Raabe J. Hydro power - the design, use, and function of hydromechanical, hydraulic, and electrical еquipment
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g.3.h.
Srhcmc of conlp~rt--
in6
tr;ll,sicnis
in
c~sc
of f:lilurc
impc!!". pulllp drivc (c.g.,
O
(P"
!nl?-turl~inc
at
pumping).
a)
I'roccdurc
by
char;~ctcristics-
rncthnd
in
t
hc
1).
c,-plane with
wi!hot~t
CIOSIITC
of tllc main
,,l~vC
on the
I>~CSSLI~C
side of tlic
.,ump
(bct\vccn
U
and
I',
scc
b))
In
th~
punlp r31!2'.
(c.
>
0)
and
ra11gc B1.(1.
<
0).
gZ
,
g
,
c~nr;l~tcristics for s~:;tesp?
c
moving wit11 cclcr
11)'
+
(1,
--
fl
In
-v
((,)-direc! ioii. Sl~bscripts
see
Jig.
8.3.5.
b) Scheme of
pump,
Fmt~~k, V~IYC;
~CIIS~OC~
loss
,.onccntr~
tcd
be~vfeen
0'
and
0.
-
L
p
=
const
llcnce for an in~peller pump, with its positive floiv from thc lower end to the upper end.
thc state on the lower end
U,,
,
at the insiant
t,,
,
((k
+
I)
T,
aftcr cutting out pump
drlvc) folloc\~\ from tllc "state"
0,
at the time
r,
on the uppt31. en3 of tlie
penstock.
In
this case
the
s~raight c11ara;tcristic according to
(S.3-53),
which crosses
0,,
intersects
thc characteri~tic of thc pump with its speed
rl,+,
(now variable in conscquel?ce
of
the
3,surned cut ollt
of
the drive) at t!ic instant
t,,
,
(Fig.
8.3.6). Whcn the throttle valve is
located
upstram of the in~peller, the intersection of the straight characteristic (8.3-53)
coming
fro111
O1.
OCCLI~S
at a point
U,
+
,
(Fig.
8.3.6
shows this for
k
-
0)
with
2
pressure
drop
due
td
the
instantaneous loss in thc valve
Ap,,
,
=
K,,
,
ci,
,
against the point
P,,
,
on
(he charnctcric~ic of the
pump
at the instant
t,,
,
with the speed
II,,
,
.
Continu~ty
rcquires that
1;.
,
and
U,,
,
have the smile velocity
c,,
,
.
The
parameter
K,..r,
of
the
Instantaneous valve cha~acteristic depends on its instantaneous position (opening) and
lhc
sense of throughflow.
Tl~c
instantaneous speed
17,.
,
of
the
purr,p results from the speed
n,,
a period
TL
earlier,
from
the braking torqlie
on
the impeller
(taken
from the characteristic or calculated from
~lf,
Q.
11,
if
available)
iM,
+
,
and
M,
due
to the instants
t,+
,,
r,
and the moment of inertia
of
the set
0,
by
means of the set's equation of motion, whlch reads in terms of angular
rclocity
w
=
x
ir/.?O
(instead of
11
rpm)
Other
exnmplc.;.
Fig.
8.3.7
ti
shows
a
failure of pump drivc togethcr with a throttling of
thr
main
'al\c during pump operation. The following is seen. Owing to the inertia of the rotating set, the
Frasurc at thc
lower
pipc clid prelimin:irily falls short of its original vnluc. This is later on followcd
Prcssurc rises
above
the origin;ll pre5surc.
By
thc actuation of the valvc yet during the normal
W~P
rcgimc, the prcssurc drop on thc lower cnd
U
of tlic pipc is enlarged cornpared with an
'Wralion without nctuating thc valve. The diagram contnins also thc lines of constant torque.
Iig8.3.7
b
shows the conscqtlenccs
of
a pump drivc klilurc
in
the
brakc
ficld of the punlp under
rnerscd
flow.
Contrary to
the
previous
example,
hcrc the prcssurc drop at thc lower pipe end
is
wmcwh;~t
smoothed.
I-ig.
8.3.7
c
shows the shut down c-f
a
turbine supplying a grid
by
itscl!'to zero
/
turbine
but
disconnecting
the
set
from
the
grid and hence raising the speed.
