Hydraulic Turbines

Hydraulic Turbine»
211 INTRODUCTION
Hydraulic (or water) turbines are the machine* which tm the
energy of water (hydro-power) and convert it into mechanical
As such these may be considered as hydraulic motor*, or prime-.
movers. The mechanical energy developed by a turbine in med in
running an electric generator which is directly coupled to the shaft
of the turbine. The electric generator thus develops electric
which is known as hydro-eUctric power.
hydro-electric power is relatively cheaper than the power generated
by other sources such as coal, oil, etc,,
hydro-electric and multipurpose projects has been undertaken in our
country in order to harness more and more power from the available
water power potential.
)
energy
power,
Since the generation of
now-a-days a number of
The idea of utilising hydraulic energy to develop mechanical
energy has been in existence for more than 2000 In the earlier
were
years.days of water-power development, water wheels made of wood,
widely applied which used either the energy of falling water (i.e.,potential energy) or the kinetic energy of the flowin
One of the types of water wheels formerly used
It consisted of a series of buckets> g stream of water.
was the overshot wheel.
the diameter of which was equal to
permitted to enter the buckets at the top, and the unbalance createdby the weight of the water caused the wheel to rotate The bucketswere destgned to empty themselves when they reached
’
,h L f
the wheel. The overshot wheel when 5S , “ed the bottom of
good efficiency, but it could not
’
be builtTo
^leT"*1
’
. Another type of water wheel (> , d,e large quantity oi
whe4, which used the kinetic
y used was the undershot
undershot wheel consisted of
a periphery of a wheel and so placed that a «, T,I
water used to strike the blades on the ,t [ Y 3treamo(
effiaency of this type of wheel was low A*
,„„1,
'
the straight blade type of undershot wheel
' ' an
injProvement on
vh.el was suggested by PonceJet,
Water was
water
energy of the water,
a series of a
An earlier type of
straight blades attached to
The

HYDRAULIC TURBINES 963
who ia^tcad of straight blades designed curved blades so that waterstrikes the blades of the wheel, practically without shock. This typeof wheels were called Poncelet wheels.
However, these water wheels utilised small heads andcapable of producing small powers.
low efficiency and they used to run very slowly and hence these «
not be directly coupled to the modern fast running electric generalfor the purpose of power generation.
were
Moreover, these wheels had a
can-ors
As such the water wheels havebeen completely replaced by the modern type of hydraulic for water)
turbines, which may operate under any head and practicallyany desired speed thereby enabling the
directly.
generator to be coupled
In general a water turbine consists of a wheel called runner (or
rotor) having a number of specially designed vanes or blades or buckets.The water possessing a large amount of hydraulic energy when
strikes the runner, it does work on the runner and causes it to
rotate. The mechanical energy so developed is supplied to the
generator coupled to the runner, which then generates electrical
energy.
21*
2 ELEMENTS OF HYDROELECTRIC POWER PLANTS
One of the essential requirements for the hydroelectric power
generation is the availability of a continuous source of water with a
large amount of hydraulic energy. Such a source of water may be
made available if a natural lake or a reservoir may be found at a
higher elevation or an artificial reservoir may
Fig. 2IT shows a general layout
which an artificial storage
The water
be formed bv
constructing a dam across a river,
of a hydroelectric power plant, in
reservoir formed by constructing a dam has been s own.
. ,
known as bead race lev'l or simplysurface in the storage reservoir is
Water from the storage reservoir is carried through pen-
Penslncl* are the pines of large
head race.
stocks or canals to the power house.
diameter, usually made ofjisel,
reservoir (<J [hfi
^carry water under pressure fiom
^ forebays are alsoIn some installations smaller reserve
^* *
ir at the head of
reinforced concrete, which
provided. A forebay is essentially a storage reservoir
the penstocks. The purpose of a forebav is to temporarily store water
when it is not required by the turbine and supply the same when
required. Where the power house is located just at the base of the
dam no forebavs are required to be provided since the reservoir melt
serves the same purpose. However, if..the power home is gjjafed
away from the storage wwvnir. then a forebaiifflayto-piuvidcd- »

HYDRAULICS AND FLUID MECHANICS964
that case water from the reservoir is first led into torebay which in
penstocks through which it is supplied to theturn distributes it to
turbines. Furthermore, where the power house is located at the end
of a canal, a forebay mav be provided oy enlarging the canal just
ahead of the power house.
OAM
r~
ENERGY UNEURACE •
C-RCSS
TAR.
RACE
1
DRAFT TU56
«
Z
*
0£TlW pi
K*
*C££ RACE ENERGY IUE
*f
i * MPutsg
ivfcs) TUR&INE
CROSS
HEAO
Mi
TAH RACE1
Lifit*
2
*
rA 1M
trt~» 7/AW'
1
//
Fig. *1 I Genera.:ayou: of a hydroelectric power plant
The water after passing through the turbines is discharged to
the tail race. The iaU.. .. ...
'
- • ••
'
. .a c.. rr - > "
;stl *
(know-n as sail water) away from the power house after it has pas**1
through the turbine. It may be a natural stream channel or
genially excavated channel entering the natural stream at some poi»
‘
+ac
%

HYDRAULIC TURBINES
downstream from the power house. The water surface in the tail
race channel
^known as tail race level or simply tail race.
DS AND EFFICIENCIES OF HYDRAULIC TUR.
965
213
ES
(a ) Heads. The head acting on a turbine may be defined in
two ways as follows :
( ? ) Gross head is defined as the difference between the head
race level and the tail race le
^el when no water is flowing. As such
the gross head is often termed as static head or total head and it may
be re
£resented by H1 as shown in Fig.21*
1.
(ii) Net or effective head is the head available at the
entrance to the turbine. It is obtained by subtracting from gross
head all the losses of head that may occur as water flows from the
head race to the entrance of the turbine,
mainly due to friction occurring in penstocks, canals etc. Thus if
H represents the net head and hf is the total loss of head between
the head race and entrance of the turbine then
H=H1— hf
The losses of head are
(21*
1)
For a reaction {or encased ) turbine as shown in Fig. 21'
1 (a) the
net head is equa 1 to the difference between (l) the pressure head at
the entrance to the turbine plus the velocity head in the penstock at
this point plus the elevation of this point above the assumed datum,
and (2) the elevation of the tail water plus the velocity head in the
draft tube at its exit. Thus
•**
V,
7 )
TVH=(
For an impulse turbine as shown in fig. 211 (b) the net head is
equal to the difference between (1) the pressure head at the entrance
to the nozzle plus the velocity head in the penstock at this point
plus the elevation of this point above the assumed datum and (2) the
elevation of the tail water. Thus
(21*
2)Z
^
~
i~
+Zi+ * * •
2g2gW
(-S+5«•)-* (21*
3)
n= ** •
Hydro-electric power plants are usually classified according to
the heads under which they work as high head, medium head and
low head plants. High head plants are those which are working
under heads more than about 250 m ; while low head
those which are working under heads less than about bC m, a t
those which are working under heads ranging
tedium head plants are
from 60 m to 250 m.

