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

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 

Frame of Reference Definitions

Frame of Reference Definitions

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

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

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

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

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

Frames of Reference

Velocity Diagrams: Another commonly seen view

Axial Compressor Velocity Diagram: 3 N 2 1

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

Relative = Absolute - Wheel Speed 1 Rotor (Blade) 3 Stator (Vane) 2

Turbine Stage Geometry Nomenclature

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

Analysis of Plane Cascade Forces Sit on frame of airfoil Fy Fx

Analysis of Cascade Forces Conservation mass, momentum

Analysis of Cascade Forces

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

Another View of Turbine Stage

Relative = Absolute - Wheel Speed 1 Rotor (Blade) 3 Stator (Vane) 2

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

Effect on increased m

Reason for including IGVs

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

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

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:

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.

Euler’s Equation

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

Compressor Stage Thermodynamic and Kinematic View

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

Axial Compressor Velocity Diagram: 3 N 2 1

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

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

Turbine Stage Thermodynamic and Kinematic View Euler’s equation

Compressor Stage Thermodynamic and Kinematic View Stage pressure ratio is

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

Work Coefficient

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.

Work and Flow Coefficients Example: Solution:

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

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

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

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

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

Nondimensional Parameters

Dimensional Analysis of Turbomachines

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: