Flow in Pipes and Closed Conduits

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4

Flow in pipes and closed

conduits

4.1 Introduction

Th

e flow

of

water, oil and gas in pipes is

of

immense practica l significance

in

civil engin ee ring. Water is conveyed from its source, norma

lly in

pressure

pipelines

Fi

g. 4.1 ,

to

water treatment plants where it enters the distribution

system and finally arrives at the consumer. Surface water drainage and

sewerage is conveyed

by

closed co ndu its, which

do not

usually operate und er

pressure, to sewage treatment plants, from where it is usually discharged

to a river or the sea. Oil and gas are often

tran

sferred fro m theif source

by

pressure pipeli n

es

to r

efi

neri

es

oi l)

or

into

3

distribution ncrwork for

supply gas) .

Surprising as it may seem, a comprehensive theory of rhe flow of fluids in

pipes was not developed until the l

ate

1930s,

and

pracrica l design methods

for the eva

lu

a

tion

of discharges, pressures and head losses d id

not appear

until 1958. Until these design tools were ava ilable, th e efficient design

of

pipeline systems was not possi ble.

This c

hapt

er describes the th eori

es

of pipe flow, beginning with a review

of the hi storica l context and en ding with the practical applications.

4.2 The historical context

Table 4.1 lists the names

of

the main

contr

ibutors,

and

their contributions,

to pipe flow theori

es

in chro nol

og

ica l order.

Th

e Colebrook

Whit

e transi

ti

on

fo

rmu la represents the culmination

of

a ll the previous work,

and ~ n

be .applied to an y

flu

id in any pipe

ope

r

ating under turbu le

nt ow

con

di t

ions.

Th

e later contributions

of

M

oo

dy,

Ackers

and

Barr are mainl y concerned with the practica l appli

cat

ion o f the

Colebrook- White equation.



92

FLOW IN PIPES

AND

CLOSED CONDUITS

Figure 4. The sy nthetic hydrological cycle.

There a re three major concepts described in the table. These are:

1. the distinction between laminar

and

turbulent flow;

2. the distinction between r

oug

h and smooth pipes;

3. the distinction between aftificially roughened pipes

and

commercial

pipes.

To undersrand these concepts the best starting point is the co ntribution of

Reynold s followed by [he laminar flow equations before proceeding to the

morc complex turbulent flow equations.



TI-lE

HISTORICAL

cmnurr 93

T a b l ~ 4.1

T h ~ chronological d ~ v c l o p m ~ n t of pipe flow

t h e o r i ~ s

Date

a m ~

1839-41

a g ~ n

and Poiseuille

1850

Darcy and

W ~ i s b a c h

1884

Reynolds

1913 Blasius

1914 Stanton and Pannell

1930

Nikuradse

19305 Prandtl and

\ 011 Karm:s n

1937 39 Colebrook and White

1944 Moody

1958 Ackers

1975

Sarr

aminar and tllrbillent flow

Contribution

laminar

flow

equation

turbulent

flow

equation

distinction betw«n laminar and

turbulent

flow

- Reynolds Number

friction factor equation for smooth

p i ~

experimental values

of

the friction

factor for smooth p i ~

experimental values

of

the friction

factor (or artificially rough pipes

equations for rough and smooth

friction factors

experimental values of the friction

factor for commercial

p i ~

and the

transition formula

the Moody diagram for commercial

pipes

the Hrdraulics Research Station

Charts and Tables for the design of

pipes and channels

dire<:t

solution of the Col

ehrook-

White equation

Reynolds experiments demonstrated that there were

twO

kinds

of

flow -

laminar and turbulent - as described in Chapter 3. He found that transition

from laminar to turbulent flow occurred at a critical velocity for a given pipe

and flujd. Expressing his results in terms of the dimensionless parameter

Re = pDV/f.L, he

found that for

Re

less than about 2000 the flow was always

laminar, and that for Re greater than about 4000 the flow was always

turbulent. For Re between 2000 and 4000,

he

found that the flow could be

either laminar

or

turbulent, and termed this the transition region.

In

a further set of experiments,

he

found thaI for laminar flow the fric

tional head

Joss in

a pipe was proportional to the velocity, and that for

turbulent flow the head loss was proportional to the square of the velocity.

These two results had been previously determined by Hagen and Poiseuille

l oo, , and Darcy and Weisbach (lJlooV1l, but it was Reynolds who put

these equations in

[h

e context of laminar and turbulent flow.



94 FLOW IN PIPES AND CLOSED CONDUITS

4.3 F

un

damental concepts of pipe flow

The momentum equation

Before proceeding to derive the laminar and rurbulent flow equations, it

is

instructive

to

consider the momentum (or dynamic) equation of flow

and

the influence of the bou ndary layer.

Referring to Figure 4.2, showing an elemental annulus of fluid, thickness

8r,Iengt

h 8/, in a pipe of radius R, the forces acting are the pressure forces,

the shear forces and the weight of the fluid. The sum of the forces acting is

equal

to

the change

of

mom

ent

um.

In

this case momentum change

is ze

r

o,

since the flow is steady and uniform. I-(en

ce

p2-rrr8r -

8/

2ttr8r T2ttr81

- (T + ;

8r

2tt{r +81') 8

+ pg21l r8/8rsin

a

=

0

Sening sine

=

- dz

/d/a

nd divid

in

g

by 2 rrr8r81

gives

p2v -

dp dT T d:z

pg

O

dl dr r dl

( t :

1ir)21l(r -) '

•

(p

.2 )2'111Sr

d/

Figure 4.2 Derivalion of Ihe momentum equation.



