Gas/Steam Medium
1
• If the medium is already a gas/steam, the phenomenon that
the flow medium will change from the liquid to the gas phase
does to occur, thus no cavitation has to be considered.
• For lower temperature, the sonic velocity is low. In order to
avoid supersonic and sonic flows the flow velocity has to
lower. But the flow velocity may be desired to be higher.
• Above sonic velocity(supersonic flow) there is an occurrence of
shock waves which represent losses.
2
• The nearness of the flow velocity to the sonic velocity a may
be expressed using the Mach Number.
M 
C
a
• IN general, flow conditions where M=1 and normally also
those with M>1 (supersonic flow) have to be avoided.
• The sonic velocity of a gas is:
a k
P

 kRT
• Since for a given gas k and R are constant, the locally
prevailing temperature determines the sonic velocity a, the
danger or reaching sonic velocity is greatest where the
temperature is the lowest.
• In general, this is at the suction end of the turbomachine.
3
Compressors
• The sonic velocity is lowest at the suction end of the impeller,
in the case of multi-stage compressors, at the suction end of
the first impeller as here the temperature is lowest.
• The criteria to limit the locally highest velocity below the sonic
velocity may be expressed as follows:
M
Woa
 0.7 to 0.8
a
The lower value refers to impeller with thick vanes
The higher value to those with thin vanes
• Similarly to avoiding cavitation, an optimum angle βoa can be
determined:
Wmax   1   Woa2
2
Where Wmax = locally highest velocity in the vane
channel near suction end
Experimental value: λ ≈ 0.2 to 0.3
4
 Assuming no pre-rotation (δr=1) and the value of λ the
optimum angle βoa is:
oa opt  32010' to32030'
 The optimum angle which avoids sonic velocity best has value
of nearly twice of optimum angle avoiding cavitation best
 By applying the above criterion the velocity in the channel
near the suction edge is kept lowest. As the velocity decreases
the frictional loss also decreases which results a higher
hydraulic efficiency.
5
The Suction Diameter-The Inlet
Number ε and the Discharge Number ε2
6
• The suction diameter has to be chosen so that βoa obtains its
desired value with regard to avoiding cavitation or sonic
velocity or with regard to obtaining lowest friction loss at the
vane suction edge.
• The following optimum values of βoa were mentioned in the
previous chapters:


0
 20 turbine : avoiding cavitation


0

 35 smallest woa;
 if  r  1

Compressor : avoiding sonic velocity

all turbomachine : obtaining lowest

friction loss at suction edge of rotor vanes
 oaopt 17 0 pump : avoiding cavitation
 oaopt
 oaopt
7
• The suction diameter D1a and the angle βoa are interrelated to
each other by the velocity Coma as follows
Com
V'

where Ao  f D1a 
Ao
Cross sectional area at point o
Perpendicular to Com
• A relation between Com and D1a can be found from the
velocity triangle at point o:
r  1
 dn 
k  1   
 Ds 
Com
Com
tan  oa 

U1a  Coua  rU1a
2
8
D1a
Coma   rU1a tan  oa   r
 tan  oa  f1Cs with D1a  f 2 DS
2
where f1 , f 2 factors of proportion ality
It follows
Coma  f1CS  f1
V'
f V'
 1
AS k  D 2
S
4
and
DS
Coma   rU1a tan  oa   r f 2  tan  oa
2
DS
  r f2
 tan  oa
 2
2
k DS
4
f1V '
8 f1V '
4V '
f1
DS  3
3
kr tan  oa f 2
k 2 n tan  oa f 2 r
9
• The suction diameter Ds may also be determined using the
following dimensionless numbers:
Pumps and
compressors
turbines
Inlet Number
Discharge Number
Com Com

CY
2Y

2
2
Com
Com
 
 2
2Y
CY
2
• Knowing the value of ε, ε2 , the suction diameter Ds follows
from
1
1
1
Com 

 2
Ao AS DS
• The values of ε and ε2 are functions of the shape Number
Nshape as can be noted from the following derivation:
10
f 2 DS
 tan  oa   r f 2 DS n tan  oa
2
4 f1 f 3V
  r f 2n tan  oa 3 2
if V '  f 3V
 k r n tan  oa f 2
Coma   r
and
C
1
  om 
2Y
2
3
4 r2 f1 f 22 f 3 tan 2oa n 2V
k
2
where
nV
2

N
shape
Y 3/ 2
  1.64 3 f1 f

2
2
Y
3
3/ 2
if Com  Coma
4f1 f 22 f 3   r

tan  oa N shape 

2
 k

 

f 3  r tan  oa N shape 
 k

2/3
2/3
11
 2  2.70  f1 f 22 f 
2/3
 r

tan  oa N shape 

 k

4/3
• The Inlet number for pumps and compressors, assuming δr=1 and
k = 1-(dn/Ds)2≈0.8 and the common range βoa =14 to 380 is
  0.70 to 1.50 3 f1 f 22 f 3 N shape 2 / 3
where 3 f1 f 22 f 3  1.1


Cavitation Sonic velocity / low friction loss at rotor inlet
As far as the slow-running rotor with Nshape < 0.1 and k =1 is
concerned the value of ε may be taken independent of Nshape as
  0.1
to
0.5


Cavitation Sonic velocity / low friction loss at rotor inlet
12
• The values δr and k of water turbines of normal designs are
δr≈1 Francis, Kaplan Turbine
K ≈ 0.8, Kaplan Turbine, k ≈1 Francis Turbine
• The following table shows some values of ε2 taken from actual
designs.
Francis Turbines
Kaplan T
Dim.
Nshape
0.063
0.065
0.123
0.210
0.34
0.52
0.70
1
ns
70
94
141
237
387
592
797
(metric)
nq
21
28
41
70
114
174
234
ε2
0.032
0.032
0.048
0.096
0.152
0.331
0.486
1
βoa
30
23.5
21.3
21.3
18.6
19.4
19.2
degree
(metric)
13
• Faster running turbines have higher values of ε2. In case of the
Kaplan Turbines ε2 may account up to 0.5.
• This means that the kinetic energy Co2 /2 with which the water is
discharged from the rotor is equal to 50 % of the available energy Y.
• All this kinetic energy would be lost if no draft tube were provided. A
draft tube, however, will ‘recover’ part of this kinetic energy.
• As ε2 of fast running turbines is large, these turbines have to be
designed with very effective draft tubes. For this reason Kaplan
turbines are mostly designed with elbow type draft tubes.
14
Number of Vanes
15
• If the number of vanes is small, each vane is loaded much and,
hence the pressure difference between both sides of the vane is
high.
• This leads to non uniform velocity distribution in the vane channel
and very high local velocities may occur. Thus also cavitation or
sonic velocity is more likely to occur.
• If the number of vanes is high the vane channel becomes narrow
and consequently the friction loss is high.
• Thick vanes do not allow to install many vanes. Generally the
thickness of the vanes should be as small as possible observing,
however the strength of the vane material, the vibration of the vane
and proper profiling if desired.
16
Z k2
• Number of vanes
rm
  2
sin 1
e
2
where e = length of the mean stream line in
meridian section measured between
vane in-and outlet
rm = radius of the middle of the line e
k = empirical factor
= 5 to 6.5 cast vanes
= 6.5 to 8(to 12) sheet metal vanes
•
Pumps and water turbines have mostly cast vanes, radial-flow blowers have
sheet metal vanes.
• Axial- Flow
e  r2  r1 and rm 
1
r1  r2 thus, Z  k r1  r2 sin 1   2
2
r2  r1
2
• The above formulas are also applicable to determine the number of
vanes of guide vanes.
17