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Blade Nomenclature
Blade Nomenclature
Axial and Radial Flow TurbinesDifferences between turbine and compressor:
Long Short
Blade 1 Last blade
Compressor Turbine
►Work as diffuser►Work as nozzle
►Direction of rotation is opposite
to lift direction
►Direction of rotation is same as
Life
►Number of stages are many►Number of stages is small <3
►Temperatures are relative low
►Temperature is high, sometimes
blade cooling is required
Axial and Radial Flow TurbinesDifferences between Radial and Axial Types.
Radial(Centrifugal)Axial
►Used for small engines►Used for large engines
►Small mass flow rates►Large mass flow rates
►Lower efficiencies►Better efficiencies
►Cheap►Expensive
►Easy to manufacture►Difficult to manufacture
Axial Flow Turbines
Most of the gas turbines employ the axial flow turbines.
The chapter is concerned with axial flow turbines.
The radial turbine can handle low mass flows more efficiently than the axial flow machines.
Axial Flow TurbineElementary Theory of Axial Flow Turbine
► Velocity Triangles.
■ The velocity triangles for one axial flow turbine stage and the nomenclature employed are shown. The gas enters the row of nozzle blades with a static pressure and temperature P1, T1, and a velocity C1, is expanded to P2, T2, with an increased velocity C2 at an angle α2.
■ The rotor blade angle will be chosen to suit the direction β2 of the gas velocity V2 relative to the blade at inlet.
■ V2 and β2 are obtained from the velocity diagram of known C2, α2, and U.
Axial Flow Turbine• Elementary Theory The gas leaves the rotor at β3, T3, with relative velocity
V3 at an angle β3. C3 and α3 can be obtained from the velocity diagram.
Axial Flow Turbine► Single Stage Turbine ■ C1 is axial → α1 = 0, and C1 = Cα1. For similar
stages (same black shapes) C1 = C3, and α1 = α3, called repeating stage.
■ Due to change of U with radius, velocity triangles vary from root to tip of the blade.
Axial Flow Turbine► Assumptions
■ Consider conditions at the mean diameter of the annulus will represent the average picture of what happen to total mass flow.
■ This is valid for low ratio of tip radius to root radius.
■ For high radii ratio, 3-D effects have to be considered.
■ The change of tangential (whirl) mass is . This amount produces useful torque.
■ The change in axial component produces the axial thrust on the rotor.
■ Also there is an axial thrust due to P2 – P3. ■ These forces (net thrust on turbine rotor) are
normally balanced by the thrust on the compressor rotor.
Axial Flow Turbine
Axial Flow Turbine► Calculation of Work
Assume Ca= constant
2 3Ca CaCa
2 2
3 3
tan tan
tan tan
UCa
2 2 3tan tan tan tane
2 2tan tanU Ca Ca
) 1(
Axial Flow TurbineApplying principle of angular momentum
2 3
2 3
(
( )(tan tan )sW U C C
U Ca
From Equation (1)
2 3(tan tan )sW UCa
Steady-state energy equation:sp oW C T
Thus:
2 3(tan tan ) /
1.148, 1.333 and 41
so p
p
T U Ca C
C
Axial Flow TurbineElementary theory of axial flow turbine
1 3,
3,
1
1
1
1 3
1
1
11/
s isent
isent
isent
o s o
s o o
os o
o
s oo o
T T
T T
TT
T
TP P
Axial Flow Turbineηs is the isentropic stage efficiency based on
stagnation (total) temperature.
1 3
1 3
o os
o o
T TT T
1 3
1 3
( ) o os
o
T Ttotal to static
T T
(used for land-based gas turbines).
Definingψ = blade loading coefficient (temperature drop
coefficient)
2
2sp oC T
U
Axial Flow TurbineThus,
2 32 (tan tan ) /aC U Degree of reaction: 0 ≤ Λ ≤ 1
2,3 2,3 2 3
1,3 1 31,3
rotor
total
h T T TT T Th
For, Ca = const. and C3 = C1
1 31 3
2 3
( )
(tan tan )p p o o
a
C T T C T T
U C
and relative to rotor blades no work, thus
(a)
Axial Flow Turbine
2 22 3 3 2
2 2 23 2
2 2 23 2
1( )21 sec sec21 tan tan2
p
a
a
C T T V V
C
C
2 213 222 3
1 3 2 3
tan tan
(tan tana
a
CT TT T U C
13 2(tan tan )
2CU
Substitute in (a):
Axial Flow Turbine
3 2
3 2
2 22 2 2 23 2
2 2
2 2 2 23 2tan tan
a w a w
w w
a a
V V C C u C C u
C u C u
C C
2 3 3 2and
3 2 3 2andV V
3 2 3 2,C C aC
U
Λ = 0.5 → Symm. velocity triangles
● Λ = 0 : Impulse turbine
● Λ = 1 :
Defining flow coefficient:
Axial Flow Turbine2 1
3 2
2 (tan tan )
(tan tan )2
Adding:3
1 1tan 22 2
21 1tan 2
2 2
From:2 2
3 3
3 3
2 2
(tan tan )
(tan tan )
1tan tan
1tan tan
a
a
U C
U C
Axial Flow TurbineIf , Λ, and are assumed, blade angles can be determined. ● For aircraft applications:
3 < ψ < s, 0.8 < < 1 ● For industrial applications:
is less (more stages) is less (larger engine size)α3 < 20 (to min. losses in nozzle)
● Loss coefficient:
1 2
1
2 22( )2
2
/ 2n nozzle statorp
o oN
o
T TC C
P PY
P P
Λ and Y: The proportion of the leaving energy which is degraded by friction.
Axial Flow TurbineExample (Mean diameter design)Given:
1
1 3
1 3
1
Single-stage turbine= 20 kg/s= 0.9= 1100 K
Temperature drop, = 145 K
Pressure ratio, / = 1.873
Inlet pressure, = 4 bar
t
o
o o
o o
o
m
T
T T
P P
P
Assumptions:Rotational speed fixed by compressor: N = 250 rpsMean blade speed: 340 m/sNozzle loss coefficient:
2 222 / 2N
p
T TC C
Axial Flow Turbine
/t rr r
2 3 1 3
1
,
0a aC C C C
Calculation:a)Λ degree of reaction at mean radiusb)Plot velocity diagramsc)Blade height h, tip/root radius,
Assume:
3
2 2
2 2 1.148 145 10 2.88340
sp oC TU
flow coefficient 0.8aCU
The temperature drop coefficient:
Assume (try):
Axial Flow Turbine
3 31tan tan
3tan 1.25
31 1tan 2
2 20.28
■ To get Λ use
This is low as a mean radius value because Λ will be low or negative at the root.
This introduce a value for α3.Take α3 = 10°
* To calculate degree of reaction Λ:
■ Get β3:α3 = 0
Axial Flow Turbine3 3 3
3
1tan tan tan 1.426
1 1tan 22 2
0.421 (Acceptable)
Reaction at root should be checked.Thus α3 = 10°, β3 = tan-1 1.426 =
54.96
2
2
1 1tan 2 0.3742 20.4212.880.8
20.49
Axial Flow Turbine2 2
2
1tan tan 1.624
58.38
3 3 2 2, , , , U
/t rr r
11 :axial aC C
With knowledge of
plot velocity diagrams.
* Determine blade height h and tip/root radius ratio, .
