chalmers university of technology discussion on design task 1 elementary axial turbine theory...
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Chalmers University of Technology
• Discussion on design task 1
• Elementary axial turbine theory– Velocity triangles
– Degree of reaction
– Blade loading coefficient, flow coefficient
• Problem 7.1
• Some turbine design aspects– Choice of blade profile, pitch and chord
Lecture 7 – Axial flow turbines
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Axial flow turbines
• Expansion occurs in stator and in relative frame of rotor
• Working fluid is accelerated by the stator and decelerated by the rotor
• Boundary layer growth and separation does not limit stage loading as in axial compressor
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Elementary theory• Energy equation for control
volumes (again):
0103
0103
21
1
23
3
00103
22TTchh
Vh
Vhwq p
gasPerfect
hh
• Adiabatic expansion process (work extracted from system - sign convention for added work = +w)– Rotor => -w = cp(T03-T02) <=>
w = cp(T02-T03)– Stator => 0 = cp(T02-T01)
=> T02= T01
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How is the temperature drop related to the blade angles ?
• We study change of angular momentum at mid of blade (as approximation)
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Governing equations and assumptions• Relative and absolute refererence
frames are related by:
23
velocityrelativefor direction
of change Assume
23
22332233
2233
radiusconstant at Flow
wwww
wwww
ww
CCUCCU
UCUCrCrCworklTheoretica
torquelTheoreticarCrC
momentumangularofchangeofRate
UCV • We only study designs where:
– Ca2=Ca3
– C1=C3
• You should know how to extend the equations!!!• We repeat the derivation of theoretical work used
for radial and axial compressors:
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Principle of angular momentum
Stage work output w:
3322
32
tantan aa
ww
CCU
CCUw
Ca constant:
32
3322
tantan
tantan
a
aa
UC
CCUw
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Combine derived equations =>
32 tantan aUCw
stagep Tcw ,0
Exercise: derive the correct expression when 3 is small enough to allow 3 to be pointing in the direction of rotation.
(7.3) tantan 32,0 astagep UCTc
Energy equation
Energy equation:
We have a relation between temperature drop and blade angles!!! :
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Dimensionless parametersBlade loading coefficient, temperature drop coefficient:
(7.6) tantan2
7.3Equation
21 32
2
,0
U
C
U
Tcastagep
Degree of reaction:31
32
TT
TT
Exercise: show that this expression is equal to =>when C3= C1 0301
32
TT
TT
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can be related to the blade angles!
C3 = C1 =>
32,0 tantan UCTcTc stagepstagep
Relative to the rotor the flow does no work (in the relative frame the blade is fixed). Thus T0,relative is constant =>
22
3222
22
3 tantan2
1
2
1 arotorp CVVTc
Exercise: Verify this by using the definition of the relative total temperature: p
relative c
VTT
2
2
,0
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can be related to the blade angles!
Plugging in results in definition of =>
(7.7) tantan2 23
31
32
U
C
TT
TT a
The parameter quantifies relative amount of ”expansion” in rotor. Thus, equation 7.7 relates blade angles to the relative amount of expansion. Aircraft turbine designs are typically 50% degree of reaction designs.
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Dimensionless parameters Finally, the flow coefficient:
5.0
0.50.3
0.18.0
Current aircraft practice (according to C.R.S):
U
Ca
Aircraft practice => relatively high values on flow and stage loading coefficients limit efficiencies
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Dimensionless parameters Using the flow coefficient in 7.6 and 7.7 we obtain:
(7.8) tantan2 32
The above equations and 7.1 can be used to obtain the gas and blade angles as a function of the dimensionless parameters
22
1
2
1 tan 2
22
1
2
1 tan 3
1
tan tan 22
1
tan tan 33
(7.9) tantan2
23
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• Exercise: show that the velocity triangles become symmetric for = 0.5. Hint combine 7.1 and 7.9
• Exercise: use the “current aircraft practice” rules to derive bounds for what would be considered conventional aircraft turbine designs. What will be the range for 3? Assume = 0.5.
Two simple homework exercises
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Turbine loss coefficients:Nozzle (stator) loss coefficients:
202
0201
22
22
2
pp
ppY
cC
TT
N
p
N
3,03
,03,02
22
33
2
pp
ppY
cV
TT
rel
relrelR
p
R
Nozzle (rotor) loss coefficients:
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Problem 7.1
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3D design - vortex theory
• U varies with radius
• Cw velocity component at stator exit => static pressure increases with radius => higher C2 velocity at root
• Twist blades to take changing gas angles into account– Vortex blading
3D optimized blading (design beyond free vortex design)
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3D design in steam turbines• Keep blade angles from
root to tip (unless rt/rr high)
• Cut cost• Rankine cycle relatively
insensitive to component losses
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Choice of blade profile, pitch and chord• We want to find a blade that will minimize loss and perform the required
deflection
• Losses are frequently separated in terms:
s losssecondary termone into Grouped
loss flowSecondaryAnnulus
Lossclearance Tip
cascadein Measured
ProfileTotal
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Choice of blade profile, pitch and chord• As for compressors - profile families are used for thickness distributions.
For instance:– T6, C7 (British types)
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Choice of blade profile, pitch and chord• Velocity triangles determine gas angles not blade angles.
– arccos(o/s) should approximate outflow air angle:
• Cascade testing shows a rather large range of incidence angles for which both secondary and profile losses are relatively insensitive
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Choice of blade profile, pitch and chord• Selection of pitch chord:
– Blade loss must be minimized (the greater the required deflection the smaller is the optimum s/c - with respect to λProfile loss)
– Aspect ratio h/c. Not critical. Too low value => secondary flow and tip clearence effects in large proportion. Too high => vibration problems likely. 3-4 typical. h/c < 1 too low.
– Effect on root fixing• Pitch must not be too small to allow safe fixing to turbine disc rim