Raabe J. Hydro power - the design, use, and function of hydromechanical, hydraulic, and electrical еquipment
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Fig
10.4.12.
Simplified
layout
of
the
impeller
@,j
one
vane.
vane
with angular velocity
o:
In
the above, r
=
radius, r,
=
mean radius at the impeller outlet and r,
=
mean radius
at
the impeller inlet.
As
fi
is constant according to equation (10.4- lo), the following applies approximately to
the
resulting force
F
perpendicular to the impeller vane:
F
=
F,
.
L
=
F0
(r,
-
r,)/(cos
p
.
sin
p).
L
signifies the mean vane length.
The
resistance force
F,
inside the vane passage can be split up into the component
F,,
originating from the opposite vane surfaces, and the component
F,,,
from the shroud and
hub
surfaces.
F,,
and
F,,,
should be related to each other in proportion to the respective
wetted surfaces. Afterwards
it
is possible to represent
F,
in the form
F,
=
i-;,
+
F,,,
=
S2
-
F,,,
in which case R, with constant
B
(eq. (10.4- lo)), is determined approxi-
mately thus:
L?
=
1
+
n
sin P(r,
+
r,)/(z b,).
(10.4- 13)
In
this case,
z
=
number of vanes and b,
=
mean breadth of the impeller passage. If, by
-
analogy to areofoil-wing theory, we were to introduce a gliding angle E,
the
specific impeller loss would be computed to the following:
hbL
=
P,/P,
=
(2
E
R/sin
/I)
.
{w/[o
(r,
+
r2)J)
.
(1
0.4-
15)
For the purpose of converting eq. (10.4-
15)
use is made of the following relationships:
Observing eqs. (10.4-8) to
(10.4- lo), sin
/?
=
c,/w
=
c,,/rv,. From eq. (10.4- 11)
-
by
application
or Pythagoras to the velocity triangle at the inlet
-
there results
W:
=
02r:
+
c;,. We proceed further to introduce the "hub to tip ratio"
N,
=
rli/rlar in
which case rli
=
radius of the inlet edge on the hub and r,,
=
radius of the inlet edge on
the
shroud (i-e. radius of the impeller eye, see Fig. 10.4.13).
]:is.
10.4.13.
r\ct\l;~l
and
ahhurncd
rotor
in
p\~~iiping
ordcr.
.
I--
r,
=--
---
I
real iniet
edge
assumed
inlet
edge
ilssunling the inlet etlse to be str~right, relation
r,
=
(r.,,,/2)(1
+
N,)
will apply.
Ir.
addition \ve define
a
flo\\l
cvcTficient
cp,,
in
\+
h~ch c,ibe
c,,,,,
=
mcridi,~ri vclocity at thc i~ilct edgc on thc shroud. Thus the specific
1mpc1lt.1. loss
It;,
can.
I'ollo\\ing
conversion
of equation
(10.4-15),
eventually
be
representcit
;I>
lollo\ic:
11;
,
=
l,
R((o
J.~,~
~.,~l)[(l
+
~1)~
+
4
.
(~rnl/'~~nlln)~
(~~111/[(1
+
'I)('
+
r2/~1)]-
(10.4- 17)
10.4.3.1.2.
Dilfuser- loss
Obser:
ills
cij.
(
10.1-
8)
ancl
(.,,,
=
(D21'D3)
c,,,~,
thc specific clifftiscr loss
Ill.,
is
computed
a>
I'ollo~\*s:
11;
=
(I,
sin'
Y,).
[c,',,!(2
11)]
.
(D,/D,)2.
(10.4-
18)
111 the
abo\-e.
a!!
chi~racters hnvc the same meaning
as
in cy.
(10.4-4).
10.4.3.1.3.
Oprimunl diameter
of
the impeller eye in respect to internal loss
c,,!
can
be
quoted [assisted
by
the continuity ccluation) as a function of the diameter
of
the in1pellt.r
eJs
Dl,
Ii'
n
straight inlet edge is assumed in accord;ince with
Fig.
