Chapter 4- Specific Static Rotor Work and cavitation.pdf

1
Wallaga University, College of Engineering and Technology
1

2
Impulse and Reaction Type of
Turbomachines
• Considering YP, the turbomachine can be grouped into:
A. “Impulse” type of Turbomachines
B. Reaction type of Turbomachines

3

4
4

5
5

6
6

7
7

8
8

9
9

10
10

11
11

12
12

13
13

14
Equal Pressure or Impulse Type of
Turbomachines
• Example A. Single-Stage Steam Turbine 0
0
0
3 =
=
− P
Y
and
P
P
The entirely available pressure difference (P3-P0) is
converted into velocity in the stationary guide vanes
Turbo machines without pressure difference in front of and beyond the rotor.

15
• The velocity existing in the clearance between the stationary
guide vanes and the rotor blades is the highest , i.e. C3 = C3max
attainable
• The absolute velocity is reduced from C3 to C0 ,While the flow
passes through the rotor.
• The specific static rotor work Yp is (for axial flow U1=U2 =
U)
( ) u
P Z
W
W
Y +
−
= 2
3
2
0
2
1
Impulse Type








−
−








−
=
−
=
2
2
2
2
2
2
2
3
2
2
0
3 1
0
U
W
U
W
P
P
YP


16
• Neglecting the hydraulic lose Zh of the
rotor, it follows because Yp = 0.
• Considering the loss:
• Where the velocity coefficient takes in to account the drop
in kinetic energy due to Zu; <1.
• The condition Wo ≈W3 demands rotor blades of the ‘hook-
form’ type, i.e. β2 > 900.
3
0 W
W =
3
0 W
W 
=
Blades of a constant-pressure steam or Gas
turbine. ‘a’ is the channel width at all points
approximately equal
Impulse Type
( ) u
P Z
W
W
Y +
−
= 2
3
2
0
2
1



17
• If blade has uniform thickness, the flow while passing the
channel is first decelerated then accelerated.
• Such change in the flow velocity is undesirable as it leads to
unnecessary losses.
• In order to obtain W≈ const. along the vane channel the blade
must be designed with strong profiling; however, such blades
are costly
Impulse Type

18
• The specific work Yblade of an impulse steam turbine stage as for a
given velocity U2 proportional to the velocity C3
• Steam turbines are designed with approximately the same angle α3=15
to 20degrees.
• As C3 of impulse steam turbines has highest possible value C3max-att.
The spec. work Yblade of these turbines has highest value
• The peripheral velocity U2 will be lowest for a given Yblade if the
turbine is designed as impulse turbine
• Impulse turbines are slow running turbines
.
max
3
3
3
3
2
3
2 cos att
U
blade C
C
C
U
C
U
Y −
=

=
=  For α0 = 900
2
.
.
max
. U
given
a
for
Y
Y att
blade
t
impulse
blade −
− =

19
Over-Pressure or Reaction Type of
Turbomachine
• Example B: Single-Stage Reaction Steam Turbine
• Part of the pressure drop occurs across the guide vanes and part
occurs across the rotor,
Turbo-machines with pressure difference in front and beyond the
rotor, i.e. (P3-P0) ≠ 0 , Yp> 0

20
• Thus C3<C3max-attainable and, hence, the spec. work Yblade
=U2C3U of the reaction turbine is smaller than that of the
impulse turbine if the same velocity U2 is assumed
• The velocity U of reaction turbines has to be higher than that
of impulse turbines if the same Yblade is to be obtained.
• Reaction turbines may be classified as fast running
turbomachines.
Comparison of Impulse and Reaction
Turbines

21
Turbines
• β1 should be small but not too small as leads to strong
whirls in the discharge flow.
• The angle β2 of reaction turbines is β2≤900 and, thus,
differs from that of impulse turbines.
• The blade of reaction turbine does not have the hook
form.
• As the relative velocity increases from W3 to W0, the
channel width decreases and no profile is necessary in
order to obtain equal channel width.
• Reaction turbine has more stages because of the lower
Yblade of its single stage.

22
❖Summary
– Impulse turbines: High-head, low flow
rate devices.
– Moving blade row changes only the
direction of the steam.
– Reaction turbines: Low-head, high-
flow rate devices.
• Moving blade row changes both the
speed and direction of the steam
Turbines

23
Y
Y
stage
the
of
outlet
and
inlet
between
work
Spec
work
rotor
Static
Spec
reaction
of
Degree P
=
=
)
(
.
.
)
1
(
1
0
0
:
0
0
:
cases
special
some
in
R
R
and
Y
machine
reaction
R
and
Y
machine
impulse
P
P




=
=
The reaction effect exists also in case of radial or mixed flow rotors
where U1≠U2 even for |W0| =|W3| as shown by the equation
( ) ( )
0
2
1
2
1 2
2
2
2
2
2
1
2
3
0
1
2

−
=
−
+
−
=
Y
Z
U
U
Y
Z
W
W
U
U
R
u
u 

Degree of Reaction








−
−








−
=
−
=
2
2
2
2
2
2
2
3
2
2
0
3 1
0
U
W
U
W
P
P
YP


24
Blade Speed Ratio
• The blade speed ratio as defined below is widely used in the
calculation of turbines especially of steam turbines.
• is the velocity which could be obtained if the spec. work
Y is converted without losses completely into velocity.
Y
U
C
U
Ratio
Speed
Blade
Y 2
=
=
Y
CY 2
=
R
C
CY
−

1
2

Where is velocity coefficient of guide vanes
(referring to velocity losses)
R
C
U h
Y −
=
1
1
cos
2 2



After some derivation


25
• Assuming the following data: ηh = 0.85; =0.98; α2= 300.
• The blade speed ratio has the value
• The following values of the blade speed ratio are obtained for
actual machines:
1
cos 2



h
5
.
0
2
1
0
2
1
5
.
0
0
=









=
=








=
=
R
for
C
U
R
for
C
U
R
Y
R
Y
 
47
.
0
44
.
0
1
47
.
0
35
.
0
1
'
'
47
.
0
35
.
0
0
0
arg
.
0
to
C
U
Turbines
Pelton
R
to
R
k
C
U
turbines
steam
reaction
k
to
C
U
turbines
steam
impuse
R
Y
R
Y
power
e
l
design
quality
high
power
small
Design
Cheap
R
Y
=








−
=
−









=
=








=

=


26
The Vane Angle β2
• Three different axial-flow vanes, namely form A, B, C for
which U2, C2m and β1 are the same but the angle β2 differ

27
• A similar sketch for three different radial-flow vanes with
β2<900 (form a), β2=900 (form b) and β2>900 (form c) is given
below.
• Vanes form b, c as ‘forward-curved’ vanes
Vane form a as ‘backward-curved’ vanes

28
The following relation exists between β2 and U2
❖ Case:α0=900
❖ Case:α0≠900
( )



+








+
=
−
=
−
=
−
=
=
=
blade
m
m
m
blade
m
U
u
u
blade
U
blade
Y
C
C
U
follows
it
and
C
U
U
Y
then
C
U
W
U
C
where
C
U
Y
and
C
U
Y
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
2
tan
2
tan
2
cot
,
cot
,




OU
blade
m
m
C
U
Y
C
C
U 1
2
2
2
2
2
2
tan
2
tan
2
+
+








+
= 



29
• The necessary peripheral velocity U2 for a given Yblade∞ can be
determined by these equation if the vane angle β2 is assumed.
• A large β2 , decreases U2 and the size of the rotor decreases,
too, if the speed n is not altered:
OU
blade
m
m
C
U
Y
C
C
U 1
2
2
2
2
2
2
tan
2
tan
2
+
+








+
= 



30
• The rotor shape is a function of n, V and Y.
• Shape number (Nshape) is a dimensionless number
and is used to define the shape of the rotor by relating n, V
and Y.
• It follows
 
0
0
2
2
3
1
2
2
3
1
1
;
1
,
1
1
s
m
s
m
s
m
s
assume
s
m
Y
s
m
V
s
n
Nshape
=


















=
=


















=






  4
3
4
/
3
2
/
1
1
1
Y
V
n
Y
V
n
Nshape =
= −
4
3
2
3
2
,
2
1
0
2
1
0
2
1
:
0
2
3
:
−
=
−
=
=
=
+
−
=
−
−
−
=
+








or
thus
or
S
m
Thus,
shape
sh N
n 1000
=
Shape Number

31
Shape Number
1. Effect of Increase in speed n on the shape
of the rotor (with unchanged β2,V and Y)
❖ The unchanged Y demands the same velocity triangle at 2.
❖ The unchanged velocity triangle can be obtained for
increased speed n but same velocity U as demanded by the
unchanged velocity triangle only at a smaller outer diam.
U
blade
blade C
U
Y
Y
Y 2
2
=

 
  4
3
4
/
3
2
/
1
1
1
Y
V
n
Y
V
n
Nshape =
= −

32
2. Effect of Increase in speed n on the shape
of the rotor
(with unchanged β2,V and Y)
  4
3
4
/
3
2
/
1
1
1
Y
V
n
Y
V
n
Nshape =
= −

33
3. Effect of Increase in V on the shape of the
slow running rotor
(with unchanged β2,n ,D2,and Y)
❖ The larger volume V can be obtained only by increasing the
channel width (b) and the eye dia. Ds
❖ The meridian component of the velocity must remain
unchanged because of the unchanged Y with same n and D2
❖ Demanding unchanged velocity triangle at 2.
  4
3
4
/
3
2
/
1
1
1
Y
V
n
Y
V
n
Nshape =
= −
m
om C
b
r
C
b
r
V 3
3
3
1
1 2
2 
 =
=

34
• A relation which is based on the head H instead on the spec.
work Y is called Specific Speed.
• Where the values has a unit of n(rpm), V(m3/s) and H(m).
• nq is not dimensionless for metric system nq has the following
unit
• For water turbines a specific speed derived from n, H and N is
often used.
4
3
H
V
n
nq =






=






=
min
.
333
min
1
60
81
.
9
4
3
4
3
2
4
3
s
m
N
N
s
s
m
n shape
shape
q
4
5
H
N
n
ns =
Specific Speed

35
Comparison of pump profile

36
Best specific Speed Range for Different Type of
Hydraulic Turbines

37
Table of design evaluation
Number of
pole pairs
criteria 1 2 3 4
n(sync) Rpm
1/s
2950
49.2
1450
24.2
950
15.3
730
12.2
Nshape 10-3 219 108 71 54
Efficiency 10 10 9 6 3
(less
efficient)
size 10 10 9 6 4

38
Values of Shape Number and
Specific Speed
Values of Nshape, nq and ns:
1000Nshape nq (water turbine)ns
Slow- running rotor 33 to 120 11 to 38 40 to 140
Medium-running rotor 120 to 250 38 to 82 140 to 300
Fast –running rotor 250 to 500 82 to 164 300 to 600
axial-flow rotor 330 to 1500 110 to 500 400 to 1800

39
Example 1
• The quantity of water available for a hydro
electric power is Q=260 m3/sec under a head of
H=1.73 m. Assuming the speed of the turbine to
be n=50 rpm & there efficiency to be 82.5%.
Find the number of turbines required.
Assume for the example , ns = 890 (metric units).

