS f D 9 Q7
D*9 C
DQ C and # #H f D
*
R
7
gS f D 9 FQ 7
7 D*:
=6>
*
Combining 0qs. , , and 6, the following equation is obtainedB D*
6..7/
* ,..F * log :.. D* D*,.9
6. D
=>
0q. is an implicit relationship for D., which has been solved by iterative method taking . and as parameters.
FIG. 1. —Error I!o"!#$ % D.
—L%#&
o' E()*" Error
FIG. +. —Error % Fr%,%o F*,or E(. 1/ The flow is smooth turbulent for . E 6 whereas it is rough turbulent for . E 6. &or these two particular cases, the relationships for the diameter are obtained as D* 6.// * 6.6D
=smooth
turbulent>
=7> D* 6.// *
6.69
=rough turbulent>
=:>
for the transition region, the following equation has been fitted so that it is in conformity with 0qs. 7 and :B
D* 6.// *,.79 *
6.6D
=D>
0q. D is a general equation that covers smooth turbulent, rough turbulent, and the transition in between. ?t is valid for 6 -F v, 6 -:, which corresponds to : % 6 : R : % 6F and 6-/ . 6-7 corresponding to 7 % 6 -/ G D 7 % 6 -7. &or these ranges of H and H, 0q. D was checked for accuracy with 0q. , and it was found to yield D within an error of I7J. &ig. shows the distribution of error involved in computation of D. using 0q. D and is indicated by the lines of equal error. The error has been defined as 66= D - Dc> /Dc in which D E the value obtained from 0q. D and Dc. E the corresponding value from the Colebrook-White equation. &or smooth-turbulent flow, the nondimensional diameter, #., varies from 6.:/-6.9 for 6-F H 6-: =i.e., : % 6 : R : % 6 F>, whereas for rough-turbulent flow, D* ranges from 6.::-6.979 for 6 -/ H. 6-7 =7 % 6-/ /D 7 % 6 -7>. 0q. D, which takes into account the effect of viscosity and pipe roughness simultaneously, computes a value of #. which ranges between 6.:/ and 6.979 for any practicable values of H and H. )eterm&!at&*! *+ )&har/e *&or the computation of discharge Q, the known data are D, , S f , and v.
7
log
:.. D
D gDS f ,..F
=9>
0q. 9 is e%plicit in @ and therefore yields a direct solution for the pipe discharge. s 0q. 9 is obtained from 0q. D it yields the same value as obtained from the Colebrook-White equation using trial-and-error method.
,..F 7 7log :.. D D gDS f
=/>
0qs. 9 and / give closed-form solution for discharge and average velocity, respectively. )eterm&!at&*! *+ Fr&t&*! S$*,e. *0q. : computes friction slope provided the friction factor is a known quantity. 4owever, since 0q. D is an implicit relationship, determination of friction factor for given relative roughness and ;eynolds number involves a trial procedure. The Colebrook-White equation was solved for the friction factor for smoothturbulent as well as rough-turbulent flow, and the following e%plicit equations are obtained by curve fittingB f
6.79 7
log :.. D f
and
=>
6.79 7
9..D log 6.- R
=F>
0qs. and F have been suitably combined to yield an e%plicit equation for friction factor for the transition 'one of turbulent flow in pipes. The resulting equation is f
6.79 7
9..D log :.. D 6. R
=>
0q. reduces to 0q. for smooth-turbulent flow and to 0q. F for rough-turbulent flow. 0q. was checked for accuracy with the Colebrook-White equation, and it was found that for 6-/ G# 6-7 and 9 % 6 : R 6F , the errors involved in friction factor are well within I.6J. The error has been defined as 66= f - f c )/f c , in which f E the friction factor computed from 0q. B and f c = the corresponding friction factor obtained from ColebrookWhite equation. The distribution of error is shown in &ig. 7 in which lines of equal error have been marked. !aking use of 0qs. : and , the friction slope is obtained as 6.76: S f
Q7 gD 9
9..D log 6. :.. D R
7
=76>
0q. 76 is an e%plicit relationship e%pressing the friction slope in terms of the known quantities. ?t computes the friction slope within I.6J of the value computed using the Colebrook-White equation.