10.4.13
nlorig with
c.,,,,
bei:~~ the
Incan
meridian speed within the inlet area, the continuity
cquation applied to the irnpcllcr inlet will reat1 thus:
In
t!le
abo1.t..
Q
=
flo~v
rate
in
the design point,
0,
=
mean angle between the radius and
'nsi;ll
\.;lnr
seer ion
\i
ithin the arcn
of
[he inlet edgc. and
,ul
=
angle between the radius and
a mean mcr id
iTlllnflo\\.llneXtlle-l'filFtl~pppp~~-~p
If
~r-;
no\\* proceed
to
insert
rS,,,,
from eq.
(10.4- 19)
into eqs.
(10.4
-
17)
and
(10.4-
18)
and4'-.
also
substitute for
r,,
in
eq.
(10.4- 17)
the expression
D1,/2,
we shall, by adding)!
eqs.
(
10.4
-
17)
and
(1
0.4-
IS).
obtain the resulting impeller and diffuser loss
hiD
8.i
function
of
the diameter of the impeller eye
D,,.
It:.,
can be represented in
the
form:
i(
1
and
K,
are desizn-conditioned constants if
r2/rl,
N1,
c,~/c,,~,
D2/D3
and
cp,,
are
ass
"med as given. As is known, the flow coefficient
q,;
defined in eq.
(10.4-
16)
is
com-
P
uted, as a matter of convenience for reasons pertaining to cavitation, according to the
Nuation:
ql0
=
tan
Dl,
=
{A/[2
(1
+
A
+
(10.4-21)
(See
[6.8]).
In this case
A
=
the pressure number for the critical point with respect to
cavitation on the vane surface,
[
=
suction-pipe loss factor.
fie optimum diameter of the impeller eye
DlaOp,,
as regards efficiency, at which the
losses
hi,
are minimized, is finally obtained from the condition
d&,/dD,,
=
0
as
follows:
Dl,
.,,
=
131
54
+
r2/r1)(1
+
N1)(D2/D3)2
.cD
sin2
a,
[/1
+
N1)2
+
4
-
(C,,,~/C,,,)~
-
qfo]
E
o
Q
g~
Q
cos
0,
'
{(I
-
~;)sin(p,
+
0,)
I3l7.
10.4.3.2.
Optimum diameter
of
the impeller eye with respect to cavitation
The
normal negative suction head
h,
is computed to
?I,
=
B'
-
NPSH,
in which case
.
B'
=
barometric pressure head minus the critical pressure head and NPSH
=
net positive
suction head of the pump. In order to minimize excavation costs,
i.e., tke amount
of
h,.
the
net positive suction head
NPSH
is required to be minimized. Proceeding from
con-swirl influx (eq.
(10.4-11))
the
follo\ving relationship applies to the net positive
suction head according to
[8.132]
and Cap.
10.2.5.1
2g-
NPSH
=
?,u:,
+
(1
+
[
+
1)c:,,.
(1
0.4
-
23)
In
the above,
ulp
=
o
Dla/2.
By
means of this, the net positive suction head NPSH
is
obtained
-
after introducing
c,,,
from
eq.
(10.4-
19)
into eq.
(10.4-23)
-
(the ratio
cm1/c,:,
being given) as a function of the impeller inlet diameter
Dl,. NPSH
may
be
represented
in
the following form:
K,
and
K,
are design-conditioned constants, at least at the bep.
The optimum diameter of the impeller eye with regard to cavitation
Dl,
opiv,,I,
at which
the net positive suction head
IVPSH
and thus too, the excavation costs are minimized,
is
then computed from the condition
dNPSHJdD,,
=
0
to the following:
10.4.3.3.