40
Solution
4
5
73
.
1
50
890
N
=
We have:
N = 1247.255MHP = 917356.05W
Ntotal=ηρQY=ηρQgH = .825*1000*260*9.81*1.73
Ntotal=3640343.85 W
Number of turbines = Ntotal/N
= 3640343.85 / 917356.05 =3.9 = 4 (Answer)
4
5
H
N
n
ns =

The rapid formation and subsequent collapse of gas
bubbles (vapor) in the liquid stream is called
cavitation

44
1. Causes and Effects of Cavitation
• The bubbles constitute cavities as far the liquid stream is concerned
thus the name ‘cavitation’
A. formation of vapor
bubbles;
B. collapse
the vapor bubble;
C. material
destruction(Assume that it
is to be impeller

45
If the pressure of the liquid stream is lowered up to vapor
pressure , evaporation will start forming small vapor bubbles in the
liquid stream.

46

47
If the bubble was located on the wall, the material of the
wall may be destroyed due to impact of the liquid particles
rushing towards the wall.
• If the bubble is entirely surrounded by the liquid, no
destruction of wall material is caused but a hard noise
can be heard and the machine may vibrate.

48
Consequently, cavitation has to be avoided. This means
it has to be ensured that the pressure along the flow nowhere
lowered up to the vapor pressure.

49
• The pressure acting on the
suction water level may be PA
(Atmospheric pressure ).
• If the suction water level is
taken as reference, the energy
content of the water at this
level is given by (PA/ρ).
2. Avoiding of Cavitation in Pumps

50
• The total energy content of the water at the suction end of the
impeller is less. The energy difference is given by:
A. The geodetic energy difference (ges [Nm/kg])
– Where es: height difference between
suction water level and the highest
point of the suction edge of the vanes.
B. The energy loss of the flow in the suction pipe up to the suction
end (pump suction flange ) Zs [Nm/kg] or Zs = gh, h=head loss

51
• Considering A. and B.,
the following total energy exists
at the height es at the pump
suction end :
  ( )
s
s
A
s Z
ge
P
kg
Nm
E +
−








=


52
• However, in regard to priming not the total energy Es matters, but the
locally lowest static pressure Esat-min occurring in the pump.
• The difference between Es and Esat-min is given by the dynamic
energy prevailing at the point of locally highest velocity and the
loss due to generating this velocity.
• The energy needed for generating the locally highest velocity may be
considered as follows:
• Firstly, the average value of the velocity at the suction end of the
vane is co according to the velocity triangle.
• The dynamic energy and the loss for generating co from the velocity
zero prevailing at the suction water level may be counted together
and expressed as :
1
2
. 2
2
0
2 

 where
C

53
• Secondly, due to the influence of the thickness of the vane and
of the pressure distribution around the suction end of the vane
the actually prevailing velocity at the suction end is greater
than co.
• The additional dynamic energy included. The additional losses
for generating this highest velocity from co may be best
expressed related to relative velocity Wo.
• Consequently,
• Experimental values: λ1≈ 0.25 to 0.35; λ2≈ 1.1 to 1.3
1
2
. 1
2
0
1 

 where
W As the additional dynamic energy
is only a fraction of the dynamic
energy of the relative velocity
( )
2
2
2
0
2
2
0
1
min
.
C
W
y
E
E stat
s 
 +
=

=
−
( )




T
s
s
A
s
stat
P
C
W
Z
ge
P
y
E
E 
+
−
+
−








=

−
=
2
2
(
2
0
2
2
0
1
min
.

Avoiding of Cavitation in Pumps
• If evaporation of the flow medium has to be
avoided, it must be
• It can be expressed also in the following
way:
• Evaporation will takes place if (Es)avail < ∆y

T
s
stat
P
y
E
E 

−
=
min
.
Where PT[N/m2] vapor pressure
at Prevailing temperature
( ) ( ) y
E
Z
ge
P
P
P
E avail
s
s
s
T
A
T
s 

=
+
−
−
=
−



55
1. Measures at the pump design
A. Suction number
B. NPSH
C. Suction Specific Speed
2. Measures outside the pump
Demands: ges, Zs and PT/ ρ as small as possible, high
PA/ρ
3. Other Measures
Measures to Avoid Cavitation