COMPARISON OF ACCURACY ccuracy of 0qs. 9 and / proposed by ;anga;anju and Aarde =F> for determination of the diameter, the discharge, and the friction slope has been studied and compared with the corresponding equations, i.e., 0qs. D, 9, and 76, respectively. &or pipe diameter, 0qs. 9 and / compute the diameter within an accuracy of -/.6J and K:.J whereas 0q. D yields the pipe diameter within I7.6J. The discharge computed from 0qs. 9 and / involves error ranging from -9.6J-K7F.6J whereas 0q. 9, being a closed-form solution, predicts e%actly the same value as that from the Colebrook-White equation. The e%plicit relationship for friction factor e%pressed by 0q. was compared with the equation proposed by Wood =6>. 0q. can be claimed to have advantages is accuracy
=Wood"s equation computes friction factor within I9J>, ease of use, and applicability over the entire turbulent 'one of pipe-flow unlike Wood"s equation which is not applicable for smooth-turbulent flow. ;egarding friction slope, 0qs. 9 and / compute it with an error ranging from -76.6J-K76.6J, whereas 0q. 76 computes within an error of I.6J. CONCLUSIONS &rom a perusal of the available methods and the proposed e%plicit equations for the solution of pipe-flow problems, the following conclusions can be drawnB . The e%plicit 0q. D permits direct and accurate computation of pipe diameter and can be used with ease. 7. 0q. / is a closed-form solution for discharge and allows for its direct determination. :. &or direct and accurate computation of the friction slope =and the head loss there from>, 0q. 76 has been obtained.
APPENDIX 1—REFERENCES . ckers. 1., L;esistance of &luids flowing in Channels and 1ipes.L Hydraulics Research Paper !. ", Her #a$es%y&s S%a%i!'ary (ffice. Mondon, 0ngland. 9F. 7. sthana. N. C., LTransformation of !oody #iagram.L !ur'al !f Hydraulics Divisi!', 2C0. 5ol. 66. )o. 48/. 1roc. 1aper 6/7D. 3une. D. pp. -F6F. :. $arr. #. ?. 4., and 2mith, . ., Lpplication of 2imilitude to the Correlation of , :D. /. 1owell. ;. W., L#iagram #etermines 1ipe 2i'ed #irectly,L ivil +'gi'eeri'g, 2C0, 5ol. 76. )o. . 2ept. 96. pp. D9-D/. . ;anga;aju. N. A., and Aarde. ;. 3., LTransformation of Colebrook-White 0quation to 8ield #irect 2olution to 1ipe &riction 1roblems.L 'iversi%y !f R!!r-ee Research !ur'al, Civil 0ngineering #epartment. ;oorkee, ?ndia. /F. F. ;ouse. 4., L0valuation of $oundary ;oughness.L "!a Hydraulic ulle%i' 01, Pr!ceedi'gs. 7nd 4ydraulic Conference,
APPENDIX II.—NOTATION 3he f!ll!i'g sy45!ls are used i' %his paper6 D = internal diameter of pipe f E pipe friction factor g E acceleration of gravity h f E loss of head L = length of pipe Q E discharge =volumetric flow rate> R E VD/v = ;eynolds number S f = h f ./L = friction slope =hydraulic gradient> V E mean velocity of flow E roughness height of pipe wall
/D E
relative roughness v E kinematic viscosity of fluid and H 0 nondimensional quantity. '12 EXPLICIT EQUATIONS FOR PIPE-FLOW PROBLEMS
N08 W+;#2 #arcy-Weisbach equationB #ischarge =water> &riction factor 4ead losses 4ydraulic properties 4ydraulics 1ipe flow and 1ipes. $2T;CTB #irect solutions of pipe flow problems are not possible because of the implicit form of Colebrook-White equation, which e%presses the hydraulic resistance of commercial pipes. The three basic and major problems encountered in hydraulic engineering practice arc the determination of pipe diameter, the discharge and the head loss. The solution of these problems on conventional lines involves many trials and tedious computations. 2ome research workers have proposed graphical solutions, which have their own inherent limitations. ;eported herein are e%plicit and accurate equations for pipe diameter and head loss and a closed form solution for the discharge through the pipe based on Colebrook-White equation. These e%plicit equations can also be utili'ed with advantage in optimi'ation studies of pipelines and water distribution systems. ;0&0;0)C0B 2wamee, 1rabhata N. and 3ain, kalank N. L0%plicit 0quations for 1ipe&low 1roblems.L !ur'al !f %he Hydraulics Divisi!'. 2C0. 5ol. 67, )o. 489, 1roc. 1aper 7D/, !ay, /. pp. /9-//D