Optimum type number with respect to efficiency and cavitation
By
equating the diameters obtained from
(10.4-22)
and
(10.4-25). DloOp.,
=
DlaOpN,,!,
the natural optimum
nb
.,,
N!s-5H
of the type number
n,
defined in equation
(10.4-
S),
is
obtained.
nb
.,
,
.,,,
is a function
of
the pump geometry (expressed by quantities
SZ,
r2/rl,
Q,,
pl, N,
,
a3),
the velocity ratio
c,,/c,,,,
loss factors (such as
[,,
E)
and cavitation-
conditioned quantities (such as
).,
(,
y,,), scc also (10.2- 10):
For
@e purpose of optimizing the impeller outside diameter in accordance with equations
(10.4-6) and (10.4-
7),
this optimum type number is required to be introduced into the
f~rmula applicable to
C
(eq. (10.4-7)).
An impeller designed on these lines represents a distinct optimum with regard to
efi,
ciency and costs of excavation.
10.4.11.
The
discharge
ratio
as
a
function of
the
speed
ratio
Applying the customary speed coefficient
Ku,
=
or,/m,
(cf.
[6.8])
we proceed to defining the speed ratio thus:
In the
above,
K
li,,
=
speed coefficient applied to the process of pumping and
K
11,
=
speed coefficient pertaining to turbining.
Xso required to bc defined is the discharge ratio at rated flows
Qp
and
QT
In the above,
Q
=
flow rate (discharge),
Q2
=
contraction factor at the pressure edge,
cm,
=
meridian speed at that point, y,
=
cm2/u2, in which case
u,
=
ct,
.
r,
=
peripheral
speed at the
impcllzr outside diameter,
=
Hp/HT
=
head ratio. Index
P
detiotes the
pumping process and
T
referes to the turbining operation at rated points.
From the velocity triangles
at
the pressure edge (Index
2)
it is possible to read off
the
following relationships applicable to the meridian speeds
c,,,
,
,
(pumping process) and
-
.i."
c,,
,
(turbining process) in accordance with Fig. 10.4.14.
3
In the above.
P,
=
vane ansle at the pressure edge,
p*
=
slip factor at that point,
c:,
=
peripheral component of the absolute speed at the pressure edge (Index
PIE
pumpingjturbining process).
By
means of these data in conjunction with Euler's fundamental equation governing
turbomachinery (Cap.
5.3)
applying in accordance with equation (10.4- ll),
it
is possibl6
to quote the discharge ratio
y
as a function of
the
speed ratio
x
as follows:
@L
P
4
=
-.
x.
;C1i2
[1
-
(1
+
PW
k'
4
P
v,,P)I
(10.4-3
@2
I-
['
-
(I
+
P?)x~vuT/(~~u;~)I
'
495
10.4.14. Impeller outlet triangle durins
Fis
,
,mp~ng
operation
(P)
and inlet triangle
P.
during
turbining operation
IT).
Rotor vane in-
dicated
by
hatched line. Values without star
torrespond to the relative flow along the vane.
Turbining operation
U
2T
U2p
Pumping operation
In
the above,
q,
=
peripheral efficiency (hydraulic efficiency) in the turbining (Index
T)
and
pumping (Index
P)
state.
Kku,,
may be computed from the speed'relationship between the type number
n,
and
the speed coefficient
K
u,,.
Taking into account eq. (10.4- 19), this relationship reads as
follows:
cos
0,
.
,'
213.
[(I
-
N:)sin(@,
+
p,)
1"'
9
The optimum value for
n;
is known from
eq.
(10.4-26)
(n,
..,
..,,).
10.4.5.
Example for the computing optimum values
of
irnpeller diameter,
type number
and
discharge ratio as compared
with corresponding quantities of an actual pump-turbine
As
an
example let us study a pump-turbine with the following data:
10.4.5.1.
Data
1)
Impeller Geometry:
r2/rlu
=
1,69;
N1
=
0,6;
f/D
=
0,Ol;
Ah/D
=
O,C?;
p2
=
27";
a,
=
20";
no. of vanes
z
=
7; D2/D,
=
0,94;
b,/(r,
+
r,)
=
0,2;
Ol=6O0; pl=7O0.
Thus, from eq. (10.4- 13),
52
=
2,019.
Assuming the inlet edge to
be
straight, r2/rl
=
2(r2/rIa)/(l
+
N,)
=
2,113 will apply.
.
499
3)
blain
Operating
Data
of
Pump-Turbine:
The data quoted undcr
I)
and
3)
agree with those on the Austrian Rodund 2 pump-
turbine as far as the laiter were available on si~nplc lines. This pump-turbine has
a
rotalio~~al speed of
n
=
375 rpm
in
both modes of operation, the outside diameter of the
iinpeller measuring
D
=
14,26 ft (4,35
m).
10.4.5.2.
Results
By
reans of these figures we obtain
cp,,
=
0,310 from eq. (10.4-21) and thus the vane
angle
of
the inlet edge on the rim
/I,,
=
arctan
cp,,
=
17,2".
From eq.
(10.4-26)
we obtain
n;
,,
,
.,,,
=
0,728, thus from eq. (10.4-7)
C
=
2,365, from
cq.
(10.4-6)
Do,
=I
l4,23 ft (4,34
m).
From eq. (10.4- 33) we obtain
K
u,,
=
1,044.
Dop,
II~,,~~,~~~,,
and
S
u,,
agree well with the actual values of the Rodund
2
pump-
turbine, which amount to
D
=
14,26
ft
(4.35
m),
n&
=
0,724;
K
u,,
=
1,037.
From eq. (10.4-
17)
Lve
compute
h:,
=
0,044, and from eq. (10.4-
18)
h:,
=
0,036. These
vaiues
require a peripheral efiiciency
q,,
=
1
-
hi.
-
Iz:,
7
0,92, which agrees with the
assurneti
qup
=
0,92
if
loss in the stlction line is omitted.
The actual speed rztio in accordance with eq. (10.4-28) reads
K
.=
0,985. Thus from
eq.
(10.4-32) we obtain
q
=
0,738, which agrees well with the actual value
q
=
0,747.
10.4.6. Special
operational features
of
pump-turbines
Compromises Resulting from Shell Diagram and Practical Requirements:
Outlined in Fig. 10.4.15 is the so-called "shell (hill)-diagram" (the
curves
with
identical
eficiency are "shell-shaped") plotted for
a
pump-turbine
in
both modes of operation
in
,
the
Y,
cp-plane.
Y
indicates the pressure coefficient,
Y
=
2
gHj(o,
r2)2
and
cp
the
flow
coeficient.
q~
=
c,,i(w,
r,)
=
Q](A2
w,
r,).
(a,
=
angular velocity (design point),
and
d
A,
=
cross sectional area at the impeller pressure edge.)
3
a
Evidently applicablr to
the
optimum points in pumping and turbining modes of opera-!
tion according to
Fig.
10.4.15 are the follo\ruling relationships
Pop
P
<
(POP
T
•
(10.4-392
f
i.e. the optimum point of the turbine zone is located further away from the
origin
ofi
coordinates than the optimum point of the
pump
zone. Compared with the pump
Fig.
10.4.15.
Hill(shel1)diagram in the pump and turbine working order, schematic.
the turbine section in the
Y,
cp-diagram is more closely limited by the runaway parabola
(where torque is zero) and the parabola of maximum discharge (this is the envelope of the
curves with identical rotational speed at maximum flow rate). From Fig.
10.4.15 it will
be
gathered that the curves of constant machine speed
n
cross through the peak of the
constant efficiency curves over a greater length in the turbine zone than in the pump zone.
This means that a deviation from the optimum point, maintaining constant rotational
speed when pumping, involves a greater drop in efficiency than when turbining.
In addition, by deviating from the optimum point when pumping, maintaining
copstant
rotational speed, the machine lands in an operating zone in which
it
is highly sensitive
to cavitation. This would necessitate increasing the negative suction head.