HYDRAULICS966
Efficiencies The various energy (or head) losses th
. .
ft*
**** tsrr!
£.“rx-1
*-*—<» <
*»••
(
lmPulSe
varioUsma
POWER
OBTAINED FROM
SHAFT
(B.H.RJ
POWER
DEVELOPED
BY RUNNER
(WH.R)
TURBINE S
RUNNER
NET POWER
FROM
RESERVOIR
GROSS POWER
FROM
RESERVOIR
(H)(H,)
SHAFT
ENTRANCE OF
SPIRAL CASINGRESERVOIR a
MECHANICAL
LOSSES-BEARWG
FRICTION
HEAD LOSS IN
PENSTOCK,hf
(0) HYDRAULIC LOSSES
(1) BLADE FRICTION
(It) EDDY FORMATION
[i’
ll) FRICTION IN DRAFT
TUBE
(i'
V)ENERGY CONTAINED
BY WATER LEAVING
DRAFT TUBE
(b) DISC FRICTION
(C) LEAKAGE LOSS
GENERATOR
LOSSES
POWER DEVELOPED
BY GENERATOR
(K.W.)
M REACTION
TURBINE
- 1Z
•
a
3 GENERATOR
*xm*xxmaEztisamnut f&rJSZJ.
GROSS POWER
FROM
RESERVOIR
NET POWER
FROW
RESERVOIR
POWER
SUPPLIED
TOWHEEL
{!” KyjiH
POWER
OBTAINED
FROM SHAFT
(B.H.P.)
POWER
DEVELOPED
6Y WHEEL
(W.H.R)(H5 mSET® I
— U i p5
l”
rl SHAFT
a33
1PENSTOCK NOZZLE p |TURBINE
IWHEEL a
sbmmifiimataaasM
RESERVOIR END END Jjcsaua sma&MwsM.ryy<>>,t ;vWyiy'jfl
i2vjS£T
HEAD LOSS IN HEAD LOSS IN
NOZZLE
MECHANICAL
L0SSES-6EARIN6
FRICTION
PENSTOCK,hf
W ii
*QB&JUC_AQS£S
(i) BLADE FRICTION
(Hi EDDY FORMATION
GENERATOR
LOSSES
POWER DEVELOPED
BY GENERATOR
(K-W.) ,
® IMPULSE TURRIMF (lii) ENERGY^
, CONTAINED
BY LEAVING WATER
(b)
AEAKAfflUGS&
1GENERATOR
Fig.21-2 Losses of energy in hydroelectri
installations
(0 Hydraulic effic;eil
turbine is the ratio of the 'y The hydraulic efficiency of the
Power developed by the runner (he-water

967
,, nrce power, W.H.P.) to the net power suDDlied k„ ..U
^ Jceto the turbine. These two powers'*ffir by
^hydraulic losses. That is by the amount of
the
W.H.P.
we+AQ)/T/75]"
where 0 is the quantity of water actually striking the runner and
AC Mteguanmy ofwaterffiat is discharged directly to t
^tailrace without striking the turbine runner: However, if AO iTneeli.gibly small, equation 21‘4 becomes — g
W.H. P.
( u'QUp'
i)
4/»= (21-4)* « *
t);.= (21-5)* * •
(n) Mechanical efficiency w The mechanical efficiency of
the turbine is the mtio of the power obtained from the shaft of the
turbine {i.e.,shaft or brake horse power, S.H.P. or B H P.) to the
power developed by the runner (i.e., W.H.P.). These two
differ by the amount of the mechanical losses vi
That is
powers
viz., bearing friction•etc.
B_H.P. (or S.H.P.)
W H.P.
Y]m
— (21.6)* * *
(in) Volumetric efficiency TJ# . The volumetric efficiency is
the ratio of the quantity of water actually striking the runner and
the quantity of water supplied to the turbine. These two quantities
differ by the amount of water that slips directly to the tail race with-out striking the runner. That is
Q
(21*
7)QT AQ
{iv) Overall efficiency y]0 . The overall efficiency of the
turbine is the ratio of the power available at the turbine shaft to the
power supplied by the water at entrance to the turbine. That is
^
B.H P. (or S. H.P.)
0
( Net power available at the turbine entrance)
•* •
(21*8)•* «
It is evident from equation 21'3 that overall efficiency of the
turbine is
(21-9)
is given by equation 21'4 in which the volumetric efficiency is
"nplied ; and
?lo — ^h X Y}m « •*
(21'9 a)* •Y]0 — 7J/* X Y]v X Y]m
IS ogiven by equation 2T5.

SLASSIFICATION OF TURBINES
Hydraulic turbines may be
siderations as indicated below.
() According to their hjdrau
^^
P jhe‘urbme5 may e class,,
^fied as impulse turbines an5m
^
turbme
^^^all the available energy of water is
. , , ,
velocity head by passing it through a contracting no p
^
ded at
the end of the penstock. The water coming out of the nozzle is mrmed
into a free jet which impinges on a series of
buckets of the rurmer
thus causing it to revolve. The runner revoives freely m air. The
water is in contact with only a part of the runnel c*. time, and
tho runner and in its subsequent now to the
A casing is however
*/ v v
214
classified according to several con-
throughout its action on
tail race, the water is at atmospheric pressure,
provided on the runner to prevent splashing and to guide the water
Some of the impulsedischarged from the buckets to the tail
turbines are Pelton wheel, Turgo-impulse wheel, Girard turbine,
Out of these turbines only Pelton,
race.
Banki turbine, Jonval turbine etc.
wheel is predominantly used at present, which has been described^
latter.
In reaction turbines, at the entrance to the runner, only a part
of the available energy of the water is converted into kinetic energy
and a substantial part remains in the form of pressure energy,
water flows through the runner the change from pressure to kineticenergy takes place gradually. As such the pressure at the inlet tothe turbine is much higher than the pressure at the outlet and itvaries throughout the passage of water through the turbine. Furtherfor this gradual change of pressure to be possible the runner in thiscase must be completely enclosed in an air-tight casing and the pass-age is entirely full of water throughout the operation of the turbine.Some of the reaction turbines are Foumevrnn Ti. me
^
rDine.
PT
,I„, Kaplan, e,c. On, „ *“”*•turbines are predominantly used at present which ha
As-
and Kaplan
ve been described
9 The turbines.. . , may als
° be classifiedjRSi
^^QaLofwater in th
(M) radial flow turbine, {in)
turbine.
_ according to the main
aXliiSrturbtleten8ential fl
°W tUrbin6
'and (fo) mixed flow
In a
to the path Jflow turbine. r‘
flows along the tangent
leIton wheel i tangentialis a