FUNDAMENTAL CONCEPTS

OF

PIPE FLOW

S

ignoring second-order terms), or

dp

1

d 1

0

dl

r

dr

where V = p+pgz is the piezometric pressure measured from the datum

4=0.

As

then

Rearranging,

1 d I

( d T

) dT T

--( '11 ')=-

r -+ 1

= -

r

dr

r dr

dr

r

dp· 1 d

----- n)=O

di r

dr

Integrating both sides with respect to

r,

dp·

1r = -& 2 +constant

At the centreline r = 0, and therdore constant = O. Hence

dp'

r

1

dl 2

4.1)

Equation 4.1) is the momentum equation for steady unifo rm flow in

a pipe. It is equally applicable to laminar or turbulent flow, and relates

the shear stress

1

at radius r to the rate of head loss with distance along

the pipe. f an expression for the shear force can

e

found in terms of the

velocity at radius r then the momentum equation may be used to relate the

velocity and hence discharge)

to

head loss.

In the case of laminar flow, this is a simple matter. Howeve r, for the case

of turbulent flow it is more comp

li

cated, as will

e

seen in the following

sections.

The dcvelopmcm of

boulldary layers

Figure 4.3 a) shows the development of laminar flow in a pipe. At entry to

the pipe, a laminar boundary layer begi ns to grow. However, the growth



96 FLOW

IN

PIPES ND CLOSED CONDUITS

boundary layer

/

la

Laminar

now

laminar

, . y

lurbulenl boundary layer

laminar boundary layer

bj Turbuknl flow

Figu

re 4.3 Boundary layers and velocity distributions.

of the bounda ry layer is halted when it reaches the pipe centreline. and

thereafter the ow consists entirely

of

a boundary layer

of

thickness r

The

res ulting velocity distribution is as shown

in

Figure 4.3 a).

For Ih e case of turbulent flow shown Ul Figure 4.3 b}, the growth of the

boundary layer is not suppressed umi l it becomes a turbulent bou nda ry

layer with the accompa nying laminar sub·layer. The res ulting velocity

profile therefore differs considerably rom the laminar case. The existence

of

the laminar sub-layer is of prime importance in explaini ng the difference

be

tween smooth and rough pipes.

Expressions relating shear stress to velocity have been developed

in

Chapter J, and these wi ll be used in explaining the pipe flow equations in

the fo

ll

owing sections.



LAMINAR

FLOW

97

4.4 Lamin

ar

flow

For the case

of

laminar flow, Newton s law

of

viscosity may be used

to

evaluate the shear stress

T)

in

terms

of

velocity

(u):

du dll,

T = ~ = ~ d r

Substituting into the momentum equation 4.1),

dlf,

d r

T ~

d, dl 2

0

dtl, 1 dp

-= - -

dr

~

dl

In

tegrating,

1 dp

I I

=

~

dT,-l

+

conStant

At the pipe boundary, I I , =0 and r =R, hence

and

1

dp

cons

tant

= ---R

4 dl

4.2)

Equation 4.2) represents a parabolic velocity distribution, as shown in

Figure 4.3 a). The discharge

(Q)

may be determined from 4.2). Returning

to

Figure 4.1 and considering the elemental discharge SQ) through the

an nulus, then

Integrating

5Q = 21Tr5m,

,

Q=21T

1

r

lI

, dr

and substituting for tI, from 4.2) gives

Q=_ 21Tdp Rr R

2

_ r

2

)dr

4jJ dl 1



98

0

FLOW IN PIPES AND CLOSED

CONDUITS

'IT dp

Q ~ R

8

dt

Also the mcan velocity (V) may be obtained directly from

Q;

0

Q 1Tdp' 4

V ~

~ R

-

A 8 df 'lTR2

I dp

V=---R

8

dt

(4.3)

4.4)

In practice, it is usual to express (4.4 ) in terms of frictional head loss by

making the substitution

Equation (4.4) then becomes

0

dp

h l ~ -

pg

b

=

32 LV

I

pgO

(4.5)

This is the Hagen-Poiscuillc equation, named aftcr the two peoplc who first

carried out (independen tl

y)

the experimental work leading to it.

The wall shear stress

TO)

may be related to the mean velocity V) by

eliminating df

(

dl from (4.1) and (4.4) to give

(4.6)

As T= T when r R then

(4.7)

Equation (4.6) shows that (for a given

V)

the shear stress is proportional

to r, and is zero at the pipe centreline, with a maximum value TO) at the

pipe boundary.



LAMINAR FLOW 99

Example

4.1 Lamil1ar

pipe flow

Oil flows through a 25

mm

diameter pipe with a mean vclocil)' of 0.3

mls.

Given

that JL = 4.8 X 10-

2

kg/m sand p = 800ks/m}, calculate

a)

the pressure drop in

a 45 m length and

b)

the maximum vclocil)', and the velocity 5 mm from the pipe

wall.

SO/lItio

First check thaI flow is laminar, i.e.

Re

< 2000.

Re

= pDV

/JL

= 800 x 0.025 x 0.3/4.8 x 10 -

2

= 125

a)

To

find

thc press ure drop, apply (4.5):

h

f

= 32JLLV pgOZ

= 32 x 4.8

x

10-

2

x 45 x 0.3)/(800 x 9.81 x 0.025

2

)

= 4.228m(of oil)

or Ap = -pgh, = -33. 18kN/m

1

.

