208
Mixed-flow and radial turbomachines
Note that the relative eddy increases/32 but decreases /31 as compared with the
vane angle/3 for the case of shock-free inflow.
From Eqn (7.51) the slip factor for shock-free inflow is
0.85 - 0.1893 tan 70 ~
= 0.687

194
7.4
Mixed-flow and radial turbomachines
Geometrical
techniques
analysis
of mixed-flow
for dealing
cascades
with
design
and
As already discussed in the introduction to Chapter 2, the fully three-dimensional
flow through axial turbomachines can be model

398
HYDRAULIC TRANSMISSION OF POWER
At the exit of the secondary the tangential velocity component of the relative velocity
is
Wau3 = Vx tan/3 s3 = 15tan(-56) = -22.24m/s
so that the component of the absolute velocity and its flow angle are
Vsu3 = Wsu2 +

322
RADIAL INFLOW TURBINES
EXAMPLE 9.3
In a radial-inflow turbine, combustion gases, with 7 = | and c p = 1148 J/(kg K),
leave the stator at the angle a2 = 67. The rotor blades at the inlet are radial, with
a radius r2 = 5.8 cm. At the outlet the shroud r

BLADE FORCES
427
V
V
Figure 12.16 Prandtl's tip loss factor.
as the diagram in Figure 12.16 shows. The sine of the flow angle at the tip is
so that
sm<pt
(l-a)V
Wt
_
~ 2
\7(R~r
R )
n(R-r)
d
Wt
(l-a)V
Following Glauert and assuming that
w1_w_
R ~ r
yields

MOMENTUM AND BLADE ELEMENT THEORY OF WIND TURBINES
403
performance. Thickness of the cross section of the blade is determined primarily by
structural considerations, but a well-rounded leading edge performs better at variable wind
conditions than a thin b

72
COMPRESSIBLE FLOW THROUGH NOZZLES
The temperature ratio across a shock is obtained by substituting the value of M 2 from
Eq. (3.23) into Eq. (3.20). The result is
Ty _ [ 2 7 M , 2 - ( 7 - l ) ] [ 2 + ( 7 - l ) M j
(3.26)
(7+l) 2 MJ
Tx
The density ratio

MOMENTUM AND BLADE ELEMENT THEORY OF WIND TURBINES
411
and downstream it yields
P - , 1~ir2
P + 2 ^
+
, 1 2 2
2
r W
Pb , 1 T A 2 , " ' 2 2
= 7 + 2 ^ +
2 ^
Adding the last two equations and using Eq. (12.17) to eliminate the pressure difference
p+ p- leads

240
Ducted propellers and fans
which has the general form of the required performance characteristic r =f(CT).
However, r cannot be completely isolated but does also appear on the right-hand
side of Eqn (8.79). Its solution must therefore be obtained by s

136
S i m p l i f i e d m e r i d i o n a l f l o w analysis f o r axial t u r b o m a c h i n e s
The last term accounts for the vortex field created at the last actuator disc AD4
which is assumed to extend to x = oo. From this discussion we may set out

COMPRESSOR ANALYSIS
269
argument to estimate the slip factor [75]. The fluid flow through the rotor is irrotational
in the laboratory frame, except for that part of the flow that moves right next to the solid
surfaces. Therefore, relative to the blade, th

18
Basic equations and dimensional analysis
1.000
Axial fans
L
r-
Radial
L compressors .a
r and fans q
Centrifugal pumps
_LJ L
,->
0.100
xial pumps
o
0
Mixed-flow pumps
=
0.010
0.001
0.00!
'
.
o'.o' o
I
!
,
l
IT
1 I
o .100
i
!
:
i
v i
,o! 0
1
00
F l o w c

TURBINE EFFICIENCY AND LOSSES
201
the factor FL depends primarily on the shape of the velocity diagrams, which, in turn, are
completely determined by ip, <p, and R. The irreversibilities are taken into account by fa
and Cs- These depend on the amount of t