m UJVNUJLI
^ 1 UJKJBINJ2S yt>y
In a radial flow turbine the water flows along the radial direc-
and remains wholly or mainly in the plane normal to the axis oftion
rotation, as it passes through the runner. A radial fl„
be either inward radial flow cvDe or „ , ,
nv turbine
inward radial flow turbine the water enterT
^d ^^
and flows radially inwards towards the c
-ntre th"C
'rCUmference
Francis turbine, Thomson turbine p- j ,. the runner* Old
“*!°m
!°f
‘he eXa
,mP
'es of
“ ward radiaTflowturbf„eW ^
mrd radial flow turbine water enters at th» , *
,
outwards towards the outer periphery of th
** ^ radially
turbine is an example of outward radial flow tur“ '
may
In an
In an ouU
In an axial flow turbine the flow of water through the
is wholly or mainly along the direction parallel to the axis of rota-
tion of the runner. Jonval turbine, Girard axial flow turbine, Pro-peller turbine, Kaplan turbine etc., are some of the examples of axial
flow turbine.
In mixed flow turbine, water enters the runner at the outer
periphery in the radial direction and leaves it at the centre in the
direction parallel to the axis of rotation of the runner. Modern
Francis turbine is an example of the mixed flow type turbine,
v
^) On the Dasis of the^
head and the quantity of water required
*
,
the turbines may be classified as (i) high head turbine, (ii) medium
head turbine, and ( Hi) low head turbine.
runner
Di*"
High head turbines are those which are capable of working
under very high heads ranging from several hundred metres to few
These turbines thus require relatively less quantity
In general impulse turbines are high head turbines. In
thousand metres,
of water.
particular Pelton wheel has so far been used under a highest head of
about 1770 m (5800 ft.)
Medium, head turbines are those which are capable of working
under medium heads ranging from about 60 m to 250 m.
turbines require relatively large quantity of water. Modern Francis
turbines may be classified as medium head turbine.
These
Low head turbines are those which are capable of woiking under
a largeThese turbines thus require
Kaplan and other propeller turbines may be
the heads less than 60 m.
quantity of water,
classified as low head turbines.
f The turbines may also be classified according to their specific
The specific speed of a turbine is the speed of a geometrically
similar turbine that would develop one horse power (metric) when
working under a head of one metre. On the basis of the specific

rJL/ KJULJ lujuv
^rmi’iiw
HYDKAUUW
various turbines may be considered in the following three
10 to 35— Pelton wheels with
970
speed the
groups :
(i) Specific speed varying from
single jet and upto 50 for double jet.
(H) Specific speed varying from 60 to 400— Francis turbines,
from 300 to 1000— Kaplan and
(Hi) Specific speed varying
other propeller turbines.
of turbines noted above may be disposed
The different types
with either vertical or horizontal shafts.
21-5 PELTON WHEEL
This is the only impulse type of hydraulic turbine now in
It is named after Lester A. Pelton (1829-1908), the
who contributed much to its development in
common use.
American engineer
about 1880. It is well suited for operating under high heads.
Fig. 21‘
3 shows the elements of a typical Pelton wheel
The runner consists of a circular disc with a numberinstallation. ^ ^
of buckets evenly spaced round its periphery. The buckets have a
shape of a double semi-ellipsoidal cups. Each bucket is divided into
two symmetrical parts by a sharp edged ridge known as splitter.
One or more nozzles are mounted so that eachTdirects a jet along a
tangent to the circle through the centres of the buckets called the
pitch circle. The jet of water impinges on the splitter, which divides
the jet into two equal portions, each of which after flowing round
the smooth inner surface of the bucket leaves it at its outer edge,
f he buckets are so shaped that the angle at the outlet tip varies
from 10° to 20° (usually kept
deflected through 160 to 17(3°. The advantage of having a double
cup shaped"
buckets is that the axial thrusts neutralise each other,
being equal and opposite, and hence the bearings supporting the
wheel shaft are not subjected to any axial or end thrust. The back
of the bucket is so shaped that as it swings downward into the jet
no water is wasted by splashing. Further at the lower lip of the
bucket a notch is cut which prevents the jet striking the preceding
bucket being intercepted by the next bucket very soon, and it also
avoids the deflection of water towards the centre of the wheel as the
bucket first meets the jet. For low heads the buckets are made
but for higher h_eads they are made of cast steel, bronze
or stainless steel.
15°) so that the jet of water getsas
In order to control the quantity of water striking the
the nozzle fitted at the end of the penstock is provided with
runner,
a spear

niUKAULIC TURBINES 971
-or needle having a streamlined head which is fixed to the end of a
rod as shown in Fig. 21 3. The spear may be operated either by a
FKOM PENSTOCK
PITCH erode
^rashsr z
2
X
iNOiCATCP
Vj
2
X
YsX
V,'XA
V,
Ys'AY,Ys
i
YsYs
I 22
i 2
AYs
i8 rs,
WWWWWl
Fig. 21* 3 Single jet Pelton wheel
hand wheel (Fig. 21‘3) in case of very small units or automatically
by a governor (described later) in case of almost all the bigger
When the shaft of the Pelton wheel is horizontal then not more than
two jets are used. But if the wheel is mounted on a vertical shaft a
larger number of jets (upto six) is possible,
units.
A casing made of cast iron or fabricated steel plates is usually
It has noprovided for a Pelton wheel as shown in Fig. 21*
3.
hydraulic function to perform. It is provided only to prevent
splashing of water, to lead water to the tail race ana also to act as
a safeguard against accidents.
usually equipped with a small brake
on the back of the
Larger Pelton wheels
nozzle which when opened directs a jet of water
buckets, thereby bringing the wheel quickly to rest after it is shut
down, (as otherwise it would go on revolving by intertia fora
are
considerable time).
WHEEL
21 6 WORK DONE AND EFFICIENCY OF PELTON
The transfer of work from the jet
takes place according to the momentum equation
of water to the buckets,
as indicated in

y tv. it A ^
the velocity triangles at the tips of th&.Fig. 21'4 showsChapter 20.
bucket of a Pelton wheel.
^-absolute velocity of jet before
F absolute velocity of jet leavir g
;_.;toh„e velocity of bucket cons.dered along the.
alrecdontangential to the ptch ctrcle
Fr=velocity of the incoming jet relative to the
. ucket
velocity of the jet leaving the bucket relative to the
bucket
Let;
striking the bucket
FK=velocity of whirl at inlet tip of the bucket
F„=velocity of whirl at outlet tip of the bucket
6=angle through which the jet is deflected by the
bucket
Since the velocities V and u are collinear, the velocity triangle
at the inlet tip of the buckets is a straight line ana thus
Vr=(V-u) and VW=V.
At the outlet tip three velocity triangles as shown in Fig. 21'4*
are possible depending upon the magnitude of u, corresponding to
which it is a slow, medium or fast runner. As the inlet and the
outlet tips of the bucket are at the same radial distance the tangential
Furthervelocity of the bucket at both the tips is same, i.e,,
the relative velocity Vrl with which the jet leaves the bucket will be
somewhat less than the initial relative velocity Vr at the inlet tip..
This is because, although the inner faces of the buckets
u— ux.
are polished
so as to minimize frictional losses as water flows over them such
losses cannot be completely eliminated.
Joss of energy will also take place as the jet strikes the splitter.These losses of energy reduce the relative
and the bucket, and hence
In addition to this some-
velocity between the je$
Vn— k(Vr) ~ k(V —u)
where k is a fraction slightly less than unity,
Now from the outlet velocity tritriangle {% ) of Fig. 21*
4
Vw1
—Vn COS <f)
—U1)=:(kVr cos
^— u)
and from the outlet velocity triangle (Hi) 0f Fig 2,.4
Kt={ul-Vncos
where ?S=(jt
-0) is the angle of the bucket
COS (f>)
at the outlet tip.However, if the losses
velocity triangle (n) Fwl=
* are neglected then k=1u*
Also for outlet

