lecture_notes_02 (1).pdf

8/11/2019 lecture_notes_02 (1).pdf

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Prof. Dr. Atl BULU1

Chapter 2

Flows under Pressure in Pipes

If the fluid is flowing full in a pipe under pressure with no openings to the atmosphere, itis called pressured flow. The typical example of pressured pipe flows is the water

distribution system of a city.

2.1. Equation of Motion

Lets take the steady flow (du/dt=0) in a pipe with diameter D. (Fig. 2.1). Taking a

cylindrical body of liquid with diameter r and with the length x in the pipe with thesame center, equation of motion can be applied on the flow direction.

Figure 2.1.

The forces acting on the cylindrical body on the flow direction are,

a) Pressure force acting to the bottom surface of the body that causes the motion of

the fluid upward is,

F1= Pressure force = ( ) 2rpp +

D

rF1

F2

xr 2

x

y

x

Flow


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Prof. Dr. Atl BULU2

b) Pressure force to the top surface of the cylindrical body is,

F2= Pressure force =2

rp

c) The body weight component on the flow direction is,

sin2 xrX =

d) The resultant frictional (shearing) force that acts on the side of the cylindricalsurface due to the viscosity of the fluid is,

Shearing force = xr2

The equation of motion on the flow direction can be written as,

( ) =+ xrxrrprpp 2sin222 Mass Acceleration (2.1)

The velocity will not change on the flow direction since the pipe diameter is kept

constant and also the flow is a steady flow. The acceleration of the flow body will be

zero, Equ. (2.1) will take the form of,

02sin22 = xrxrpr

rx

p

= sin

2

1 (2.2)

The frictional stress on the wall of the pipe 0with r = D/2,

2sin

2

10

D

x

p

= (2.3)

We get the variation of shearing stress perpendicular the flow direction from Equs.(2.2) and (2.3) as,


3/24

Prof. Dr. Atl BULU3

20D

r= (2.4)

Fig. 2.2

Since r = D/2 y,

=

210

D

y (2.5)

The variation of shearing stress from the wall to the center of the pipe is linear as can

be seen from Equ. (2.5).

2.2. Laminar Flow (Hagen-Poiseuille Equation)

Shearing stress in a laminar flow is defined by Newtons Law of Viscosity as,

dy

du= (2.6)

Where = (Dynamic) Viscosity and du/dy is velocity gradient in the normal directionto the flow. Using Equs. (2.5) and (2.6) together,

r

y

0

y

D/2


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5/24

Prof. Dr. Atl BULU5

=

==

22

2

1

2

22

rD

Dy

Duu

rD

yyD

r

= 2

22

4r

D

D

uu

(2.10)

Equ. (2.10) shows that velocity distribution in a laminar flow is to be a paraboliccurve.

The mean velocity of the flow is,

A

udA

A

QV A

==

Placing velocity equation (Equ. 2.10) gives us the mean velocity for laminar flows as,

8

2

=Du

V (2.11)

Since

02 =u

And according to the Equ. (2.3),

2sin

2

10

D

x

p

=


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Prof. Dr. Atl BULU6

2sin

2

12 D

x

pu

=

Placing this to the mean velocity Equation (2.11),

=

sin

32

2

x

pDV (2.11)

We find the mean velocity equation for laminar flows. This equation shows that

velocity increases as the pressure drop along the flow increases. The discharge of the

flow is,

VD

AVQ4

2==

=

sin

128

4

x

pDQ (2.12)

If the pipe is horizontal,

x

pDQ

=

128

4

(2.13)

This is known as Hagen-Poiseuille Equation.

2.3. Turbulent Flow

The flow in a pipe is Laminar in low velocities and Turbulent in high velocities.

Since the velocity on the wall of the pipe flow should be zero, there is a thin layer

with laminar flow on the wall of the pipe. This layer is called Viscous Sub Layer andthe rest part in that cross-section is known as Center Zone. (Fig. 2.3)


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Prof. Dr. Atl BULU7

Fig. 2.3.

2.3.1. Viscous Sub Layer

Since this layer is thin enough to take the shearing stress as, 0and since the flow

is laminar,

dyudu

udy

du

2

2

0

=

===

By taking the integral,

consyuu

dyuu

+=

=

2

2

Since for y =0 u = 0, the integration constant will be equal to zero. Substituting

=/gives,

yu

u

2

= (2.14)

Viscos Sublayer

Center Zone

y

0


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Prof. Dr. Atl BULU8

The variation of velocity with y is linear in the viscous sub layer. The thickness of the

sub layer () has been obtained by laboratory experiments and this empirical equation

has been given,

=u

6.11 (2.15)

Example 2.1. The friction velocity u*= 1 cm/sec has been found in a pipe flow with

diameter D = 10 cm and discharge Q = 2 lt/sec. If the kinematic viscosity of the liquid

is= 10

-2

cm

2

/sec, calculate the viscous sub layer thickness.

mmcm

u

2.112.0

1

106.11

6.11

2

==

=

=

2.3.2 Smooth Pipes

The flow will be turbulent in the center zone and the shearing stress is,

( )vudy

du+= (2.16)

The first term of Equ. (2.16) is the result of viscous effect and the second term is the

result of turbulence effect. In turbulent flow the numerical value of Reynolds Stress)( vu is generally several times greater than that of( )dydu . Therefore, the

viscosity term ( )dydu may be neglected in case of turbulent flow.

Shearing stress caused by turbulence effect in Equ. (2.16) can be written in the similar

form as the viscous affect shearing stress as,

dy

duvu T == (2.17)


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Prof. Dr. Atl BULU9

Here Tis known as turbulence viscosityand defined by,

dy

dulT

2 = (2.18)

Here l is the mixing length. It has found by laboratory experiments that l = 0.4y for

0 zone and this 0.4 coefficient is known as Von Karman Coefficient.

Substituting this value to the Equ. (2.18),

y

dyudu

u

du

y

dy

dyduyu

dy

duyu

dy

duy

dy

du

dy

duy

T

T

=

=

=

==

==

=

5.2

4.0

4.0

16.0

16.0

16.0

2

220

2

2

0

2

Taking the integral of the last equation,

consLnyuu

y

dyuu

+=

=

5.2

5.2 (2.19)

The velocity on the surface of the viscous sub layer is calculated by using Equs.(2.14) and (2.15),


10/24

Prof. Dr. Atl BULU10

=

==

=

uu

uy

yu

u

6.11

6.11

2

Substituting this to the Equ. (2.19) will give us the integration constant as,

=

+=

=

=

uLnuucons

uLnLnuucons

uLnuucons

Lnyuucons

5.25.5

6.115.26.11

6.115.26.11

5.2

Substituting the constant to the Equ. (2.19),

+=u

LnuuLnyuu

5.25.55.2

5.55.2

5.55.2

+=

+=

yuLn

u

u

uyu

Lnuu

(2.20)

Equ. (2.20) is the velocity equation in turbulent flow in a cross section with respect to

y from the wall of the pipe and valid for the pipes with smooth wall.

The mean velocityat a cross-section is found by the integration of Equ. (2.20) for areA,


11/24


A

udA

A

QV

A

==

75.15.2

75.12

5.2

+=

+=

DuLn

u

V

uDu

LnV

(2.21)

2.3.3. Definition of Smoothness and Roughness

The uniform roughness size on the wall of the pipe can be e as roughness depth. Most

of the commercial pipes have roughness. The above derived equations are for smoothpipes. The definition of smoothness and roughness basically depends upon the size of

the roughness relative to the thickness of the viscous sub layer. If the roughnesess are

submerged in the viscous sub layer so the pipe is a smooth one, and resistance and

head loss are entirely unaffected by roughness up to this size.

Fig. 2.4

Since the viscous sub layer thickness () is given by,

=u

6.11 pipe roughness size

e is compared with to define if the pipe will be examined as smooth or rough pipe.