3
18
ccd
by
mcitns of
a
synchronous closing
of
gate
and main vitlvc (indicatetl
by
the hatchcd parabola
''
-
whcrc
I.
ilcpcods on the timing of shut down) without disconnecting the set from ths grid.
~~~r;llly
prcssllrc risc at thc lover pipc end, speed dropping.
Fig,
8.3.7d sliowr
it
clos~~rc
of
gales
under constant spcccl. This cnsc
may
hilppcn in
a
tcrnary set.
%,hen
chiingin!: cluickly from turbini~lg to pumpillg. Also herc gcncrally prcssurc rises at the lo\\;er
psns(~~k end.
8.3.7e shows thc samc as in Fig. 8.3.7~ but with disconnection of the set from the electric pid.
speed rising.
8.3.8.
Characteristic
lr~ctliod
in
the
c,
w-planc
TI,^:
disadvantage of
a
water hammer computation in the
p,
c- or the related
YJ,
cp-
or
11,
Q-plane exists for pul-r~ps in the fxt, that the brake and turbine zone overlap each
other-
~hcrefore a point of
the
H,
Q-plane is not always assigned unequivocally to
a
definite
operating
poi~lt of a machine. This troilble
can
be avoided
by
the application of the
w;
c-
(or
11,
Q-)graph (Fig. 8.3.8). This is especially advisable, when the machine operates at
quickly varying speed at start up or run down.
In
this graph any point of the plane is
definitely
assigned to
a
special operation of the machine.
The question arises: How the straight characteristics of they, c-plane should be
replaced?
For this purpose consider again the positive characteristic in the
c,
p-plane
in
its dil'feren-
tiat
form without losses (8.3
-
21):
dp.,
+
~adc+
=
0.
(8.3
-
56)
The increments
dp+
and
dc+
due to
dx,
=
(a
+
c)dt can be written down in
a
more
expanded manner
Asp
is a function of
c
and
o
lnscrting this in
(8.3-57)
the rclation (8.3-56) can be written as
Fig.
8.3.8.
Characteristics-method in the velocity
(c)
speed
(w)
P~C. Curved cliaracteristics
(-,
(,
for states moving with
elcrity
-
a,
-+
a
alo11g a pipc
(x)-axis.
Lines
of
constant pres-
'We.
Preferably
uscd, if transients cross the 1st brake regime of
Pump
and the following rcgimc of
a
cen!ripetal turbine
'=Francis turbine) since there is
no
overlapping of machine
ch"ra~tcristics as cxisting in the
p,
c-system.
Obviously, for
a
suhst:~nrial incrcmcnt
of
:I
vasi:iblc within thc systc111 of chal-;lcte
due to
tls,
=
(tr
-1
(.)tit
anti
with respect to
cis
=
c7.u..
(1s
-
(1s.
the I:\st
cclual
convcstcd into
tlo
+
+
([((7/)/$c),,
+
q
n]
/((?p/i)~)~)
dc
,
=
0.
In
this relation
tito,
and
tlc,
are substantinl
increments
of
to
and
c
in the charnct
systcrn, which moves with celerity
a
relativc
to
the fluid in
+
s-direction. It is s
also in the simple case of constant ccleritjl
cr.
the inclinatiotl
(tlro/dc)
of the water ha
line in thc
to,
c-plane depends on machine-cot~clitioncd values such
as
(t?p/ac),
(?p/(;(o),,
from which thc first usually can be neglected compared with
Q
a.
For calculating water haminer and resonance due to
it
in pipe systems, see Cap.
8.3.9.
Remedies
against
water
hammer
--
By-pass outlet on FTs with long penstock: Here the water hanimer is limite
eiinlinated by a valve through which the flow can be by-passed in the case of
shut down. This is done by a valve, operated, in
!he case of a rapid shut down by the
ring (pro~lided the machine has one). The valve is designed
so
as to function as an
en
dissipater
in its downstream area (Fig.
8.39
a). (Usually two halves of a pump spiral
ca
~vound against each other with a diffuser joined to it, sometimes also
a
conical
known as
a
Howell Bi~iiger valve
in
North America).
Afterwards this valve
is
smoothly shut off by a spring or oil pressure-induced
to limit pressure
surge in the
penstock.'^
lower end.
?'he control scheme and valve design (Fig.
8.3.9)
show a valve, that is operated by
pressure loaded piston
19,
actuated from the spced governor through the gate
motor.
It
operates as follows.
A
quick shut down raiscs the prcssurc by
a
dash pot
4
linkcd to the gate scrvornotor piston.
relieves
the closing piston
19
of the by-pass valve and loads an auxiliary opening piston
20
on-
valve rod. M'hcn the rapid shut down of the distributor is conipletcd, the by-pass closes smoot
by tl~e action of a throttle
17
and the spring
5
in the dash pot.
Due to the flow
passi:~g tne throttle
34,
the opening piston
20
is relieved of its oil load.
Sim
neousiy the relief vi3lvc
8
of
the dash pot piston
5
closes the by-pass outlet
19.
Thc design has a safe[!; clcvice. When the motion of the by-pass valve
19
is blocked, c.g. by jam
a
iurthcr quick sh~t down of the gate
is
also blocked. This follows from the oil now displac
'
the dash pot
7
and its valve piston 6. This closes the oil outflow from the closing side of the gate
motor by closing the spring-loaded
needlc valve
16
in the duct of. this outflow.
In Pelton turbines water hammer is limited by
a
jet deflector (jet presser, jet
c
(Fig.
11.2.9),
which in the case of
a
quick shut down, deflects at least
a
part oft
ihe bucket. Sin~ultaneously the needle closes slowly by mcans of a throttle in
admission line of the needle servo motor. Then under steady
state condition, the de
is brought to the ready position. Usually this position is next to the jet edge. This req
a
cam
or a corresponding device to assign
a
certain opening (needle position) to
a
pOs
of the jet deflector.
-
Therefore two control loops
are
required for needle and jet
deflector.
They
may
bc
connect
parallel or
in
series, the first being preferred nowadays. The control of the recently
260
M
W. 6-nozzle
PTs
Silz (Austria)
does
not use the
cam
dcvice. Hcre under steady stn
the ready position of thc
jet
deflector is always the same and given by a stop collar.
'The
320
~i~.
s
3.9.
Oil- and das11-pot-
apcr.l!s'l
by-pass c?utlet of
I.-rnnc~b t~lrbine. Against ciccs-
~)~CSSLI~C
risc during thc
Sh,lr down with long pcnstoclts
or
311
expcctcd fast rcgulntion.
safety nccdlc valvc
16
closes if
,al~~
19
of the by-pass outlet
jams.
response required now results from the vcry short time needed to move the deflector to the edgc of
rhc
jet.
Water hamrncr and cavitation: As shown with
(8.2-21)
and expoundcd in Cap.
8.2.4,
the collapsc
01
3
yaporous cavity causes water hammer. And water hammer waves may propagate the pressure
pulsc from
21
bubblc collapsing within thc liquid to a distant wall. Hencc water hammer connects tlic
hydrodynamic cavitation wit11 its erosion. 14owever. the linkage of a dibtinct hydrodynamic cnvitn-
tion of known duration. e.g on the runner vane surface of a Kaplan turbinc to the resulting pitting.
is
one of the major unsolved problems the operator of hydro turbincs 1s confronted wi!h. This has
scveral reasons. Firstly, at
a
full-size machine hydrodynamic cavitation cannot be visualized. Only
an ultrasonic noise
duc to cavitation could be rccordcd. Secondly, for economic reasons ar.y
observation of cvcntual pittings can only bc made aftcr a longer period with an usually strongly
varying mode of operation and hencc also of cavitation.
Hence thc pitting then observed is a
timc-averaged value
Am,,.
Provided the gate and runner vane position relative to each other and
according to the draft are also occurring in
the.real turbine, a record of dischargc and nct head vs
clapsed
t,
in connection with known cylindrical runner vane section, may help to recalculate
atso
a
or
w,,,
(along the critical vane section) vs time elapscd. Thus according to Knapp's law.
wherc
T
is
the observation period,
K,
a
parameter
for the given material but usually due to a ccitain
Icsts dcvicc (e.g.,
magneto
strictive apparatus), whose pitting generation mechanism diflers greatly
from that in the turbine and probably is a function of
w,,,.
According to Knapp's test also
n
depends
On
the type and dcgrcc of cuvitation and hencc probably also on
w,,.
