This document discusses the thermodynamics of axial compressors and turbines used in gas turbines. It provides an overview of the Brayton cycle and discusses how to calculate the shaft power required for compression, temperature rises in the compressor, and outlet temperature as a function of pressure ratio, inlet temperature, and isentropic efficiency. Higher compression ratios are desirable because they increase turbine efficiency and overall gas turbine efficiency, though the rate of efficiency improvement decreases at higher ratios. Designing high-efficiency machines requires balancing increased efficiency from high compression against diminishing returns.

1. Gas Turbine Axial
Compressors andCompressors and
Turbines
Thermodynamics calculationsThermodynamics calculations
3rd December 2009
Prepared by: Cheah CangTo
Supervised by: James Richard Bryan

2. TURBO GROUP – Thermodynamics of Axial Compressor and Turbine
Objectives of this discussion:
a) Overview of Brayton cycle.
b) Is to provide the insight of prediction on thermodynamics performances of
gas turbine axial compressor and turbine.
c) Note when considering the fundamentals of axial compressor and turbine
design, it should be emphasized that successful compressor / turbine design
is very much an art, and all the major engine manufacturers have developed
a body of knowledge which is kept proprietary for competitive reasons.
2Thermodynamics of axial compressor
a body of knowledge which is kept proprietary for competitive reasons.

3. TURBO GROUP – Thermodynamics of Axial Compressor and Turbine
Overview of Brayton cycle
The Brayton cycle was first proposed by George Brayton for use in the
reciprocating oil-burning engine that he developed around 1870. Today, it isreciprocating oil-burning engine that he developed around 1870. Today, it is
used for gas turbines only where both the compression and expansion
processes take place in rotating machinery.
Picture taken from Wikipedia
1 to 2: Isentropic air compression1 to 2: Isentropic air compression
2 to 3: Isobaric (constant pressure) heat addition
3 to 4: Isentropic expansion of heated air
3Thermodynamics of axial compressor
4 to 1: Isobaric heat rejection (for closed-cycle)

4. TURBO GROUP – Thermodynamics of Axial Compressor and Turbine
With input parameters:
a) air mass flow ratea) air mass flow rate
b) compressor compression ratio (note turbine expansion ratio is
proportional per Brayton’s P-v diagram discussed earlier)
c) inlet temperaturec) inlet temperature
d) polytropic efficiency
Basically, there are two parameters needed to be evaluated:
a) shaft power requires to perform compression work
b) temperature rises at compressor outlet
4Thermodynamics of axial compressor

5. TURBO GROUP – Thermodynamics of Axial Compressor and Turbine
Formula for gas power
Power (kJ/s = kW) = mass flow rate (kg/s) x specific heat capacity at constant pressurePower (kJ/s = kW) = mass flow rate (kg/s) x specific heat capacity at constant pressure
(kJ/kg.K) x delta temperature (K)
Known / given: mass flow rate
To find:To find:
a) specific heat capacity at constant pressure, Cp
b) delta temperature = T_out – T_in (note: Inlet temperature, T_in is given)b) delta temperature = T_out – T_in (note: Inlet temperature, T_in is given)
5Thermodynamics of axial compressor

6. Specific heat capacity at constant pressure, Cp is a function of temperature.
TURBO GROUP – Thermodynamics of Axial Compressor and Turbine
Specific heat capacity at constant pressure, Cp is a function of temperature.
What should the temperature be? Inlet temperature or outlet temperature?
The answer is mean temperature, i.e. T_mean = (T_in + T_out)/2 should be used in order
to evaluate Cp.
But, how do we calculate T_mean since T_out is an unknown parameter.But, how do we calculate T_mean since T_out is an unknown parameter.
This problem can be solved by iteration method, i.e. take T_mean = T_in (i.e. without
T_out) for the first pass of iteration loop until values of T_out converge.
A0 0.992313
A1 0.236688
A2 -1.852148
A3 6.083152
A4 -8.893933
A5 7.097112
A6 -3.234725
A7 0.794571
A8 -0.081873A8 -0.081873
8765432
1000
8
1000
7
1000
6
1000
5
1000
4
1000
3
1000
2
1000
10 





×+





×+





×+





×+





×+





×+





×+×+= meanmeanmeanmeanmeanmeanmeanmean
p
T
A
T
A
T
A
T
A
T
A
T
A
T
A
T
AAC
6Thermodynamics of axial compressor


7. TURBO GROUP – Thermodynamics of Axial Compressor and Turbine
Outlet temperature, T_out is a function of:
a) pressure ratioa) pressure ratio
b) gama = Specific gas constant / (Specific gas constant – specific heat capacity at
constant pressure)
c) inlet temperaturec) inlet temperature
d) isentropic efficiency of compressor it is defined on following page
inTratiopressure






×








−
−
γ
γ
1_
1
in
isentropic
in
out TT +







 


=
η






7Thermodynamics of axial compressor
b) gama = Specific heat capacity at constant pressure / (Specific heat capacity at constant
pressure - Specific gas constant)

8. TURBO GROUP – Thermodynamics of Axial Compressor and Turbine
Isentropic efficiency of compressor is a function of:
a) pressure ratioa) pressure ratio
b) gama
c) polytropic efficiency
1_
1
−
−
ratiopressure γ
γ
1_
1
−
= −
ratiopressure
isentropic γ
γ
η
1_ −
× polytropic
ratiopressure
ηγ
8Thermodynamics of axial compressor

9. 1−γ
TURBO GROUP – Thermodynamics of Axial Compressor and Turbine
1_
1_
1
1
−
−
=
×
−
−
polytropic
ratiopressure
ratiopressure
isentropic
ηγ
γ
γ
γ
η
1_ −ratiopressure
Isentropic efficiency of compressor falls as pressure ratio is
increased for the same polytropic efficiency.
9Thermodynamics of axial compressor

10. TURBO GROUP – Thermodynamics of Axial Compressor and Turbine
In the previous slide, we learned that isentropic efficiency of compressor falls as
compression ratio increases.
But why GT manufacturers are building high pressure ratio machines (e.gBut why GT manufacturers are building high pressure ratio machines (e.g
LMS100, compression ratio of 40:1)?
Reasons are:Reasons are:
a) Isentropic efficiency of turbine increases as expansion ratio increases.
b) Higher OVERALL (combination of compressors, combustors, diffusers, turbineb) Higher OVERALL (combination of compressors, combustors, diffusers, turbine
expanders, etc) efficiency. Overall efficiency means net shaft output power
divided by fuel input power. Compression work is taking energy from the fuel, in
contrast turbine is extracting work from it. Turbine work is defined by:contrast turbine is extracting work from it. Turbine work is defined by:




∆××=
•
TCpmPower turbineturbine










−××=∆ −
γ
γ 1_
_
1
1
ratioExpansion
EffTT turbineisentropicinturbine
10Thermodynamics of axial compressor
 _ ratioExpansion

11. TURBO GROUP – Thermodynamics of Axial Compressor and Turbine
In order to maintain maximum temperature drop across turbine, it is necessary to
have higher expansion ratio, this means the pressure shall be kept high at
turbine inlet and low at turbine outlet. This leads to the reason why high powerturbine inlet and low at turbine outlet. This leads to the reason why high power
rating GT requires high compression ratio. Although it is true that a higher T_in
(combustion outlet temperature) will improve efficiency (refer Brayton cycle), for
the purposes of this discussion T_in is assumed constant since all turbinethe purposes of this discussion T_in is assumed constant since all turbine
manufacturers have similar combustor outlet temperatures to maximize
efficiency.
c) Mismatch (in term of air density) between compressor and turbine affecting
overall GT efficiency occurs on high compression ratio machine at high ambientoverall GT efficiency occurs on high compression ratio machine at high ambient
temperature. This effect can be clearly seen when comparing temperature de-
rate curves between low pressure ratio industrial gas turbines versus high
pressure ratio aero-derivative gas turbines. However, the topic is out of scope ofpressure ratio aero-derivative gas turbines. However, the topic is out of scope of
this discussion.
11Thermodynamics of axial compressor