Flow

e

Pipe Center

Pipe wall


12/24


a)

=u

e 70

The height of the roughness e is higher than viscous sub layer. The flow in the center

zone will be affected by the roughness of the pipe. This flow is named as Wholly

Rough Flow.

c)


13/24


305.25.2

305.2

305.20

eLnuLnyuu

eLnuconst

conste

Lnu

=

=

+=

The velocity distribution at a cross section for wholly rough pipes is,

e

yLn

u

u

e

yLnuu

305.2

305.2

=

=

(2.22)

The mean velocity at that cross section is,

73.42

5.2

73.42

5.2

+=

+=

e

DLn

u

V

e

DLnuV

(2.23)

2.4. Head (Energy) Loss in Pipe Flows

The Bernoulli equation for the fluid motion along the flow direction between points(1) and (2) is,

Lhg

Vpz

g

Vppz +++=+

++

22

2

2

2

2

1

1

(2.24)

If the pipe is constant along the flow, V1= V2,

( )12 zzp

hL

=

(2.25)


14/24


Figure 2.5

If we define energy line (hydraulic) slope J as energy loss for unit weight of fluid

for unit length,

x

hJ L

= (2.26)

Where x is the length of the pipe between points (1) and (2), and using Equs. (2.25)

and (2.26) gives,

sin

12

=

=

x

pJ

x

zz

x

pJ

(2.27)

g

V

2

2

pp +

hL

p

Z1

Z2

Horizontal Datum

Energy Line

H.G.L

Flow

1

2

x


15/24


Using Equ. (2.3),

JD

x

pD

D

x

p

4

sin4

2sin

2

1

0

0

0

=

=

=

(2.28)

Using the friction velocity Equ. (2.8),

JgD

JD

u

u

u

44

2

2

0

0

==

=

=

gD

uJ

24= (2.29)

Energy line slope equation has been derived for pipe flows with respect to friction

velocity u*. Mean velocity of the cross section is used in practical applications insteadof frictional velocity. The overall summary of equational relations was given in

Table. (2.1) between frictional velocity u* and the mean velocity V of the cross

section.


16/24


Table 2.1. Mathematical Relations between u*and V

Laminar Flow

(Re2000)

Smooth Flow

u

e 70

+= 73.4

25.2

e

DLnuV

After calculating the mean velocity V of the cross-section and finding the type of low,

frictional velocity u*is found out from the equations given in Table (2.1). The energyline (hydraulic) slope J of the flow is calculated by Equ. (2.29). Darcy-Weisbach

equation is used in practical applications which is based on the mean velocity V to

calculate the hydraulic slope J.

g

V

D

fJ

2

2

= (2.30)

Where f is named as the friction coefficient or Darcy-Weisbach coefficient. Friction

coefficient f is calculated from table (2.2) depending upon the type of flow where

VDVD==Re .


17/24


Table 2.2. Friction Coefficient Equations

Laminar

Flow(Re2000)

Smooth Flow

ue

70 73.4

25.2

8+

=

e

DLn

f

Transition Flow


18/24

Fritur

dia

sho

Su

2. For tra

rough

3.

For w(ff =

tion coeffiulent flow

ram has b

s the func

mary

a) Energy

For a pipe

sition flo

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olly roug)De

cient f is c, the calcul

en prepare

tional relat

loss for un

ith length

s, f depend

ipe (e/D),

flows,

alculated fation of f

d to overc

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t length of

L, the ener

18

s on Reyn

(Re,ff =

is a fun

om the eq

will alwayme this di

n f and Re,

Figure 2

pipe is calc

D

fJ =

gy loss will

lds numbe

)D

ction of t

uations gi

s be doneficulty. It i

e/D as cur

.6

ulated by

g

V

2

2

be,

(Re) of th

he relativ

en in Tabl

by trial ans prepared

es. (Figur

arcy-Weis

Prof. Dr. A

flow and

roughnes

e (2.2). In

d error meby Nikura2.6)

ach equati

tl BULU

elative

s (e/D)

case of

thod. A

dse and

on,


19/24


JLhL = (2.31)

b) The friction coefficient f will either be calculated from the equations given in

Table (2.2) or from the Nikuradse diagram. (Figure 2.6)

2.5. Head Loss for Non-Circular Pipes

Pipes are generally circular. But a general equation can be derived if the cross-section

of the pipe is not circular. Lets write equation of motion for a non-circular prismatic

pipe with an angle of to the horizontal datum in a steady flow. Fig. (2.7).