(No

te: the negative sign indicates that pressure

reduces in Ihe direction of flow.)

b)

To

find the elocilies, apply (4.2):

Idp 22

=--- R -r)

,

4JL

dl

The maximum velocity U ... occurs at the pipe centreline, i.e. when r = 0, hence

1 33. 18x10

J

2

U

x

=- 48 02

x

5 (0.025/2)

4 x.) ) - 4

=0 .6m/s

N

nte:

fln

.. _2x mean

veloc

i

ty

(colllpare (4.2) and (4.4).))

To

find

the velocity 5 111m from the pipe wall Us), usc (4.2) with r =

(0.025/2)-

0.005, i.e.

r

= 0.0075:

1/ =_ I )(_33.18XIO)(0.01252_0.00752)

J 4x4.8 ) 10-1 45

=0.384m/s



100

FLOW IN PIPES AND CLOSED CONOUITS

4.5 T urbule

nt fl

ow

For rurbulent fl ow, Newton s viscosity law ~ s not apply and, as described

in

Chapter

3

semi-empirical relationships for TO were derived

by

Prandt

l

Also, Reynolds experiments, and the ea

rl

ier on

es

of

Darcy and Weisbach,

indicated that head loss

waS

pr

opo

rti

ona

l to mea n

ve

locity squared. Using

the momentum equation (4. 1 then

dp R

o=

dl

2

,nd

dp·

hfpg

dl

L

hence

Assuming h KV

2

based on the experimental results cited above, then

0

1o,

h, KV ).

Returning

[0

the momentum eq uation

and

making the substitution

TO

K. V

.t

then

hence

0



TIJRBULENT FLOW

101

Making the substitution X= 8Kdp then

(4.8 )

This is the Darcy-Weisbach equation, in which is

ca

lled the pipe friction

factor and is sometimes referred to as

f

(America n practice) or

f

ea rly

British practice). In current practice, A is the normal usage and is found, for

instance, in the Hydraulics Research Stalion charts and tables. It should be

noted that A is dimensionless, and may be used with any system of units.

The original investigators presumed that the friction factor was constant.

This was subsequently found

to

be

incorrect (as described

in

secrion 3.6).

Equations relating A to both the Reynolds Number and the pipe roughness

were developed later.

Smooth pipes and the Blasius equation

Experimenta l investigations

by

Blasius and others early in the 20th century

l

ed

0

[he equation

X= 0.316/Re°.2J

(4.9)

The later experiments of Stanton and Pannel, using drawn brass tubes,

confirmed the validity of the Blasius equation for R

ey

nolds' Numbers up to

10

5

• However, at higher values

of

Re (he Blasius equation underestimated

X for these pipes. Before further progress could be made, the distinction

between smooth and 'rough pipes had to

e

established.

Artificially rough pipes mId Nikllradse s experimental results

Nikuradse made a major contribution 0 the theory of pipe flow by objec·

tively differentiati ng between smooth a nd rough turbulence in pipes. He

carried out a painstaking

se

r

ies

of

ex

periments to determine both the fric·

tion factor :lTld the velocity distributions at various Rey nolds Numbers

up to 3 x 10

6

• In th

ese

experiments, pipes were artificially roughened by

sticking uniform sand grains on

to

smooth pipes. He defined the relative

roughness

(k,/D)

as the ratio of the sand grain size to the pipe diameter.

By

using pipes of different diameter and sand grains of different size, he

produced a set of experimenta l resu lts of and Re for a range of relative

roughness

of 1130

[ 1/1014.



He plotted his results as log ~ against log Re for each value of K /D as

shown in Figure 4.4. This figure shows that there are five regions of flow,

as follows:

(a) Laminar flow. The region in which the relative roughness has no in

flu

ence on lhe

fr

iction factor. This was assumed

in

deriving the I-I

agen-

Poiseuillc equation (4.5). Equating this to the Darcy-Weisbach equation

(4.8) gives

3 ~ V L ~ L V l

=

pglY

2gD

0

~ = 6 4 ~ = 6 4

pDV Re

(4.10)

Hence, the Darcy-Weisbach equation may also e used for laminar

flow, provided that

~ is

evaluated by (4.10).

(b)

Transition from laminar to turbulent

fl

ow.

An

unstable region

between Re

=

2000 and 4000. Fo rtunately, pipe flow normally lies

outside this region.

(c) Smooth turbulence. TIle limiting line

of

turbulent flo

w,

approached

by

a ll values of relative roughness as Re decreases.

(d)

Transitional turbulence. The region in which}., varies with both Re

and k /D. The limit of this region varies wit h k /D. In practice, most of

pipe flow

lies

within th is region.

0. 116

\

ransitional rough I U r b u l e ~ e

DIlks

J1

turbulence

•

,

'

•

'

.1ll2

' ~ /

'

11J 6t

,

I 2$2

,

'

,

00t6

, ,

00

00

'

00

00

Re (log

loCa

le)

Figure

4.4

Nikuradsc s

experimental

results.



TIJRBULENT FLOW

103

e)

Rough turbulence. The region

in

which

A

remains constant for a given

k

D and is independent

of

Re.

An

explanation of why these five regions exist has already been given

in

section

J 6

t

may

be

summarized as follows:

Laminar flow. Surface roughness has no influence on shear stress trans

miSS ion.

ransitional turbulence. The presence

of

the laminar sub-layer smooths

the effect of surface roughness.

Rough tu rb

ul

ence. The surface roughness

is

large enough to break up the

laminar sub-

lay

er giving turbulence right across the pipe.

The rOllgh aud smooth laws

o

tlO

Komia

and Pra dtl

The publication of Nikuradse s experimental results (particularly his

VelOCity

distribution measurements) was used

by

von Karman and Prandtl

to supplement their own work on rurbulcnt boundary layers.

By

combining

their theories of turbulent boundar} layer flows with the experimental

results, they derived the semi-empirical rough and smooth laws. Th

ese

were:

for smooth pipes

for rough pipes

1 ReJ i:

=21 g -

J i: 2.51

_1_ 2 log 3.7D

J i: k

(4.

11

)

4.

12)

h

e smooth law

is

a bener

fit

to the experimental data than the Blasius

equation .

The Colebrook-White tra s;t;ol1 fomlllia

The experimental work of Nikuradse and the theoretica l work of von

Karman and Prandtl provided the framework for a theory of pipe fric

tion. However, th

ese

results were not of direct use to engineers because

th

ey

applied on

ly

to artificia

ll

y roughened pipes. Commercial pip

es

have

roughness which is uneven both in size and spacing, and do not, therefore,

necessari ly correspond to Ihe pipes used in Nikuradse s experiments.




Colebrook and White made nyo major contributions to the development

and application of pipe friction theory to engineering design. Initially, they

carried out experiments to determine the effect of non·uniform roughness as

found in commercial pipes. They discovered that in the turbulent transition

region the A-Re curves exhibited a gradual change from smooth to rough

turbulence in contrast to Niku radse s S -shaped cu rve s fo r uniform rough

ness, size and spacing. Colebrook then went on to determine the effective

roughness size

of

many commercial pipes. He achieved [his by studying

published results of frictional head loss and discharge for commercia l pipes,

ranging

in

size from

4

inches (10

1.

6mm) to 61 inches (l549.4mm), and

for materials, including

drawn

brass, galvanized, cast and wrought iron,

bitumen-lined pipes and concrete-lined pipes.

By

comparing the friction

factor of these pipes with Nikuradse s results for uniform roughness size in

the rough turbulent zone, he was able to determine an effective roughness

size for the commercial pipes equivalent to Nikuradse s results. He was thus

able to publish a list of

k.

values applicable to commercial pipes.

A second contribution of Colebrook and White stemmed from their

experimental results on non-uniform roughness. They combined the von

Kamlan-Prandtl rough and

smooth

laws in the form

\ \ k, 2.5\

.J).

=

2 og

3 7D

Rc.J).

(4.13)

This gave predicted results very dose to the observed transitional behaviour

of commercial pipes, and is known as the Colebrook-White transition

formula. It is applicable to the whole of the turbulent region for commercial

p

ip

es using an effective roughness val ue determined experimentally for each

type

of

pipe.

The practical application o the Colebrook-Wlhite tra lsition formula

Equation (4. 13 was

not

at first used very widely by engineers, mainly

because it was not expressed directly in terms of the standard engineering

variables of diameter, discharge and hydraulic gradient.

In

addition, the

equation is implicit and requ ires a

rr

ial-and-error solution.

In

the 19405,

sl

ide-rules and logarithm tabl

es

were the main computational aids

of

the

engineer, since pocket calculators and computers were not then available.

So these objections to the use of the Colebrook-White equation were not

unreasonable.

The first attempt to make eng in

ee

ring calculations easier was made by

Moody. He produced

a

h-Re pl

ot

based

all

(4.13) for commercial pipes, as

shown

in

Figure 4.5 which is now known as the Moody diagram. He also

presented an explicit formula for h:

/



~

MIRe

I I . m · : I · ~ ; c

2

_

01<

tun

IH)64

O.INS

1 1

0,(136

0,10

2

). 0.02S

IUlZ

4

IUl2

0

W

•

o

II

2

001

0

U fJOK

:

\

;

\1

\ 1

Re

eT

/

,

10 2W }

-

,

,

; ,

.,

.,

\

\

'.

- -

Iriln 'llOnallUrbulencc

k_1r.msitional

I

lurbulence

rough

turbulence

,

I

,

t ..

-

~

r-:.----

I

~

,

,

I::::-

'

-

sm

OO l

pipes

O <MXI 001

; ,, 0.000 005

I '

I I I I I

I I '

10

'

10

.

10·

10

'

Re

Figure 4.5 The Moody diagram.

0.05

0.04

0.03

0.02

0.015

0.01

0.008

•

0

•

•

.002

.001

.0008

.0006

0 0004

0

0

0

.0002

.0001

.00005

0

lXIOO1

10'



106

FLOW IN PIPES AND CLOSED CONDUITS

which gives A co rrect to ±5% for 4 x 10

3

<

Re <

I

x 10

7

and for k./D <

0.01.

In a more recent publication, Barr 1975) gives another explicit formula

tion for X

1 2 1 k

,

5.1286

. j .= og 3.7D Reo.n

4.15)

In th is formula the

smoot

h law component 2.5 lI

Re

A has been replaced

by an

approx

imation 5. 1286/ Rco.

89

). For Re > l Os this provides a solution

for S,{hf/L) to an accuracy better than

±

I%.

However, the basic engineering objections

to

the use

of

the

Co

l

ebrook

White equation were not overcome until rhe publication of Charts for the

Hydralt ;c

Desigll

o(ClJalllleis a

nd

Pipes in 958

by

the Hydraulics

Re

sea rch

Station. In this publication,

the

three dependent engineering variables

Q,

D

and

Sf) were presented in the form of a series of charts for various k,

values, as shown in Figure 4.6. Additional information regarding su itable

design values for

k,

and

other

matters was also included. Table 4.2 lists

rypical values for various materials.

These chans are based on the combination of the Co lebrook-White equa

tion 4.13) with the

Darcy-Wei

sbach formula 4.8 , to give

r:;:;:;;;

(k 2.51.

