1. Turbomachinery
Lecture 2b
Efficiency Definitions

3. Gas-Turbine Brayton Cycle
Most Gas Turbines Use Ideal Brayton Cycle as 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. Gas-Turbine Brayton Cycle3 2 4 9
Compressor
Combustor
Turbine
Ambient
For constant (or nearly constant) velocity
Neglect fuel mass flow effect

5. Gas-Turbine Brayton Cycle3 2 4 9
Compressor
Combustor
Turbine
Ambient
Gas-Turbine Brayton Cycle

6. Compressor Adiabatic (Isentropic) Efficiency
Poout
h02
Usually in terms of stagnation
properties, but in centrifugal and
industrial machines
 inlet use Po, exit Ps
h02i
V2/2
If inlet/outlet K.E. small,
Poin
h01 s

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

8. Compressor Adiabatic Efficiency
Caution - Different pressure ratios are used in this definition
Usually Pt exit and Pt inlet (for total-to-total efficiency)
Pr = P02/P01
Sometimes Ps 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. Efficiency Definitions & Relations
Compressor & Turbine Adiabatic Efficiency
Temperature level effect on Pr
Stage Efficiency related to overall
Polytropic Efficiency

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 pvn = 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. Compression System
Consider Compression Part of Cycle – Because Constant Pressure Lines Diverge as Entropy Increases,
(h1a - h1) + (h1c - h1b) + (h1e - h1d) +(h1g - h1f) > (h2i - h1)
so,

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 h0 rise to make a compressor of higher pressure rise. The actual h0 =(h0 ), but the isentropic h0 > (h0 )isent,

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. Compressor Polytropic Efficiency

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

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

17. Compressor Efficiency
And the Overall Compressor Efficiency is
Where

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

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

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

21. Example

22. Example

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

24. Turbine Adiabatic (Isentropic) Efficiency

25. Turbine Adiabatic Efficiency
Turbine People Usually Use Expansion Ratio - P1/P2
Consider Cooling Air Later (Just 1st Law Bookkeeping)

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 Pr increases.

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

28. Turbine System
Now Consider Turbine Part of Cycle – Because Constant Pressure Lines
Diverge as Entropy Increases,
(h1a - h1) + (h1c - h1b) + (h1e - h1d) +(h1g - h1f) > (h2i - h1)
so,

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

30. Turbine Efficiency Analysis: Dixon 2.1
small stage efficiency = poly = 86%
overall Pr = P02/P01= r = 4.5 to 1 = 4.5
mean  =
 = and
ad-overall = %

31. LPT
HPT
SS-Fan
HPC

32. Component
Ideal
Actual
Figure of Merit
Inlet
Adiabatic & rev. [isentropic]
PR=1, TR=1
Adiabatic, not rev.,
PR<1, TR=1 PR
Compressor
Adiabatic & rev.
Adiabatic, not rev.
Turbine
Nozzle

33. Efficiency Comparison1 Pr