HYDRAULIC TURBINES
If w kg
°f wa
‘er per second strikes the buck
^
Chapter 20, the work done per second
Work done
973
ets then, as shownon the wheel is given as
g KUi
w r~
cos u
W r
ff
- u)(--k cos tj,)
Ju (21-10)•••
Y
M L
l
Jra
—^
us
a
PLAN
100-
fr
SECTION YV
v*vw —*=1» Yf*KV-«)
INLET VELOCITY TRIANGLE
u
XIT.
(Ill)
tfo M
V !%
A$ !
Vr,
.**<•
&i
*sifeo*«lL:'ia 6oa< c5 <
&'
( jB -KCT) Vw!« ©)
OUTLET VELOCITY TRIPLES
Fig. 21*4 Pelton wheel bucket and velocity triangles
energy supplied to
§7 of the jet which is equal to ( WV2
/2g) per second.
<&>S0*>VW1 POSITIVE)
^<90*
^, fCGABVE)
the wheel is in the form of kineticThe'
®ner

FLUID MECHANICS
of the Pelton wheel is giver
*
HYDRAULICS AND
Therefore the hydraulic efficiency
974
by
( W /a)(V — n)( l -f k cos
" "
TWV^Pg )
2u{V — u){ Jrk cos 4> )
f
h=
(2Ml)••4
V2or
For a given jet velocity V and the bucket tip angle <f> at outlet
condition for maximum efficiency may be obtained by consider-Thus if
^ is assumed constant
the
ing {drih/du)=0.
j(fo) _2( l +& cos 4)
(27— 4w)=0
V2
du
-(f ) (21'
12)*=0-5 V •••or
Thus the hydraulic efficiency is maximum when the bucket
Substituting thisspeed is equal to half the velecity of the jet.
condition in equation 21*
11, the maximum hydraulic efficiency is
obtained as
(21*
13)(%)ma*=£0+& COS 4> ) •«
•
Now if Jc=1, then equation 21*
13 indicates that the maximum
value of V}h will be equal to 1 or 100% when
^=0 or 0=180° i.e.f
the buckets are so shaped that the jet gets deflected through 180°.This is however, theoretical maximum value of The actualy
h-maximum value of TJA will be slightly less and it varies from 0 9 to
0‘94 (or 90 to 94%). This is so because the actual value of k is not
equal to one but it is slightly less. Further in actual practice 0 can
not be made equal to 180 , because in that case the jet leaving onebucket will strike the back of the bucket just following it, thus exert-ing a retarding force on it. Hence im order that the outgoing jetkeeps clear of the following bucket, the bucket tip angle $ at outletis usually kept ranging from 10” to 20”, (the average value being 15»
),so that the jet gets deflected through 160° to 170°
also known as side clearance angle. The angle $ is
If there is no loss of energy as the water flowthen the work done per second on the
output of the Pelton wheel) may also be
3 over the buckets
Pciton wheel (or the power
expressed as
Work done= ^ (V 2 ~V 2 )
The hydraulic efficiency of Pelton
29 (21*
14)* ••
wheel raay also be expres-sed as
( W /2g)V2
(21*15)V2 ••«

g substituting the value of V1 obtained from the
triangie’
equation 21 15 becomes exactly similar
SSS1•
outlet velocity
to equation 21*
11
with k
—The loss of head as the waler flows through the buckets of the
pelton wheel may be obtained by applying the Bernoulli’s equation
between the inlet and the outlet tips of the bucket.
-VwU ±Vmu, F-,2
Thus
--llL
*9 2g9
where hi is the loss of head in the buckets.
From the velocity triangles at the inlet and the
the bucket
outlet tips of
FwudrV F2
— FA
L 2g
^ [ V2~
Vl2
, Vrl2 -~VJ
L 5
9 29 2g
j(since u=u1 )29 2g
Thus by substitution, we get
Vr2 ~
Vri2
_Vr2
2g 2g
(V ~u)2
JlL= d ~ B)= (1-F)
2g
(21*
16)
The other efflciencies of Pelton wheel may be expressed by the
equations 2T6, 21' 7 and 2T8 (or 2T9).
The overall efficiency of 85— 90% may usually be achieved for
large Pelton wheels. The volumetric efficiency for Pelton wheel,
ranges from 0*97 to 0*
99.
«•••
2P7 WORKING PROPORTIONS OF PELTON WHEEL
(i) The ideal velocity of jet usually known as spouting velocity
={V 2gH ) where H is the net head. Plowever, the actual velocity
of the jet is slightly less, due to friction loss in the nozzle. Thus
V=Cvi/ 2gH (21*
17)
where Gv is the coefficient of velocity for the nozzle with its value
ranging from 0 97 to 0*
99.
(wj As obtained above for maximum ?)h the velocity of wheel
u at pitch circle is equal to 0*
5 V . However, in actual practice the
Maximum efficiency occurs when the value of u is about 0 46 V .
Moreover, it is convenient to express u in terms of Hf in the form ol
^expression u=4>(/ 2gB) where is known as speed ratio.
Coiffidering F=0*
98 ( V 2gH )
Thus
W =Q*
46 F=0'
45 (V2gH )
In Practice the value of
^ranges from 0*
43 to 0*
47.
...(2H8)
u

HYDRAULICS AND FLUID MECHANICS
(in) Angle through which jet of water gets deflected in buckets
=165°, unless otherwise stated.
Least diameter d of the jet is given by
=0-542
976
(iv)
( Q 1/2
W H 'LnCv (V2QH )A
metres
(21-19)•••
where Q is the discharge through the jet in m3
/sec.
(u) Mean diameter or the pitch diameter D of the Pelton wheel
If the wheel rotates at N r.p.m., thenmay be obtained as follows.
u={nDNj60). Thus
„ 60« _60((£/ 2gH )
D~
” izN
(21-20)•••
7zN
(vi) The ratio of pitch diameter D of the wheel to the jet
diameter d is known as jet ratio and is represented by m i.e,}
m— (Djd). The jet diameter is an important parameter in the
design of a pelton wheel. For maximum efficiency the jet ratio
should be from 11 to 14 and normally a jet ratio of 12 is adopted
in practice. A smaller value of m results in either too close a
spacing of the buckets or too few buckets for the whole jet to be
used. A larger value of m results in a more bulky installation.
However, in extreme cases a value of m as low as 7 and as high as
110 has been used.
(vi) Some of the main dimensions of the bucket of a Pelton
wheel as shown in Fig. 21'4 expressed in terms of the jet diameter
are as noted below :
5=(4 to 5)d ;
CMO‘81 to TO5)d ;
M=( l -1 to T25)<J ;
^=5° to 8°.
£=(2-4 to 3-2)d ;
Z=(T2 to T9)d ;
Angie
^=10° to 20°
Again
(vii) The number of buckets for a Pelton wheel should be
such, that the jet is always completely intercepted by
that volumetric efficiency of the turbine is very close to unity. The
number of buckets is usually more than 15. Certain empirical
formulae have been developed for determining the number of
ouckets. One such formula which is widely used has been given by
faygun according to which the number of buckets 2 is approxi-mateiy given by
the buckets so
G1+15 )z= ~ (0'5 m+15)
good for all values of
(2T21)••*
This equation has been found to hold
ranging from 6 to 35.

