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Chalmers University of Technology
Axial compressors 1 + compEDU tutorial
• Elementary axial compressor theory– Velocity triangles– Limitations for compressor performance
• relative Mach number limitations• deflection limitation
– Degree of reaction
• compEDU– Axial compressor tutorial– Overhaul and maintenance lab specification
Chalmers University of Technology
Axial flow compressors
• Working fluid is accelerated by the rotor and decelerated by the stator– Boundary layer growth and
separation (stall) limits the rate of allowable diffusion
• Diffusion (decrease of velocity and increase of static pressure) occurs in stator and in relative frame of rotor
Chalmers University of Technology
Elementary theory• Energy equation for control
volumes (again):
• Adiabatic compression process (work added to system - sign convention added work = -w)– Rotor => -(-w) = cp(T02-T01)
<=> w = cp(T02-T01)– Stator => 0 = cp(T03-T02) => T03= T02
0103
0103
21
1
23
3
00103
22TTchh
Vh
Vhwq p
gasPerfect
hh
Chalmers University of Technology
How is the temperature rise related to the blade angles ?
• We study change of angular momentum at mid of blade (as approximation)
Chalmers University of Technology
Theory 9.1 – Stage temperature rise• Relative and absolute refererence frames are related by:
C = V + U
• Many compressors have been designed assuming Ca=Ca1=Ca2. We will assume this for the following derivation
• We repeat the derivation of theoretical work used for radial compressors and axial turbines:
12
11221122
1122
radiusconstant at Flow
ww
wwww
ww
CCU
UCUCrCrCworklTheoretica
rCrCtorquelTheoretica
momentumangularofchangeofRate
Chalmers University of Technology
Theory 9.1 – Stage temperature rise• Combining that flow occurs at a constant radius (=>U2=U1) and that the axial velocity is assumed to be constant
(design assumption) we get:
2112
2211
22
11
tantantantan
tantantantan
tantan
tantan
22
11
ww
ww
V
a
C
a
V
a
C
a
CCU
CCU
Chalmers University of Technology
Theory 9.1 – Stage temperature rise• Using the trigonometric relation from above together with the work relation we get:
tantan
relation rictrigonomettantan
21
1212
a
aww
mUC
mUCCCmUW
• Introducing the derived relation for work into the energy equation finally yields relationbetween air angles and temperaturerise:
210102 tantan p
a
c
UCTT
Chalmers University of Technology
Conclusions
210102 tantan p
a
c
UCTT
To obtain a high temperature rise we should:
• High blade speed (U)• High axial speed (Ca)• High fluid deflection (β1- β2)
Blade stresses and aerodynamic considerations limit these design selections
• The isentropic efficiency then relates the blade angles to the pressure rise of the stage
Chalmers University of Technology
Axial velocity and Mach numbers
• Relative Mach number greatest at blade tip. Assuming axial and constant velocity over rotor entry:
• The static temperature is: –
• Speed of sound is: –
pc
CTT
2
21
011
• Typical (high) values : C=200 m/s, U = 450 m/s => Mrel,tip= 1.5 (check this yourself)
22tt UCV
1RTa
Chalmers University of Technology
• The relative velocity decreasesin the rotor. – Too rapid retardation => separation and excessive losses. – A design criteria to limit the retardation is set by the de Haller number:
72.01
2 V
V
• High fluid deflection => high rate of diffusion.
Fluid deflections and limitations
Chalmers University of Technology
• A better estimate of diffusion can be derived if pitch and chord is taken into account.– Greatest diffusion from Vmax to V2 on
suction side. Here, boundary layer growth will be most severe => largest part of losses created in this region
– We approximate the diffusion, D, by this velocity change according to:
Fluid deflections and limitations
c
s
V
C
V
V
V
VcsC
V
VforresultsTestV
VVD
w
numberHallerde
w
11
2
1
21
max1
2max
212
Chalmers University of Technology
Blockage
210102 tantan a
p
UCc
TT
21
21
tantan
tantan
a
aa
CUmU
CCUmUW
• Boundary layer growth at annulus walls creates a peaky flow profile
• For a fixed design (α 1 and β 2) can no longer be varied within the diffusion constraints), increasing Ca leads to a decrease in work output:
• This is approximated by the use an empirical factor - the work done factor λ according to:
Chalmers University of Technology
Degree of reaction• Diffusion takes place in both rotor and stator.
– The division characterizes the design– The quantity measuring this division is the
degree of reaction - Λ :
stageinriseenthalpystatic
rotorinriseenthalpystatic
• We will derive Λ assuming:– variation in cp over temperature ranges is
neglible => we can use temperatures– Ca constant– C3 = C1 => Δ TStage= Δ T0,Stage
Chalmers University of Technology
Degree of reaction• Let Δ TA and Δ TB denote the static temperature rise in the rotor
and stator respectively. Then: 1221 tantantantan aaSpBAp UCUCTcTTcW
• Use that all work input occurs in the rotor, i.e.
21
22
21
1
22
20102 2
1
22CCTc
c
CT
c
CTcTTcW Ap
pppp
• Combining the two relations yields:
2
1
2
212
11
22
21
2212
21
22
coscos2
1tantan
cos,
cos
2
1tantan
2
1
aaa
aa
aAp
CCUC
CC
CC
CCUCCCWTc
Chalmers University of Technology
Degree of reaction
Apa
a
aa
aa
TcC
UC
CUC
CUC
12
22
2
12
12
12
12
22
22
222
12
22
12
22
2
12
tantan2
tantan
cos
sincos
cos
sincos
2tantan
1sincos that Usecos
1
cos
1
2tantan....
• From the definition of Λ we then have:
12
12
1212
2
12
12
12
22
2
12
tantan2
1
tantan
tantantantan2
tantan
tantan
tantan2
tantan
U
C
UC
CUC
UC
CUC
TT
T
a
a
aa
a
aa
BA
A
Chalmers University of Technology
Degree of reaction
equationsSum
2121
2211
tantan2
tantan2
1
tantantantan2
U
C
U
C
C
U
aa
a
Basic velocity triangles again:
22
11
tantan
tantan
a
a
C
U
C
U
2112 tantan2
tantan2
1 U
C
U
C aa
Thus, we get:
Chalmers University of Technology
Learning goals• Know how to relate blade angles to stage
temperature rise• Understand how fluid mechanics limits the
performance of axial compressor design: – Mach number limitations– Blockage
• Have a basic insight of gas turbine overhaul and maintenance as given by compEDU tutorial