56
• The outer point a of the vane
suction edge is exposed to the
cavitation most as here , U and
therefore Wo has the highest value
along the vane suction edge.
The ratio between Coa and Woa is given by the angle βoa.
To avoid cavitations-the minimum of Δy-is given by the
following equation :
( )
( ) 2
.
1
3
.
0
'
32
17
2
1
tan
2
1
0
2
1
1
=
=
=
+
=







and
if
opt
oa
opt
oa
Case: α0a=900
Measures at the pump design to Avoid Cavitation

57
• Case:α0a≠900 :- (βoa)opt- values are given by the following
diagram
a
ou
a
ou
r
u
W
u
C
1
1
1 =
−
=

Pre-rotation factor
(Swirl number)
( )
2
1
2
2
1 1
1
2
1
tan






+








−
+
= r
opt
oa
Measures at the pump design to Avoid Cavitation

58
A. Suction Number Sq
• It is a dimensionless number in regard to prevailing lowest static
pressure.The design of the pump, the locally lowest static pressure
is influenced by:
a. The value of Δy: i.e. by the friction coefficients λ1 and λ2
actually existing and the relation of co to wo as given by the angle
βoa
b. The value of the speed n: ( if for the same pump the speed
n is increased, the locally lowest static pressure is decreased)
c. The volume v: (if for the same values of n and βoa the volume
V is increased, the absolute value of co and wo increase; hence
the locally lowest static pressure in the eye decrease).
4
3
y
V
n
Sq

=
( )
2
2
2
0
2
2
0
1
min
.
C
W
y
E
E stat
s 
 +
=

=
−

59
• As for same n and V, a
higher Sq value refers to
a lower Δy.
Suction number
The diagram indicates that a pre-rotation δr=0.8 (slight pre-rotation in
direction of impeller rotation) avoids cavitation.(The values indicate
the design point of pump).
4
3
y
V
n
Sq

=
Experimental values,
Sq=0.3(low) to 0.4 to 0.45(Normal)
and above avoid cavitations

60
• If the hub is extended through the eye of the impeller a decrease in free
cross section of the eye with an increase of the meridian velocity for same
V is noted. ( A, Cm for the same V)
• In order to take account of this increase in velocity due to existence of a
hub, the suction number is defined as
• The value of Δy may be determined by assuming λ1 and λ2 if Co and Wo are
known using :
• or by assuming the suction number
2
4
3
1
: 







−
=

=
s
n
q
d
d
k
Where
y
k
V
n
S
Suction number
( )
2
2
2
0
2
2
0
1
min
.
C
W
y
E
E stat
s 
 +
=

=
−

61
A. Suction number
B. NPSH
Demands: ges, Zs and PT/ ρ as small as possible, high
PA/ρ
3. Other Measures

62
B. Net Positive Suction Head (NPSH)
• (NPSH) is the head required at the pump inlet to
keep the liquid from cavitation or boiling.
• There are two values of NPSH that we work with.
1. NPSH required(NPSHR)
2. NPSH available(NPSHA)

63
NPSHR and NPSHA
❖ Required NPSH, denoted NPSHR, that must be maintained, or
exceeded, so that cavitation will not occur. Since pressure
lower than those in the suction pipe will develop in the impeller
eye, it is usually necessary to determine experimentally, for a
given pump, the required NPSHR.
❖ Available NPSH, denoted NPSHA, represents the head that
actually occurs for the particular flow system. This value can
be determined experimentally, or calculated if the system
parameters are known.

64
• A useful parameter is the available suction head at
entry to a pump or at exit from a turbine.
• This is usually referred to as the net positive suction
head, NPSH, defined as :
• Where po and pv are the absolute stagnation and
vapor pressures, respectively, at pump inlet or at
turbine outlet.

