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10 Analysis of pipe flows

The Moody chart

Experiments were carried out to plot f against ReD and/D.This results in a chart which depict lines of constant /D in

a f-ReD plots. This is the well known Moody chart as

shown in Figure 8.20

Important features of the Moody chart are:

1. Both axes are in log scale.2. Laminar flow region is found in Re < 2000,which

represent a straight (in log plot) for all roughness.

3. The transition region is found for Re between 2000and 4000. Here the flow may jump betweenlaminar line and one of the turbulent lines.

4. For Re > 4000 the flow is turbulent and f dependson the roughness ratio /D.

5. There is an envelope where the lines on its righthand side are horizontal. That implies that for each

/D the value of f is constant for further increase in

Re.

.

In the design of pipe

flow it is best to stay

out of the transition

region.

If the flow is

turbulent, it is prefer

to have the pipe

operates at the right

hand side of the

envelope. In this

case the pipe flow

will be steady andeasier for the pump

to maintain the flow.

Note that for smooth

pipe, /D=0, the linenever approaches

horizontal.

The following equations apply to turbulent pipe flow:

( ) 212

p Df

V

=

l

( ) 21* 2L

Vp fh

g D g

= =

l

1/

1

n

c

u r

V R

=

( ) ( )* *

2 10 Lp p g h=

.


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ENG243 Fluid Mechanics 2010 page: 2

The Colebrook formula

Using Moody chart to read off the value of f was done in

the pre-calculator days. There are many equations that

curve-fitting the Moody chart and allow more accurate

value of f to be calculated. Amongst these, the Colebrook

formula is believed to give the best result.

10

1 / 2.512.01

3.7log

D

D

f Re f

= +

note that f appears on both sides of the equation hence an

iteration process is required to obtain its value.

To begin with the iteration, the first value of f is derived

from the Miller formula:2

0 10

/ 5.740.25

3.7log

0.9

D

Df

Re

= +

or from Eqn 8.35b

1.11

10

1 / 6.91.8

3.7log

D

D

Ref

= +

Note that the

logarithm is base 10

Actually f from one

of these two

equations is

sufficient in most

practical purposes.

The iteration process begins with a given value of ReD and

/D.

The first estimated value of f is derived from Eqn 8.35b orMiller formula.

This value of f is then substituted into the right hand side of

the Colebrook formula so that the f in the left hand side is

the updated value.

We continue to use the Colebrook formula in a loop until a

desirable accuracy is achieved.

Three iterations will

produce sufficient

accuracy in most

situations.

Minor Losses -- The Loss Coefficient -- K

There are losses of total pressure (related to energy) for

flow over various pipe connections such as pipe bends,

nozzles, pipe junctions, diffusers, ... etc.

For a drop of total pressure (p) over these rather short

connections, we define a loss coefficient, K, such that:

21

2p K V = and

2

.minor2

L

Vh K

g=

In the case where the velocities are different between the

upstream and downstream station, the value of K will

depend on the choice of the velocity in one of the two

sides. For most situation, the higher velocity is chosen to

define K

Section 8.4.2

Here V is the

average velocity;

that is Q/A.


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Consider a pipe system as shown above. Let the subscripts

of the minor lost coefficient Kn be related to the numeric as

indicated in the diagram and let the lengthab

l be the length

of the pipe between (a) and (b).

The energy equation is given by:

( )* * 2 2121 2 . .minor 1 2L major Lp p gh gh V V o + =

where

( ) [ ]

.

212 1 1 3 3 4 4 5 5 6 6 7 7 8 /

L majorgh

V f D

=

+ + + + +l l l l l land

( ) [ ]212.minor 1 1 3 4 5 6 78L Lgh V K K K K K K K = + + + + + +

the total head loss in this system is:

( ) ( ) ( )* * 2 2121 2 . .minor 1 2/ /L major LH p p g h h V V g = = +

the power consumed by the system is: ( )Q g H Watt

Example 8.8

Note that KL is based

on the pipe velocity

V1.

K1 is referring to the

entrance loss (if

any).

Velocities appear in

the energy equation

because the velocity

at inlet and exit are

not the same.

Piezometricpressures are use in

the equation.

The problem will be

simpler if the friction

factor, f, is given.

On the other hand,

the pipe roughness is

required and

together with Q (to

get Re) to allow f beevaluated from the

Colebrook formula.

Section 8.4.2 contains values of K for various pipefittings

and configurations.

The minor loss coefficient, K, can be converted to an

equivalent length using the relationship:

/e

K f D= l that is: /e

KD f=l

Hence the minor losses in the problem above may be

written in terms of equivalent lengths:

( ) [ ]212.minor 1 1 3 4 5 6 78 /L Lgh V f D = + + + + + +l l l l l l l

The equivalent

length is meaningful

if the system has a

constant pipe

diameter.


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Section 8.5 illustrates a wide range of pipe flow

applications.

Section 8.5; 8.5.1

Multiple pipe systems

(1) Pipes in series

For pipe of different diameters connected in series, there

will be minor losses at each pipe junction. These minor

loss coefficients can be found from commercial catalogues

for pipefittings and these coefficients may be base on the

upstream velocity of the junction or downstream.

