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Lecture 8 – Axial turbines 2 + radial compressors 2. Axial turbines Turbine stress considerations The cooled turbine Simplified 3D axisymmetric inviscid flow Free vortex design method Radial compressors 2 Diffuser and vaneless space Compressor maps. - PowerPoint PPT Presentation
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Chalmers University of Technology
Lecture 8 – Axial turbines 2 + radial compressors 2
• Axial turbines– Turbine stress considerations– The cooled turbine– Simplified 3D axisymmetric inviscid flow
• Free vortex design method
• Radial compressors 2– Diffuser and vaneless space– Compressor maps
Chalmers University of Technology
Choice of blade profile, pitch and chordRotor blade stresses:
1 centrifugal stress:
2 gas bending stresses reduce as cube of chord:
3 centrifugal bending stress
Annulus area
ns)interactio estator wak toduen fluctuatio to
subject(MN/m 93 ... exampleour
1
22
3
velocityin whirl Change
32max
zc
h
n
CCm mwmwgb
bar 2000MN/m 200]example fromgeometry [
3
4 taperNormal
2
22
max
alloy
CoCrNi
b
t
rr
bct ANardr
a
Steady stress/Creep Combination steady/fluctuating
Chalmers University of Technology
The cooled turbine• Cooled turbine
– application of coolant to the nozzle and rotor blades (disc and blade roots have always been cooled). This may reduce blade temperatures with 200-300 K.
– blades are either: • cast - conventional, directionally solidified, single crystal
blade• forged
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The cooled turbineTypical cooling
distribution for stage:
Distribution required for operation at 1500 K
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The cooled turbine - methods• Air cooling is divided into the following
methods– external cooling
• Film cooling
• Transpiration cooling
– internal cooling
Techniques to cool rotor blade
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The cooled turbine - methods
Techniques to cool stator blade
• Stator cooling– Jet impingement cools the hot leading
edge surface of the blade.
– Spent air leave through slots in the blade surface or in the trailing edge
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3D axi-symmetric flow (inviscid)• Allow radial velocity components.
– Derive relation in radial direction– Balance inertia, FI, and pressure forces
(viscous forces are neglected)
• Derived results can be used to interpret results from CFD andmeasurements
onaccelerati
SS
SS
Sw
massdirectionblade
inwidthunit
streamlinealongonacceleratiRadial
iii
streamlinecurvedtodueforceRadial
ii
forcelCentripeta
iI
dt
dC
r
C
r
Cdrrd
FFFF
sincos22
)(
)()(
Chalmers University of Technology
3D flow (inviscid)• Pressure forces FP balancing the
inertia forces in the radial direction are:
rdpdpdrdprdrdpdpdrdprd
termsorderhighneglect
drddp
pprdddrrdppF
dd
P
222)(
directionradialin actingelement of
sideson force Pressure
22sin
• Equating pressure forces and inertia forces yields:
SS
SS
Sw
dt
dC
r
C
r
C
dr
dp
sincos1 22
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3D flow
• The above equation will be usedto derive an energy relation.
r
C
dr
dp w21
• For many design situations rs can beassumed to be large and thus αs small.These approximations give the radial equilibrium equation:
Chalmers University of Technology
3D flow
• The radial variation is therefore:
222
0 2
1
2 wa CChC
hh
• The stagnation enthalpy at any radius is (neglecting radial components):
dr
dCC
dr
dCC
dr
dh
dr
dh ww
aa 0
• We have the thermodynamic relation:
which produces:
dp
dhTds
2
1
11
dr
dp
dr
dsTtermsorderhigherneglect
dpdr
d
dr
dp
dr
dTds
dr
dsT
dpTds
dr
d
dr
dh
Chalmers University of Technology
3D flow• We now have:
dr
dCC
dr
dCC
r
C
dr
dsT
dr
dCC
dr
dCC
dr
dp
dr
dsT
dr
dh
ww
aa
w
ww
aa
termmequilibriuradialThe
2
0 1
• If we neglect the radial variation of entropy we get the vortex energy equation:
dr
dCC
dr
dCC
r
C
dr
dh ww
aa
w 2
0
Chalmers University of Technology
Theory 8.1 – The free vortex design methodUse:
and design for:– constant specific work at all radii– maintain Ca constant across the annulus
11222
1
1
1
221
1
212
2
1
nIntegratio2
lnln)ln( ln)ln()ln(
)ln( 0
rCrCr
r
r
rr
C
CCC
Cr
dr
C
dC
dr
dCC
r
C
www
www
ww
www
w
Thus Cwr must be kept constant to fulfill our design assumption.This condition is called the free vortex condition
– Designs based on free vortex principle sometimes yields a marked variation of degree of reaction with radius
dr
dCC
dr
dCC
r
C
dr
dh ww
aa
w 2
0
Chalmers University of Technology
Design methods (Λ m = 0.50)
Free vortex blading (n = -1)gives the lowest degree of reaction in the root region!
• For low root tip ratios a high degreeof reaction is required in the mid to ensure positive reaction in the root
Chalmers University of Technology
Free vortex design - turbines• We have shown that if we assume
– constant specific work at all radii, i.e. h0 constant over annulus (dh0/dr=0)
– maintain Ca constant across the annulus (dCa/dr=0)
• We get– Cwr must then be kept constant to satisfy
the radial equilibrium equation
• Thus we have Cw r = Ca tanα r r = constant. But Ca constant => tanα r r = constant, which leads to the radial variations:
mm
mm
r
r
r
r
33
3
22
2
tantan
tantan
33
33
3
22
222
22
tantan
tantantan
a
m
mm
m
a
m
mm
m
a
C
U
r
r
r
r
C
U
r
r
r
r
C
U
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Radial compressor 2 - General characteristics
• Suitable for handling small volume flows– Engines with mass flows in this range will have very small geometrical
areas at the back of an axial compressor when operating at a pressure ratio of around 20.
– Typical for turboshaft or turboprop engines with output power below 10MW
• Axial compressor cross section area may only be one half or a third of the radial machine
• Better at resisting FOD (for instance bird strikes)
• Less susceptible to fouling (dirt deposits on blade causing performance degradation)
• Operate over wider range of mass flow at a particular speed
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Development trends
3
2
• Pressure ratios over 8 possible for one stage (in production – titanium alloys)
• Efficiency has increased around % per year the last 20 years
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Axial centrifugal combination - T700
Chalmers University of Technology
The vaneless space - diffuser
!!!Constant
0
rC
space
vanelessinTorque
w
)(γ
)(γ
r Mγ
MAP
RTm 12
1
2
0
0
2
11
Use Cw and guessed
Cr => C => T => M, Mr
Perform check on area (stagnation properties
constant):
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The diffuser• Boundary layer growth
and risk of separation makes stagnation process difficult
• Diffuser design will be a compromise between minimizing length and retaining attached flow
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Shrouds• Removes losses in
clearance.
• Not used in gas turbines– Add additional mass– Unacceptable for high
rotational speed where high stresses are produced
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Non-dimensional numbers - mapsWe state that:
design of scalelinear
viscositykinematic
speed rotational
),,,,,,,,(
),,,,,,,,(
01012
0101102
D
n
DdesignRnTPmf
DdesignRnTPmfP
c
based on the observation that we can not think of any more variables on which P02 and ηc depends.
Chalmers University of Technology
Non-dimensional parameters
• Nine independent parameters
• Four primary variables– mass, length, time and temperature
• 9 - 4 = 5 independent non-dimensional parameters– According to pi teorem.
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Non-dimensional numbers• Several ways to form non-dimensional
numbers exist. The following is the most frequently used formulation:
),,,,(
),,,,(
2
012
01
012
2
012
01
011
01
02
designnD
RT
nD
DP
RTmf
designnD
RT
nD
DP
RTmf
P
P
c
Chalmers University of Technology
Non-dimensional numbersFor a given design and working fluid we obtain:
),,(
),,(
number Re
2
012
01
012
number Re
2
012
01
011
01
02
nD
RT
nD
DP
RTmf
nD
RT
nD
DP
RTmf
P
P
c
Compressors normally operate at such high Reynolds numbers that they become independent of Re!!!
Chalmers University of Technology
Non-dimensional numbersWe arrive at the following expressions:
),(),(
),(),(
0101
012
012
01
012
0101
011
012
01
011
01
02
T
n
P
Tmf
RT
nD
DP
RTmf
T
n
P
Tmf
RT
nD
DP
RTmf
P
P
c
Compressors normally operate at such high Reynolds numbers that they become independent of Re!!!
Chalmers University of Technology
Compressor maps• Data is usually collected
in diagrams called compressor maps– What is meant by surge– What happens at
right-hand extremities of rotational speed lines
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SurgeWhat will happen in point D if mass flow drops infinitesimally– Delivery pressure drops– If pressure of air downstream of
compressor does not drop quickly enough flow may reverse its direction
– Thus, onset of surge depends on characteristics of compressor and components downstream
Surge can lead to mechanical failure
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Choke• What happens for increasing
mass flow?– Increasing mass flow
– Decreasing density
– Eventually M = 1 in some section in impeller (frequently throat of diffuser
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Overall turbine performance
• Typical turbine map– Designed to choke in stator
– Mass flow capacity becomes independent of rotational speed in choking condition
– Variation in mass flow capacity below choking pressure ratio decreases with number of stages
– Relatively large tolerance to incidence angle variation on profile and secondary losses give rise to limited variation in efficiency with rotational speed
Chalmers University of Technology
Learning goals• Have a basic understanding of how cooling is introduced
in gas turbines• Be familiar with the underlying theory and know what
assumptions the radial equilibrium design principle is based on
• Have some knowledge about – the use and development of radial compressor
– the physics governing the diffuser and
vaneless space • Understand what are the basis for compressor and
turbine maps.– Know about limitations inherent to the maps
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