Direct solution for problems in pipe friction

(1)

NOTULES HYDRAULIQUES

DIRECT SOLUTION fOR PROBLEMS IN PIPE fRICTION

lBY K. G. RANGA RAJU

*

AND R. J. GARDIE

*

' ) 1 '

ID)

=

~"OgI0^'._\9-J'_....

\.

1 18.7 (D /2 k)

+

^{1.74 -} ^{2 Lo"}t:>10 \^{( 1}

^{-1- .-.}

û'"\/1'

^.. ^..-

1

vT

Ist 1IIETHOD:

The resistance of a commereial pipe is expressed '[1

J

by the Colebrook-'White equation, namely

Equation(3) has been found to he asymptotic both to the smoothancl the rough-pipe equations and to follow clos el y the trend of experimental points in the transition region.

Equatiün(:3) artel' necessary manipulations can be written as:

of trial and error for problems in the thircl category.

By forming a set of dimensionless parameters difIerent l'rom the ones in the above c1iagrams, the authors have shown that a direct solution can be obtained for all the three categories of problems.

(l)

(2) 11[= . _ - -lLV2

2gD

where

r:

friction factor; and V: mean velocHy of now.

For the case of laminaI' How, the friction factor is given []] by the equation :

l = -

_ût64

Pipe Hovv problems, in general, involve the 1'01- lovving variables-the discharge of the Huid Q, the loss of head h[ (or pressure b..p) in the length of pipe L, the kinematic viscosity of the lluid v, the pipe diameter D and the equivalent sand grain roughness of the material of the pipe k. These problems fall into three difl'erent categories. They are (l) '1'0 Iind the head loss (2) to determine the discharge (:i) to iind the pipe diameter, when the other variables are given.

The basic equation for the solution of these pro- h!ems is the Darcy-\Veisbach resistance equation,

VIZ.

By substitution of variolls values of A and C in Equation (4), the parameter 13 has heen found and 917 wh cre ût= (VD/v) and problems belonging to aH

the three categories can be solved di rectly by the use of equation (2). For turbulent Hows, one has to make use of the diagrams prepared by Moody [2]

or by Rouse [1]. These diagrams have been prepared with help of Colebrook-\Vhite equation (1) for the resistance of commercial pipes. Prohlems of the Iirst category can be solved direcUy with the help of the above diagrams, while the form present- ed by Rouse enables a direct solution even for the second category of problems; however, using the ahove diagrams, one has to go through a process

- - - _..

_

^..^{_ - - - -}

Header in Civil Engineering, University of Boorkee, Hoorkee, Indi a.

*. PI'ofessor of Civil Engineel'ing, University of Hoorl,ee, Hoorkee, India.

[1] BOUSE (H.): Elementary mechanics of Iluids. Jo1ln lVilcU and sons, Ine. New Yol'!"

[2] MOODY (L.F.): Friction factors for pipe flow. Trans.

ASME, Novemher 1944.

0.H03B

AG

=

2 Log10(A2C)

( 6.60 \ 1.74 - 2LOf!10~.. \ 1

+

^-A-). !

where:

~. \/gD~

A=

v

'>

v- C

= --,---)

... J{:'glo..¹;:;--s-.

and:

s = !3.L

L

(4)

(6)

(7)

(8)

Article published by SHF and available at http://www.shf-lhb.org or http://dx.doi.org/10.1051/lhb/1966059

(2)

K. G. RANGA RAJU and R. J. GARDE

which can be put in the fonu:

By dimensional analysis, the above relation can be written as:

where Ap = "(h{, "( being the specifie weight of fluid;

p : Mass density of fluid;

!..\ : Dynamic viscosity of fluid.

(12) (11)

4

-, 0 1

Log

(*)(ij'!"

-2

,

c

1 !

.

!

1

!

/ 1

!

j

1 1/

V /

ⁱ

1

ⁱ

/

1

il

¹

1

,. if

/

^-,---

~ ₁

1 Il

¹¹

Ji

^! ⁱ¹ ¹

V

¹^!

1 1

-10 1

-5 -4 -3

-9 -8 -7 -6 -5 -2 4

o

-4 -3

o Log~

,2

-,

The relationship between the above parameters can be obtained with the help of the Moody's diagram. For dill'erent values of al and k/D,

l

was obtained from the Moody's diagram and thus the parameters klgS/v'" and Q/vk computed. A study of the l)]ot between Q/vk and klgS/v'" with k/D as the thin} variable over a range of smooth, transition and rough regions, indicated that with slight ap- proximation, a unique relation can be obtained bet- ween the parameters (Q/vk) (k/D)Sf;l and kJgS/v'" . The relaliol1ship between these two parameters has heen shown in Figure 2 and can he expressed hy the following equations:

(I __Q.\ k \)S/:l _ '),~ k')qS\o.[jj,

\ vi;) J)J - ~.;)1 "....~:;-) . /;')uS ( 0 ) (/. SI:) 101' - - " ; -

<

0.:35 or \-"'-- -~-

<

l.GO

v- vk J)

(9)

1000

(10)

'00

r':-

!1p

=

<P,(k, p, Q,D,!J.)

0·25 la

'6 ^. :

A0

'4

, .

RY

i ..

. ^. . _i ^.

P# Rf

'2

pg'

'A

. , .

YZm·

^:

.&

~VÎ

^!

~V

-YI

^•

8 Y~~

VI

,

W$'

.... ^•

14 ///1

.

1Ü"~A/ _':'

1)0'

AJ; W,

^...^{, .}

COl~ ,n··

~7¹ ^la" ^,I(f i

2

';///^VI •••n'⁾⁰ ^,v ^! ^,

[/:;:;

_//i'0' ^,

0

_2~ ~~

^.10' ¹

^,,"0

^. ¹ⁱ ^j _,

2 3 4 6 810A'~

1/

Diagl'am fol' solution of Colebl'oo]('s equation.

Graphique pour la ,';0III !ion de l'éqlla!ion de Cole brook.

the relation between the se parameters has been shown as Figure 1. The diagram eovers the nov;

in the transition and the rough boundary region.

Extension of the diagram to smooth boundary Jlo,vs was felt to be unneeessary as the lnethod suggested by Powell

[in

yields direct solutions in such a case.

Figure 1 yields a direct and accu rate solution for the pipe diameter, given Q, v, Sand k. AIso, for known values of D, v, Sand k, Q can be found directly l'rom the above diagram, but the diagram will have to be eonstrueied to an exaggerated 13 scale for accurate solutions of problems in this eategory. Direct solulion for S, however, cannot be obtained with the help of this diagram, if Q, v, D and k are known; but this category of problems can be solved directly using the Moody's diagram.

IInd METHOJ):

The functional relationship between the various parameters involved in pipe frieiion problems ean be written in the form:

CD

[3] PQWELL (R.W.) : Hesistance to How in smooth pipes found dil'eetly. Civil Engineering, Novembel' 1954, p. (i2.

2/Helatio]] hetween k:lgS/v'" and (Q/vk) (k/D)SI:;.

Relatioll entre k3gS/v~ et (Q/vk) (kID) 813.

918

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The data used in developing the above relationship cover a range of Reynold's number VD/v from 4000 to 7.10⁷ and from the completely smooth to the completely rough region [up to (kID) = 4.10-2 ]

Equations (12) and (13) yield direct solutions for aIl categories of problems concerning pipe friction.

Despite the fact that the above equations are empiri- cal ant not exact, they have the advantage over Figure 1 that in this method the inherent errors involved in interpolating and reading Figure 1 are absent. Further, equations (12) and (3) are valid

(Q)

_{V f}₁ ^(~\_D₎⁾^{8/3 _}^- ^2.40 ¹^/./_" ^k3gS_V-⁰ ^0.,,0,

k³gS

(Q \

¹^k\818

for - 0 -

>

0.35 or - j - ! 1]))

>

1.60

v- vfJ \ / (13)

LA HOUILLE BLANCHE/N° 8-1966

for the smooth, transition and rough boundaries, which is not the case for Figure 1.

The errors in the diameter (for given Q, v, k and S) obtained from equations (12) and (13) as com- pared to the value obtained using Moody's diagram are found to be very small (about ± 3

%).

^In

view of the fact that a standard pipe diameter has to be used ultimately and also since the method of successive trials by using the Moody's diagram is seldom carried to the exact value, equations (12) and (13) seem to be suited for problems of this category. The errors in the value of Q obtained from the ab ove equations seem to be of the order of ± 5 %' However, the en"ors in the values of S computed from these equations are of the order of

± 10

%

and in some stray cases as high as ± 15

%.

919

Direct solution for problems in pipe friction

NOTULES HYDRAULIQUES