Lets assume that the minor losses of pipe junction are

based on the upstream velocity. Let n be the subscript that

denotes the properties associate with the nth pipe. For

three in series as shown below the total head loss (major +

minor) is expressed as:

( ) [ ]

( ) [ ] ( ) [ ]

212 1 1 1 1 1 2

2 21 12 22 2 2 2 2 3 3 3 3 3

/ /

/ / / /

LH V f D K g

V f D K g V f D g

= +

+ + +

l

l l

and again there is a mass conservation equation:

( ) ( ) ( )2 2 24 4 41 1 2 2 3 3D V D V D V = +

(2) Pipe in parallel as shown above

In this case, the head loss for all the three pipes are the

same and all equal toA B

z z (i.e., pressure at A & B are

atmospheric). Hence:

[ ]21

/2

L n n n n n entrance n exith V f D K K

g

= + +l

for n = 1, 2 & 3

Section 8.5; 8.5.1

The energy equation

between A and B is

given by:

2 2

2 1

*

1

2L

p

g

V V Hg

=

+

neglecting theentrance and exit

losses.

Note that velocities

at A and B are zero.


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(3) Y- connections Figure 8.36

Here the flow in pipe 1 is spilt into pipes 2 & 3. Strictly

there will be minor loss between the flow from pipe 1 to

pipe 2 and from pipe 1 to pipe 3. However if these pipes

are very long the minor losses are small in comparison withthe major losses and hence neglected in this study.

The first equation is the mass conservation: 1 2 3Q Q Q= +

Nest is to consider the head loss equations. There are two

loops: from A to B via pipes 1 & 2 and from A to B via

pipes 1 & 3. These two head loss equations are:

2 2

1 1 1 2 2 2

1 2

02 2

A BA B

p p V f V fz z

g g g D g D

+ + + =

l land

22

3 2 31 1 1

1 3

02 2

A BA B

V fp p V fz z

g g g D g D

+ + + =

ll

Rewrite the conservation equation as:

1 1 2 2 3 3AV A V AV= +

These three equations enable the solution for the three

unknown velocities.

Section 8.5.2

Note that here we

write out pressureand height instead of

the piezometric

pressure.

Usually the pressure

at A and B are

atmospheric and are

set to zero.

If the friction

coefficients are not

given but have to bederived from

Colebrook formula

through the use of

pipe roughness, the

problem becomes

more difficult to

solve.


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Pipe flow rate measurement

One of the traditional methods of measure flow rate in pipe

is to use an obstruction in the pipe and calibrate the flow

rate against the pressure drop over the obstruction.

Flow through an obstruction, as shown above, is

asymmetric: On the contraction side (side 1) the flow

follows the boundary. However, at the expansion side

(side 2) the flow separates from the wall and become a jetwithin the fluid in the pipe. Hence that is a large loss of

pressure head over the relatively short section.

The most common method is the use of an orifice plate.

The orifice plate is a simple device and can be manufacture

in a mechanical workshop. The plate has a concentric

circular hole. The plate is sharpened as shown above. In

according to flow measurement standard, pressure tappings

are located at D upstream and D/2 downstream of the plate.

The volume flow rate is related to the difference of

piezometric pressure by the formula: ( )2 *Q CA g p=

and the calibration constant, C, is given by:

2.52.1 8

0.75

91.710.5959 0.0312 0.184

Red

C

= + +

where= (d/D)2

Section 8.6

The exit flow will

follow the wall if the

wall expands at an

angle not more than

70

as in the case of

venturi meter.

This type of flow

measurement is

economical to make

but it may incurunacceptable loss of

pressure head.

Detail for flow

measurement in pipe

flow can be found in

Australian standardAS2360 and British

standard BS1042

D

D/2

d

Flow

D

p*

D


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The Venturi meter

The venturi meter is well contoured such that flow are

contracted and expanded with very little head loss.

However there is a pressure drop between the inlet section

and at the throat of the venturi. This is due to the increase

of kinetic energy. For an ideal case with no head loss, the

piezometric pressure drop between the inlet and the throat

is given by: ( )2

2 2 221 1* 12 2

dd D D

D

Vp V V VV

= =

As

2

d

D

V D

V d

=

hence

42

4

1* 1

2D

Dp V

d

=

the flow rate ( ) 24 dQ D V=

In practice, there may be a need for a discharge coefficient,

C, to cover all the losses etc. Hence:actual

Q CQ= .

The venturi meter

incurs very little

head loss to the flow.

However, it takes up

more space which

may not be practicalin many cases. In

addition it cost much

more than an orifice

plate.

The Rotameter

The rotameter is an ingenious device. It directs flow

vertically upward along a slightly diverging cylinder. A

float will be suspended in the cylinder at a position where

the fluid momentum on it is balanced by its weight. See

Figure 8.48. The height of the float indicates the flow rate.

Other flow meters and sensors

Recently many elector-mechanical flow meters have been

developed. In general the flow rate are calibrated with

voltage output of the device and can be interfaced with

computer data logging systems.

2

4

4

2 *

41

pQ D

D

d

=


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