1. Turbomachinery Lecture 5 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. Frame of Reference Definitions

5. 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

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

7. 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 Cx 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
C1 = C3

8. Velocity Diagrams Velocity Scales For axial machines
Cx = u >> Cr
For radial machines
Cx << Cr at outer radius but Cx may be << or >> Cr 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

9. 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

10. Frames of Reference

11. Velocity Diagrams:
Another commonly seen view

12. Axial Compressor Velocity Diagram:3 N 2 1

13. Flip from previous page, i.e. rotor going DOWN

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

15. Turbine Stage Geometry Nomenclature

16. Flip from previous page, i. e
Flip from previous page, i.e. rotor going DOWN and rotor second airfoil row

17. Analysis of Plane Cascade Forces Sit on frame of airfoilFy Fx

18. Analysis of Cascade Forces
Conservation mass, momentum

19. Analysis of Cascade Forces

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

21. Another View of Turbine Stage

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

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

24. Effect on increased m

25. Reason for including IGVs

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

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

28. 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:

29. 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.

30. Euler’s Equation

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

32. Compressor Stage Thermodynamic and Kinematic View

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

34. Axial Compressor Velocity Diagram:3 N 2 1

35. Compressor Stage Thermodynamic and Kinematic View
Across rotor, power input is
Across stator, power input is
From mass conservation,

36. Compressor Stage Thermodynamic and Kinematic View
Euler’s equation
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

37. Turbine Stage Thermodynamic and Kinematic View
Euler’s equation

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

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

40. Work Coefficient

41. 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.

43. Work and Flow Coefficients
Example:
Solution:

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

45. 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

46. 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

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

48. 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

49. Nondimensional Parameters

50. Dimensional Analysis of Turbomachines

51. 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: