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Chapter 5 Flow Analysis Using Control Volume (Finite Control Volume Analysis )

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Flow Analysis Using Control Volume ... non-deforming control volume ---Control volumes containing a gas turbine engine on an aircraft in flight, and ... – PowerPoint PPT presentation

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Title: Chapter 5 Flow Analysis Using Control Volume (Finite Control Volume Analysis )


1
Chapter 5 Flow Analysis Using Control Volume
(Finite Control Volume Analysis )
2
  • Many practical problems in fluid mechanics
    require analysis of the behavior of the contents
    of a finite region in space (a control volume).
  • for example,
  • to determine the amount of time to
    allow for complete
  • filling of a large storage tank.
  • to estimate of how much power it would
    take to move
  • water from one location to another
    at a higher
  • elevation and several miles away
    may be sought.

3
  • The bases of this analysis method are some
    fundamental principle of physics , namely ,
  • Conservation of mass
  • Newtons second law of motion , and
  • the first and second laws of
    thermodynamics.
  • The finite control volume formulas are easy to
    interpret physically and are not difficult to use.

4
5.1 Conservation of Mass-The continuity
Equation
  • 5.1.1 Derivation of the continuity Equation

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t t-dt t
t t tdt
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5.1.2 Fixed, non-deforming Control Volume
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5.1.3 Moving, Non-deforming Control Volume
  • Example of moving, non-deforming control volume
  • ---Control volumes containing a gas turbine
    engine on an
  • aircraft in flight, and gasoline tank of an
    automobile
  • passing.

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5.1.4. Deforming Control Volume
  • A deforming Control volume
  • Changing volume size control surface
    movement.

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5.2 Newtons Second Law --The linear momentum
and moment-of-momentum equations.
  • Newtons second law of motion for a system is

gtAny reference or coordinate system for which
this statement is true is called
inertial. A fixed coordinate system is
inertial. A coordinate system that moves in
a straight line with constant velocity and
is thus without acceleration is also
inertial.
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  • When a control volume is coincident with a system
    at an instant of time, the forces acting on the
    system and the forces acting on the contents of
    the coincident control volume are instantaneously
    identical, that is,

20
  • Furthermore, for a system and the contents of a
    coincident control volume that is fixed and
    non-deforming, the Reynolds transport theorem
    allows us to conclude that

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  • For a control volume that is fixed (inertial) and
    non-deforming, Eq.(5.19),(5.20), and (5.21)
    suggest that an appropriate mathematical
    statement of Newtons Second law of motion is
  • Eq(5.22) is the linear momentum equation for a
    fixed, non-deforming control volume.
  • Body force -- gravity only
  • Surface force -- exerted on the contents
    of the control
  • volume by material just outside the
    control volume
  • in contact with material just inside
    the control volume

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Several important notes
  • (1) 1-D flow problem when the flow is uniformly
    distributed
  • over a section of the C.S.
  • (2) Linear momentum is directional three
    orthogonal
  • coordinate directions.
  • (3) Flux term is linear momentum-
  • for Steady flow (In the
    textbook ,it is aussmed

  • Steady flow for the
    momentam problem)
  • (5) If control surface ? direction of flow
  • ?Surface force exerted at these locations by
    fluid outside the C.V. on fluid inside will be
    due to pressure.

26
  • (6) Uniform pressure on control volume
  • (7) Positive external force if the force is in
    the assigned
  • positive coordinate direction. Negative
    otherwise.
  • (8) Only external forces acting on the contents
    of the control
  • volume are considered in the linear
    momentum
  • equation.(Eq.5.22)

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  • For a system and
  • an inertial , moving , non-deforming control
    volume that are both coincident at an instant of
    time , the Reynolds transport theorem leads to
  • This is the linear momentum equation for an
    inertial, moving, non-deforming control volume
    that involves steady flow.

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5.3 First Law of Thermodynamics-The Energy
equation
  • 5.3.1 Derivation of the Energy Equation
  • The first law of thermodynamics for a system is,
    in word

----(5.56)
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  • Eq.(5.55) is valid for inertial and non-inertial
    reference system
  • For the control volume that is coincident
    with the system at an instant of time

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5.3.2 Application of the Energy Equation
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J/kg(N?m)/kg (kg?m/s?m)/kg m2/s2
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5.3.3 Comparison of the Energy Equation with
the Bornoulli Equation
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