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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