364
HYDRAULIC TURBINES
Figure 10.3 A six-jet Pelton wheel. (Drawing courtesy Voith Siemens Hydro.)
the diameter of the jet, as well as the recommended number of blades, and (c) the
mechanical efficiency.
Solution: (a) The shaft power
W0 ripQgHe
solved for

102
COMPRESSIBLE FLOW THROUGH NOZZLES
Substituting and simplifying gives
from which M2 may be determined.
EXAMPLE 3.12
Steam with 7 = 1.3 flows from a low-pressure nozzle shown in Figure 3.17, with
nozzle angle a = 65. The throat at the exit plane is cho

380
HYDRAULIC TURBINES
0.60
0.65
0.70
0.75
0.80
rlrt
0.85
0.90
0.95
1.0
Figure 10.12 Flow angles for the turbine.
Small turbines have been excluded from Figure 10.2. The single-jet Pelton wheel with low
flow rate extends past the left margin of the graph.

BLADE FORCES
421
drag-to-lift ratio is 0.005. Calculate the axial and angular induction factors at r/R =
0.7.
Solution: For given a the lift and drag coefficients are calculated from
CL = 2-n- sin a
C D = ECL
The rest of the solution is carried out by ite

150
STEAM TURBINES
1.00
0.95
0.90
0.85
r|s 0.80
0.75
0.70
0.65
0.60
0.55
0.50
0.3
0.4
0.5
U/V2
0.6
0.7
0.8
Figure 5.6 Stage efficiencies of single stage impulse turbines with nozzle angles in the range from
60 to 78 with (N = 0.02 and R = 0.14; the exit k

394
HYDRAULIC TRANSMISSION OF POWER
relative velocity at the exit of the primary are both axially directed. Evaluate the
power developed by the secondary element in this torque converter.
W^
Primary inlet
Primary exit
Figure 11.7 Velocity triangles for Ex

INDEX
blade element theory development by S. Drzewieci,
402
blade element theory of W. Froude, 402
blade forces for a nonrotating wake, 415
capacity factor, 5
ducted turbine, 408
Glauert theory for an ideal turbine, 424
history, 8
induction factors for an

EXERCISES
309
The integral ^3 now can be written as
b(r)
dr = 2R
TT/2+5
sin
-d6 = 2R
R
^sin
2
!
-d6
c + cos (
Evaluation of this leads to
2R
(^
+S
)
-cosS-2Vc2
1 1 + tan(<5/2)
T 1 - tan(<5/2)
- ltan"
Collecting the results gives the design formula
27rr 2

116 Simplified meridional flow analysis for axial turbomachines
Approximate solution matching Cx at the root mean square radius rms
Let us assume that Cx = Cx at the r.m.s, radius, namely rms = V ' ~1( r 2h + r 2) . T h u s E q n
(5.19) yields directly an

234
Ducted propellers and fans
10.0
8.0
fa
6.0
~ - - - - - r = 0.5
4.0
- . ~ 1 ~ 0.6
-gE- 0.7
0.8
1.0
1.2
2.0-
0.0
0
i
~
~
i
5
6
)
8
9
~0
CT
Fig. 8.14 Duct loss weighting coefficient fd as a function of C T and r
where the weighting coefficient fd is a fu

298
Appendix H
Fig. 11.14 Menu for handling files and other tools
rawdata
This file contains (x,y) coordinates of the last blade
profile but set at zero stagger. Sample data for the
default design, Fig. 11.14, are given in Section II.10.
testdata This fil

182
Mixed-flow and radial turbomachines
derivation of the property rothalpy relevant to energy transfer in rotating systems
with radial flow, Section 7.2. Dimensionless velocity triangles will then be considered
in Section 7.3 to link stage duty based on

164
Vorticity p r o d u c t i o n in turbomachines and its influence u p o n meridional f l o w s
and introducing Eqns (6.58)
dp
Cr ~p
Cx @
dn
Cs Ox
Cs Or
Thus finally, the compressibility terms on the right-hand side of the governing
equation (6.57) beco