1. Non-Ideal Cycle Analysis
MAE 5350: Gas Turbines
Non-Ideal Cycle Analysis
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk

2. NON-IDEAL (“REAL”) CYCLES
Principal deviations from ideal behavior:
Imperfect diffusion of free-stream flow in engine inlet
Non-isentropic compression and expansion in the turbomachinery (compressors and turbines)
Stagnation pressure change in combustor
Incomplete combustion in combustor
Variation of gas properties (specific heat, g) through the engine
Incomplete expansion, or over expansion, in the nozzle
Extraction of compressor discharge air for turbine cooling and for airframe use (bleeds)

3. SOME COMMENTS ON THE WORKING FLUID
We have assumed that the working fluid can be approximated as a perfect gas with constant specific heats
In reality the specific heats, and specific heat ratio, , vary through the engine
The effect of pressure is small (on the order of 0.1% for 20kPa to 45 Mpa [Cumpsty]) but the effect of temperature is appreciable
The variation in cp and  is given in the next chart, which shows the dependence on temperature, for different values of the equivalence ratio, f, which is the ratio of the fuel air-ratio to the fuel air ratio for stoichiometric combustion
For simplicity, the values of  will be taken to be different but constant in the different components. They will be denoted by subscripts
The value for the compressor is denoted by c, the value for the turbine by t.
Appropriate values are 1.4 and 1.3, respectively

4. VARIATION IN SPECIFIC HEAT AT CONSTANT PRESSURE, cp, AND SPECIFIC HEAT RATIO, , WITH TEMPERATURE FOR AIR AND FOR COMBUSTION PRODUCTS OF KEROSENE;  is the Equivalence Ratio [Cumpsty]

5. DEPARTURE FROM IDEAL BEHAVIOR: LOSSES IN ENGINE COMPONENTS
Component efficiency has a large impact on cycle performance
Characterizing losses in components (departures from ideal reversible processes) is a key aspect of real cycle analysis
We will examine basic mechanisms and measures developed for assessing loss
In this, it will be seen that entropy generated due to irreversibility is the most useful measure of loss (inefficiency)
For an adiabatic flow the entropy increase translates to a stagnation pressure change
pt can thus often be used as a loss indicator

6. LOSS SOURCES
Viscous dissipation
Boundary layers
Shear layers (mixing)
Heat transfer across a finite temperature difference
Shocks

7. THERMODYNAMIC CYCLES [Walsh and Fletcher]
Carnot Cycle
Non-Ideal Brayton
Cycle for turbojet,
turboshaft, turboprop,
ramjet

8. LOSSES AND STAGNATION PRESSURE CHANGES
Consider a medium that undergoes an irreversible process 1--->2
Example: flow through a screen or throttle
No work done, no heat transfer, therefore ht = constant
Represent the states on a T-s diagram
It is “conventional” to think about losses in terms of changes in stagnation pressure. Why?

9. FLOW THROUGH A SCREEN, THROTTLE, OR BLADE ROW

10. LOSSES IN A THROTTLING PROCESS (I)
Losses are often expressed as a decrease in stagnation pressure
Directly measurable quantity
Related to the minimum work required to reverse the process to its original state 2 ---> 1
Lost work
For the flow through a screen, examine the work required to reverse the process using an ideal process
First Law of Thermodynamics: e = q - w, neglecting changes in all forms of energy except internal energy
For perfect gas e = e(T) only, so for our example of the screen: e(Tt1) = e(Tt2)  q = w
For a reversible process, the heat received per unit mass is: dqrev=Tds

11. LOSSES IN A THROTTLING PROCESS (II)
Thus for the screen, qrev=Tt1s So
wrev=Tt1s
THE WORK REQUIRED TO RETURN THE SYSTEM TO INITIAL STATE (THE “LOST WORK”) IS DIRECTLY RELATED TO THE CHANGE IN ENTROPY
Now we relate this entropy change to the change in total pressure
If TT = constant and cp and cv are constant then

12. LOSSES IN A THROTTLING PROCESS (III)
The minimum work required to restore the fluid to its initial state is thus directly connected to the change in stagnation pressure for a flow with constant stagnation temperature

