Axial compressor theory - stage-wise isentropic efficiency - 18th March 2010
-
Upload
cangto-cheah -
Category
Documents
-
view
71 -
download
4
Transcript of Axial compressor theory - stage-wise isentropic efficiency - 18th March 2010
Axial compressor
theory
Stage-wise isentropic efficiencyStage-wise isentropic efficiency18th March 2010Prepared by: Cheah CangTo
TURBO GROUP – Axial compressor theory - Stage-wise isentropic efficiency
From previous works, air deflection angle and air outlet angle are calculated. Moving on, in this chapter we will find:
a. Spacing (pitch) between blades
b. Chord
2Axial compressor theory - Stage-wise isentropic efficiency – Cheah CangTo
TURBO GROUP – Axial compressor theory - Stage-wise isentropic efficiency
Design deflection curves
40
50
S/C = 1.5 from CRS
S/C = 1.0 from CRS
S/C = 0.5 from CRS
S/C = 1.5 (curve fit)
S/C = 1.0 (curve fit)
S/C = 0.5 (curve fit)
to find
Analysis of the values of nominal deflection determined from a large number of tests covering different forms of cascade, has shown that its value is MAINLY dependent on the pitch/chord ratio and air outlet angle.
s/c = 0.5
s/c = 1.0
s/c = 1.5
3Axial compressor theory - Stage-wise isentropic efficiency – Cheah CangTo
0
10
20
30
-10 0 10 20 30 40 50 60 70
Air outlet angle, degrees
Air
de
fle
ctio
n,
de
gre
es
to find
prediction based on S/C
s/c = 1.5
TURBO GROUP – Axial compressor theory - Stage-wise isentropic efficiency
Determination of chord length will now depend on the pitch, which itself is dependent on the number of blades in the row. When making a choice for this number, the aspect ratio of the blade, “h/c” has to be considered because of its effect on secondary losses.
number of blades, n = 2 x pi x mean radius / pitch
note: h/c = 3 for initial guess, iterative calculations require. (h = blade height)
4Axial compressor theory - Stage-wise isentropic efficiency – Cheah CangTo
Blade heightChord
TURBO GROUP – Axial compressor theory - Stage-wise isentropic efficiency
Referring to the diagram of forces acting on the cascade, the static pressure rise across the blades is given by:
( )
( )
_22
_2
2
2
1
_
0201
0102
2
2
2
1
2
101
2
20212
coscos2
2
:
2
2
1
2
1
wVV
p
wVVp
wppdefine
ppVVp
VpVpppp
aa −
−
=∆
−−=∆
=−
−+−=∆
−−
−=−=∆
αα
ρ
ρ
ρ
ρρ
a
a
a
5Axial compressor theory - Stage-wise isentropic efficiency – Cheah CangTo
( )_
2
2
1
2
2
2
2
1
2
2
2
1
2
_
2
2
1
2
2
21
tantan2
tantancos
1
cos
1:
cos
1
cos
1
2
coscos2
wV
p
note
wV
p
a
a
−−=∆∴
−=−
−
−=∆
ααρ
αααα
αα
ρ
αα
a
( ) ( )( )
21
21
121
1221
1
tantantan2:
tantan2
1tan
tantan2
1
tan
tantantan2
1
tan
ααα
αα
ααααα
α
+=
+=
+
=
+−
= −−−
m
a
aa
a
aaa
m
or
V
VV
V
VVV
TURBO GROUP – Axial compressor theory - Stage-wise isentropic efficiency
Force acting along the cascade (from consideration of momentum changes) per unit length is given by:
( ) ( )21
2