MULTIPLE JET PELTON WHEEL
2l'8
the jet velocity, wheel speed and ikTZt"!*
** r
“tn
“lons
ot be made big enough to develop any desired p'ow"8
The
amount of power developed by a single runner of a Pelton wheel
byp
——*•» «
*
ced so closely that water issued f , r
. . r • i
uea lrorn one jet after striking the
interferes with another jet
s
jet is
of
C3B11
runner. The nozzles must never bespace
spa
As such the maximum numberrunner
0f jets so far used with a single runner of some large units is six. A
pelton wheel having more than one jet spaced around its runner is
called multiple jet Pelton wheel. If P is the power developed by a
pelton wheel when working under head H and having one jet only,
"then the power developed by the same Pelton wheel will be (nP), if n
used for its working under the same head.jets are
Sometimes even if by using more number of jets for a single
the required power is not developed then a number of runners
runner,
mounted on a common shaft may be used. In some cases a combi-
nation of the above two systems may be used, i.e., a number of
multiple jet wheels may be mounted on the same shaft.
219 RADIAL FLOW IMPULSE TURBINE
For a radial flow impulse turbine the inlet velocity triangle is
not a straight line and hence
W
Further if there is no loss of energy in the runner vanes then
also be expressed by equation 2T14. Thus
Work done
— —g
the work done may
equating the two, we get
J=27
inlet and the outlet tips of a radial
From the velocity triangles at the
vane it can be shown that
Thus by substituting
Frl2
— Fr2
J^ 2
rv2-Vi2
L 2
this value in the above expression it
becomes
F_,2
— y 2
y
n y r Qw2
— tq2
2g2g
Vr2
. (U2~UX2
2g 2g J
„,(21*
22)Fri2
^or
2g

FLUID MECHANICSHYDRAULICS'AND978
The second term on the right hand side of equation 2122 represents.
the centrifugal head impressed on the water as it flows through the
radial flow impulse turbine. For an outward flow
ion 21*
22, Vrl>Vr ; and for
of arunner
wardlwmrbini6
;
^ZndCce Vn<Vr- That is the centrifugal
the relative velocity of water in an outward flow
inward flow turbine. As such a better
an m-
head increases
turbine and decreases it in an
be enforced in the case of an inward flowcontrol of speed can
turbine.
21 10 REACTION TURBINES
As stated earlier the principal distinguishing features of a
reaction turbine are that only part of the total head of water is
converted into velocity head before it reaches the runner, and that
the water completely fills all the passages in the runner. Thus the
pressure of water changes gradually as it passes through the runner.
The two reaction type of turbines which are predominantly used
these days are Francis turbine and Kaplan turbine, which are
described below.
2111 FRANCIS TURBINE
Fig. 21‘5 illustrates a Francis turbine which is a mixed flow
type of reaction turbine. It is named in honour of James B. Francis
11815— 92), an American Engineer, who was the first to develop an
inward radial flow type of reaction turbine in 1849. Later on it was
modified and the modern Francis turbine is a mixed flow type, in
which water enters the runner radially at its outer periphery and
leaves axially at its centre.
The water from the penstock enters a scroll case (also calledspiral case) which completely surrounds the
of the case is to provide
the circumference of the turbine
runner. The purpose
even distribution of water around
an
runner, maintaining an approxi-water so distributed. In order to
mately constant velocity for the
keep the velocity of water constant throughout its path around therunner, the cross-sectional area of the casincr n J JTKo • , uie casmg gradually decreased.1he casinS ls made of cast steel, plate steel,steel depending upon the
these a plate steel spiral (or
turbines
concrete or concrete andpressure to which it is subjected.
ODeratina on
Case is c
°mmonly. provided foroperating under 30 m or higher heads.From the scroll
Out of
case the
comprising of a series of fixedscroll case all around its inner
functions to perform.
wafer passes through a stay ring,
vanes, which is assembled with the
periphery. The stay ring has two
water from the case to the
ft directs the

HYDRAULIC TURBINES
979
:Ae vanes or wicket gates
it by the case, under intend pr
^Tu‘reTf
^t^ 'T* imposed
electrical generator and transmit the samJ7*? thewei&t
stay ring may be produced either by c!st;“ 16 found«ion.
case; or casting or fabricating it sepaSy ^
gui
upon
0fthe
The
the
with the case.
rh SCROLL CASINGSHAFT
-h- ‘
I"
I
V
GUIDE VANE
WICK$* RUNNER VANEGATE
DRAFT TUBE
TAIL RACETL--T 2T.
X 3L
sr
r.rrr---H
*
FROM
PENSTOCK
I
I
REGULATING RING
T
GUIDE VANE
Oo .
o9
o
UNKoo
SCROLL-CASING
oo
o
Fig. 21*
5 Sectional arrangement of Francis turbine
from the stay ring the water passes through a series of guide0r wicket gates provided all around the periphery of the tur-ne runner.
^Ues
bi
The function of the guide vanes is to regulate the

quantity of water supplied to the runner and to direct water on to
the runner at an angle appropriate to the design. The guide vanes
airfoil shaped and they may be made of cast steel or stainless
steel or plate steel. Each guide vane is provided with two stems,
the upper stem passes through the head cover and the lower stem
seats in a bottom ring. By a system of levers and links, all the guide
vanes may be turned about their stems, so as to alter the width of the
passage between the adjacent guide vanes, thereby allowing a variable
quantity of water to strike the runner. The guide vanes are operated
either by means of a hand wheel (for very small units} Gr automatic
cally by a governor.
The main purpose of the various components so far described
is to lead the water to the runner with a minimum loss of energy.
The runner of a Francis turbine consists of a series of curved vanes
(about 16 to 24 in number) evenly arranged around the circum-ference in the annular space between two plates. The vanes are
so shaped that water enters the runner radially at the outer peri-
phery and leaves it axially at the inner periphery. The change in
the direction of flow of water, from radial to axial, as it passes
through the runner, produces a circumferential force on the runner
which makes the runner to rotate and thus contributes to the useful
output of the runner. The runners are usually made up of cast
iron, cast steel, mild steel or stainless steel. Often instead of making
the complete runner of stainless steel, only those portions of the
runner blades, which may be subjected to cavitation
made of stainless steel. This reduces the cost of the runner and at
the same time ensures the operation of the runner with a minimum
amount of maintenance. The runner is keyed to a shaft which is
usually of forged steel. The torque produced by the
transmitted to the generator through the shaft which is usually con-nected to the generator shaft by a bolted flange connection.
980
are
erosion, are
runner is
The water after passing through the_ , , _ runner flows to the tail
race through a draft tube. A draft tube is a pipe or passage of
gradually increasing cross-sectional area which connects the runnerexit to the tad race It may be made cf cast or plate steel orconcrete. It must be airtight and under all conditions of operationlts
IoWlend,mfUSt bKeStubmerSed below the level of water in the tailThe draft tube has tworace. purposes as follows ;
(i) It permits a negative or
, . , , . . sucti
°n head to be established at
the runner exit, thus making it possible to install the turbine
the tail race level without loss of head.
above

