1. Turbomachinery Lecture 5a Airfoil, Cascade Nomenclature
Frames of Reference
Velocity Triangles
Euler’s Equation

2. Airfoil Nomenclature 
Chord: c or b = xTE-xLE; straight line connecting leading edge and trailing edge
Camber line: locus of points halfway between upper and lower surface, as measured perpendicular to mean camber line itself
Camber: maximum distance between mean camber line and chord line
Angle of attack: , angle between freestream velocity and chord line
Thickness t(x), tmax 

3. Frame of Reference Definitions

4. Cascade Geometry Nomenclature
s pitch, spacing laterally from blade to blade
 solidity, c/s = b/s
 stagger angle; angle between chord line and axial
1 inlet flow angle to axial (absolute)
2 exit flow angle to axial (absolute)
’1 inlet metal angle to axial (absolute)
’2 exit metal angle to axial (absolute)
 camber angle ’1 - ’2
turning 1 - 2
Concave Side
-high V, low p
- suction surface
Convex Side
-high p, low V
- pressure surface b bx
Note: flow exit angle does not equal
exit metal angle
Note: PW angles referenced to normal
not axial

5. Compressor Airfoil/Cascade Design
Compressor Cascade Nomenclature:
Camber - "metal" turning
Incidence  +i more turning
Deviation  + less turning
Spacing or Solidity

6. Velocity Diagrams Apply mass conservation across stage
UxA = constant, but in 2D sense
Area change can be accomplished only through change in radius, not solidity.
In real machine, as temperature rises to rear, so does density, therefore normally keep Ux constant and then trade  increase with A decrease
same component in absolute or relative frame
Rotational speed is added to rotor and then subtracted
If stage airfoils are identical in geometry, then turning is the same and
V1 = V3

7. Velocity Diagrams Velocity Scales For axial machines
Vx = u >> Vr
For radial machines
Vx << Vr at outer radius but Vx may be <<
or >> Vr at inner radius
Velocity Diagrams
Velocity Diagram Convention
Objectives: One set of equations Clear relation to the math
Conclusion: Angles measured from +X Axis
U defines +Y direction
Cx defines +X direction

8. Velocity Diagrams: Compressor and turbine mounted on same shaft
Spinning speed magnitude and direction same on both sides of combustor
Suction [convex] side of turbine rotor leads in direction of rotation
Pressure [concave] side of compressor rotor leads in direction of rotation

9. Frames of Reference

10. Velocity Diagrams:
Another commonly seen view

11. Axial Compressor Velocity Diagram:3 N 2 1

13. Turbine Stage Geometry Nomenclature

14. Relative = Absolute - Wheel Speed1
Rotor (Blade) 3
Stator (Vane) 2

16. Analysis of Cascade ForcesFy Fx

17. Analysis of Cascade Forces
Conservation mass, momentum

18. Analysis of Cascade Forces

19. Analysis of Cascade Forces
L, D are forces exerted by blade on fluid: Fy  Fx L D

20. Another View of Turbine Stage

21. Relative = Absolute - Wheel Speed1
Rotor (Blade) 3
Stator (Vane) 2

22. Combined Velocity Diagram of Turbine Stage
Work across turbine rotor
Across turbine rotor

23. Effect on increased m

24. Reason for including IGVs

25. Euler’s Compressor / Turbine Equation
Work = Torque X Angular Velocity
Angular Velocity of the Rotor
Torque About the Axis of the Rotor
B & D, integer # of blades pitches apart
 Identical flow conditions along B & D

26. Euler’s Equation
Only tangential force produces the torque on the rotor. By the momentum equation:
Since flow is periodic on B & D the pressure integral vanishes :

27. Euler’s Equation Moment of rate of Tangential Momentum is Torque []:
rate of work = F x dU = F x rd = [angular momentum][]
torque vector along axis of rotation
Work rate or energy transfer rate or power:
Power / unit mass = H = head
1st Law:

28. Euler’s Equation Euler's Equation Valid for:
Steady Flow
Periodic Flow
Adiabatic Flow
Rotor produces all tangential forces
Euler's Equation applies to pitch-wise averaged flow conditions, either along streamline or integrated from hub to tip.

29. Euler’s Equation
Euler Equation applies directly for incompressible flow, just omit “J” to use work instead of enthalpy:

30. Compressor Stage Thermodynamic and Kinematic View

31. Compressor Stage Thermodynamic and Kinematic View
Variable behavior
- P0, T0, K.E.

32. Compressor Stage Thermodynamic and Kinematic View
Across rotor, power input is
Across stator, power input is
From mass conservation, and if cx = constant, then
Euler’s equation

33. Analysis of Stage Performance
Geometry = velocity triangles
Flow = isentropic relations [CD]
Thermodynamics =Euler eqn., etc.
All static properties independent of frame of reference
All stagnation properties not constant in relative frame

34. Compressor Stage Thermodynamic and Kinematic View
Euler’s equation continued
Large turning (1 - 2) within rotor leads to high work per stage, but this is in reality limited by boundary layer effects
for constant U, the work per pound of air decreases linearly with increasing mass flow rate. Thus slight increases in m leads to decreased W, decreased pressure ratio leading to lower m

35. Compressor Stage Thermodynamic and Kinematic View
Stage pressure ratio is

36. Turbomachinery Lecture 5b Flow, Head, Work, Power Coefficients
Specific Speed

37. Work Coefficient Define Work Coefficient:
Applying Euler's Equation to E

38. Work Coefficient

39. Work Coefficient
This equation relates 2  terms to velocity diagrams and applies to both compressors & turbines. The physics, represented by Euler’s Equation, matches the implications of Dimensional Analysis.

41. Work and Flow Coefficients
Example:
Solution:

42. Work and Flow Coefficients
Solution continued: W1 C1 U
Cx1 1 1

43. Work and Flow Coefficients
Note:
Similar velocity triangles at different operating conditions will give the
same values of E (work) and  (flow) coefficient
Since angles stay the same
and Cx/U ratio stays the same,
E is the same
W1A 1 1
C1A
Cx1 UA UB

44. Work and Flow CoefficientsPr
Flow, Wc A E 
A,B B B1 Pr
Flow, Wc E  B1 B2 A1 B2 A1
Nc1 A2 A2
Nc2

45. Work and Flow Coefficients
Effect on velocity triangles
Low E High E
W1A
C1A
Cx1 1 1
W1A
C1A
Cx1 1 UA 1 UA

46. Work and Flow Coefficients
Effect on velocity triangles of varying E = (cu2 - cu1)/U is design
low E results in low airfoil cambers
high E results in higher cambers
Effect of varying  = cx/U in design
low  results in flat velocity triangles, low airfoil staggers, and low airfoil cambers
high  results in steep velocity triangles, higher airfoil staggers, and higher airfoil cambers
Prove these statements by
sketching compressor stage and
sketching corresponding 3 sets of velocity triangles

47. Nondimensional Parameters

48. Dimensional Analysis of Turbomachines

49. Returning to Head Coefficient
Also "Head" is P/ (Previously shown), P2 can be a pressure coefficient. Incompressible form:
Compressible form:
Remembering compressor efficiency definitions, for incompressible flow:

50. Power Coefficient

51. Power Coefficient Power Coefficient =
Head Coefficient * Flow Coefficient

52. Returning to Head Coefficient
Now that has been shown to be corrected speed, return to

53. Flow and Head Coefficients
Many compressor people use  &  to represent stage performance scaled to design speed.
where "des" refers to the design point corrected flow etc. for the stage.

54. Specific Speed
Ns is a non-dimensional combination of so that diameter does not appear.