65

66
Thomas cavitation parameter is defined
by

67
A. Suction number
B. NPSH
Demands: ges, Zs and PT/ ρ as small as possible,
high PA/ρ
3. Other Measures

68
C. Suction specific speed, nqs
• It uses for cavitation check
• It is a non-dimensional version of 𝑁𝑃𝑆𝐻 required
• To prevent cavitation for centrifugal pumps.
4
3
4
3
)
( R
qs
NPSH
V
n
h
V
n
n =

=
3
)
( 4
3
4
3

=

=
R
qs
NPSH
V
n
h
V
n
n

69
nqs and NPSH
Where:- Δh= Δy/g ,
n is measured in rpm
4
3
h
V
n
nqs

=
( )
( )
( )required
available
avail
s
NPSH
g
y
NPSH
g
E
=

=

70
Home take activity
• How high can the following pump
be installed above the water level of
the suction tank which is under
atmospheric pressure? Pump data:
V=0.1m3/s, n=1450rpm, dn/Ds=0.5
i.e k=0.75, Zs=13(m2/s2 assumed),
Sq=0.42 assumed or given by testing
the already existing pump.
A. For the case pumping cold water at
t=200, PT=0.0234bar,PA=0.932bar
B. For the case pumping hot water at
t=800, PT=0.474bar,PA=0.932bar

72
Demand: ∆y as small as possible, is
obtained using:-
a. Low Shape Number:- (Large pump, low speed, multi-suction, thus Co
and Wo kept small) Shape number (Nshape)
b. Favorable βoa (assumes favorable ratio of Co to Wo)
c. Low vane loading by means of increasing the vane length, vanes drawn
into the impeller eye (influences λ1).
2
2
2
0
2
2
0
1
C
W
y 
 +
=

d. Good Streamlining of impeller approach i.e.
especially avoiding or rapid changes of
the flow direction.
  4
3
4
/
3
2
/
1
1
1
Y
V
n
Y
V
n
Nshape =
= −
( )




T
s
s
A
stat
P
C
W
Z
ge
P
E 








+
−
+
−








=
2
2
2
0
2
2
0
1
min
.

73
e. Smooth Surfaces of walls of pump eye and vanes.
f. Use of entrance guide vanes in order to generate a pre-rotation in direction
of the impeller rotation (δr =0.8).
Demands: ges, Zs and PT/ ρ as small as possible, high PA/ρ
a. small ges: pump is to be located as close to the suction water level as
possible
b. Small Zs: loss of the suction pipe is to
be kept small: low flow velocity
(Cs = 1 to 2 m/s normally);
– short suction line.
– Avoiding of unnecessary bends;
– Smooth inside surface of pipe
– Leak proof suction line to avoid air intake;
– Avoiding of pre-rotation in suction pipe;
( )




T
s
s
A
stat
P
C
W
Z
ge
P
E 








+
−
+
−








=
2
2
2
0
2
2
0
1
min
.

74
C. Small PT/ρ: Pump is to be located where the liquid has its
lowest temperature. Circulating pumps of heating installations
should be installed in the runback.
( )




T
s
s
A
stat
P
C
W
Z
ge
P
E 








+
−
+
−








=
2
2
2
0
2
2
0
1
min
.

75
d. Rise in PA/ρ: Use of a feeding pump which is located
deeper than the main pump and runs with low speed. Often,
also ejector pumps with no rotating parts are used if height
es or temperature of liquid are very high.
( )




T
s
s
A
stat
P
C
W
Z
ge
P
E 








+
−
+
−








=
2
2
2
0
2
2
0
1
min
.

76
3. Other Measures
• Cavitation can also be avoided by using proper pump materials
with high fatigue resistance and high ductility and providing very
smooth surfaces
• Preferable pump materials are: Chrome-steel, stainless steel
• The destruction effect of cavitation may also be eliminated partly of
fully by inducing air into the liquid at the suction line.
• The air will fill partly the cavities which consequently do not
collapse anymore so suddenly

79
Wollega University
End of
CHAPTER

Chapter 4- Specific Static Rotor Work and cavitation.pdf

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