Assumption:
Calculation of area at Section 2 (exit of nozzle)
Axial Flow Turbine2
2
2 1
2
2 2 2
22
2 2
340 0.8 272 m/s
cos 519 m/s
1100 K
5.9 K2
a
a
o o
op
C U
C C C
T T
CT T T
C
22
2 2
2
0.05 117.3 5.9 K2
976.8 K
Np
CT TC
T
1 1 1
2
/ 14
22
2.49 baroPo o
o
P TP
P T
Axial Flow TurbineFor the nozzle:
1
1
21 2
1 1
1
1/(2 ) 1 112 2
1 4 2.162 1.853
o p
oc
c
MT T C CM
T T
PP
P
P2 > Pc, the nozzle is not choked. 2, 2.49throatThus P P 32
2 22
22 2 2 2
2
22 2
22 2 2 2 2 2 2
0.833 /
, , m , 0.0833
throat area of nozzles; A
, m 0.0437 , also A cos
aa
P kg mRTmA or C A A mC
mNC
or C A N A N m A N
Axial Flow TurbineCalculate areas at section (1) inlet nozzle and (3) exit rotor.
3
1 1
1
1 1
1
1 1 3 33
21
1 1
11 1
1
311 1
1
21 1 1
, but C C , 276.4 /cos
10672
3.54
1.155 /
0.626
aa a
op
o o
a
CC C C and C m s
CT T T Kc
P T P barP T
P kg mRT
m C A A m
Axial Flow Turbine
3 1 5
3
3 3
o
23
3 3
13 3
3
333 5
5
Similarly at outlet of stage ( rotor)T 1100 145 955 ,
9222
1.856
0.702 /
o o
op
o o
T T K given
CT T T K
c
P T P barP T
P kg mRT
3
23 5 5
23 3 3
3/ 0.702 /
0.1047
Blade height and annulus radius ratio a
P RT kg m
m C A A m
Axial Flow TurbineMean radius
m
3402 0.2162 (250)
for known (A); A 2 r
m m mu Nr r m
also h
t r , 2 2 2m r m
m
A h hh then r r rr
using areas at stations 1,2,3 thus
21mAmh1
/t rr r
Location123
0.06260.08330.1047
0.040.06120.077
1.241.331.43
Axial Flow TurbineBlade with width WNormally taken as W=h/3Spacing s between axial blades
t
r
a t
space 0.25, should not be less than 0.2 Wwidth
r* should be 1.2 1.4r
unsatisfactory values such as 0.43 can be reduced by changing axial velocity through .increasing C reduce r check has to
sw
will
v be made for mach number M .
Axial Flow TurbineVortex TheoryThe blade speed ( u=r) changes from root to tip, thus velocity triangles must vary from root to tip.
Free Vortex designaxial velocity is constant over the annulus.Whirl velocity is inversely proportional to annulus.
,C ,tan
tanC ,tan
33
22
constrtconsC
tconsrtconsC
a
a
Along the radius.
2 3 2 3( ) tansW u C C C r C r cons t
Axial Flow TurbineFor variable density, m is given by
t
r
r
ra
a
rdrCm
Crrm
2
2
2
2
2
)2(
2 2
2
2
a 2
2 22
3 33
tan tan
C cosntant, thus changes as
tan tan (a)
tan tan (b)
a
mm
mm
C r cons t r C
but is
rr
similarlyrr
Axial Flow Turbine2 2
2
2
s 3
3
2 2 2 2
m2
2
a 3 3
3 33 3
tan tan , , tan tan
r tan (c)r
for exit of rotor u C tan tan
tan tan (d)
a aa
m
m a
a
mm
m a
uu C C thusC
urmr C
C
r r uthusr r C
Ex: Free vortexResults from mean diameter calculations
2 2m 3
3 2
3
58.38, 20.49, 10 ,54.96, 0.0612, 0.216,
0.077,2
om m
m m
r m
h rhh r r
Axial Flow Turbine