12. 
TURBO GROUP – Thermodynamics of Axial Compressor and Turbine










−××=∆ −
γ
γ 1_
_
1
1
ratioExpansion
EffTT turbineisentropicinturbine
Delta T across turbine vs expansion ratio
800




γ
_ ratioExpansion
700
750
800
DeltaTemperature(K)
600
650
DeltaTemperature(K)
500
550
DeltaTemperature(K)
400
450
0 5 10 15 20 25 30 35 40
Expansion ratio
12Thermodynamics of axial compressor
Expansion ratio

13. •
TURBO GROUP – Thermodynamics of Axial Compressor and Turbine
turbineturbine TCpmPower ∆××=
•
Turbine power output vs expansion ratio
90000
95000
80000
85000
Turbineshaftpower(kW)
70000
75000
Turbineshaftpower(kW)
55000
60000
65000
Turbineshaftpower(kW)
50000
55000
0 5 10 15 20 25 30 35 40
Expansion ratio
13Thermodynamics of axial compressor

14. TURBO GROUP – Thermodynamics of Axial Compressor and Turbine
2.00
Specific work output vs pressure ratio T3/T1 = 2
T3/T1 = 3
T3/T1 = 4
T3/T1 = 5
1.50
T_in)]
T3/T1 = 5
1.00
Specificworkoutput[W/(Cp*T_in
0.50
Specificworkoutput[W/(
0.00
0 5 10 15 20 25 30
-0.50
Pressure ratio
14Thermodynamics of axial compressor

15. TURBO GROUP – Thermodynamics of Axial Compressor and Turbine
2.00
Specific work output vs pressure ratio T3/T1 = 2
T3/T1 = 3
T3/T1 = 4
1.50
T_in)]
T3/T1 = 5
PR=17
1.00
Specificworkoutput[W/(Cp*T_in
PR=17
Optimum pressure ratio for a
0.50
Specificworkoutput[W/(
PR=11
Optimum pressure ratio for a
given temperature ratio, T3/T1.
0.00
0 5 10 15 20 25 30
Specificworkoutput[W/(
PR=7
PR=3
-0.50
Pressure ratio
15Thermodynamics of axial compressor
Pressure ratio

16. OVERALL (combination of compressors, combustors, diffusers, turbine expanders, etc) efficiency
TURBO GROUP – Thermodynamics of Axial Compressor and Turbine
OVERALL (combination of compressors, combustors, diffusers, turbine expanders, etc) efficiency
increases as compression ratio increases.
Overall GT efficiency versus compression ratio
OverallgasturbineefficiencyOverallgasturbineefficiencyOverallgasturbineefficiency
( )rationcompressiooverall _ln09979.007641.0 ×+=η ( )rationcompressiooverall _ln09979.007641.0 ×+=η
Note: Overall GT efficiency is derived from machine manufacturers’
published heat rate..
Compression ratio
Gas turbine heat rate data courtesy of James Bryan
[GSGnet.net (2009)]
16Thermodynamics of axial compressor
Compression ratio

17. TURBO GROUP – Thermodynamics of Axial Compressor and Turbine
GT thermal efficiency versus pressure ratio:
comparison between Brayton and actual cycle
0.6
0.7
1
0.5
0.6
Brayton
Actual
γ
γ
η 1
1
2
1
1 −






−=
P
P
Brayton
0.4
0.3






×+= 2
ln09979.007641.0
P
P
actualη
0.1
0.2



 1P
0.0
0.1
17Thermodynamics of axial compressor
0 5 10 15 20 25 30 35 40

18. TURBO GROUP – Thermodynamics of Axial Compressor and Turbine
GT thermal efficiency versus pressure ratio:
comparison between Brayton and actual cycle
0.6
0.7
0.5
0.6
Brayton
Actual
Reduction of thermal efficiency due
to irreversible losses.
0.4
to irreversible losses.
0.3
0.1
0.2
0.0
0.1
18Thermodynamics of axial compressor
0 5 10 15 20 25 30 35 40