JL
lUKtslNES
981
(ii) It converts a large proportion of velocity
the runner into useful pressure
energy rejected
as a recupe-energy i.e., it actsfrom
rator of pressure energy.
Kg. 21-6 shows the different types of draft tubes which
employed in the field to suit particular conditions of installation.
are
{Q) STRAIGHT DIVERGENT TUBE.
(b) MOODY SPREADING TUBE (OR HYDRACONE).
(c) SIMPLE ELBOW TUBE.
(d) ELBOW TUBE HAVING CIRCULAR CROSS SECTION
AT INLET AND RECTANGULAR AT OUTLET.
Fig. 21'6 Different types of draft tubes
Of these the types (a) and (b) are the most efficient, but the types
(c) and (d) have an advantage that they require lesser excavation for
their installation.
type draft tube the central cone angle should not be more than 8°.
This is because if this angle is more than 8° the water flowing
through the draft tube will not remain in contact with its inner
surface, with the result that eddies are formed and the efficiency of
the draft tube is reduced.
2112 WORK DONE AND EFFICIENCIES OF FRANCIS
TURBINE
. If W kg of water per second strikes the
*n Chapter 20 the work done per second
Passed as
It has been observed that for straight divergent
then as derivedrunner
the runner may be ex-on
W
[V„u-VmvJWork done= 9

Evidently the maximum output under specified conditions is
obtained by making the velocity of whirl at exit FW1 equal to zero.
Then
— (P„«)Work done= •
—
or Work done per kg of water
(?)
Now if H is the net head then the input energy per second for the
runner~{WH ).
Therefore hydraulic efficiency of the Francis turbine is given
VyjU
However, if VW1 is not equal to zero, then
Vu)U V
by
(21’
23)%
••
(21-23 a)Vh=
The value of th ranges from 85 to 95%.
Again if P represents the horse power developed by the
shaft then the mechanical efficiency is given by
•••
gH
runner
P
(21-24)1)m
W ( V
^
u
75 U
Further the overall efficiency is given by equation 21'9 asf
P
(21-25)*
*]o X — •••(WHJ75)
The overall efficiency of a Francis turbine ranges from 80 to
90%.
2113 WORKING PROPORTION OF FRANCIS TURBINE
(i) The ratio of the width B of the wheel to the diameter D(Fig. 21’8) of the runner is represented by n, that is
n—{B/D)
The value of n ranges from 0T0 to 0’ 45.
(ii) The ratio of the velocity of flow Vf at the inlet tip of
the vane to the spouting velocity {V 2gh) is known as flow ratiothat is
(21*
26)•••
V,
The value of 41 ranges from 015 to 0*
30.
also the speed ratio <f> is defined as
(fr
—uflVZgH ). The value of <f> ranges from 0’60
(21-27)•••
{in) In this case
to 0*90.

HYDRAULIC TURBINES
DESIGN OF FRANCIS TURBINE RUNNER
983
2VU
A Francis turbine runner is required to be designed to develon
a known power P, when running at a known speed iV r p m under a
known head H e probable values TQ
*, 7)0, n and <|> are assumed
The design of the runner which involves the determination
size and the vane angles is carried out as follows :
(*' ) Determine the required discharge Q from the relation
P=T)O( WH/75)=TI()(WQH/75).
of its
(ii) If Z is the number of vanes in the runner, t is the thick-
ness of the vane at inlet and B is the width of the wheel at inlet
then the area of flow section at the wheel inlet=(rrD— Zt )B=kKBD,
where k is a factor which allows for the thickness of the vanes.
Then
Q =kuBDVf =kKnD2Vf
B— nD.
Thus assuming a suitable value of k the diameter D and the
width B of the runner can be determined. For the first approxima-
tion the vane thickness may be neglected in which case k may be
assumed to be equal to unity.
( Hi) The tangential velocity of the runner at inlet may be
determined from the equation u — ( nDN/60).
(iv) The velocity of whirl Vw at inlet of the runner can be
determined from the expression y
h=(VwujgH ).
(v) From the inlet velocity triangle, the guide vane angle a
and the runner vane angle 6 at inlet can be calculated from the
expressions tan ct=(Vf /Vw) and tan 0 ==Vf /{VW — u).
(vi) The runner diameter Dx at the outlet end varies from
(1/3) D to (2/3) D and usually it is taken equal to (1/2)D. Thus the
tangential velocity of the runner at outlet may be determined from
the expression «1=(rcZ)
1JVr/60).
(vii) If tx and B1 are respectively the thickness of the vane
and the width of the runner at outlet, then
Q=(nD1— Zt1) B1 xVfl=k1nDiBiVfi
(21-28)•••
•since
(21-29)•••
From equations 2T28 and 21 29
kxnB1D1 (21-30)V f » •»
vfl knBD
-Normally it is assumed that V f — Vfx and k — kv then Bx 2 (
A=(l/2)D.

"
m
FLUID MECHANICS
HYDRAULICS AND
(yin) Generally the runner is designed to have the velocity 0f
whirl Vm at outlet equal to zero, i.e., Vwl=0 and (3=90°. Then,
from outlet velocity triangle the runner vane angle $
n
be determined from the expression tan
^=(Vfl/u1).
(ix) The number of runner vanes should be either one more or
less than the number of guide vanes, in order to avoid setting
up of periodic impulse.
984
at outlet may
one
DRAFT TUBE THEORY2T15
which points 1, 2 and 3 have been consider-
exit and at the outlet end of the
Bernoulli’s equation between
heads at the inlet and the
Refer Fig. 21*
7 in
ed at the runner entrance, runner
draft tube respectively. By applying
points 1 and 2 the pressure and velocity
outlet ends of the runner may be obtained.
Applying Bernoulli’s equation between points 2 and 3, we get
+
V
^+z*+h
'Pa Ek2
w * 2(
/
(21-31)•••
and velocities at points 2
where p2, Pz and V2,V3 are the pressures
and 3 respectively, and Jif is the loss of head in the draft tube. From>
equation 21'31
— =~ — fe— 23> —w w
n=w w +hBut
(H)ttw w 2•9
Since (z2— z2— h)=H„the height of runner exit above tail race-
level, thus
i?
= Pi _rHa+vJ-i£]+hf
w w L J
It may be seen from equation 21*
32 that the pressure at the
runner exit is suction pressure, that is below atmospheric. Hs lS
known as static suction head and [F22—F32)/2g] is known as dyna-mic
then
(21*
32)•••
suction head. Generally hf is expressed as hf=fc(F22—Pa )l
^’
f =?-[*+<!-*> ] (21*
33)
Now if the draft tube is made cylindrical then F2=F3 and if
the frictional loss is neglected, then equation 21'32 indicates that the
pressure head at the runner exit would be less than atmospheric
•••
2ff

HYDRAULIC TURBINES
pressure by an amount H the height of the runner exit above the-
ta11 race. Hence with such a draft tube the turbine would not loose.
98&
f
J I t
l
1
I2
H
(D AT RUNNER ENTRANCE
® AT RUNNER EXIT
d) OUTLET OF DRAFT TUBE
I*2
| Pc
JLrh
3
23
DATUM
Fig. 21*
7 Draft tube theory
head Ha because of equal reduction in pressure head at the runner*
exit. On the other hand if a draft tube of gradually increasing
cross-section is used, then the value of pressure at runner exit is
further reduced by an amount equal to (1 — &)[(F22— F32)/2gf]. In
other words such a draft tube causes a large portion of kinetic head
to be converted into pressure head.
The efficiency of a draft tube is defined as
Actual regain of pressure head
Velocity head at entrance to draft tube
The actual regain of pressure head
F22-F321 F22-F32
)1c) ([ -
*/ =(1- 2g2g
UV
^-rfllW -hf (l -fc)r(F22-F32)/2g1 (21-34)»ld= •••
(F,*/2g)(F,*/2g)•
FRANCIS TURBINE RUNNER AND2116 SHAPE OF
DEVELOPMENT OF KAPLAN TURBINE RUNNER
A Francis turbine runner of given diameter, when required
to develop certain power P, under a low head, should be so designed
that it admits comparatively large quantity of water. This can bo