13. CONNECTION BETWEEN ENTROPY CHANGES AND TURBOMACHINERY COMPONENT EFFICIENCY
Compressor
Turbine
Efficiency for compressor: For given pt2 / pt1 , how much shaft work is done
Shaft work / unit mass flow rate = ht2 - ht1 (assuming adiabatic)

14. ht2 = ht2 + ht2 - ht2 = ht2 + Dht
Along pt2 = const. curve
ht2 = ht2 + ht2 - ht2 = ht2 + Dht
Isentropic compression
At const pt
But or
Thus at const pt

15. CONNECTION BETWEEN ENTROPY CHANGES AND COMPONENT EFFICIENCY
ht2 = ht2 + Tt2 Ds
or 2 ;
Entropy rise directly tied to efficiency, similarly for turbine ;
Turbine

16. SUMMARY: LOSSES AND STAGNATION PRESSURE
Entropy is the basic measure of loss
- Entropy is not measured directly
For adiabatic processes, we can relate entropy changes and
changes in stagnation pressure
- Stagnation pressure is measured
Stagnation pressure is often used as the figure of merit for component
loss (or component efficiency)
The next several slides show the application for an inlet/diffuser
combination
d = pt out / pt in

17. INLET AND DIFFUSER LOSS
Subsonic diffusers
Need to supply air to the engine at the Mach number the compressor demands
Need to be efficient over range of free-stream Mach numbers from take-off to cruise
Modern computational tools enable efficient inlets with stagnation pressure recoveries greater than 0.95
Supersonic diffusers
Shock waves exist and introduce a loss mechanism
Very large variations in capture stream tube area
Inlet compression is a larger fraction of the overall compression process and overall cycle efficiency is thus more sensitive to inlet design
References provide detailed information about inlet design

18. SCHEMATIC DIAGRAMS OF SUBSONIC AND SUPERSONIC INLETS AND DIFFUSERS [Kerrebrock]

19. REPRESENTATIVE VALUES OF INLET/DIFFUSER STAGNATION PRESSURE RECOVERY AS A FUNCTION OF FLIGHT MACH NUMBER [Kerrebrock]

20. EFFICIENCIES IN TURBOMACHINERY COMPONENTS: COMPRESSOR AND TURBINE
Consider the compression process through a compressor stage
The goal is to achieve a given stagnation pressure ratio, and to do this at minimum work
We need a relation involving dh and dp to capture this
dh = Tds + dp/r
Apply this to the stagnation conditions:
dht= Ttds + dpt/t
The flow in the compressor is essentially adiabatic
The second law says that for a fluid particle ds > 0 for all real processes
For given change in pt, as ds increases, so does ht , the stagnation enthalpy
Thus, for a given change in stagnation pressure, the change in stagnation temperature, which is a direct measure of the work we must do, reflects how “good” we are at compression

21. THE ADIABATIC (OR ISENTROPIC) EFFICIENCY
For a compressor, the comparison of ideal to actual work for a given stagnation pressure rise or ratio furnishes the metric known as adiabatic (or isentropic) efficiency
Pt2
Ideal work for pt change:
Ideal Dht = ht2s - ht1 2s 2
Tt2
Actual
Actual work for pt change: Actual Dht = ht2 - ht1
T or h
Ideal
ht
Pt1
Tt1
h = Ideal work/actual work 1 s
Ideal work for pt change: Ideal ht = ht25 - ht1
Actual work for pt change: Actual ht = ht2 - ht1

22. COMPRESSOR ADIABATIC EFFICIENCY
The adiabatic (sometimes called isentropic) efficiency is the ratio of the ideal work for a given pt to the actual work needed
Definition:  = Ideal work / Actual work
There is a difference between this quantity and the cycle efficiency:
- The cycle efficiency can describe an ideal situation
- Cycle efficiency is set by the second law --a fundamental limitation on the conversion of heat to work
- The adiabatic efficiency is a measure of “how well we did
the design” and reflects our capabilities