19. TURBO GROUP – Thermodynamics of Axial Compressor and Turbine
But, hang on a second…previous slide tells us overall GT efficiency goes up as
compression ratio increases (or temperature drop across turbine increases ascompression ratio increases (or temperature drop across turbine increases as
expansion ratio increases), then why GT manufacturer don’t produce high
efficiency machine, let say more than 50% at the expense of high compression
ratio?ratio?
The reason behind this, at least what I have in mind is explained on the following
pages.
19Thermodynamics of axial compressor

20. TURBO GROUP – Thermodynamics of Axial Compressor and Turbine
Let us revisit “Delta temperature versus expansion ratio” curve presented earlier.
Delta T across turbine vs expansion ratioDelta T across turbine vs expansion ratio
750
800
650
700
DeltaTemperature(K)
3
25
Temperature drop is more sensitive on expansion
550
600
DeltaTemperature(K)
65
Temperature drop is more sensitive on expansion
ratio of low pressure range compared to high
pressure range. It means rate of change of
efficiency is decreasing with increasing expansion
ratio.
450
500
3
From this point onwards, please be absolute clear
that (don’t confuse):
a) Overall GT efficiency increases as expansion
ratio increases.
b) Rate of change of overall GT efficiency
decreases as expansion ratio increases.
400
0 5 10 15 20 25 30 35 40
Expansion ratio
decreases as expansion ratio increases.
20Thermodynamics of axial compressor

21. TURBO GROUP – Thermodynamics of Axial Compressor and Turbine
In order to quantify sensitivity of efficiency as a function of expansion ratio, the following
expression is derived: by differentiating the curve function described previously to obtain
slope gradient.
Rate of change of efficiency = delta temperature turbine / delta expansion ratioRate of change of efficiency = delta temperature turbine / delta expansion ratio




−××=∆
1








−××=∆ −
γ
γ 1_
_
1
1
ratioExpansion
EffTT turbineisentropicinturbine
These are nearly constant
( )24812.0
_1 −
−=∆ ratioExpansionT
Gama = 1.33 (for turbine)
24812.1
_24812.0 −
×= ratioExpansion
dT
( )24812.0
_1 −
−=∆ ratioExpansionT
( )
24812.1
_24812.0
_
−
×= ratioExpansion
ratioExpansiond
dT
It is proposed that this new function is used as the basis for gas turbine
performance prediction and comparison within a database environment.
21Thermodynamics of axial compressor
performance prediction and comparison within a database environment.

22. TURBO GROUP – Thermodynamics of Axial Compressor and Turbine
Rate of change of overall GT efficiency as a function of expansion ratio is now defined:
( )
24812.1
_24812.0
_
−
×= ratioExpansion
ratioExpansiond
dT
( )ratioExpansiond
dT
_ ( )_ ratioExpansiond
0.12
This curve (dT/d[expansion ratio]) enables us to visualize how sensitive is
overall GT efficiency as a function of expansion ratio.
( )ratioExpansiond _
0.08
0.1
overall GT efficiency as a function of expansion ratio.
LM 6000
Compression ratio = 28.1
LMS 100
Compression ratio = 40
0.06
0.08
0.04
0
0.02
0 10 20 30 40 50 60
22Thermodynamics of axial compressor
0 10 20 30 40 50 60
Expansion
ratio

23. TURBO GROUP – Thermodynamics of Axial Compressor and Turbine
Overall GT efficiency merely
increased by 3.43%increased by 3.43%
OverallGTefficiencyOverallGTefficiencyOverallGTefficiency
23Thermodynamics of axial compressor
Compression ratio

24. TURBO GROUP – Thermodynamics of Axial Compressor and Turbine
It’s time to put all theories discussed in the previous section into practical work.
Attempt the following example:
Calculate the isentropic efficiency, outlet temperature and power input for a
compressor (of gas turbine) of 20:1 pressure ratio, polytropic efficiency of
91.07%, with a mass flow of 100 kg/s and an inlet temperature of 35 deg. C.
Also insert a parametric table to investigate the effects (on the outlet
temperature and power input) of changing inlet temperature to 30 and 38 deg. C
respectively.respectively.
24Thermodynamics of axial compressor