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ratJO
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êooml
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987HYDRAULIC TURBINES
'
M
of water is required to be passed, it is so designed that the
purely axial right from the inlet section to the outlet sectionntity
qua"
flow is
jj
'jr *KAPLAN TURBINE
A Kaplan turbine is a type of propeller turbine which was
developed by the Austrian engineer V. Kaplan (1876— 1934). It is
SCROLL
GUIDE VANE CASING
RUNNER VAN
OR
BOSS
DRAFT TUBE
RUNNER VANE
GUIDE VANE
Fig. 21-9 Sectional arrangement of Kaplan turbine

H I
an axial flow turbine, which is s
"'* ®
ef t0 develop large amount off
hence requires a large quantity
turbine and hence it operates in,
It is also a reaction
^ race to the tail race,
entirely closed conduit from tne
988 relatively low heads, and)
power.
an
e
nfINLET VELOCI
TRIANGLES Vf u=u
l ', /!/
1 _L
f NO SHOCK AT
INLET
BLADE POSITION
AT PART LOAD
pTIVOT
reRTAmETOWpAfiTBSw'oPERAnOI*
4 v*
4
_L
^££..
Uj*=U
) OUTLET VELOCITY
TRIANGLES0
Uf=U
BLADE ANGLES 9 AND 0 AT FULL LOAD
CHANGE TO 6' AND 0' AT PART LOAD
Velocity triangles for a Kaplan turbine runner bladeFig. 21*10
From Fig. 21-9 it will be seen that the main components of at.
Kaplan turbine such as scroll casing, stay ring, arrangement of guide
vanes, and the draft tube are similar to those of a Francis turbine.
Between the guide vanes and the runner the water in a Kaplan (or
propeller) turbine turns through a right-angle into the axial direction
and then passes through the runner. The runner of a Kaplan (or
propeller) turbine has four or six (or eight in some exceptional
cases) blades and it closely resembles a ship’s propeller. The blades
(or vanes) are attached to a hub or boss and are so shaped that water
flows axially through the runner. Ordinarily the runner blades of a
propeller turbine are fixed, but Kaplan turbine runner blades can be
turned about their own axes,
be adjusted while the turbine is in motion
runner blades is usually carried out
so that their angle of inclination may
• This adjustment of the
. . t
automatically by means of a
servomotor operating inside the hollow coupling of turbine andgenerator shaft. When both guide-vane ormi
- j ,
angle may thus be varied a high efficiency can be maintffined"' * "
wide range of operating conditions. rn other wordswhen a lower discharge is flowing through theciency can be attained in the case of a
explained with the help of Fig.21*
10,
over a
even at part load,
runner, a high effi-a Kaplan turbine. It may be
ln w
^ch inlet and outlet vela-

. triangles for a Kaplan turbine runner working at constant speed
under constant head at full load and at part load are shown. It will
observed that although the corresponding change in the flow
through the turbine runner does affect the shape of the velocity
trangles, yet as the blade angles are simultaneously adjusted, the
under all the working conditions flows through the runnerwater
blades without shock. As such the eddy losses which are inevitable
in Francis and propeller turbines are almost completely eliminated in
a Kaplan turbine.
2118 WORKING PROPORTIONS OF KAPLAN TURBINE
In general the main dimensions of Kaplan turbine runner are
established by a procedure similar to that for a Francis turbine
However, the following are the main deviations :
(i) Choose an appropriate value of the ratio n— (d/D), where
d is boss diameter and D is runner outside diameter. The value of
usually varies from 0*
35 to 0*60.
(ii) The discharge Q flowing through the runner is given by
Q=
^(D*-d?) Vf =~ ( D*-d>)ty V 2JH
The value of flow ratio ty for a Kaplan turbine is around 0-70.
(Hi) The runner blades of Kaplan turbine runner are warped
or twisted, the blade angle being greater at the outer tip than at the
hub. This is because the peripheral velocity of the blades being
directly proportional to radius, it will vary from section to section
along the blade, and hence in order to have shock free entry and exit
of water the blades with angles varying from section to section will
have to be designed.
The expressions for the work done and the efficiencies of
Kaplan turbine are same as those for Francis turbine.
2IT9 GOVERNING OF TURBINES
All the modern hydraulic turbines are directly coupled to the
electric generators. The generators
constant speed irrespective of the variations in
•tant speed N r.p.m., of the generator is given by expression
runner.
n
(21*35)••*
are always required to run at
the load. This consr
pN (21*36)f= © « •
60

990
enerated in cycles per second and
where / is the frequency of power g
p is the number of pairs of poles.
N= —V
The speed of the generator can be maintained constant only if the
speed of the turbine runner is constant equal to the one given by
equation 21*
37. It is then known as the synchronous speed of the
turbine runner for which it is designed.
Usually /=50 and hence
60/ 3000 (21*
37)•••
V
will however, go on varying and
then the speed of theThe load on the generator
now if the input for the turbine remains
turbine runner will tend to either increase or decrease depending
on the load on the generator is reduced or increased.
the speed of the generator also to vary accordingly, which
is, however, not desirable because it may result in varying the fre-
As such the speed of the turbine
same
This in turn
will cause
quency of power generation,
runner is required to be maintained constant so that the generator
always runs at constant speed under all conditions of working. It
is usually done by regulating the quantiry of water flowing through
the runner in accordance with the variations in the load. Such an
operation of regulation of speed is known as governing and is usually
done automatically by means of a governor. One of the common
types of governor which is predominantly used with modern turbines
is oil pressure governor. As shown in Fig. 21*
11 its component parts
are as follows :
(£) Servomotor also known as relay cylinder.
(ii) Relay valve also known as control valve or distribution
valve.
(in) Actuator or pendulum which is belt or gear driven from the
turbine main shaft.
(iv) Oil sump.
(v) Oil pump which is driven by belt connected to turbinei r*
main shaft.
(vi) A system of oil supply pipes connecting the oil sump with
the relay valve and the relay valve with the servomotor.
The working of the governor is explained below :
When the load on the generator drops, the speed of turbine
Now since the actuator or pendulum is driven bythe turbine main shaft, due to increase in the speed the balls moveupward, resulting in an upward movement of the sleeves shown inFig.21T1 (a). As the sleeve moves up, the left hand end of the maim
runner increases.

