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1
Turbomachinery
Class 12
2
CompressorTurbine
3
/ 1
1
1
1
1
p
p
T T
Ta T
T
/ 1
1
1
1
1
p
p
c c
ca c
c
Compressor Turbine
4
2 2 1 10 02 01
U UU C U Ch Cp T T
gJ
2 12 1
1tan tan
2 2u u x
W W CR
U U
2 2 2
0 02 2R
W W Ch h h
gJ gJ
/ 12 2
02 2 1
01 01
12
Ri
R p R
P U U
P gJC T
5
Radial Inflow Turbines
• Radial turbines largely used in power system applications
• Primitive design, easy to fabricate• Capable of large per stage shaft work with
low mass flow rate• Low sensitivity to tip clearance• Bulky / heavy
Low Cost Radial Compressor – Ex. 4, p. 83
6
01
01
1
1
02
02
2
2.3
378
0.38
0
6.1
533.7
0.50
42,000
1.4
p bar
T K
M
p bar
T K
M
N rpm
Low Cost Radial Machines
7
1 21 2
1 1 01 1
2 21 1 1 1 1 1
1 1 1 1 1
11 1 1
01 1 11 1
01
2 2263.9 439.8
60 60( , ) 363.3 382.1
145.2 220.4
0 220.4
263.9 tan 263.9 / 220.4 47.90
2 cos( , ) 2.29 / s
r u r
u
Nr NrU mps U mps
T f M T K a
C M a mps W U C mps
C C C W C mps
W U
p rbm f M kg
T
1 022 2
02 2 2
ec
1cos 79.7
2 ( , )
m T
p r b f M
Low Cost Radial Machines
8
2 2 02 2
2 2 2 2 2 2 2
2 2 2 2 2 2
2 22 2 2
12 2 2
1 1 1 1
( , ) 462.1 430.9
379.2 cos 62.4
sin 374.0 65.8
90.7
tan / 46.5
2
r r
u u u
r u
u r
r
T f M T K a
C M a mps C C W
C C W C U
W W W mps
W W
m C rb
Low Cost Turbine – Ex. 5 p. 85
9
10
Axial vs. Radial Machines
Need to determine what type of turbine is most efficient for application - function of Ns for both compressors and turbine
11
2 23/ 4 3/ 4
1 /0
0201
01
/ /
1
s
ideal
p
N m N mN
h pc T
p
Need to determine what type of turbine is most efficient for application - function of Ns for both compressors and turbine
12
Radial Inflow [90Radial Inflow [9000 IFR] Turbines IFR] Turbines
Kinematic view Thermodynamic view
Exit part of rotor (exducer) is curved to remove most of tangential component of velocity
Advantage of IFR turbine: efficiency equal to axial turbine, greater amount of work per stage, ease of manufacture, ruggedness
13
Radial Flow Turbines
• Radial Inflow Turbine with Scroll or Distributor
14
Radial Flow Turbines
• Radial Inflow Turbine Stator/Rotor
15
Radial Flow Turbines
• Radial Inflow Turbine Stator/Rotor [No shroud]
16
Radial Flow Turbines• Radial Inflow Turbine Scroll
Scroll or distributor - streamwise decreasing cross flow area - provide nearly uniform properties at exit
17
Radial Flow Turbines
18
Radial Flow Turbines
• Scroll Design Principles– Mass balance rVr=constant
– Free vortex rV=constant
1 12 1 2 1
2 2
2 12 1
2 1
cos
tan
tan tan
r
r
r r
r r
V V
V
V
r rV V V V
r r
V V
V V
19
Radial Flow Turbines• Radial Inflow Turbine Scroll - Stator
20
Radial Flow Turbines• Radial Inflow Turbine Impeller
Note - direction of rotation - rotor rearward curvature
21
Radial Flow Turbine Design• Nominal Stator / Rotor Design:
Station 1 – Inlet to StatorStation 2 – Exit of Stator, Inlet to Rotor
[Radially inward]Station 3 – Exit of Rotor
[Absolute velocity is axial]Station 4 - Exit of Diffuser
• Rotor inlet relative velocity is radially inward
- For Zero Incidence at Rotor Inlet, W2=Cr2 and C2=U2