Hence any prediction of
~irting on the basis of the usual tcst rcsults from the lab and over a longer period of continuously
operation at a
rcal turbine, is rathcr unsafe.
9.
Similarity
taws,
characteristics, research
9.3.
Introduction
In hydro power machines, model tests are needed to predict thc behaviour of proto
by means of similarity laws either known or yet to be found. The strict appIicati
mcdel tests to a prototype requires also knowledge of efficiency scale effects due
t
iiol~ls
numbcr, rou~hness and probably other effects such as structure and level of
lence.
A
survey of
a
fluid machine's behaviour may be obtained by the so-called hill diag
FIcrc the limits of operation in consequence of surging, due to working away from the
of cavitation index required, gate limit, runaway and generator capacity are of intc
Scaling of efficiency also needs knowledge of loss in the different elements of a
machine.
The measurement of loss and its components can be made on the model or the
pro;
by
conventional methods using flow meters, strain gauge and dynamometers. Th
mo<lynamic method limited to higher heads and prototypes saves test time.
The unsteady behaviour of machines especially under off-bep conditions may be in
gated by model tests carried out by instrumentation with a quick response. This req
a
kno\\~ledge of probes and manometers for measurement of pressure and veloci
n~;lgnitude
rind
direction afid also very close to the wall. This includes also informa
11l)out sources and magnitude
of
measuring errors.
Tests
rnny
be carried out in water or in air. The measurement on rotary devices req
special
sisnal transmission devices. The visualization of flow phenomena merits
s
attection. Actual test results of quasi-steady and dynamic measurements in air and
on stationary and rotary elements of hydro turbomachines by means of air
inje
multi port vectorial probes of quick response and laser anemometer are shown
to
the problerils.
The
references
on
this
topic span a wide range. [9.1; 9.21 dcal with similarity
la\&,
[9.3 to
9.
the related problem of technical roughness on the wetted surface of hydraulic machines,
[9.
thc u~~ccrtainty of their cl13ractcristics. Comparative nieasurenients of characteristics between
and
pratotype are :cported
on
pumps and turbines [9.7], on Francis turbines (9.81. on pump-
(9.91. The thermodynamic method for eficicncy
is
treated
in
[9.10 to 9.121, tests on proto
high-head pump-turbiucs in [9.13; 9.141, on large pump-turbines in [9.15;
9.161,
general
pr
tests
in
[9.17],
and
flow
measurement on prototype in [9.18]. Experiences on large Pelton
t
arc given in (0.191.
Acceptance tests on large pump-turbines (9.201 were compared
with
model tests instead of
tes~s
[9.21
to
9.231. Test rips were described for high head (9.24; 9.251. for
1,5
MW
output
impulsc turbines i9.271. for
different
reaction machines [9.25 to 9-31), and for the pur
hydr:lulic engineering
[9.32].
322
rests Ilave bcen carried out in water and air
[0.33;
9.341,
in
different air test rigs [9.35], by air tests
",,dcr v;~i;lblc dcnsity 19.361, and thus for sc;lling eficicncy.
,,,,,olig;ltions have been rn:ide on 2-stage punip-turbines with adjustable gates [9.37]. rcsearcl~
\vns
G,rricd out on components of pump-turbines, such
as
the components of their stagcs [9.38]. the
,,,ti,ke [9.39]. the gates [9.40], thc turning passages i9.411.
hle3surcnl~nts are
discussed,
bcsidcs the bcp on radial impeller pumps [9.42] and on Kaplan turbincs
19.
431, on the pump torque under zero flow (9.441. and on Francis turbines during unstable opera-
,ions
(draft tube vortex) [9.45 to 9.511.
Syhicni oscillations wcre i!ivcstig:~ted in [9.52
to
9.541, the noise in power stations [9.55]. tllc radial
forces
on the rotor [9.56]. The instrumcntation of tcst rigs is described in [9.57], their a~~tonintion
in
[g.sS],
speed measurement in [9.59], acoustic velocity nieasurcnicnt in [9.60]. the use of strain
*cs on Kaplan runncr vanes in [9.61], flow control in [9.62], the utilization of lasers in the Iabo-
p311r
r.lory
in [9.63 to 9.671.
scaling in general is discussed in [9.68], tlie scaling down of efficiency in [9.69], the scaling
up
of
efi-
,-iCncy in connection with roughness in [9.70]. the scaling of performance data in [9.71]. Influences
wall nearness and
orientation
of tlie head of a directional probc have been mcasurcd
by
Kiilrrlcl
the pressure distribution on the amplified probe head [9.72]. hleasurernents of the
3-diniensional flow in
a
diagonal pump arc described in (9.731, those of the 3-dimensional nbsolutc
and
relative flow
in
axial machines with variable runner vane numbers by
Kiii~tlel
in [9.74].
~hc
relative flow field
Oil
the runner vane of a Francis turbine has been measured by
Uiir
(9.751.
The
mcasllrcd and
predicted
flow in such a turbine is compared in [9.76] and by Pjber~ner
in
16.221.
Other effccts like the influence of a variable geometry of runner outlet [9.77], the
yon
Karn~un vortex
street [9.78], the runner inlet geometry [9.79], the geometry of the ciraft tube inlet [9.Y] on the char-
acteristics of Francis turbines, have bcen measured.
Dy~amic measurements of the absolutc and rela-
live flow field in Kaplan turbines were carricd out by Castorpll (9.811 and on Kapl'ln
and
Francis
turbines close to tlie wall of runner vanes were reported by
Furrrler
[9.82]. The relativc and absolute
flow field of a Francis turbine under
scveral modes was obse~vcd dynamically by
Srlrletil~ilc~r
and
G~riclt
[9.83].
9.2.
Similarity
laws
and
characteristics
of
machines
9.2.1.
Criteria
of
similarity,
numbers
of
Fronde, EIIIc~, Rej~noI[Is
In
hydraulic fluid machines, similarity laws arc required for the interpretation of model
tests
and hence predicted behaviour of full sized prototypes
[9.84].
The
first supposition for ally prediction of this kirld is the geometric similarity of the flow
passages. Strictly speaking this includes also the technical roughness of the wetted
surfaces. According to experience the latter
illfluences the loss only, when the ReynoIds
numbcr due to this
roughness
elevation
k,
defined by
kcjv
does not exceed
50
to
80,
c
being
thc mean velocity and
\I
the kinematic viscosity
[9.3].
Time-averaged steady flow is considered. The observations on the model then transferred
to
the prototype have to be referred
to
a
certain load factor with respect to the bep. Here
a
great problem exists, since the bep itself is subjected to scale effects usually not clearly-
known. This difficuliy in the case of water turbines usually is lessened as the Reynolds
number of real water-operated prototypes
is
in the range, where the change of eficien'c~
with
tlie Reynolds nu~nbcr becolnes
a
problem of the second order.
Therefore
to a first
Order approxin~ation thc load factor referred to the bep might be used as a criterion for
or homologous operation of model and prototype.
For
fhc ,~lmvc rc.,~~on~. In tlic lollo\\ing consic!oration
of
fir.)t ordcr cll'ccts, thc sc;llc c[fcct
ir:tcrn.~l lo\<
.~nd
cl5c1cnc.j cl11c
to
\i/c
311~1
1:c;td is ~icglcctcJ.
AIl
r
lw \~ni~l;~r~t! laws
;ire
rclatcd to nnn-cavi rating flow. sincc wall-fiIt;~~hcd
cavities
di
sinl~larity
of
plofils\ consltlcrcd. Strictly spcaklng, thcy would :~lso inducc
an
unstcudy
flow,
no
side1 cd hcrc.
Any simi1;irity
is
related to homologous stations in model and prototype, e.g
nominal rotor diameter
D
(Fig.
9.2.1).
Neglecting
the boundary layers
as
Second
effects.
steady flow in gco~netrically similar
ducts
implics also
a
similar
flow
prltt
homologous stations and similar velocity triangles tl~crc
011
modcl and prototype.
all
the velocity tcrnls in the triangle at
a
given station under given load factor
proportional to each othcr, namely
c,,,
-
11
-
c
-
w.
-
fror~tle's
law: Here only the kinetic energy
c2/2
and the potential energy
gll
at
a
de
station under definite load factor in the model and prototype are proportional to
other. Hence
Fr
=
,/~'/(~k)
=
const.
In geometrically similar plants without losses,
the
net head
ti
and the elevation
h
proportional to e3ch other. Thus the last relation leads to a
Froirde
!lumber in
a
wide
meaning:
c~,'(~
H)
=
const. This brings the important relation
c
=
const
(y
H)'12.
