Velocity Triangles. ⡠The velocity triangles for one axial flow turbine stage and ... direction β2 of the gas velocity V2 relative to the blade at inlet. ⡠V2 and β2 ...
Blade Nomenclature
Blade Nomenclature
Axial and Radial Flow Turbines Differences between turbine and compressor: Compressor
Turbine Blade 1
Long
Last blade
Short
► 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 Pc, the nozzle is not choked. Thus, Pthroat = P2 = 2.49 P ρ 2 = 2 → ρ 2 = 0 .8 3 3 kg / m 3 R T2 m A2 = , o r , m = ρ 2 C a A 2 , A 2 = 0 .0 8 3 3 m 2 ρ 2C a m th ro at area o f n o zzles; A 2 N = ρ 2C 2 o r , m = ρ 2 C 2 A 2 N ⇒ A 2 N = 0 .0 4 3 7 m 2 , also A 2 co s α 2 = A 2 N
Axial Flow Turbine Calculate areas at section (1) inlet nozzle and (3) exit rotor. C a1 = C 1 , b u t C 1 = C 3 a n d C 3 = T1 = T o1 −
C a3 cos α 3
, → C a 1 = 2 7 6 .4 m / s
C 12 → T1 = 1 0 6 7 K 2c p γ
T γ −1 P1 = 1 → P1 = 3 .5 4 b a r To Po1 1 P1 ρ1 = ⇒ ρ 1 = 1 .1 5 5 k g / m 3 R T1 m = ρ 1 C a1 A1 ⇒ A1 = 0 .6 2 6 m 2
Axial Flow Turbine S im ila rly a t o u tle t o f s ta g e ( ro to r) T o 3 = T o1 − ∆ T o 5 = 1 1 0 0 − 1 4 5 = 9 5 5 K , g iv e n T3 = To3 −
C 32 ⇒ T3 = 9 2 2 K 2c p γ γ −1
T P3 = 3 ⇒ P3 = 1 .8 5 6 b a r To Po 3 3 P ρ 3 = 3 ⇒ ρ 5 = 0 .7 0 2 k g / m 3 R T5
ρ3 = P3 / RT5 ⇒ ρ5 = 0.702kg / m 2 m = ρ3Ca3 A3 ⇒ A3 = 0.1047 m 2 Blade height and annulus radius ratio
Axial Flow Turbine Mean radius 340 = 0.216 m 2π (250) also for know n (A); A = 2 π rm h
u m = 2π N rm ⇒ rm =
⇒h=
A 2π rm
h h then rt = rm + , rr = rm − 2 2
using areas at stations 1,2,3 thus Location
A1 m h1m rt / rr
2
1
2
3
0.0626
0.0833
0.1047
0.04
0.0612
0.077
1.24
1.33
1.43
Axial Flow Turbine Blade with width W Normally taken as W=h/3 Spacing s between axial blades
space s = = 0.25, should not be less than 0.2 W width w r * t should be 1.2 → 1.4 rr unsatisfactory values such as 0.43 can be reduced by changing axial velocity through φ . increasing Ca will reduce rt check has to be made for mach number M v .
Axial Flow Turbine Vortex Theory The blade speed ( u=ωr) changes from root to tip, thus velocity triangles must vary from root to tip. Free Vortex design axial velocity is constant over the annulus. Whirl velocity is inversely proportional to annulus.
C a2 = cons tan t , Cω 2 r = cons tan t C a3 = cons tan t , Cω 3 r = const, Along the radius.
