Turbomachinery

1

• Conservation Laws
• Bernoulli Equation

2
Eulerian Control Volume Approach

• Eulerian Lagrangian
• Extensive property dependent on mass in volume
  under consideration N
• Intensive property independent of mass in volume
  under consideration n

3
Lagrangian Approach

• Lagrangian Approach
• Follow fluid particle or droplet on path called
  path line. Fluid particle is an aggregate of
  discrete particles of fixed identity. Focus on
  particles as they move through the flow.
• Each particle is labeled by its original
  position.
• Examples are particle tracking for sprays and
  coatings, continuum mechanics, oceanographics
  flow meters drift along prevailing currents

4
Lagrangian Approach

• Scalar convection
• Substantial derivative for a coordinate system
  of an extensive property N
• Consider some arbitrary extensive property N of
  the fluid associated with the CV
• n N per unit mass and ?dV is the elemental mass
• Differentiate along particle path ? a
  Lagrangian process
• where the total or Stokes derivative reflects
  the change in N over time and space

5
Lagrangian Approach
6
Lagrangian Approach - Example
7
Eulerian Control Volume Approach

• Elemental Fixed Volume Cube

8
Eulerian Control Volume Approach

• Eulerian or Control Volume Approach
• Watch a fixed point in space, not one particle,
  as time proceeds.
• CV is designated in space and the bondary known.
  The amount and identity of the matter in CV may
  change with time, but shape is fixed.
• Property field, e.g. VvelocityV(x,y,z,t)
  streamline is defined.
• Path lines and streamlines are identical in a
  steady flow.

9
Scalar Conservation
Timet Timet?t
Time rate of change convection
10
Conservation Laws
11
Conservation Laws2D Steady Flow
p
No. equations 5 No. unknowns ?, u, v, w, p,
h0 Therefore need additional relations to close
system
12
Conservation Laws2D Steady Flow
13
Conservation Laws2D Steady Flow
u1
14
Bernoullis Equation
15
Bernoullis Equation
16
Inviscid Momentum Equation

• Neglecting other force terms (gravity, magnetism,
  etc.) on the flow, the inviscid, integral
  momentum equation is
• Basis of control volume approach to many problems
• Note - for steady flow calculate force on
  immersed object from flow variables on the
  surface of the control volume!!!!
• To solve unsteady and/or viscid flows must
  integrate throughout the volume - orders of
  magnitude more difficult!

17
Steady Inviscid Momentum Equation

• Integral form of Inviscid Momentum Equation
• is outward normal from the surface area. In
  2 dimensions

18
Steady Inviscid Momentum Equation
For cylindrical surface
y
r
?
x
19
Steady Inviscid Momentum Equation

• Substituting into vector equation
• Writing this as two scalar equations

20
Steady Inviscid Momentum Equation

• Pressure force is positive to right, acceleration
  is positive to right, so

21
Steady Inviscid Momentum Equation

• Examples
• Circular cylinder in flight Compressible flow
  homework
• Circular cylinder in duct Compressible flow
  homework
• Jet Engine In Flight

22
Inviscid Momentum Equation

• Example Application to Jet Engine in Flight
• AB 0 cancels IJ
• BC 0
• CD 0 cancels EF
• DE 0
• EF 0 cancels CD
• FG 0
• GH 0
• HI 0
• IJ 0 cancels AB
• JA 0

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Inviscid Momentum Equation

• Summing terms
• Jet engine control volume chosen to eliminate
• Same result for any control volume fully
  enclosing the engine
• Generally cannot eliminate for
  internal flows

24
Uninstalled (Ideal) Thrust

• So From Force Considerations (Control Volume
  Analysis), the Uninstalled (Ideal) Thrust for an
  Engine is
• Why now pressure term?

25
Specific Fuel Consumption

• The Rate of Fuel Used by the Propulsion System
  per Unit of Thrust
• uninstalled
• installed
• So

26
3D Steady Flow Energy Equation
If flow velocity brought to zero adiabatically,
apply Gibbs equation at stagnation properties
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Other Important Equations