• Rotor exit absolute flow is axial
- For Axial Flow at Rotor Exit, C3=Cx3 and C3=0
C2
Cm2=Cr2=W2
U2
Cm3=C3=Cx3
U3
W3
22
Radial Flow Turbine Design- 900 IFR
• For adiabatic irreversible [friction] processes in rotating components
• From the Alternate Euler Equation:
• and
2 2 2 2 2 22 3 2 3 2 3
0 2Rotor
U U W W C Ch
gJ
2 2 22 2 2C W U 2 2 2
3 3 3W C U
2 3
2 22
2 32 2 2orel orel orel
U UUI h h h
23
Radial Flow Turbine Design
• substituting:
• Thus from Alternate Euler’s Equation :
gJ
UUWUCUWCh Rotor 2
23
22
23
22
22
23
23
22
0
22
0 01 03 02 03Rotor
Uh h h h h
gJ
24
Specific Speed & Diameter Indicates Flowpath Shape
14
dim 12
s non
D gHD
Q
1/2
3/4s US
N QN
gH
Specific Speed Indicates Flowpath Shape(Cordier Diagram)
From Wright and Balje
From Logan
Ns is dimensionless
26
Radial Flow Turbine Design• Example: Dixon 8.1
• The rotor of an IFR turbine, designed to operate at nominal condition, – Diameter is 23.76 cm and rotates at 38,140 rev/min.
– At the design point the absolute flow angle at the rotor entry is 72 deg.
– The rotor mean exit diameter is ½ the rotor diameter
– The relative velocity at the rotor exit is twice the relative velocity at the inlet.
27
Radial Flow Turbine Design• Example: Dixon 8.1
2
3 2
3 2
02
22
2 2 2
2 2 2
23.76
38,140
/ 2 12.88
2
72
38,140 0.2376 / 60 474.560cot 154.17
/ sin 498.9
mean
Given
D cm
N rpm
D D cm
W W
rotor inlet design point flow angle
NDU mps
W U mps
C U mps
28
Radial Flow Turbine Design• Example: Dixon 8.1
2 22 2 2 2 23 3 3
2 2 2 2 22 3 2
2 2 2 2 23 2 2
2 2 2 22 3
2 2 2 2 2 22 3 2 3 2 3 2 2
0
2 154.17 0.5 474.5 38,786 /
1 0.25 168,863 /
3 71,305 /
210,115 /
225,142 /2
C W U m s
U U U m s
W W W m s
C C m s
Examing relative sources of specific work
U U W W C CW h m s
W
2 2 20 2
0.375( ) 0.158( ) 0.476( ) [% ]
225,142 /
U W C of total
Also
W h U m s
29
Radial Flow Turbine DesignExample: Baskharone p. 434-8
•0=inlet 1=stator exit 2=rotor inlet 3=rotor exit
•Stator / nozzle exit Mach number M1=0.999
00 0
4.16 / s 71,600 1.33
8.84 1205 0.038stator
m kg N rpm
p bar T K
30
Radial Flow Turbine Design• Example: Baskharone p. 434-8
• 0=inlet 1=stator exit 2=rotor inlet 3=rotor exit
• Stator / nozzle exit Mach number M1=0.999
00 0
4.16 / s 71,600 1.33
8.84 1205 0.038stator
m kg N rpm
p bar T K
1 1 1 1
01 00 00
010 1 1
01 1
1156.71
0.999 1034.4 628.35 590.6
[ ] 73.0cos
pc R
M T a C
p p p
m T RFP f M
p A
31
Radial Flow Turbine Design• Example: Baskharone p. 434-8 cont’d
• In constant area interstage duct, apply free-vortex condition to flow from stator exit to rotor inlet
1 1 1
1 1 1
cos 172.6
sin 564.8r
u
C C mps
C C mps
2 1 1 2
2 1 1 2
2
22 02 2
2 2
2
( / ) 184.8
( / ) 605.1
632.69
/ 2 1032
627.62
1.008
r r
u u
p
C C r r mps
C C r r mps
C mps
T T C c K
a RT mps
M
32
From Centrifugal Compressor Notes
• Slip: flow does not leave impeller at metal angle [even for inviscid flow]
• If absolute flow enters impeller with no swirl, =0.• If impeller has swirl (wheel speed) , relative to the impeller the
flow has an angular velocity - called the relative eddy [from Helmholtz theorem].