On a facility purely designed as a pumped storage plant the pressure
coeficient
'Po,
belonging to the average level-difference between the head and tailrace level, is normally
the same in the pumping and turbining modes of operation.
If, as is usually the case, provision is made for the same adapted rotational speed
n,,,,
for
both modes of operation, the above circumstance calls for two courses of action to be
taken (see Fig. 10.4.15).
First, it is necessary for selection of the constant rotational speed
n,,,,
to be lower in the
pumping mode and higher in the turbining mode than the speed due
to
the bep.
Secondly, owing to the same pressure
coefficient
Yo
=
Yo,
=
Yo,
in both modes
of
opera-
tion, the actual turbine operating point
T,,,,
must be transferred into the partial-load
region (see Fig.
10.4.15) where eficiency is lower and swirl occurs inside the suction pipe.
This is mainly necessary in order to warrant optimum conditions as regards cavitation
when pumping.
From Fig.
10.4.15
it
will also be noted that, even in the case of a two-speed motor-generator
set
(rarely used), which permits optimum rotational speeds
no,
,/no,,
in both modes of operation, not
both optimum points
Po,
and
T,,
are attained if the level-difference (and thus
too,
the pressure
coefficient
Yo)
is the same
in
pumping and turbining modes of service.
Fig.
10.4.16.
Effect of wakes behind the impeller vanes at pumping at the inlet zone of gate.
The discharge
ratio
q
must (aside from the hydraulic optimum value) be determined with due
consideration to the average duration of the base- and peak-load period in the course of a working
cycle. Ftrrthermore, attention must be paid to balancing thc water quantities in the turbining and
pumping modes of operation, also taking into account thc processes of inflow and outflow during
the storage cycle provided the system is supplied
by
a river.
&,laximum turbine cfficicncy is attained only at the beginning
of
turbining in the storage cycle
owing
to the presence of the maximum level-difl'erence and thus too, the highest pressure coefficient
Y
(see
Fig.
10.4.15).
Maximium output will then be yielded.
10.4.7.
Sources
of
troubles
and
remedies
'
10.4.7.1.
Normal
pumping
In the course of pumping, wakes occur on the impeller outlet (see Fig. 10.4.16).
This
produces a periodically highly varying angle of attack, on the guide vanes.
If
a wake
passes
a
guide vane, the unfavourable angle of attack generates a momentary stall
past
the leading inlet edge of the vane (see Fig. 10.4.16).
As a result of this the vane channel is temporarily blocked. At the same time a
torque,
snbject to periodic change, acts upon the guide vane in the opening direction.
If this causes rupture to a safety member (provision for which is
usualiy made on
gates
actuated using a central servomotor), the guide vane may strike against the adjacent
vane*
with its end projecting into the impeller. This causes destruction of the vane's ends
andb*
possibly to the coupling bolts'as well when gate lever stroke limiters are not provid%
Such damage can be prevented by arranging for each of the guide vanes to be acthd-
by an individual servomotor, these motors being s~bjected to synchronous controL
$
The individual servomotor must provide a torque suficient to drive a vane even in
ce
the vane is temporarily in
a
misaligned position, see [10.106].
..
*a
In
the course of pumping, care should
be
taken not to regulate the gate. It will
be
to block the guide vanes. If, for example, the upper vane pin is provided with a
fhctl?
i
Fig.
10.4.17.
Section of
a
pump-hrbine
0
11
the guide vane drive mecha-
pism
is placed in
a
hollow, ribbed head
Author's proposal.
collar, this collar can be pressed against the housing (by pressure applied to the front side
of
the lower trunnion) so as to block the vane.
~mploying short upper vane stems (corresponding to Fig. 10.3.17), the means of driving
which are in the head cover, it is possible
to
prevent dangerous torsional vibrations from
lever, stem and vane.
10.4.7.2.
