Turbomachinery Summary Equations

2 Important Equations

4 Gibbs Equation

Efficiency

7 Similarity – Compressible Flow

8 Total Pressure Mass Flow Parameter Defines common flow parameters. Corrected flow to standard day [eliminate effect of outside ambient conditions].

9 Frame of Reference Definitions

10 Cascade Geometry Nomenclature b bx 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 Note: flow exit angle does not equal exit metal angle Note: PW angles referenced to normal not axial Concave Side -high V, low p - suction surface Convex Side -high p, low V - pressure surface

11 Compressor Airfoil/Cascade Design Compressor Cascade Nomenclature: Camber - "metal" turning Incidence Deviation Spacing or Solidity

12 Axial Compressor Velocity Diagram: N

Frames of Reference 13

14 Analysis of Stage Performance – Compressor Rotor

15 Rotor (Blade) Stator (Vane) Relative = Absolute - Wheel Speed N

16 Relative Flow Conditions T 0R - changes with wheel speed across a rotor T 0 - no change with radius across a stator No work done by a stationary object! Using Isentropic relation between P & T, Ideal Exit Relative Total Pressure is: P 02Ri is ideal, assuming 100% efficiency

17 Reaction Definition: 1=rotor inlet 2=rotor exit 3=stator exit For axial machines For Cx constant

18 Relationships Between Work Coefficient, Flow Coefficient, Reaction and Flow Angles for Constant Cx, U Machines

Losses in Compressors and Turbines Compressors Turbines Alternative form 19

20 Impact of Deviation on Airfoil Shape Carter’s Rule for compressor cascades Carter’s Rule for turbine cascades Metal angle decreased to achieve design exit angle goals

21 Compressor Design for Lift, Min Loss, Max Range & Choke Margin Avoid flow reversal Ideal Avoid separation Avoid leading edge sep. bubble Can also view this in terms of p s /p 0

22 Compressor Loss Analysis - Lieblein's Dfactor Correlation of cascade data [Velocities in Relative Frame] or de Haller [ 0.72<W2/W1<1] Momentum thickness [  ] correlated to Dfactor [0 < Dfactor <.7] Loss coefficient related to cascade wake momentum thickness Efficiency related to loss coefficient

23 Turbine - Zweifel Coefficient Area  F  ideal Area  F  Solidity play important role in turbine efficiency: (1) spacing small, fluid gets maximum turning force with large wall friction forces; (2) spacing large, fluid gets small turning force with small wall friction losses.

Component Matching Criteria 24