= - 2 2gDS

f

log -

r>::n<

3.7D D,/2gDS,

4.16)

where Sf

=

, IL, the hydrau l

ic

gradient. Note; for further details

concerning the hydraulic gradient refer to Chapter 12.) In this equation

the velocity and hence discharge) can be computed directly for a known

diameter and frictional head loss.

More recently,

the

Hydraulics Research Station have also

produced

Tables fOT the Hydraulic Design of Pipe

s.

In practice, any two of the three variables Q, D and

Sf)

may be known,

and

therefore

the

most

appropriate so

lution technique depends

on

circum

stances. For instance, in the case of an existing pipeline, the diameter

and

available head are known and hence the di

sc

harge may be found direcrly

from 4. 16). For the case

of

a

new

insta llation, the available head and

required discharge are

known and

the requisite diameter must be found.

This witJ involve a trial-and-error procedure unless the HRS charts or

tables are used. Finally, in the case

of

analysis

of

pipe networks, the

required discharges and pipe diameters

are known and

the head loss must

be computed. This

problem

may be most easily solved using an exp

li

cit



DiamelC:r (m)

Figurt 4.6 Hydraulics Research Station chart

for k

0.03

rum

Tablt 4.2 Typical k valuts.

Pipt maltrial

brass, copptr, glass, I ersptx

> e f o s ccrntnl

wrought iron

galvanized iron

plastic

bitumen-

lin

e ductile iron

spun concre

le lin

e ductile iron

slimed concrtte sewer

k

mm)

0.003

O oJ

0.06

0.15

0.03

0.03

0.03

6.0




Examples illustrating the applica tion

of

the various methods

to

the so

lu

tion of :1 simple

pi

pe friction problem now follow.

Examp

le

4.2

Estimati

Olt

of

discharge

givell

diameter

Qltd

head loss

A pipeline IOkm long,

300mm

in diameter and with roughness size 0.03 mm,

conveys water from a reservoir top water level

850111

300ve datum) 10 a water

tr

ea

tment plant inlet water l

evel

700 m above datum). Assuming that

th

e

~ r v o i r

remains fuJI, rslimate the discharge, using the following methods:

a) the Colebrook-Whi tt formula;

b) the

Moody

diagram;

e)

the

HRS

chartS.

Note

Assume

v =

1.13 x 10

-

ml/s.

So/lltion

a) Using 4. 16),

D=O.3m

k.=O.OJmm

f = 850- 700)/10000 = 0.015

hence

v

=

-2,flg)

0.3

)

0.01510g 0.03

x

IO-J

+

; ~ 2 . ~ 5 F 1

X ~ I ~ . I F J F X ~

~ ~

)

3.7) 0.3

O.3J2g

x 0.3 x 0.015

=

2.5

14

m

ls

Q

V

2.5

14 )

'II xO.3

2

0

78 J

A 4 lm /s

h) The same solution should be obtainable using the Moody diagram; however,

it is less accurate since

it

involves interpolation from a graph. The solution method

is as follows:

(1)

ca lculate ksl D

2) guess a value for V

3) calculate

Re

4) estimate A using the M

oody

d iagram



11}RBUl.ENT Fl.OW

109

Th

is

is a tedious solution technique, but it shows why the HRS charts were

produced

(I )

kslD = 0.03 x 10 -)/0.3 = 0.0001.

2)

As

the solution for

V

has already been found

in

part (a) take

V

=

2.5

m/s.

3)

DV

0.3 x 2.5

Re = -

~ x : : ;

=

0.664 x 10

6

4)

Referring to Figure 4.5, Re = 0.664 x

10

6

and

k

I

D

= 0.000 1 confir

ms

that

the

flow

is in

the transitional turbulent region. I:ollowing the

k, ID

curve until

it intersects with Re

yiel

ds

A::=0.014

Note: Interpolation is difficult due 10 the logarithmic scale.)

5) Using 4 .8 ),

J

ALV

1

=0.0 1

x l

4

x2.5

1

2gD 2gxO.3

= 148.7m

6)

H- 850-700)-150, .

148.7.

7)

A bener guess for

V

is obtained

by

increasing

V

sligh

tl

y. This wi ll not

significantly alter

A

but will increase hr In this instance, convergence

to

the

solution

is

rapid because the corree solution for

V

was

ass um

ed initia

ll

y

c)

f

the HRS chart shown in Figure 4.6 is used, then the solution of the equa tion

lies at the interseetion of the h)'draulic gradient line (sloping downwards right to left)

with the diameter (venical

),

reading off the corresponding discha r

ge

(line sloping

downwards left to right).

s, = 0.0 15 1ODS, = J.5

and

D=300mm

giving

Q

= 180 lIs = 0. 1

8m

1

/5

Examp le 4.3 EstimatiOll of pipe diameter give discharge al1d head

A

di

scharge of 400Vs

is 10 be

conveyed from a headworks at

1050m

above datum

to a treatment pla nt at 1000m above datum. The pipeline length is 5 km. Estimate

the requi red diameter. assuming that

ks

= 0.03

mOl.



1\0

FLOW IN PIPES

AND

CLOSED

CONDU

ITS

Solutio

This (( quires an iterative solution

if

methods

(a) or (b

of

Ihe

previous example are

used. However, a direct solUlion can be obta ined using the HRS charts.

S

SO/SOOO 100S,,,,, I

and Q

=

400 lIs

giving D = 440 mm

In practice. the nearest (larger) available diameter would be used (450 mill in this

case).

Example 4.4 Estimation of head loss

gillen

discharge nd diameter

The known outflow from a hranch

of

a disuiburion

syStem is

30

lis.

The

pipe

diameter is ISO mm, length 500 III and roughness coefficient estimated at 0.06 mm.

Find the head loss

in

the

pipe, uSlOg

the explicit formulae

of

Moody and Barr.

Solutioll

Again, the HRS charts could be used directly. Howc\ er, if Ihe analysis is being

carried out

by

computer, solution

is

more efficient using an equ:ltion.

Q 0.03 mJ/s, D _

O.

15m

V

=

L7m

/ s

Re=0.15x 1.7/

1.1

3

x 10-

Re=0.226 x 10

Using the

oody fo

rmula (4.14)

(

(

20oooXO.06

X

10

-

1

)

)

~ = 0 O O 5 5

1+ 0.15

+0.226

0.0182

Using the Barr formula (4.15)

1

(0.06 X 10

-

1

5 . 1 2 8 6 )

./):.=

-

21

08

3.7xO.15

+

(0.226 x

1

0 )



TURBULENT FLOW

111

The accuracy of Ihese formulae may

e

compared

by

substituting in Ihe

Colebrook-While equarion (4.13) as follows:

1/ ,ff;

0.0182 7.415

ks 2.51

I

-2log

3 7D

+

Re.... A

=.... A

7.441

Thi s confirms Ihat both formu l

ae

are accurale in Ihis case.

The head loss may now

be

compulcd using the Darcy-Weisbach formula (4.8):

= 0.01

82x500x

1.72

f

2gxO 15

=

8.94m

The z n \Villiams formula

The emphasis here ha s been placed on the development and use of the

Colebrook-White transition formula. Using the charts

or

t l e

it is simple

to apply to s

in

gle pipelines. However, for pipes

in

series or parallel or for

the more ge neral case

of

pipe networks it rapid ly becomes impossible

to use

for hand calculations. For this reason, simpler empirical formulae are sti

in common use. Perhaps the most notable

is

the Hazen

-W

illiams formula,

which takes the form

or,

alternatively,

_ 6.78L (V)

hf - D1.I65 C

where C is a coefficient. The va lue of C varies from about 70 to 150,

dependi ng on pipe diameter, material and age.

This formula gives reasonably accurate results over the range

of

Re

commonly found

in water distribution systems, and because the value

of

C is assumed to

be

constant,

it

can be easily used for hand calculation.

In

reality, C should change with Re, and caution shou ld e exercised in its

use

.

An interesting problem

is

to compare the predicted discharges as calculated

by the Colebrook-White equation

and

by the Hazen-Williams formula

over a large range of Re fo r a given pipe. The use

of

a microcomputer is

recommended

fo

r th

is

exercise.



112 FLOW IN PIPES AND CLOSED

CONDUITS

4.6 local hc ad losses

Head losses, in addition to those due to friction, are always incurred at

pipe bends, junctions and valves, etc. These additional losses are due to

eddy formatio n generated in the fluid at Ih e fining, and for completeness,

they must

e

taken into accou nt. in the case of long pipelim:s e.g. severa l

kilometres) the local losses may e negligible,

but

for sho

rt

pipelines, they

may be greater than the frictiona l losses.

A general theoretical treatment for local head losses is not available. It is

usual to assume rough turbulence

si

nce this leads to the si mple eq uation

4. 17)

where hl is the local head loss and

kl

is a cons tant for a particular fitting.

For the particular case of a sudden enlargement for instance, exit from a

pipe to a rank) an expression may be derived for kL in terms of the area of

the pipe. This result may be extended to the case of a sudden

contr::J c

tion for

in stance, entry to a pipe from :\ tank).

Fo

r

all

other cases e.g. bends, valves,

junctions, bellmouths, etc.)

val ue

s for

kl

must be derived experimenta

lly

Figure 4.7 a) shows the case

of

a sudden enlargement. From position

1)

to 2) the velocity decreases and therefore the pressure increases . At position

I

I

D

®

II

ta)

AI

• tvckkn r t t t

/f

I

D

®

II

I)) AI. s\tdckft conlraction



LOCAL HEAD LOSSES

113

(I ) turbulent eddies are formed, which gives rise to a local energy loss. As

the pressure ca

nn

OI change instantaneously at the sudden enlargement, it is

usua lly assum

ed

that at position (1 ) the pressure

is

the same as at po

si

tion

(1 ). Applying Ihe momentum equation between I) and

2).

The continuity eq uation (Q

=

A2V

2

is now used to eliminate

Q.

so, with

some rearrangement,

.)

The local head loss may now be found

by

applyi ng the enerb > equation

f<om I)

ro (2),

or

b)

If

(a)

and

(b)

are combined and rearranged,

Th

e continuity equation may now be used again to

ex

press the result in

te

rm

s of th e areas. Hence, substituting

V A A

2

for

V

2

0

(4.18)



114

FLOW IN PIPES AND CLOSED CONDUITS

Equation (4.18 ) relates hl to the areas and rhe upstream ve loc ity.

Comparing th

is

equation with (4 . 17) yields

For the case

of

a

pipe

discharging in

to

a

tank A2 is

much grea ter tha n

AI>

and hence kl

= 1. In

o th er words,

fo r

sudden

large

expansion, th e head

loss equals the velocity head before expansion.

Figure 4.7(b) shows the case

of

a sudden contraction. Fro m posi

ti

o n (1)

to

( 1)

the flow contracts, forming a vena co

ntr

acta. Experimenrs indicate

that the contraction of the ow area

is

generally about 40 . f the energy

loss from {I) to {1

' )

is

assumed

to

be negligibl

e,

then the rema ining head

loss occurs

in

th e expansion from (I') to 2). Since an expansion loss gave

ri

se to (4.18),

that equat

ion may n

ow be

applied here. As

then

0 '

b

L

=

0.44Vi 2g

(4. 19)

i.

e. kL =

0.44

.

Typical kL

va

lu

es for

other important

local losses (bends, tees,

bdlmouths

a

nd

valves)

are

given in

Table 4.3.

Table

4.3 Local

head loss coefficients.

kL va lu

e

Item Theoretical Design practice

bdlmoUlh entrance

0.05

0.10

ex

il

0.2

0.5

90· bend

0.4 0.5

90

in-line flow

0.35 0.4

branch to lint

1.20

1.5

gate valve (open) 0.12

0.25

Comments

v

=

velocity

in

pipe

(for equa l di ameters)

(for equal diameters)



LOCAL

HEAD LOSSES

11 5

Example

4.5

Discharge caiclilation (or a simple pipe system ifreluding

local losses

Solve Example 4.2 allowing fo r local head losses incurred by the following items:

20 90° bends

2 gate valves

1

bdlmouth entry

bdlmouth exit

SO/lllio

The available static head 150m) is dissipated bOlh

by

friction and local head losses.

Hence

Using Table 4.3,

III =

20 x 0.5)+ 2)

0.25)+0.1

0.5]yl 2g

= 11 1 ylj2g

Using the Colebrook-White formula as in Example 4.2) now requires an iterative

solution, since

h

is initially unknown.

A

solution procedure is

as

follows:

I) assume h,

1 1

i.e. ignore lid

2) calculate Y

3) calculate hl using V

4) calculate

h +hl

5)

i

h +hl

F

1 1

,

set h

'

H -

III

and return to 2)

Using Example 4.2, an initial solution fo r

Y

has already been found, i.e.

V=2.5 14mjs

Hence,

Adjust

II,

11 = 1

50-3.58_

J46.42m

f

=

146 42j10

000

=

0.01464




Substitute in (4.1 6),

(

0.03 x 10 -1 2.51 x 1.13 x 10- ' )

V =

2/2g

x 0.3 x 0.0146410g ; - ; = : ; ; ~ ~ E ~ ~ '

3.7xO. 15

O.3j2gxO.3xO.O I 64

=2.386m j s

Recalculate hL'

Check

hL

h

= 1

46.42 3.22

= 149.64 150

This

is

sufficiently accurate

to

e acceptable.

Hence,

Note; Igno ring L gi

ves

Q = 0. 18

m1

/ s.

4.7 Partially

full

pipes

Pipe systems for surface water drainage and sewerage are normally designed

to flow full, but not under pressure. This contrasts with water ma

in s

which

are normally full and under pressure. The Colebroo

k-Whit

e equati

on

may

be used for drainage pi pes by noting that, because the pipe flow is n

ot

pressurized, the water su rface is parallel to the pi pe inv

er

t, so the hydraulic

gradient equals the pipe gradien t:

where So is the pipe gradient.

Additiona

ll

y an estimate of the di scharge and velocity

fo

r the partially

fu ll

condition

is

required. This enables the engineer

to

check if self-cleansing

velocities a

re

maintained

at

the minimum discharge. Self·cleansing veloci

ties are of crucial importance in the des ign of surface water drainage and

sewerage networks, where the flow may conta in a considerable suspended

solids load.

A free surface flow has one more variable than full pipe flow, namely the

height

of

the

fre

e surface. This

can

introduce considerable complexity (refer

to Chapter 5). However, for th e case of circular conduits the Colebrook



PARTIAUY FULl PIPES

11

7

Starting from the assumption that the frictio n facto r for the partially fu

ll

condition

be

haves similarly to that for the

fu

ll

con

di ti

on, it remains to

fi

nd

a parameter for th e partially fu

ll

pipe whi ch

is

equiva lent

to

the dia meter

for the full pi

pe

case. The hydraulic radi us

R is

such a parameter:

where

A is

the water cross-sectiona l area and

P is

the wetted peri meter. For

a pipe flowing full,

A/ P= I D2

/

411 D = D/4

0

Hence the Colebrook-White transition law applied to partia

ll

y fu

ll

pipes

becomes

I

21

,

2.51

)

fi = - og 3.7 x 4R + Refi

4.20)

where Re = V/

II

Figure

4.8

shows a pipe with partia

ll

y fu

ll

fl

ow al a depth

d). Sta

rting

from the Darcy-Weisbach equation 4.8) and repl

aci

ng h /

I

by So gives

V'

=

2g5,DI

Hence, for a gi

ven

pipe with partia

ll

y fu

ll

flow,

V = 2g5 , 4RI )'

D



118 FLOW IN PIPES ND CLO

SED

CONDUITS

0 '

= constant· 1

2IX 1/2

Forming

the

ratio

V,/ Y

D

=

Vp

gives

V

_}..

12R / l / X il

D

,. II

4.21)

where the subscripts p, D and d refer, respectivdy,

to

the proportional

value, the full depth D) and the partially

full

depth

eI). Simi

larl y,

For a circular

pipe,

and hence

Q

_} I / lA R / Z/} I/z

. D p .

i

Ad =

( I> - n

<I>

)

D

P

d

=

,0/2

R,

=

I

_

')

A,=

( '-2: ')

R,=( I

_

S

; ; ' )

4.22)

4.23)

4.24)

Substitution of 4.23) and 4.24) into 4.2 1) and 4.22) allows calculation

of the proportional velocity and discharge for any proportional depth dID).