21tantantantan ααρααρ −×=−×= aaaa VsVVVsF
6Axial compressor theory - Stage-wise isentropic efficiency – Cheah CangTo
TURBO GROUP – Axial compressor theory - Stage-wise isentropic efficiency
Drag force,DPm cCVD
2
2
1ρ=
( ) ( )[ ]
( ) ( )
m
mmama
mmmamamm
swD
swVsVsD
swVsVspsFD
α
ααααραααρ
αααααραααραα
cos
cossintantansintantan
coscostantantansintantancossin
_
_
21
2
21
2
_
21
2
21
2
=∴
+−−−=
+−−−=∆−=
a
3____
_2
coscoscoscos
cos2
1
αααα
αρ mDPm
wswswssw
swcCV =
EquatingDPm cCVD
2
2
1ρ= with
mswD αcos_
= yields:
( by definition of CDP )
7Axial compressor theory - Stage-wise isentropic efficiency – Cheah CangTo
1
2
3
2
1
_
1
22
1
3_
2
3_
2
2
_
2
_
2
_
cos
cos
2
1
cos
cos
2
1
cos
2
1
cos
cos
2
1
cos
2
1
2
1
cos
α
α
ρ
α
α
ρ
α
ρα
α
ρ
α
ρρ
α
mDP
mDP
a
m
m
a
m
m
m
m
mDP
V
w
c
sC
V
w
c
sC
V
w
c
s
V
w
c
s
V
w
c
s
cV
swC
××=∴
××=
××=
××=××==
a
a
TURBO GROUP – Axial compressor theory - Stage-wise isentropic efficiency
Lift force,
Equating with yields:
( by definition of CL )Lm cCVL2
2
1ρ=
( ) ( )[ ]
( )[ ] mmmma
mmmamamm
swVsL
swVsVspsFL
ααααααρ
αααααραααραα
sinsintancostantan
sinsintantantancostantansincos
_
21
2
_
21
2
21
2
−+−=
−−+−=∆+=
a
Lm cCVL2
2
1ρ= ( )[ ] mmmma swVsL ααααααρ sinsintancostantan
_
21
2 −+−=
( )[ ]
( )[ ]
( )[ ]
−
+−=
−+−=
cV
sw
cV
VsC
swVscCV
m
m
m
mmmaL
mmmmaLm
_
2
_
2
21
2
_
21
22
2
1
sin
2
1
sintancostantan
sinsintancostantan2
1
ρ
α
ρ
αααααρ
ααααααρρ
a
8Axial compressor theory - Stage-wise isentropic efficiency – Cheah CangTo
( )[ ]( )
( )[ ]
( )[ ]( )
( )[ ]
−
+−=
−
+−=
−
+−=
−
+−=
cV
sw
c
sC
cV
sw
cV
sVC
cV
sw
cV
sVC
cV
sw
cV
sVC
a
mmmmmmL
a
mm
a
mmmmaL
m
a
m
m
a
mmmaL
m
m
m
mmmaL
2
2_
23
21
2
2_
2
23
21
2
2
2
_
2
2
21
2
2
_
2
21
2
2
1
cossincossintancostantan2
2
1
cossincossintancostantan2
cos2
1
sin
cos
sintancostantan2
2
1
sinsintancostantan2
ρ
αααααααα
ρ
αααααααα
αρ
α
α
ααααα
ρ
αααααα
a
a
a
a
continue on next page...
TURBO GROUP – Axial compressor theory - Stage-wise isentropic efficiency
( )mmmmm
mmmm
c
a
c
ac
c
baa
c
aba
ac
ab
c
a
ac
ab
c
a
c
b
a
b
c
a
ααααα
αααα
coscossintancos
cossintan,cos
3
2
3
22
3
23
3
22
3
3
23
3
22
2
2
2
3
3
3
===+
=+
=+=+∴
=××==
m
m
mm
mmm
mmm
a
b
a
c
c
ba
c
a
c
ba
c
a
c
b
αα
αα
ααα
ααα
tancos
cossin?
cos,cossin
?coscossin
3
3
3
2
3
2
3
3
3
3
2
2
2
2
32
==×==
==×=⇒
×=
a
( )[ ]
−+−
=
cV
sw
c
sC mmmmmm
L2
2_
23
21
1
cossincossintancostantan2
ρ
αααααααα
9Axial compressor theory - Stage-wise isentropic efficiency – Cheah CangTo
( )[ ]
( )[ ] ( )
( )mDP
mL
mmmmmmL
a
mmmL
Cc
sC
V
w
c
s
c
s
cV
sw
c
sC
cV
sw
c
sC
αααα
α
αα
ρ
ααα
αρ
ααααα
ρ
ααααα
tancostantan2
cos
tancos
2
1
costantan2
cos2
1
costancostantan2
2
1
costancostantan2
21
1
2
3
2
1
_
21
1
22
1
3_
21
2
3_
21
−−
=∴
××−−
=
−
−=
−
−=
a
a
cV
ca
2
2
1ρ
TURBO GROUP – Axial compressor theory - Stage-wise isentropic efficiency
CDP and CL can be calculated by using data from the following two curves.