Abnormal
operating conditions
The transitional operating conditions specified below and capable of occurring following
load
rejection while turbining or failure of the pump drive, might possibly invo:\e
extrelnely varying forces and moments acting on the guide vanes and thus too, heavy
vibrations of the machine shaft and cover
[10.75; 10.1571.
a)
Slight opening of the gate when pumpirlg at full rotational speed.
b)
Pump brake operation
at
low reversed
flow
((p
<
O), lower ro!ationril speed and slight
opening of the gate:
In
this case the jet, partially deflected outwards
by
the runner vanes,
exercises a fluctuating closing moment on the
guidc vanes.
c)
Centripetal turbine
(=
Fi-ancis turbine) operation with extensive reversed flow
(y
<
0)
and
reversed (negative) sense of rotation.
d)
Runaway of turbine
(cp
<
0): when accelerating
a
radial machine up to the point of
runaway, the flow will be
throltled owing to centrifugal forces, this causirig water
hammer. Runaway whose operating point on the
Y,
cp-diagram (Fig.
10.4.15)
ib
located
on
the runaway parabola
=
const.
(S,Z,,
is a dangerous condition owing to a high
rotational speed being present without
normal speed control. While occurrence of the
operating conditions described under a) through
d)
may be "permanent" in
a
case of
emergency,
tile followi:~g two conditions occur only ten~porarily in the transitional
Process.
e)
Turbinc brake operation above the runaway speed with reversed flow
(q
<
0):
this
zone, located on the
If/,
v-diagram
(Fig.
10
4.1
5)
between the runaway parabola and
Ordinate, is passed through when the flow rate is reduced due
to
closure of thc gate, while
[he speed is almost equal owing to the inertia of the rotating pilrts.
4
Reversed pumping
(y,
>
0)
with reversed (negative) sense of rotation: this condition
Occurs, proceeding from the turbining proccss, when water hammer is caused due to
Ut
+ZT~
A
Reversed
Pump
Fig.
10.4.18.
Water
hammer characteristics
in
the turbine, brake and reversed
pump
flow
regime.
For notation see Fig.
8.35.
closure of the gate, and the rotational speed decreases only slowly owing to inertia.
Fig. 10.4.18
i~ldicates, for example, the water Iiarnrner characteristics in the event of the
load being switched off at a point of time
t
in the tlirbining mode of operation and the
gate continuing to close. Registered in Fig.
10.4.18 are the conditions at tile upper and
lower end
oC
the pressure pipe (Of,.,TL or
U,+,,J
as well as on the pump-turbine
(?;+,TL,
BI.~+,,~~, Prf+,,,,) this being done at the times indicated in the index
t
+
nT,
(n
=
0,
1,2
. .
.).
T,
=
Lln
=
transit time of the pressure wives in the pipeline
(L
=
pipe
length between
0
and
U,
n
=
sonic speed,
T
=
turbining,
Br
=
braking,
Pr
=
reversed
pumping).
10.4.7.3.
Begirrnifig
of
pumping
Up
to
80%
of the full-load input is required to speed up a water-filled radial type
centrifugal, pump with closed valve.
Hence, speeding up into pumping action of the pump-turbine presents a problem.
The
-
technical challenge and objective in this case is to attain as short a start-up time
as,i
possible employing the smallest auxiliary set possible. To arrive at this solution numerous
4
methods have already been devised, a few
of
which are mentioned below.
i3
The pump-turbine can be accelerated
by
the main motor-generator set directly provided
this is capab!e of functioning as
an
asynchronous motor. Owing, however, to the
low
permissible motor temperature (80
"C),
this necessitates the use
of
short-circuited full
i
pole pieces as for zxample at the Vianden
2
kciiity [lO.ltil] iowering alternator efficien
011
radial pump-turbines with extensive head, the rcquired start-up power has, in
m
cases, so far been reduced by filling the impeller with air. Howeber, this solution exte
the
start-up time and transfers the ensuing problems to the subsequent filling pro
During the process of filling, the turbulent, vortex-affected two-phase mixture
with
its compre
air-pockets may stimulate dangerous machine-shaft vibrations with frequencies below the
504