1. 1 Turbomachinery Lecture 2b - Efficiency Definitions

2. 2

3. 3 Gas-Turbine Brayton Cycle Most Gas Turbines Use the Ideal Brayton Cycle as the Basis for Design –Isentropic compression (2 to 3) –Constant pressure heat addition (3 to 4) –Isentropic expansion (4 to 9) –Constant pressure heat rejection (9 to 2)

4. 4 Gas-Turbine Brayton Cycle For constant (or nearly constant) velocity Neglect fuel mass flow effect T S 3 2 4 9 Compressor Combustor Turbine Ambient

5. 5 Gas-Turbine Brayton Cycle T S 3 2 4 9 Compressor Combustor Turbine Ambient

6. 6 Compressor Adiabatic (Isentropic) Efficiency Usually in terms of stagnation properties, but in centrifugal and industrial machines  inlet use Po, exit Ps ss Poin Poout V 2 /2 If inlet/outlet K.E. small, h 02i h 02 h 01

7. 7 Compressor Efficiency Compressor Efficiency is a Function of –Compressor Pressure Ratio –Pressure Ratio of Each Stage –Isentropic Efficiency of Each Stage Consider the Special Case Where Each Stage Pressure Ratio and Each Stage Efficiency are the Same where where

8. 8 Compressor Adiabatic Efficiency Caution - Different pressure ratios are used in this definition –Usually P t exit and P t inlet (for total-to-total efficiency) Pr = P 02 /P 01 –Sometimes P s exit is used when considering total-to- static efficiency –There is no “right” definition of efficiency –Only the ideal work is affected by the choice of exit pressure Reference pressure for  

9. 9 Efficiency Definitions & Relations Compressor & Turbine Adiabatic Efficiency Temperature level effect on Pr Stage Efficiency related to overall Polytropic Efficiency

10. 10 Why Polytropic Efficiency? Polytropic efficiency: arises in context of a reversible compressor, compressing a gas from an initial state to a final state, but obeying pv n = constant, where n is called the polytropic index Comparison of isentropic  for 2 machines of different pressure ratio (Pr) is not valid, since for equal  poly, the compressor with the highest Pr is penalized with a hidden temperature effect  see next chart

11. 11 Compression System Consider Compression Part of Cycle – Because Constant Pressure Lines Diverge as Entropy Increases, (h 1a - h 1 ) + (h 1c - h 1b ) + (h 1e - h 1d ) +(h 1g - h 1f ) > (h 2i - h 1 ) so,

12. 12 Why Polytropic Efficiency? Adiabatic efficiency makes thermodynamic sense for cycle analysis. Changing adiabatic efficiency with varying number of identical stages does not describe fluid mechanics –  ad is a function of Pr and losses –Suppose we combine 2 compressors of equal  ad and equal  h 0 rise to make a compressor of higher pressure rise. The actual  h 0 =  (  h 0 ), but the isentropic  h 0 >  (  h 0 ) isent,

13. 13 Polytropic Efficiency - "Small Stage Efficiency" Compare fluid mechanical performance of different machines using  poly. Compressor Composed of large number of “small stages” Compressor: Compressor Polytropic Efficiency

14. 14 Compressor Polytropic Efficiency

15. 15 Compressor Stage Efficiency Mattingly uses the notation: Each Stage of a Multi-Stage Compressor Has an Adiabatic Efficiency Let  sj and  sj Represent the Pressure and Temperature Ratio of the jth Stage

16. 16 Compressor Efficiency From the Stage Efficiency We Have So for N Stages

17. 17 Compressor Efficiency And the Overall Compressor Efficiency is Where

18. 18 Compressor Efficiency Only for  s = constant Really want  s =constant

19. 19 Compressor Efficiency  s can be constant but  s is not constant

20. 20 Compressor Efficiency So for this Special Case (Constant Stage Pressure Ratio and Efficiency)

21. 21 Example

22. 22 Example

23. 23 Compressor Adiabatic Efficiency Actual work can be from temperature or measured rotor torque & mass flow

24. 24 Turbine Adiabatic (Isentropic) Efficiency

25. 25 Turbine Adiabatic Efficiency Turbine People Usually Use Expansion Ratio - P 1 /P 2 Consider Cooling Air Later (Just 1 st Law Bookkeeping)

26. 26 Consequences of Molier Diagram Constant Pressure Lines Diverge on Molier Diagram – Looks Slight, but it Matters More Work for given Pr as T increases Less Pr in Latter Stages of Compressor & Turbine Lower overall compressor efficiency as Pr increases Higher overall turbine efficiency as P r increases.

27. 27 Polytropic Efficiency - "Small Stage Efficiency" Turbine: Turbine Polytropic Efficiency Turbine Adiabatic Efficiency

28. 28 Turbine System Now Consider Turbine Part of Cycle – Because Constant Pressure Lines Diverge as Entropy Increases, (h 1a - h 1 ) + (h 1c - h 1b ) + (h 1e - h 1d ) +(h 1g - h 1f ) > (h 2i - h 1 ) so,

29. 29 Turbine Efficiency Analysis: Dixon 2.1 Calculate the overall efficiency of turbine  ad continued

30. 30 Turbine Efficiency Analysis: Dixon 2.1 small stage efficiency =  poly = 86% overall Pr = P 02 /P 01 = r = 4.5 to 1 = 4.5 mean  = 1.333  = 0.6868 and  ad-overall = 88.16%

31. 31 HPC SS-Fan LPT HPT

32. 32 ComponentIdealActualFigure of Merit InletAdiabatic & rev. [isentropic] PR=1, TR=1 Adiabatic, not rev., PR<1, TR=1 PR CompressorAdiabatic & rev.Adiabatic, not rev. TurbineAdiabatic & rev. NozzlePR=1, TR=1PR<1, TR=1PR

33. 33 Efficiency Comparison Pr 1 ss pp