THEORY OF PROPULSION 10' Blade Element Theory

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THEORY OF PROPULSION 10. Blade Element Theory

• P. M. SFORZA
• University of Florida

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Quasi-two-dimensional flow in an infinitesimal
annular cylinder
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Force field on blade element
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Circulation-based lift
Frictionless flow in straight cascades gave the
following result
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Combined velocity diagram
w2 wm w3 ca
c2 c3
b2 bm b3
a2 a3
w3u
wmu
w2u
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Force field geometry
e
bm
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Turning and axial forces from lift and drag forces
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Momentum and blade theory compared
Blade element theory
Cascade momentum analysis for frictionless flow
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Combination of momentum and blade element analyses
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Power and the blade element results
Assuming a constant average density for the flow
in the stage yields
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Degree of reaction
The parameter fca/u is a typical measure of mass
flow through the stage
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R and the mass flow function f
where
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Influence of the trailing edge
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Influence of the trailing edge (cont.)
constantB
f ca/u determines r
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The pressure coefficient y
where
The pressure coefficient is
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Pressure coefficient vs mass flow coefficient
Typical range for lightly loaded stage
compressors
fans
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Non-dimensional combined velocity diagrams
rrfrictionlessfcotbm
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Rotor and stator forces
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The deflection coefficient t
From the combined velocity diagram we see that
Adding or subtracting yields
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The theoretical degree of reaction r
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Rotor and stator forces
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Rotor and stator forces (cont.)
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Pressure change across rotor and stator with
friction
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Power required
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Efficiency of compression
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Maximum efficiency
erese
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Contour plot of efficiency
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Problem 12 50 stage turbine
12. A 50 reaction stage axial flow turbine with
u366m/s at the mean line of the rotor where
d76.2cm has Tt1144K and pt687kPa entering the
stator from the combustor The blade angle a425o.
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Problem 12 combined velocity diagram
WuDcuu(c4u c5u)(1/2)(c42-c52) (w42-w52)
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Problem 12 Work extracted by turbine
Because c4w5 and c5w4 either equality holds
W (c42-w42) u(c4cosa4-w4cosb4)
And since w4cosb4u-c4cosa4
W u(2c4cosa4-u) c422(u/c4)cosa4 (u/c4)2
W/c42 2hbcosa4 hb2
hbu/c4
Where the blade efficiency
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Problem 12 Temperature leaving rotor
Maximum efficiency occurs when hbcosa4 or u
c4cosa4
Wmax. blade eff. c42cos2a4 u2
Wmax. blade eff. (366m/s)2134kJ/kg
Wcp(Tt4-Tt5) (1.16kJ/kg-K)(1144K-Tt5)134kJ/kg
Tt51029K
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Problem 12 Flow properties
T4 Tt4 c42/2cp 1144K (404m/s)2/2(1160m2/s
2-K) 1074K a4 (gRT4)1/2 (1.33287m2/s2-K1
074K)640m/s M4 (404m/s)/(640m/s) 0.631 M4
(171m/s)/(640m/s) 0.267 (relative to rotor at
rotor exit) T5 Tt5 c52/2cp 1029K
(171m/s)2/2(1160m2/s2-K) 1016K a5 (gRT5)1/2
(1.33287m2/s2-K1016K)622m/s M5(171m/s)/(622
m/s)0.275 M5(404m/s)/(622m/s) 0.649
(relative to rotor at rotor exit)
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The isentropic relation yields
p4 pt410.5(g-1)M42 -g/(g-1) 685kPa1M42/6
-4 532kPa
The equation of state yields
r4p4/RT4(532kPa)/(0.287kJ/kg-K)(1074K)
1.73kg/m3
The conservation of mass requires
mr4Aca4 r5Aca5
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Problem 12 Mass flow and power
p5 r5RT5(1.73kg/m3)(0.287kJ/kg-K)(1016K) 504kPa
pt5 p510.5(g-1)M52 g/(g-1) 504kPa1M52/6 4
529kPa
Flow area A0.95p(ro2-ri2)0.95p(0.3810.0635)2-
(0.381-0.0635)20.304m2 Mass flow through the
turbine mr4Aca4 r4(0.304m2)ca(1.73kg/m3)(0.30
4m2)(171m/s) m 85.3 kg/s Power developed by
the turbine is PmW (134kJ/kg)(85.3kg/s)11.45M
W
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Problem 12 Flow properties through the blade
passage
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GE CF6-50 Characteristics
Low pressure turbine P43.6MW High pressure
turbine P51.7MW Core mass flow
m125kg/s Fan mass flow m540kg/s Burner exit
temperature Tt42940R
Used on Boeing B747 and Airbus A300