1 turbomachinery lecture 2b - efficiency definitions
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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
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