23. BEHAVIOR OF STAGNATION PRESSURE AND TEMPERATURE IN A COMPRESSOR STAGE
Stagnation pressure and temperature rise in rotor
Shaft work is done on fluid
Stagnation temperature is constant in the stator
Forces on fluid, and angular momentum changes, exist,
but no shaft work is done
Stagnation pressure falls in stator
Constant stagnation temperature, but entropy rises;
Tds = dh - dp/r again
Ideal work/unit mass is cp(Tt3’ - Tt1) [Using notation below]
Actual work/unit mass is cp(Tt2 - Tt1) > cp(Tt3’ - Tt1)
hcompressor = Ideal work/Actual work for same pressure ratio
= Adiabatic/isentropic efficiency

24. THERMODYNAMIC STATES IN A COMPRESSOR STAGE [Cohen, Rodgers, Savaranamutoo]

25.  is stagnation temperature ratio: p is stagnation pressure ratio
COMPRESSOR ADIABATIC EFFICIENCY IN TERMS OF PRESSURE AND TEMPERATURE RATIOS
The adiabatic efficiency (and the corresponding quantity for turbines) is a metric of how effectively we are able to raise the stagnation pressure
It is useful to put it in terms of the stagnation pressure and temperature ratios, which are the quantities actually measured
 is stagnation temperature ratio: p is stagnation pressure ratio

26. ACTUAL AND IDEAL WORK FOR A TURBINE
pt1 1
ht actual
pt2 h
ht ideal 2 2s s

27. ADIABATIC EFFICIENCY FOR A TURBINE
For a turbine the “reverse” situation occurs
For a given pressure ratio (expansion ratio), the work extracted in the real process is less for the actual process than for the reversible, adiabatic (isentropic process)
Adiabatic (isentropic) efficiency for the turbine is defined as the ratio of Actual work / Ideal work, i.e., the ratio of the amount we actually received, compared to the amount we could have received in an isentropic process
In terms of temperature and pressure ratios:

28. PARAMETERS AFFECTING CYCLE POWER AND EFFICIENCY
The ratio Tt4/Tt2, the ratio of turbine entry temperature to compressor inlet temperature is an important parameter
For a given , i.e., given cycle pressure ratio, increasing Tt4/Tt2 brings a rapid rise in net power (this is effectively how the engine is controlled)
Compressor and turbine efficiencies have a marked effect on overall cycle efficiency and work. This is because for a Brayton cycle, much of the turbine work goes to drive the compressor
The next two pages show plots of net power per unit of enthalpy flow and cycle efficiency for different values of the temperature ratio Tt4/Tt2 as well as the effects of component efficiency on cycle efficiency

29. ENGINE CYCLE (THERMAL) EFFICIENCY VARIATIONS [Philpot]

30. POWER AND CYCLE EFFICIENCY TRENDS WITH TURBINE TEMPERATURE AND COMPONENT EFFICIENCY [Cumpsty]
Tt4 / Tt2
Tt4 / Tt2
Tt4 / Tt2

31. BRAYTON CYCLE FOR SIMPLE GAS TURBINE
Pressure ratio 40, inlet temperature =288K, turbine temperature 1700K, turbine and compressor adiabatic efficiencies both 0.9 [Cumpsty]

32. BRAYTON CYCLE FOR GAS TURBINE WITH SEPARATE POWER TURBINE
Pressure ratio 40, inlet temperature =288K, turbine temperature 1700K, turbine and compressor adiabatic efficiencies 0.9 [Cumpsty]

33. ENGINE PERFORMANCE SET BY
Basic cycle selection
Pressure ratio
Turbine inlet temperature
Bypass ratio
Technology levels available
Achievable component efficiencies
Achievable work per stage
Mechanical design and materials selection

34. SUMMARY
A gas turbine engines can be regarded as “a core with different loads fitted to it” [Cumpsty]
It can be analyzed in an approximate and useful manner by replacing the combustion by an equivalent heat transfer
The thermodynamic cycle efficiency, and the cycle power, are strongly dependent on:
Adiabatic component efficiencies
Ratio of turbine entry stagnation temperature (max temperature in the cycle) to compressor inlet stagnation temperature