• Recap: Lecture 2: 24th July 2015, 1530-1655 hrs.

– Applications of gas turbine engines

– Review of thermodynamics concepts

• Energy, Enthalpy, Entropy

• T-s diagram

• Isentropic processes

• Tds equations

Energy analysis of steady flow systems

• For single entry and exit devices,

)(2

mass,unit per or

)(2

12

2

1

2

212

12

2

1

2

212

zzgVV

hhwq

zzgVV

hhmWQ

2

Turbines and compressors

Turbine

WT

1

2

Control surface

Insulation

m

m

3

Turbines and compressors

• For a turbine for eg., the energy equation would be:

)(

,negligible are PE and KE If

)2

()2

(

21

2

2

221

2

11

hhmW

gzV

hmWgzV

hm

out

out

4

Stagnation properties

• Enthalpy represents the total energy of a fluid in the absence of potential and kinetic energies.

• For high speed flows, though potential energy may be negligible, but not kinetic energy.

• Combination of enthalpy and KE is called stagnation enthalpy (or total enthalpy)

h0 = h + V2/2 (kJ/kg) Stagnation enthalpy Static enthalpy Kinetic energy

5

Stagnation properties

• Consider a steady flow through a duct (no shaft work, heat transfer etc.).

• The steady flow energy equation for this is: h1 + V1

2/2 = h2 + V22/2

or, h01=h02

• That is in the absence of any heat and work interactions, the stagnation enthalpy remains a constant during a steady flow process.

6

Stagnation properties

• If the fluid were brought to rest at state 2,

h1 + V12/2 = h2 =h02

• The stagnation enthalpy represents the enthalpy of a fluid when it is brought to rest adiabatically.

• During a stagnation process, the kinetic energy of a fluid is converted to enthalpy (internal energy + flow energy), which results in an increase in the fluid temperature and pressure.

7

Stagnation properties

• When the fluid is approximated as an ideal gas with constant specific heats,

cpT0 = cpT +V2/2

or, T0 = T +V2/2cp

• T0 is called the stagnation temperature and represents the temperature an ideal gas attains when it is brought to rest adiabatically.

• The term V2/2cp corresponds to the temperature rise during such a process and is called the dynamic temperature.

8

Stagnation properties

• The pressure a fluid attains when brought to rest isentropically is called the stagnation pressure, P0.

• For ideal gases, from isentropic relations,

)1/(1

00

)1/(

00

have, density wefor Similarly,

T

T

T

T

P

P

9

Stagnation properties

s

h

Isentropic stagnation state

h

h0

Actual stagnation state

Actual state

P

P0 P0,actual

V2/2

The actual state, actual stagnation state, and isentropic stagnation state of a fluid on an h-s diagram.

10

Compressor/fan performance

• Compressors are to a high degree of approximation, adiabatic.

• Compressor performance can be evaluated using the isentropic efficiency, ηc

0102

0102

ratio pressuregiven for n compressio of work Actual

ratio pressuregiven for n compressio of work Ideal

hh

hh

w

w s

c

ci

C

11

Compressor/fan performance

T

02s

P01

P02 02

01 T01

T02s

T02

Actual and ideal compression processes

s

1: Compressor inlet

2: Compressor exit

12

Compressor/fan performance

1

1

1

1/

1/

1/

/1

/1

0102

0102

0102

0102

0102

0102

0102

C

C

C

s

ssC

PP

TT

TT

TT

TT

hh

hh

• The isentropic efficiency is thus a function of the total pressure ratio and the total temperature ratio.

13

Compressor/fan performance

• Besides isentropic efficiency, there are other efficiency definitions, stage efficiency and polytropic efficiency that are used in assessing the performance of multistage compressors.

• Stage efficiency will be discussed in detail during later lectures

• The three efficiency terms can be related to one another.

14

Compressor/fan performance

• The polytropic efficiency, ηpoly, is defined as

0

0

/)1(

0

0

00

/)1(

0

0000s

/)1(

00

0

0

0

0

111

,1/for expansion binomial Using

1dT And,

constant.

gives,relation isentropic the,compressor idealan For

change pressure aldifferenti afor n compressio of work Actual

change pressure aldifferenti afor n compressio of work Ideal

P

dP

P

dP

PdP

P

dPPT

PT

dT

dT

dh

dh

dw

dw

s

sss

poly

Compressor/fan performance

.efficiency

polytropicconstant a assuming ratio pressure with the

efficiency isentropic therelatesequation above The

1

1

1

1,

)/(/

02, and 01 statesbetween gIntegratin

1

equation, above theRewriting

/

/1

/

/

1 Therefore,

)/()1(

/1/1

)/()1(

01020102

0

0

0

0

00

00

00

00

0

0

0

0

0

0

poly

poly

C

C

C

CC

poly

sspoly

s

or

PPTT

P

dP

γ

γ

T

dT

TdT

PdP

γ

γ

TdT

TdT

dT

dT

P

dP

γ

γ

T

dT

Turbine performance

• The flow in a turbine is also assumed to be adiabatic, though in actual engines there could be turbine blade cooling.

• Isentropic efficiency of the turbine is defined in a manner similar to that of the compressor.

/)1(

0201

0201

1

1

ratio pressuregiven for expansion of work Ideal

ratio pressuregiven for expansion of work Actual

t

t

sts

t

t

hh

hh

w

w

Turbine performance

s

T

01

P02

P01

02 02s

T02 T02s

T01

Actual and ideal turbine processes

1: Turbine inlet

2: Turbine exit

Turbine performance

• The polytropic efficiency, ηpoly, is defined as

00

00

00

00

0

0

0

0

0

0

/)1(

00

0

0

0

0

//)1(

/

/

/

1

Therefore, constant.

gives,relation isentropic the turbine,idealan For

change pressure aldifferenti afor work turbineIdeal

change pressure aldifferenti afor work turbineActual

PdP

TdT

TdT

TdT

dT

dT

P

dP

γ

γ

T

dT

PT

dT

dT

dh

dh

dw

dw

ss

poly

s

s

sss

poly

Turbine performance

.efficiency

polytropicconstant a assuming ratio pressure with the

efficiency isentropic therelatesequation above The

1

1

1

1,

02, and 01 statesbetween gIntegratin

/1

/1

/1

)1(/

t

t

t

t

tt

t

poly

poly

poly

or

Lect-2

Thermodynamics of compressors

• Simplified aero-thermodynamic analysis

• Optimised cycle design to precede the detailed component design

• Prediction of work requirements

• Efficiency of the compressor

• Enables faster design modifications

Thermodynamics of compression

(i) Adiabatic (process 1-2’) , Pvγ=c

(ii) Isothermal process (1-2’’), Pv=c

(iii) Isochoric (Process 1-2’’’), Pv =c

Lect-2

Thermodynamics of compressors

i) Isentropic process (1-2’)

ii) Polytropic process (1-2)

iii) Isothermal process (1-2’’)

iv) Isochoric Process (1-2’’’)

Lect-2

Thermodynamics of compressors

X1 , X2 are the losses in the rotor and the stator respectively

Compression in terms of static parameters

Lect-2 Thermodynamics of compressors

Compression in terms of total parameters

Basic operation of axial compressors

• Axial flow compressors usually consists of a series of stages.

• Each stage comprises of a row of rotor blades followed by a row of stator blades.

• The working fluid is initially accelerated by the rotor blades and then decelerated in the stator passages.

• In the stator, the kinetic energy transferred in the rotor is converted to static pressure.

• This process is repeated in several stages to yield the necessary overall pressure ratio.

Thermodynamics of multi-stage compressors

• The flow at the rotor exit with high kinetic energy is still to be converted to static pressure through diffusion.

• The exit kinetic energy of a compressor is of the same order as the entry kinetic energy and the entire work input is expected to be converted to pressure.

Rotor isentropic, stator isothermal Rotor polytropic, stator isothermal

Averaged T-s characteristics