1 Turbomachinery Lecture 5 Airfoil, Cascade NomenclatureFrames 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 Nomenclatures 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 DesignCompressor 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 machinesCx = 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 shaftSpinning 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. eFlip 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 ForcesConservation mass, momentum

19 Analysis of Cascade Forces

20 Analysis of Cascade ForcesL, 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 StageWork across turbine rotor Across turbine rotor

24 Effect on increased m

25 Reason for including IGVs

26 Euler’s Compressor / Turbine EquationWork = 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 ViewVariable behavior - P0, T0, K.E.

34 Axial Compressor Velocity Diagram:3 N 2 1

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

36 Compressor Stage Thermodynamic and Kinematic ViewEuler’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 ViewEuler’s equation

38 Compressor Stage Thermodynamic and Kinematic ViewStage 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.

42

43 Work and Flow CoefficientsExample: Solution:

44 Work and Flow CoefficientsSolution continued: W1 C1 U Cx1 1 1

45 Work and Flow CoefficientsNote: 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 CoefficientsEffect on velocity triangles Low E High E W1A C1A Cx1 1 1 W1A C1A Cx1 1 UA 1 UA

48 Work and Flow CoefficientsEffect 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 CoefficientAlso "Head" is P/ (Previously shown), P2 can be a pressure coefficient. Incompressible form: Compressible form: Remembering compressor efficiency definitions, for incompressible flow: