1

Turbomachinery Lecture 2b

- Efficiency Definitions

2

:First Law of Thermo in other forms

dE dQ dW

dE dQ dW

dt dt dtor

dE dQ dW

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)

T

S

3

2

4

9

Co

mp

ress

or

Combustor

Tu

rbin

e

Ambient

T

S

3

2

4

9

Co

mp

ress

or

Combustor

Tu

rbin

e

Ambient

T

S

3

2

4

9

Co

mp

ress

or

Combustor

Tu

rbin

e

Ambient

T

S

3

2

4

9

Co

mp

ress

or

Combustor

Tu

rbin

e

Ambient

S

3

2

4

9

Co

mp

ress

or

Combustor

Tu

rbin

e

Ambient

4

32 3 2

43 4 3

49 4 9

92 9 2

4 9 3 2

( )

( )

( )

( )

( )

c p

in p

t p

out p

out t c p

W m h mc T T

Q m h mc T T

W m h mc T T

Q m h mc T T

Net W W W mc T T T T

Gas-Turbine Brayton Cycle• For constant (or nearly constant) velocity• Neglect fuel mass flow effect

T

S

3

2

4

9

Co

mp

ress

or

Combustor

Tu

rbin

e

Ambient

5

1

3 2 3 2 4 9

1 14 3

3 2

/ / / (

1 11 1

( )( / ) Pr

out outthermal

in fuel p

p p T T T T isentropic along constant p lines

netW netW

Q mc T Tp p

Gas-Turbine Brayton Cycle

T

S

3

2

4

9

Co

mp

ress

or

Combustor

Tu

rbin

eAmbient

6

Compressor Adiabatic (Isentropic) Efficiency

450

650

850

1,050

1,250

1,450

1,650

-0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06

S

T

IdealReal

P in

P out

ad

ideal work input

actual work input

02 01

02 01

iad

h h

h h

1 / 1

1r

adr

P

T

Usually in terms of stagnationproperties, but in centrifugal andindustrial machines inlet use Po, exit Ps

s

Poin

Poout

V2/2

If inlet/outlet K.E. small, 2 2

1 2V V

02 1

02 1

iad

h h

h h

h02i

h02

h01

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

p

c c

s s

N

s stage efficiency polytropic efficiency

number of stages

compressor pressure ratio compressor temperature ratio

stage pressure ratio stage temperature ratio

Pr π Tr 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

2

2 0/02

3

3 0

s s

o

s s

o

p rotor exit p

p rotor exit pp

p stator exit p

p stator exit p

Reference pressure for

/02p

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,

450

650

850

1,050

1,250

1,450

1,650

-0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06

S

T

IdealReal

P in

P out

1

2i

1g

1f1e

1d1c

1b1a

2

450

650

850

1,050

1,250

1,450

1,650

-0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06

S

T

IdealReal

P in

P out

1

2i

1g

1f1e

1d1c

1b1a

2

1 1 1 1 1 1 1 1 2 1

2 1 2 1

( ) ( ) ( ) ( ) ( )

( ) ( )a c b e d g f i

p a

h h h h h h h h h h

h h h h

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,

01 02

01, 02,, 01, 02,

01 02

''02, 02,

''01, 02,

,2 ,101 02,

s sad stage s s

s s

s sad stages ad stage

s

h h

h hh h

h h

But h h

h h

h h

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

0

0

0 00 01 0

0 0

, 0,

ideal ipoly

actual

i p

dW dh

dW dh

for ideal compressor s

dP dPdh c dT RT

P

0 0

0 0

/1

/poly

dP P

dT T

0 0

0 0

1

p

dT dP

T P

p

rr PT /1

14

Compressor Polytropic Efficiency

0 0

0 0

1 /0 0

0

0

:

1

p

ideal ideal ideal

ideal ideal

idealp

polytropic or small stage efficiency

ideal work for differential p change

actual work for differential p change

dw dh dT

dw dh dT

From adiabatic gas law T CP

dT

dT

1 /0 0

0 0

/

/p

c c

dP P

dT T

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

1

1

1sj

sjsj

Tr

πPr 1 / 1 /1 1

1 1r

adr

P

T

16

Compressor Efficiency• From the Stage Efficiency

We Have

• So for N Stages

1

, ,

, 1 , 1

11 1t j t j

t j sj t j

T P

T P

1

,

10 , 1

11 1

Nt jtN

jt sj t j

PT

T P

17

Compressor Efficiency• And the Overall Compressor

Efficiency is

• Where

1

1

0

0

11

1 1

tN

c tadcomp tNc

t

PPTT

,

10 , 1

Nt jtN

cjt t j

PP

P P

18

Compressor Efficiency

11

1/

1 1

1

,

10 , 1

1 11 1

1

1 11 1 1 1

11 1

sj

sj sj ssj sj

NNc s sj c

N N

Nc sj c

sj sj

Nt jtN

jt sj t j

PT

T P

Only for s

= constant

Really wants=constant

19

Compressor Efficiency

1

1 /

1 1ln ln

ln1

ln 1 1

1ln

1ln 1 1

Nc s

p Nc

sjsj

sj

p

sjs

s can be constant but s is not constant

20

Compressor Efficiency• So for this Special Case (Constant Stage

Pressure Ratio and Efficiency)

1

1

1

1

1

11 1 1

1

11 1 1

cad Ncomp

Nc

sj

N

sj

N

ssj

21

Example

1/16

1 /

11/3.5

11

16 25

25 1.223

0.93

1ln

0.9321

ln 1 1

1 25 1

11 1 1.2231 1 1 0.932

c

sj

s

sj

p

sjs

cad Ncomp

Nc

s

Consider stage compressor PR

If is assumed

/ 3.5

0.8965

1 1N

22

Example

1

1

10.8965

1p

cadcomp

c

Other way

23

Compressor Adiabatic Efficiency

• Actual work can be from temperature or measured rotor torque & mass flow

Torque

PTCJm rtinp 1/1

24

Turbine Adiabatic (Isentropic) Efficiency

800

900

1,000

1,100

1,200

1,300

1,400

1,500

1,600

1,700

1,800

0.040 0.045 0.050 0.055 0.060 0.065 0.070 0.075 0.080

S

T

P in

P out

Ideal

Real

ad

actual work output

ideal work output

1 2

1 2ad

i

h h

h h

2 1

1 /

2 1

1 /

1 /ad

T T

P P

25

Turbine Adiabatic Efficiency

• Turbine People Usually Use Expansion Ratio - P1/P2

• Consider Cooling Air Later (Just 1st Law Bookkeeping)

1 2

1 /

1 2

11

/1

1/

ad

T T

P P

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

P

dPRT

dPdh

dh

dh

i

ipoly

dP

P

RT

dTC pp

1pdT dP

T P

/1 p

rr PT

1 /

1

1r

ad

r

T

P

28

Turbine SystemNow Consider Turbine Part of Cycle – Because Constant Pressure Lines Diverge as Entropy Increases,

(h1a - h1) + (h1c - h1b) + (h1e - h1d) +(h1g - h1f) > (h2i - h1)

so,

ai

fgdebcap

hh

hh

hhhhhhhh

hh

)(

)(

)()()()(

)(

12

12

11111111

12

800

900

1,000

1,100

1,200

1,300

1,400

1,500

1,600

1,700

1,800

0.040 0.0450.050 0.0550.060 0.0650.070 0.0750.080

S

T

P in

P out

Ideal

Real

1

2i1e

1d

1c

1b1a

2

800

900

1,000

1,100

1,200

1,300

1,400

1,500

1,600

1,700

1,800

0.040 0.0450.050 0.0550.060 0.0650.070 0.0750.080

S

T

P in

P out

Ideal

Real

1

2i1e

1d

1c

1b1a

2

29

Turbine Efficiency Analysis: Dixon 2.1Calculate the overall efficiency of turbine ad

01 02 02 01

01 02 02 01

1

01 01

02 02

1

02

01

0 0 0 0 00

0 0 0 0 0 0

0 0 02 02

0 0 01 01

1 /

1 /

exp

/ 1

1po

ads s

s

s

poly ps

poly

h h T T

h h T T

P T1r ansion pressure ratio

Pr P T

Tr

T

dh dh dT dTRc

dh dp p dp RT p

dT dp T p

T p T p

1ly

poly

continued

30

Turbine Efficiency Analysis: Dixon 2.1

01 02 02 01

01 02 02 01

1 / 1

1 / 1

poly

ads s

h h T T

h h T T

• small stage efficiency = poly = 86%

• overall Pr = P02/P01= r = 4.5 to 1 = 4.5

• mean = 1.333

= 0.6868 and

ad-overall = 88.16%

Therefore if axial flow turbine has

31

Adiabatic Efficiency @ 90% Polytropic

82%

84%

86%

88%

90%

92%

94%

96%

0 10 20 30 40 50

Pressure Ratio

Ad

iab

atic

Eff.

Turbines

Compressors

HPC

SS-Fan

LPT

HPT

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 Adiabatic & rev.

Nozzle PR=1, TR=1 PR<1, TR=1 PR

ad

p

ad

p

1 /

02 01pW c T T

1 /

02 01pW c T T

/ 1

1

1

1

1

p

p

ca t

c

/ 1

1

1

1

1

p

p

ca c

c

33

Efficiency Comparison

1 / 1 /03 01

1 /03 01

1 /

11 /

1 ln

ln

1 1

1 1

1 1

lim1 1

p

p p

p

is

p

p p

T T

T T

Pr1

s

p