note: These two curves are plotted based on extensive cascade tests
10Axial compressor theory - Stage-wise isentropic efficiency – Cheah CangTo
TURBO GROUP – Axial compressor theory - Stage-wise isentropic efficiency
Mean deflection
25
30
35
40
11Axial compressor theory - Stage-wise isentropic efficiency – Cheah CangTo
0
5
10
15
20
-20 -15 -10 -5 0 5 10
Incidence
TURBO GROUP – Axial compressor theory - Stage-wise isentropic efficiency
Mean stagnation pressure loss
0.05
0.06
0.07
0.08 2
1
_
2
1V
w
ρ
12Axial compressor theory - Stage-wise isentropic efficiency – Cheah CangTo
0.00
0.01
0.02
0.03
0.04
-20 -15 -10 -5 0 5 10
Incidence
TURBO GROUP – Axial compressor theory - Stage-wise isentropic efficiency
DSDADPD
DA
LDS
CCCC
h
sC
CC
++=
=
=
02.0
018.02
=3
2
_
cos1 αρ m
D
s
C
V
wCoefficient of loss:
Let ,theoretical static pressure rise:
( )
( )
( )22
2
2
2
1
2
2
2
2
1
2
2
_
2
2
1
2
2
secsec
cos
1
cos
1
2
tantan2
tantan2
ααρ
αα
ρ
ααρ
ααρ
−=∆
−=∆
−=∆
−−=∆
Vp
Vp
Vp
wV
p
a
altheoretica
altheoretica
a
a
a
a
0_
=w
13Axial compressor theory - Stage-wise isentropic efficiency – Cheah CangTo
1
2
2
1
cos
cos
2 α
αρ m
c
sV ( )
2
2
1
2
2
1
2
2
1
22
1
1
2
2
22
1
1
2
2
2
1
22
2
2
1
2
cos
cos1
2
cos
cos1
2sec
sec1
2
sec
sec1
2
sec
secsec2
α
α
ρ
α
αρ
α
αρ
α
ααρ
ααρ
−=∆
∴
−=
−=∆
−=∆
−=∆
V
p
VVp
Vp
Vp
ltheoretica
ltheoretica
altheoretica
altheoretica
a
a
a
∆
−=
2
1
2
1
_
2
1
2
1
1
V
p
V
w
ltheoretica
isentropic
ρ
ρ
η
TURBO GROUP – Axial compressor theory - Stage-wise isentropic efficiency
Example: Solar Mars 90 Calculated isentropic efficiency for every single stage.
14Axial compressor theory - Stage-wise isentropic efficiency – Cheah CangTo
TURBO GROUP – Axial compressor theory - Stage-wise isentropic efficiency
Example: LM2500Calculated isentropic efficiency for every single stage.
15Axial compressor theory - Stage-wise isentropic efficiency – Cheah CangTo
TURBO GROUP – Axial compressor theory - Stage-wise isentropic efficiency
Example: LM2500+Calculated isentropic efficiency for every single stage.
16Axial compressor theory - Stage-wise isentropic efficiency – Cheah CangTo
TURBO GROUP – Axial compressor theory - Stage-wise isentropic efficiency
Example: LM2500+G4Calculated isentropic efficiency for every single stage.
17Axial compressor theory - Stage-wise isentropic efficiency – Cheah CangTo
TURBO GROUP – Axial compressor theory - Stage-wise isentropic efficiency
Parameter LM2500 LM2500+ LM2500+G4 Unit
Overall performance
ISO power 23262 31076 33679 kW
heat rate 9611 8782 8782 kJ/kW.hr
Eff_thermal 37.46 40.99 40.99 %
gas power 29.93 40.06 44.03 MW
Comparison between LM2500, LM2500+ and LM2500+G4
18Axial compressor theory - Stage-wise isentropic efficiency – Cheah CangTo
Compressor
(ISO conditions)
gas power 29.93 40.06 44.03 MW
mass flow rate 68.50 83.80 89.50 kg/s
pressure ratio 17.90 21.50 23.00 -
number of stages 16 17 17 -
rotational speed 6885 6278 6124 rpm
t_out 711.13 749.12 761.90 Kelvin
delta T 422.98 460.97 473.75 Kelvin
Eff_isentropic * 84.26 84.19 84.44 %
End of note
* based on stage-wise isentropic efficiency