Figure 2.7

( ) onacceleratiMassxAxPpAApp =+ sin0

Where P is the wetted perimeter and since the flow is steady, the acceleration of theflow will be zero. The above equation is then,

=

=

sin

0sin

0

0

x

p

P

A

xAxPpA

(2.32)

A

xAW =

0

+p

p

Flow

P=Wetted

Perimeter

x


20/24


Where,

P

AR= = Hydraulic Radius (2.33)

Hydraulic radius is the ratio of wetted area to the wetted perimeter. Substituting this

to the Equ. (2.32),

=

=

sin

sin

0

0

x

pR

xpR

Since by Equ. (2.27),

sin

=

x

pJ

Shearing stress on the wall of the non-circular pipe,

RJ =0 (2.34)

For circular pipes,

RD

D

D

D

P

AR

44

42

=

===

(2.35)

This result is substituted (D=4R) to the all equations derived for the circular pipes to

obtain the equations for non-circular pipes. Table (2.3) is prepared for the equations

as,


21/24


Table 2.3.

Circular Pipes Non-Circular pipes

JD

40 =

g

V

D

fJ

2

2

=

( )eDff Re,=

VD=Re

RJ =0

g

V

R

fJ

24

2

=

( )eRff 4Re,=

RV4Re=

2.6. Hydraulic and Energy Grade Lines

The terms of energy equation have a dimension of length [ ]L ; thus we can attach a

useful relationship to them.

Lhzg

Vpz

g

Vp+++=++ 2

2

22

1

2

11

22 (2.36)

If we were to tap a piezometer tube into the pipe, the liquid in the pipe would rise in

the tube to a height p/ (pressure head), hence that is the reason for the name

hydraulic grade line (HGL).The total head

++ z

g

Vp

2

2

in the system is greater

than

+z

p

by an amount

g

V

2

2

(velocity head), thus the energy (grade) line (EGL)

is above the HGL with a distanceg

V

2

2

.

Some hints for drawing hydraulic grade lines and energy lines are as follows.


22/24

1.

2.

3.

4.

5.

6.

7.

By definiti

the velocit

HGL andHead loss

downward

when a pu

rise in thethe pump.If energy i

EGL and

In a pipe o

the water i

used to locthe outlet e

the upstrea

For stead(diameter,

unit of len

HGL will

If a flow p

size, the veEGL and

because th

larger velo

If the HG

atmospheri

on, the EG

head. Th

GL will coor flow in

in the dire

p supplie

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abruptly t

GL will dr

channel w

the syste

ate the HGnd of a pip

end, whe

flow inoughness,

th will be

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ssage chan

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e head per

ity.

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22

Figure

L is positi

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here the pr

because

at certai, where th

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Fig.2.8).

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points in tliquid cha

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t has uniso on) alo

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along the

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will be l

p is ne

the HGL

ro, as in la

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Prof. Dr. A

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23/24


Figure 2.9

If the pressure head of water is less than the vapor pressure head of thewater ( -97 kPa or -950 cm water head at standard atmospheric pressure),cavitation will occur. Generally, cavitation in conduits is undesirable. It

increases the head loss and cause structural damage to the pipe from

excessive vibration and pitting of pipe walls. If the pressure at a section inthe pipe decreases to the vapor pressure and stays that low, a large vapor

cavity can form leaving a gap of water vapor with columns of water on

either side of cavity. As the cavity grows in size, the columns of watermove away from each other. Often these of columns of water rejoin later,

and when they do, a very high dynamic pressure (water hammer) can be

generated, possibly rupturing the pipe. Furthermore, if the pipe is thin

walled, such as thin-walled steel pipe, sub-atmospheric pressure can causethe pipe wall to collapse. Therefore, the design engineer should be

extremely cautious about negative pressure heads in the pipe.


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