The expression for}.. equation 4.20)) is, however, rather awkwa

rd to

manipulate . Consider

first the

case of rough turbulence. Then,

Hence.

_1_ =2

1

g

(3.70)

,f A

ks

Fo

210g 3.7 x

4R,/ks

;;;; = 210g 3.70/k

s

)

This may

be

expressed by its

equ

iva l

ent:

Fo 10gR,



PARTIALLY

FULL

PIPES

11 9

as

10gR, l

og 3.7D/ks)+logR,

1+ =

log 3

.7D/ k

s

} log 3.7D/k

s

}

log[

3.

7D/ ksllR

, /

R

o)

I

=

log 3.7D/ k

s

}

=

log[3.7 x 4R

, /

ksl

log 3.7D/ k

s

}

Equation 4.25) may be substituted

into

4.21) and 4.22) to yield

10gR, )

I

V

=

1+ log 3.7D/ k

s

) R,

and

10gR, ) f

Q

p

= 1 log 3.7D/k,} pRp

The equivalent expressions for the transition region as derived in

Hydraulics Research Paper No .

2,

published in 1959) are

_ 10gR, )

l { l

V,_

1+ 10g3.70 R,

4.26)

and

Q = 1 p

ARI /l

10 gR

P

log 3.70

p p

4.27

)

where

a ks + 1

r

D 3 6 0 0 D S ~ l

4.28)

These

re

suhs for 0

=

1000 are ploned

in

Figure 4.9. Tabulated values for

various 0 may

be

found

in

Hydraulics Resea rch Ltd 1983a). Neither Vp

nor

Q,

are very

se

nsitive

to O

Figure 4.9 shows that the di

sc

harge

in

a pa rtially full pipe may

be

greater

than the di

sc

harge for a

fu

ll

pipe. This is because the wetted perimeter

reduces rapidly immediately the pipe ceases to

be

full whereas the area

does not, with a consequent

in

crease in

ve

locity. However, this condition

is usually ignored for d

es ign

purposes because, if the pipe runs

full at

any



120

flOW IN IPES AND CLOSED CONDUITS

0.0

0.8

0.'

e

~

0.'

0.3

2

0. 1

U

o

1 01 0.1

Q

0-4 OS 06 0.7 0.8

Qd/Of)

and

V VD

v

09 10 1 I 12

Figure 4.9 I'roporlionai discharge and velocity for pipes flowing partially full

(with

a

1000).

section (e.g. due to wave action or unsteady conditions), Ihen the discharge

will rapidly reduce to the full pipe condition and cause a 'backing up of

the flow upstream.

Example 4.6

Hydrauli

c desigl1 of a selver

A sewerage pipe is to be laid at a gradient

of

I

in

300. The design maximum

discharge is 75 Us and the design minimum now is estimated

to

be IOUs. Determine

the required pipe

i m e u

10 bOlh carry the maximum discharge and maimain a

sel f·cleansing velocity of 0.75 mls at the minimum

di

scha

rge.

Soilltio

ll

The easiest way 10 solve this problem is

to use

the HRS design ch:ms or tables. For

a sewer,

ks

= 6.00mm

(Table 4.2). Howt:ver, to illustratt: tht: solution, Figurt: 4.6

is

uSt:d

(for which ks

=

0.03mm :

Q=

75 1/5

IOOh L =

100

/300 =

0.333

Using Figure 4.6



REFERENCES AND FURTHER READING 121

Next check the velocity for Q

=

10 l/s

Q = 10/75=0 133

Using Figure 4.9 (neglecting the effect

of

0),

dID =O.25 for Q, = 0.133

Hence V

_ 0.72 and

" , ,=0

.72

)

1.06 = 0.76m/s

Th

is val ue

excttds

the

sel

f·cleansing velocity, and hence Ihe solution is D=

300mm.

In cases where Ihe se lf-cleansing \'c ocity is nOI maimaincd, it is necessary 10 increase

Ihe diameter

or

the pi pe gradiem.

Note: Th

e solution using

ks = 6mm

and accounting for 0 gives the following

values:

D

375mm

for

Q=

811/s and V _ O.73m/s

0 = 45

Q, = 1

0/8

1= 0.123

d/D=O 024

V

=

O.67m/s

V =0.49m / s

Hence It would be n«essary to increase

D

or So. In this case, increasing So would

be p ~ f e r a b l e

Rcfcrences

~

n furth er reading

Ackers, P.

J9

58)

ResistallU of

Fillids Fl

owing

ill

Cllmmels and

Pip

es.

H

yd

raul ics

Research )laper No. I, HMSO, London.

Barr, D. I. H. (1975) Two additional methods

of

dir

«

solution of Ihe Colebrook-

White function.

Pr

oc. Instn Giv. Engrs, 59, 827.

Colebrook, C.

F.

(1939)

Tur

bulent nows in pipes, with particular reference to the

tra nsi tion region between the smooth and rough pipe laws. J In

stn Gi

ll.

Engrs,

t

I

133.

Colebrook, C.

F

and White, C. M (1937) Experiments with nuid friction in rough.

en

ed

pipes.

Proc.

R

oy.

Soc.,

A16 1, 367.

Hydraulics Research Limited (1983a) Tables for 'he Hydraulic Design of Pipes,

4th edn, Thomas Telford, London.

Hydraulics Research Limited (1983b)

CI,arts for the Hydraulic Des

ign of

Gllalln

e/s

and Pipes,

51h edn, Thomas Telford, London.

Moody,

L. F.

(1944)

Fri

ction factors for pipe flows.

Trans. Am. Soc.

Mew.

Engrs.,

Documents

Flow in Pipes and Closed Conduits