• Effect of superimposing relative eddy and through flow at exit is one basis for concept of slip.
Relative eddy Relative eddy with throughflow
33
• Static pressure gradient across passage causes streamline to shift flow towards suction surface
• In reality, the incidence to the rotor varies over the pitch of the rotor as:
due to – Potential and wake interaction with the vane.– Relative eddy effect seen at exit of compressor– Effect produces a LE slip factor
This variation over the pitch leads to an - optimal incidence and - optimal number of blades
where the efficiency of the rotor is a maximum.
Radial Flow Turbine Design
2 2 , rCfU
P=pressureS=suction
34
• Rotor Inlet Velocity Triangle (with incidence):
– Average relative velocity W and avg. relative incidence 2
• If we define an incidence factor, [like slip factor in compressors]:
Radial Flow Turbine Design
2
2
U
CU
U2
W2CR2=CM2
C2
CU2
35
Radial Flow Turbine Design
• From the work of Stanitz regarding slip factors:
– Note: More Blades, goes to 1 and inflow becomes radial
• Then from the rotor inlet velocity triangles, the inlet flow angle to the rotor is:
0.63 21 1 where Z=Number of Rotor Blades
Z Z
2 22
2 2
2tan where = M
M
U C
Z C U
36
Radial Flow Turbine Design
• Criteria for the Optimal Number of Blades:
2
2
min
min
2
1 12
2
1
2
0;
r
T T T T T T
From particle physics analogy
dWF ma f r f r W
dtp W p
W r and Wr r
WGets r implying that
W at given r is not constant across passage p s
W W r
At r r if W U r and W r U U
min
min 2
2
2 2 tan
T
T
T
Z
UZ
W
Jamieson model
37
Radial Flow Turbine Design
• Criteria for the Optimal Number of Blades:
Optimum blade number balances loading & friction
• Rohlick model uses (quantities at the inlet to rotor):
• Jamieson model
min2
2 min
2
min 2
2
0.03 57 12
UC Z
U Z
Z
min 22 tanZ
38
Radial Flow Turbine Design
• Other Correlations for Optimal Number of Blades (Rohlick results similar to Jamieson):
from Dixon
39
This is to clarify some of the confusing notation in Dixon regarding blade count
• Stanitz correlation
– uses blade number and flow coefficient to calculate the relative radial turbine exit flow angle.
• Other correlations – uses semi-empirical expressions for calculating the optimum
[minimum] blade count Z for an optimum efficiency design, where
– For such a design the exit flow will be radial [in the absolute frame], therefore 2=0 and the correlations are in terms of the corresponding absolute frame air angles [2], e.g.
22
2
2 2tan
M
U
ZC Z
2 2 2tanrU C
40
This is to clarify some of the confusing notation in Dixon regarding blade count
• Jamieson
• Rohlik
min 22 tanZ
2
min 20.03 57 12Z
41
Radial Radial OutflowOutflow Turbine Turbine
Ljungstrom Steam Turbine: Dixon - steam turbine design - No stator blades counter-rotating blades - radial outflow - large amount of work per stage - rugged
42
Radial Outflow Turbines
• Ljungstrom Turbine arrangement
Compatible with expanding steam, more area with same blade height as density drops
Vaneless - Counter rotating
Old Configuration recently re-invented for gas turbines
– axial counter-rotating
43
Counter-Rotating Turbines
• Counter Rotation High Stage Work
Compare: – Conventional Axial Stage, 50% Reaction & 90 Gas Turning
vs. – Counter Rotating, Vaneless Stages with 90 Gas Turning
Cx1 = Cx2, U1 = U2 = Cx/U= 0.6
Repeating Stages
Counter Rotation U changes direction
44
Radial Flow Turbine Analysis
• Remember from Class:
2
2tan 1
RE
2
22tan 1
RE
2
2tan 2
RE
2
22tan 2
RE
0
1
2
1
2
3
45
Radial Flow Turbine Analysis
• In this problem, for the axial stage
=0.6, R=0.5, and 1212
- Iteration:
Guess 1
From Calculate E.
From Calculate 2
Iterate until turning (12 is correct
• For the counter-rotating stage…..match turning
2
2tan 1
RE
2
2tan 2
RE