(
)
Ws = u Cω2 + Cω3 = ω (Cω2 r + Cω3 r ) = cons tan t
Axial Flow Turbine For variable density,
m is given by
δ m = ρ 2 ( 2π r δ r )C a rt
m = 2π C a 2
∫
2
ρ 2 rdr
rr
(C ) r = cons tan t = r (C ω2
a2
tan α 2
)
but Ca 2 is cosntant, thus α 2 changes as r tanα 2 = m tan α 2 m r 2 similarly r tan α 3 = m tan α 3m r 3
(a)
(b)
Axial Flow Turbine u = Ca2 tan α 2 − Ca2 tan β 2 , thus, tanβ 2 = tan α 2 −
u Ca2
r u r = m tan α 2 m − m (c) r 2 rm Ca2 for exit of rotor u = Cas tan α 3 + Ca3 tan α 3 r u r thus tanβ3 = m tan α 3m + (d) r 3 rm 3 Ca3 Ex: Free vortex Results from mean diameter calculations α 2 m = 5 8 .3 8 , β 2m = 2 0 .4 9 , α 3 m = 1 0 o , β 3 m = 5 4 . 9 6 , h 2 = 0 . 0 6 1 2 , rm = 0 . 2 1 6 , h
3
h 2
= 0 . 0 7 7 , rr = rm −
Axial Flow Turbine r r r r ⇒ m = 1.164, ( m ) 2 0.877, m = 1.217, m = 0.849 rt rt 2 rr 3 rt 3 u 1 u also m = = m = 1.25, Results are Ca 2 φ Ca3
α2
β2
α3
β3
Tip
54.93
0
8.52
58.33
Root
62.15
39.32
12.12
51.13
mean
58.38
20.49
10
54.96
Axial Flow Turbine U = tan α 2 − tan β2 = tan β3 − tan α3 Ca & cp∆Tos == m & cp (To1 − To3 ) = m & UCa (tan α 2 + tan α3 ) == m & UCa (tan β2 + tan β3 ) W=m & UCa (tan α 2 − tan α1) = m & UCa (tan β2 − tan β1) =m T' p ∆Tos = To1 − To3 = ηsTo1(1 − o3 ) = ηsTo1(1 − ( o3 ) γ /(γ −1) ) To1 po1 T −T where ηs = o1 o3 To1 − To' 3
EES Design Calculations of Axial Flow Turbine Known Information To 1 = 1100
[K]
P ratio =
1.873
DelTs =
145
Etta turbine
=
0.9
Assumptions U = 340 N rps
=
φ =
0.8
α3 =
[m/s]
250
10
Loss nozzle
=
0.05
EES Design Calculations of Axial Flow Turbine cp =
1148
DelTs =
φ =
γ
=
1.333
Po 1 Po 3 C2 · cos ( α 2 )
Ca U γ γ – 1
Gamr =
Epsi
0.287
To 1 – To 3
P ratio = Ca =
R =
=
DelTs
2 · cp ·
2
U Epsi = Reaction
2 · φ · ( tan ( β 2 ) + tan ( β 3 ) ) φ · ( tan ( β 3 ) – tan ( β 2 ) ) 2
=
U =
Ca · ( tan ( α 2 ) – tan ( β 2 ) )
U =
Ca · ( tan ( β 3 ) – tan ( α 3 ) )
EES Design Calculations of Axial Flow Turbine Calculate A2 Loss nozzle
T2 – T2dash
=
C2
2
2 · cp To 2 =
To 1
To 2 – T2 = Po 1
Po 1
Pth = Rho2 =
A2
=
Gamr
T2dash γ + 1 2
=
Pc
2
2 · cp
To 1
=
P2
C2
Gamr
P2 Pth R · T2 m Rho2 · Ca
A2 · cos ( α 2 ) =
A2N
EES Design Calculations of Axial Flow Turbine Calculate A3 Calculate A1 To 1 – T1 = Po 1
To 1
=
P1
A1
=
2
C3
To 3 – T3 =
2
2 · cp
2 · cp Gamr
T1 P1
Rho1 = C1 =
C1
Po 3
To 3
=
P3
T3 P3
Rho3 =
R · T3
R · T1
C3 =
Ca m Rho1 · Ca
A3
Gamr
Ca m
=
Rho3 · Ca
EES Design Calculations of Axial Flow Turbine Blade height at section 2
Blade height U =
A2 =
2 · π · r m · h2
r t2 =
rm +
r r2 =
rm –
rratio 2
=
2 · π · N rps · r m
Blade height at section 1 A1 =
2 · π · r m · h1
r t1 =
rm +
r r1 =
rm –
rratio 1
=
h1 2 h1 2
2 h2 2
r t2 r r2
Blade height at section 3 A3 =
2 · π · r m · h3
r t3 =
rm +
r r3 =
rm –
rratio 3
=
r t1 r r1
h2
r t3 r r3
h3 2 h3 2
EES Design Calculations of Axial Flow Turbine
A1 = 0.06345
A2 = 0.08336
A2N = 0.04372
A3 = 0.1046
α 3 = 10
β2 = 20.49
β 3 = 54.97
C1 = 272
α 2 = 58.37 C2 = 518.7
C3 = 272
Ca = 272
cp = 1148 [J/kgK]
DelTs = 145
Epsi = 2.88
Ettaturbine = 0.9
γ = 1.333
Gamr = 4.003
h1 = 0.04666
h2 = 0.06129
h3 = 0.07692
Loss nozzle = 0.05
m = 20 [kg/s]
Nrps = 250 [rev per sec]
P1 = 355.1
P2 = 248.8
P3 = 186.1
Pc = 215.9
φ = 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