Tip54.9308.5258.33
Root62.1539.3212.1251.13
mean58.3820.491054.96
3 3
2 3
232 3
m
a
1.164, ( ) 0.877, 1.217, 0.849
u 1also 1.25, Results areC
m m m m
t t r t
m
a
r r r rr r r r
uC
2 2
Axial Flow Turbine
'3o1o
3o1os
)1/(
1o
3o1os
1o
'3o
1os3o1oos
12a12a
32a32a3o1oposp
3322a
TTTT where
))pp(1(T)
TT1(TTTT
)tan(tanUCm)tan(tanUCm
)tan(tanUCm)tan(tanUCm)TT(cmTcmW
tantantantanCU
EES Design Calculations of Axial Flow Turbine
Known Information
To1 = 1100 [K]
P ratio = 1.873
DelTs = 145
Etta turbine = 0.9
Assumptions
U = 340 [m/s]
N rps = 250
= 0.8
3 = 10
Loss nozzle = 0.05
EES Design Calculations of Axial Flow Turbinecp = 1148 R = 0.287 = 1.333
DelTs = To1 – To3
P ratio = Po1
Po3
Ca = C2 · cos ( 2 )
= CaU
Gamr =
– 1
Epsi = 2 · cp · DelTs
U 2
Epsi = 2 · · ( tan ( 2 ) + tan ( 3 ) )
Reaction =
2 · ( tan ( 3 ) – tan ( 2 ) )
U = Ca · ( tan ( 2 ) – tan ( 2 ) )
U = Ca · ( tan ( 3 ) – tan ( 3 ) )
EES Design Calculations of Axial Flow TurbineCalculate A2
Loss nozzle = T2 – T2dash
C2 2
2 · cp
To2 = To1
To2 – T2 = C2 2
2 · cp
Po1
P2 =
To1
T2dash
Gamr
Po1
Pc =
+ 12
Gamr
Pth = P2
Rho2 = Pth
R · T2
A2 = m
Rho2 · Ca
A2 · cos ( 2 ) = A2N
EES Design Calculations of Axial Flow Turbine
Calculate A1
To1 – T1 = C1 2
2 · cp
Po1
P1 =
To1
T1
Gamr
Rho1 = P1
R · T1
C1 = Ca
A1 = m
Rho1 · Ca
Calculate A3
To3 – T3 = C3 2
2 · cp
Po3
P3 =
To3
T3
Gamr
Rho3 = P3
R · T3
C3 = Ca
A3 = m
Rho3 · Ca
EES Design Calculations of Axial Flow Turbine
Blade height
U = 2 · · N rps · rm
Blade height at section 1
A1 = 2 · · rm · h1
r t1 = rm + h12
r r1 = rm – h12
rratio1 = r t1
r r1
Blade height at section 2
A2 = 2 · · rm · h2
r t2 = rm + h22
r r2 = rm – h22
rratio2 = r t2
r r2
Blade height at section 3
A3 = 2 · · rm · h3
r t3 = rm + h32
r r3 = rm – h32
rratio3 = r t3
r r3
EES Design Calculations of Axial Flow Turbine
A1 = 0.06345 A2 = 0.08336 A2N = 0.04372 A3 = 0.1046 2 = 58.37
3 = 10
2 = 20.49 3 = 54.97 C1 = 272 C2 = 518.7
C3 = 272 Ca = 272 cp = 1148 [J/kgK] DelTs = 145 Epsi = 2.88
Ettaturbine = 0.9 = 1.333 Gamr = 4.003 h1 = 0.04666 h2 = 0.06129
h3 = 0.07692 Lossnozzle = 0.05 m = 20 [kg/s] Nrps = 250 [rev per sec] P1 = 355.1
P2 = 248.8 P3 = 186.1 Pc = 215.9 = 0.8 Po1 = 400 [kPa]
Po3 = 213.6 Pth = 248.8 Pratio = 1.873 R = 0.287 [kJ/kgK] Reaction = 0.4211
Rho1 = 1.159 Rho2 = 0.8821 Rho3 = 0.7029 rratio1 = 1.242 rratio2 = 1.33
rratio3 = 1.432 rm = 0.2165 rr1 = 0.1931 rr2 = 0.1858 rr3 = 0.178
rt1 = 0.2398 rt2 = 0.2471 rt3 = 0.2549 T1 = 1068 T2 = 982.8
T2dash = 977 T3 = 922.8 To1 = 1100 [K] To2 = 1100 [K] To3 = 955
U = 340 [m/s]
Axial Flow Turbine
Axial Flow Turbine
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