-
Errlei-'s
law: Here only the pressure energy
p/~
and the kinetic energy
c2/2
at a
ce
station under a definite load factor are proportional to each other. Hence
Ert
=
II/(Q
c2)
=
const.
This type of number is also used as cavitation index, (see
8.2).
-
Strotrlrirl
number: This number appcars in unsteady flow, put aside in this se
is proportional to a
time interval
r'
required, to cover
a
certain distance
L,
e.g.
the dist
of
neighhouring vortices in
a
v.
Knritlnrl
vortex street, with a certain velocity
c.
H
Sr
=
Ll(ct)
=
const.
-
Rej.t~ciltls
number: This determines the portion of viscosity
-
conditioned loss,
ol the
second
order, put aside in this section. For its sirict definition Cap.
5.4.
In
it
is defil~ed as the ratio of inertia force to viscous force
Re
=
L
c/v
=
const,
where
L
is
a
length n~ostly responsible for energy dissipation, e.g. chord of
axi
vanes, and
c
the velocity due
to
it.
9.2.2.
The
unit
value5
of
speed,
flow, power
-
The
unit speed: The angular velocity (in radls) of a rotor with
a
reference dia
(Fig.
9.2.1)
and
a
blade speed
11
at
D
is given by
o
=
2
1110.
With respect to
(9.2-
the si~nilarity of all the veiocity components of
a
velocity triangle, including
the
speed
u
and the meridional velocity
c,,
the followi~lg holds true
I1
-
cmrr
C
-
(gH)'I2.
Hence:
w
-
(gH)'\"D
or with
n;,,
the
unit
speed,
o,
=
n;
iu
H)'I2/D.
strictly speakins, the unit spced is an angular velocitv of a sirnilfir machine, operating at
kp
with
H
=
1
mL,'s2, h::ving the nolnin;~l roior di:~metcr
D
=
1
111.
For pmctic:ll
pur-
pses
9
is oniitted and the rotational speed is capressed
in
11
rpln Hcnce
Contrary to
I]',,
the unit speed
n,,
depends up011 thc system of units used.
Its
n~cnning
,,the speed in rpm for
a
mschine nsith
I-1
=
1
nl and
D
=
1
111
undcr a definite
load,
usu-
ally
the rated
one.
-
The
unit discharge: The diwharge (flow)
C),
for
a
certain load can be determined from
a
ccrtain area
A
of
a
flow passage with the mean rncridional velocity
c,,,,,
by
Q
=
c,,,,,
A.
,\ssuming
A
-
L)'
and
from
(9.2-
6)
c,,,,,
-
(~lN)"2, then at bep
where
Q;,.,,
is the unit
flow
at
the bcp,
a
nail-dimensionnl
figure, valid for
a
i~nilar
machine with
D
=
1
m, when
gll
=
1
111';s~. For practical purpose:; sometimes the rated
discharge
Q
(usually above
Q,,,)
is
taken
at
a
point in the overload ral;ge, which has
a
defi-
nite efficiency clr013 co~nparcd with the bep (t1st1;111y about
2
%).
Morzo~er
g
is onlitted.
I
lcnce
Q
=
Q,
~~11'1'.
(9.2
--
10)
Q,,
being thc unit flow of a geometrical similar machine of
D
=
1
ri
operating under
If
=
1
m
at rated load. Contrary to
Q',
,
.,
and
I!',,
.
diinensionless numbers of
a
certain
machine
lype.
the values
Q:,
and Ir,,, both dimension-linked, depend on the system of
units for
the
same machine.
-
Unit powcr: Thc power follo\vs from the 1.e1ation
P
=
eQgH
t1",
where
11
is
the
cfliciency at
reference
load, tile positive sign for
a
turbine, the negative for
a
pump. Thus
P;,
beill3 thc non-dimensional unit output (input) for a geometricaliy similar machine of
D
-
1
m, operating under
{j
tl
=
1
ln2,/s2 at thc reference load point with a fluid of density
@rind
the irnasini~ry
efficiency
11
=
1.
Elirnin:iti~~_c
Ijl-I
in
(9.2
-
8), (9.2
9),
(9.2-
11)
yields
lhc
relations
11
-c
Q,
P
-
113
DS,
also used.
9.2.3.
