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FULLY DEVELOPED PIPE AND

CHANNEL FLOWS

KUMAR DINKAR ANAND

3rd YEAR, MECHANICAL ENGG.

IIT-KHARAGPUR

GUIDANCE : PROF. S CHAKRABORTY

INDO-GERMAN WINTER ACADEMY-DECEMBER 2006

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: THE OUTLINE :

Hydraulically developing flow through pipes and channels and

evaluation of hydraulic entrance length. Hydraulically fully developed flows through pipes and channels .

Hydraulically fully developed flow through non-circular ducts.

Definition of Thermally fully developed flow and analysis of thermally

fully developed flow through pipe and channels.

Analysis of the problem of Thermal Entrance: The Graetz Problem.

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Fully Developed Flows

There are two types of fully developed flows :

1.) Hydraulically Fully Developed Flow

2.) Thermally Fully Developed Flow

Contd

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Hydraulically Fully Developed Flow

Definition: As fluid enters any pipe or channel , boundary layers keep on growing

till they meet after some distance downstream from the entrance region. After this

distance velocity profile doesn't change, flow is said to be Fully Developed.

Analysis of fluid flow before it is fully developed:

Velocity in the core of the flow outside the boundary layer increases with

increasing distance from entrance. This is due to the fact that through any cross

section same amount of fluid flows, and boundary layer is growing.

This means

hence

0>dxdU

0

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Schematic picture of internal flow through a pipe :

Velocity Profile ,Using the boundary conditions :

1.) At

2.) At

3.) At

We get the velocity profile as :

Contd

2

)( cybyayu ++=

=y Uu =

0=u

=y 0=dydu2

)()(2

)(

yyU

yu

=

0=y

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Where Free stream velocity of entering fluid

Free stream core velocity inside the tube

Core velocity of fully developed flow

Radius of pipe

Now from the principle of conservation of mass :

Hence ,

Contd

=U=U

+=

R

R

R

UrdrurdrRU

0

2

22*

rRy ==R=eU

2)/(6/1)/(3/21

1

RRU

U

+=

2

2

)/(6/1)/(3/21

)/()/(2

RR

yy

U

u

+

=

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+=

0 0

2 })/1()/1(/{ dyUudx

dUUdyUuUuUdxd

w

Boundary Layer momentum integral equation:

Where, Shear stress at wall,

From Bernoulli's Equation for free stream flow through core:

Using Navier-Stokes equation at the wall

Contd

0==

y

wy

u

x

p

dx

dUU

=

1

0

2

2

=

=

yy

u

x

p

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Solving for boundary layer thickness :

Integrate momentum Integral Equation

Using the boundary condition at

For determination of Entrance Length :

putting at

We get the expression for Entrance Length as:

Contd

)(

0= 0=x

)( eL

R=eLx =

)( eL

De

D

LRe03.0=

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Analytical expression for Entrance Length :

Hence it can be observed that our expression for Entrance Length differsfrom the analytical expression due to the following reasons:

1.) We have assumed parabolic velocity profile in the boundary layer

2.)We have not used the Navier-Stokes boundary equation at wall for velocity

profile determination

3.) We are doing boundary layer analysis which gives approximate results

Contd

)( eL

De

DL Re06.0=

2)/()/(2 yyUu =

0

2

2

==

yy

u

x

p

2)/(6/1)/(3/211

RRUU

+=

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Schematic picture of internal flow through a channel:

Velocity Profile

Using the boundary conditions :

1.) At

2.) At

3.) At

We get the velocity profile as :

Contd

2)( cybyayu ++=

0=y 0=u

=y Uu ==y 0=dydu

2)()(2)( yyU

yu =

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Here , Distance between the parallel plates of channel

Width of the Channel

Free stream velocity of entering fluid

Free stream velocity inside channel

Core velocity of fully developed flow

Entrance Length

Hydraulic Diameter

Contd

=D

=W

=U

=U

=eU

=eL

=HD DW

WD

P

AH 22

44 ===

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From the principle of conservation of mass:

Hence when flow is fully developed

Contd

+=

)2/(

00

22*

D

UdyudyDU

)/(3/21 1 DUU =

)/(3/21)/()/(2 2

D

yy

U

u

=

)2/( D=

= UUe 5.1

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From Boundary layer momentum integral equation :

Where, Shear stress at wall,

From Bernoulli's Equation for free stream flow through core:

Using Navier-Stokes equation at the wall

Contd

+=

0 0

2 })/1()/1(/{ dyUudx

dUUdyUuUuUdxd

w

0==

y

wy

u

x

p

dx

dUU

=

1

0

2

2

=

=

yy

u

x

p

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Solving for boundary layer thickness :

Integrate momentum Integral Equation

Using the boundary condition at

For determination of Entrance Length :

putting at

We get the expression for Entrance Length as:

OR

Contd

)(

0= 0=x

)(e

L

eLx = R=

)( eL

D

e

D

L

Re025.0=HD

H

e

D

L

Re00625.0=

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Analytical expression for Entrance Length :

Hence it can be observed that our expression for Entrance Length differs

from the analytical expression due to the following reasons:

1.) We have assumed parabolic velocity profile in the boundary layer

2.) We have not used the Navier-Stokes boundary equation at wall for

velocity profile determination

3.) We are doing boundary layer analysis which gives approximate results.

Contd

)( eL

De

D

LRe05.0=

2

)/()/(2 yyU

u

=

0

2

2

=

=

yy

u

x

p

)/(3/21

1

DU

U

=

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Analysis of fully developed fluid flow:

Fully Developed Flow Through a Pipe:

From Equation of continuity in cylindrical coordinates:

for an incompressible fluid flowing through a pipe

Contd

0)(1 =+

xuru

rrr

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Here, radial velocity

axial velocity

radius of pipe

No fluid property varies with ,

,at wall of the pipe

hence it is zero everywhere.

Hence Equation of continuity reduces to :

Momentum Equation in radial coordinate:

Contd

=ru=u

=a

0=r

u

,0=xu )(ruu =

,0=

r

p )(xpp =

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Momentum Equation in axial direction :

)( dr

du

rdr

d

rdx

dp

=

Solving above differential equation in (r) using the boundary conditions:

1.) Axial velocity (u) is zero at wall of pipe (r =R)

2.) Velocity is finite at the pipe centerline (r=0).

We get the fully developed velocity profile:

Contd

=

22

14 a

rxpau

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Shear Stress Distribution :

Shear stress ,

Maximum shear stress at wall ,

=

=

x

pr

dr

durx

2

=

x

pa

20

Contd

Hence it can be observed that

Shear stress decreases from

maximum to zero at pipe

centerline and then increases

to maximum again at wall.

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Volume Flow Rate :

volume flow rate ,

== xpaurdrQ

a

0

4

82

Now in a fully developed flow pressure gradient is constant ,

Hence ,( )

L

p

L

pp

x

p entexit

=

=

LpaQ 84

=

Contd

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Average Velocity :

Average velocity ,

=== x

pa

a

Q

A

QV 8

2

2

Maximum Velocity :

At the point of maximum velocity , 0=dr

du

This corresponds to core of pipe , 0=r

Hence VxpaUuu r 2

4

2

0max =

=== =

Contd

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Fully Developed Flow through Channel :

From equation of continuity within the entrance length : 0=

+

y

v

x

u

In entrance length boundary layers growing , 0xu 0v

It means flow is not parallel to walls in entrance region

Contd

)(

a

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Equation of Continuity for an incompressible fluid in fully developed region :

0=

x

u

)(yuu =

Momentum equation in y-direction (transverse direction) :

0=yp )(xpp =

Momentum equati