25. Input parameters
TURBO GROUP – Thermodynamics of Axial Compressor and Turbine
Input parameters
a) Compression ratio 20:1
b) Polytropic efficiency of 91.07%
c) Mass flow of 100 kg/sc) Mass flow of 100 kg/s
d) Inlet temperature of 35 deg. C.
But, something is missing...inlet pressure?
Then make assumption, machine is located at mean sea level, i.e. atmospheric
pressure = 101325 pascal
e) Inlet pressure = 101325 pascale) Inlet pressure = 101325 pascal
Parameters pass 1 pass 2 pass 3 pass 4 pass 5 pass 6 pass 7 pass 8 pass 9 Unit
Cp 1004.33 1039.11 1036.61 1036.78 1036.77 1036.77 1036.77 1036.77 1036.77 J/(kg.K)
gama 1.4002 1.3817 1.3830 1.3829 1.3829 1.3829 1.3829 1.3829 1.3829 -
T_out 788.16 764.59 766.21 766.10 766.10 766.10 766.10 766.10 766.10 K
Outlet temperature is converging from 4th
T_out 788.16 764.59 766.21 766.10 766.10 766.10 766.10 766.10 766.10 K
T_mean 308.15 548.16 536.37 537.18 537.12 537.13 537.13 537.13 537.13 K
Isentropic efficiency 0.8694 0.8694 0.8694 0.8694 0.8694 0.8694 0.8694 0.8694 0.8694 -
Outlet temperature is converging from 4th
iteration onwards.
25Thermodynamics of axial compressor

26. TURBO GROUP – Thermodynamics of Axial Compressor and Turbine
Parameters pass 1 pass 2 pass 3 pass 4 pass 5 pass 6 pass 7 pass 8 pass 9 Unit
Cp 1004.33 1039.11 1036.61 1036.78 1036.77 1036.77 1036.77 1036.77 1036.77 J/(kg.K)
gama 1.4002 1.3817 1.3830 1.3829 1.3829 1.3829 1.3829 1.3829 1.3829 -gama 1.4002 1.3817 1.3830 1.3829 1.3829 1.3829 1.3829 1.3829 1.3829 -
T_out 788.16 764.59 766.21 766.10 766.10 766.10 766.10 766.10 766.10 K
T_mean 308.15 548.16 536.37 537.18 537.12 537.13 537.13 537.13 537.13 K
Isentropic efficiency 0.8694 0.8694 0.8694 0.8694 0.8694 0.8694 0.8694 0.8694 0.8694 -
Outlet temperature, T_out = 766.10 K or 492.95 deg. C
Shaft power = 100 (kg/s) x 1036.77 (J/kg.K) x [766.10 - 308.15] (K)
= 100 x 1036.77 x 457.95= 100 x 1036.77 x 457.95
= 47478882.15 J/s
= 47479 kW
26Thermodynamics of axial compressor

27. TURBO GROUP – Thermodynamics of Axial Compressor and Turbine
Parametric table
Effects on shaft power and outlet temperature as inlet temperature
changes.
T_in (C) Power (kW) T_out (deg.C) Isentropic efficiency
30 46749 481.65 0.8693
35 47479 492.95 0.8694
changes.
35 47479 492.95 0.8694
38 47917 499.71 0.8694
Shaft power vs inlet temperature Outlet temperature vs inlet temperature
47600
47800
48000
500.0
Outlettemperature(deg.C)
47000
47200
47400
Shaftpower(kW)
490.0
Outlettemperature(deg.C)
46600
46800
30 31 32 33 34 35 36 37 38 39
Inlet temperature (deg. C)
480.0
30 31 32 33 34 35 36 37 38 39
Inlet temperature (deg. C)
Outlettemperature(deg.C)
27Thermodynamics of axial compressor
Inlet temperature (deg. C) Inlet temperature (deg. C)