The
type
~~tlrnber
(specific
speed)
'.
As
a function of
tllr
warking
d;n;l
g
H,
Q,
:I)
or
n:
Tlle type number (specific spced)
,I,
as
a
non-dimension:t1 figure is the angular velocity (radls) of
a
similar machine, whose
-3
*
flo\v
Q,,,
-
I
I'
I
tx.~~.
hc.11
I1
=
i
I.
I:l.o~n
(0.2
-
7)
11;
=
11;
ill,, where
tg
;
11,
,I
I
I.
01
:is
D,
-
I
,'Q;
',
,>.
l~i\cr~ing tliis
ill
tl~c brcgoing relation
yields thc
I)
pc
nunlbcr In terms
ol'
thc unit v;~lucs
11;
,
,
Q',
.,
ti:,
11;
,
QaI ','flJ. (9.2-
12)
With the siniil,~l-ity lii\vs
(0.2
-7),
(9.2--0)-
from which
II~,.~
=
UJ
D
11)ll2
;
Q;
,Jp
=
Q,,~/~I~(~
II)~~~], (9.2-
13)
tlic relatiorr
(9.2-
12) gives the typc numbcr ils
a
function of
the
working data of set
ij
ll?
Q",,,
(JI
as
a
nh
=
(,JQ~;~;($~ 1d)314.
.
(9.2-
14)
11;
is n din~ensio~lles; lipure for
a
certain ni:ichine type, and hence independent of
the
I
system of linils. wlien
Q,,,,
yN
and
(9
are measurecl
in
correspon(ling units. For practical
4
purposes in Eirr-or<, the following unit-linked p:lranieter
11,
is still used. It is defined
the speed
ir.
rpm of
u
similar machine under
n
head !I
=
1
m with the flow
Q
=
1
rn31s
at the reference point (bep or
overload)
IT,,
=
IIQ'~'/H~'~
=
11~~
Q:{2.
The values
11;
and
11,
are re1:lted to each other by
11,
=
(333/2 n)
nil.
For
a
n,,
an
cnerload point of Q
=
1,17
Q,,,
(indicated by the subscript
1/1
j:
r~,
!,,
=
57,3
above the values
rl,
and
II:,
are linked to
a
p:~rticular turbomachine w~th delinit
of design eithcr
for
a
pump or a tubine (see also below).
In
the design of
\i
a
ter turbines also the spccific speed introduced in 1905 by Cnnlere
used
[9.85].
This is defined
as
the 5pecti in rpm of
it
similar turbine, which develops
a
load
output
P,
,
=
1
1111
under
a
hcad of
H
=
1
rn.
The full load output i~sually co
sponds to
a
point with a certain efficiency drop of about
2
O/o
coalpared with the bep.
T
11,
=
II~,
el;:,
=
11
~~t;~/~i~'~
=
nq(gQ~1111/736)11?
With a
full
loil~l efficiency q!,,
=
0,9:
a
density
Q
=
lo3 kg!m3:
11,
=
3,5
11,
=
180
Actually
P,
rncusurcd in
kW.
15
used. The following relation exists between t~,(hy)
a
ll.(kCV): rt,(l~p)
=
1.17
n,(kW). The formulas (4.2-
14)
to (9.2-
16)
yield the type num
as
a function of the ivorking data
(j~
(or
;I),
Q,,
(or Qlil) and hence P,, (or ell).
11.
The type number as a function of design features: Next the type number
is
e
by the geometry
of
a
certain turbornachine design and its velocity coefficients
K
Klr
=
~q'(2~H)'~*
=
~oD/[2(2~H)'/~],
Kc,,
1
op
=
C,
op/(2gH)112
=
Qop/[(n14) D:(1
-
N:)
(2gM)1121,
where
D,
is about the runner throat dian;etel- (strictly speaking the outermost runner
e
diameter),
N,
the hub/'tip diameter ratio there. Comparisoll of (9.2-17), (9.2-18)
a
(9.2
-
7),
(9.2-
9)
brings
IT;
,
=
K
I~,
Q;
1
01'
=
x2-311
(1
-
~f)
-
KC,,,,.
(Ogi):
Using the unit speed
II,,
and unit flow
Q,,
from (9.2-8) and
(9.2-
10)
gives
nl,
=
(60!~)(2~j'l~K1r,