ttXUKAULlC TURBINES
1 vef is ra*se<
^’
which causes the bell crank lever to move downward6
A simultaneously pushes the piston of the control valve down in
SYMBOL
991
ACTUATOR OR
PENDULUM OPENING
CLOSING
i
FLY BALL
SLEEVE
i
t BELL CRANK
LEVERMAIN LEVER’si/
i
RELAY OR
CONTROL !1 f.FULCRUM o
CONNECTED
TO TURBINE
MAIN SHAFT
V
£s
I FULCRUM
}ROLLER
OIL
I
I
PUMP SPEAR
'-Z CAMOLy
NOZZLE
-«
OIL
SUMP
or (oSERVO-MOTOR OR
RELAY CYLINDER / OEFlECTOR (V
(0) FOR IMPULSE TURBINES SPEAR
FROM v
PENSTOCK
ROD
CONNECTED TO
RELAY VALVE
SCROLLt I
I CASING
!
SERVO-MOTOR
i
mREGULATING.>*SHAFT . «
GUIDE VANE -
REGULATING ROD
REGULATING
LEVER
*
FROM
PENSTOCK REGULATING
RING
8)} FOR REACTION TURBINES
Fig. 21*
11 Governing mechanism of turbines
In the case of a Pelton turbine the downward motionits cylinder.
of the bell crank lever brings the deflector in front of the jet thereby
diverting a portion of the jet away from the buckets. On the other
hand in the case of a reaction turbine such as Francis or Kaplan
turbine the downward motion of the bell crank lever (or other suit-
able arrangement) operates the relief valve shown in Fig. 21*
12, thus
allowing a portion of water to flow directly from the spiral casing to
Thus both these devicesthe tail race without striking the runner,
deflector and relief valve have similar function to perform,viz. 3

FLUID MBV^
HYDRAULICS AND
nf rapid closure of the nozzle 0penin
the same time the quantity of w
i The rapid closure of the no,,,
iS
^isCenotdesirable because a sudden reduc!, the penstock may result in serious water
992
§
the necessity
eliminateThese
or the guide vanes, at
the runner
the guide
striking
opening or
tion of the rate
hammer proble
With the downward
for pipeline 2 opens
ms.
motion of the piston of the control valVe
and oil under pressure is
admitted
the passage
1 PILOT VALVE
LI'
OIL OR WATER .
UNDER PRESSURE
1
>WATER FROM
TURBINE
CASING
SPEAR
BYE PASS
TO
TAIL RACE
Fig. 2112 Relief valve
from the control valve cylinder to the servomotor on the left side ofthe piston. The servomotor nistnri ,, , “ tne lelt
,
In the case of Pelton wheel the s ’ therefore’ moves to the right-to the spear rod, thus causes the
^
rV
°motor Pist°n being connectedward motion of the spear reduces ti
*^ l
° m
°Ve forward. The fer-tile rate of flow (as required for ’!ozz
'e outlet and thus decreasesnormal turbine speed is restored H
decreased load) whereby the
turbine the forward motion 0f the °Wever
-in the case of a reactionto the regulating ring as shown in 1?.Se
°motor piston is transmittedguide vanes to move simultaneously8’/* 11 (6> whichrea of flow passage between tl,e J- °ne direction, and thus thethe rate of flow 0f water striki
‘
'Jacei« guide vanes is reduced
runner is also reduced*
all thecauses

w HYDRAULIC TURBINES 993
the normal turbine speed is restored theonce
its initial normal position and through
main lever
a suitable cam
WheIJ
* ngement the deflector is brought back to its original position or
] jef valve stops byepassing of water to the tail race.the re
When the load on the generator increases, the speed of the
turbine runner decreases. Due to this the balls move downward
resulting in the downward movement of the sleeve. The left hand
end of the main lever is lowered which pulls the piston of the control
valve up in the cylinder. With the upward motion of the piston of
the control valve the passage for pipeline 1 opens and oil under
pressure rushes from the control valve cylinder to the servomotor
the right side of the piston. The servomotor piston then moves to
the left. This increases the nozzle outlet or the passage between
the adjacent guide vanes, thereby allowing a larger quantity of water
to strike the runner (as required for the increased load) and the
normal speed for the turbine runner is thus restored.
on
In the case of Kaplan turbine since in addition to guide vanes
the runner vanes are also adjustable, the governor is required to
operate both sets of vanes
also operated by a separate servomotor and a control valve and the
and the control valves for both the runner and the
simultaneously. The runner vanes are
servomotors
guide vanes are interconnected to ensure that for a given guide vane
opening there shall be a definite runner vane inclination.
However, the large hydroelectric units are nowadays provided
In general, an electrohydraulicwith electrohydraulic governors.
consists of a hydromechanical actuator and an electrical
the machine room. The
governor
cabinet, both mounted in
cabinet contains the electrical part of the
cabinet contains the hydromechani-
equipment
electrical equipment
governor only, while the actuator
cal devices. A special tachogenerator
transmits currents at a frequency oi 50 cycles
.
ing circuits of the governor. When the speed oi the unit changes,
the frequency of the tachogenerator supplies also changes, where-
upon a discriminator responding to frequency vanat.ons emits an
electric signal. After being amplified, this signal is transmitted to
the electromechanical final-control element, where it is transformed
into a mechanical displacement transmitted through the lever system
of the amplifier valve to the main governor valve. Being moved
from its mid (or normal) position in direct proportion to the strength
of the electric signal, the main governor feeds oil under pressure
t0 the servomotor of the turbine distributor, lhe spear
connected to the main shaft
to the electric measur-

FLUID MECHANICS
in the direction corresponding
HYDRAULICS AND
thus turn in
994
nozzle or the guide vanes
to the sign of electric signal.
21 20 RUNAWAY SPEED
head and full gate
to almost zero value
of the turbine also
and it will attain
or limiting speed
under maximum
opening,
and at the same time the governing
fails, then the turbine runner will tend to race up
the maximum possible speed. T
“ Obviously for
of the turbine runner is known as . ,w?«med for
safe design the various rotating
^""'"
runaway speed normally
runaway speed. For a Pelton wheel . tnru-
ranges from 18 to 1'9 times its normal speed, for a Francs turbine
it normally ranges from 2 to 2'2 times its normal speed an for a
Kaplan turbine it normally ranges from 2 5 to > times its normal
speed.
21-21 SURGE TANKS
the generator decreases
the governor reduces the rate of flow of water striking the runner
in order to maintain the constant speed of the runner. But the
sudden reduction of the rate of flow in the penstock may lead to
setting up of water hammer in the pipe, which may cause excessive
inertia pressure in the pipeline due to which the pipe may burst.
Two devices viz.,deflector and relief valve as described earlier are
thus provided to avoid the sudden reduction of the
the penstock. But neither of these devices
As indicated earlier when the load on
rate of flow in
are of any assistance
when the load on the generator increases and the turbine is in needof more water. Thus in order to fulfil both the above notedments, in addition to the above noted devicessuch as sufcjc tank and fovsbay
are employed in the
plants where the penstock is
for medium and low head
the penstock is short.
require-certain other devices
usually employed. Surge tanks
and medium head hydro-power
hvr|
er
^ °n
^’ an<
^ ^orekays are suitable
hydropower plants wi
are
case
wnere the length of
An ordinary surge tank i
reservoir, as shown in Fig. 21T3
at a point as close
is kept well above the
When the load
wL?:rdnCai0pen'
toPPed storage
as possible to the turbine
^T^ ^ th® penst0ck
~
maximum water level •
' 1 U p p e r ilp of tank
on the turbine is stearlv ,
m the suPPty reservoir.
no velocity variations in the pipeline J ^ normal and there are
gradient oaav The water surfaced th?* Wi
“ be a normal pressure
surge tank will be lower

to the friction head
surge tank. When
he 1oad on the generator is reduced, turbine gates are closed and
tjie Nvater moving towards the turbine has to move backward. The
ected water is then stored in the surge tank in the space between
levels a and b and a rising pressure gradient obb1 is developed.
SURGE TANK
the reservoir surface by an amount equal
I*1
s
^n the pip® connecting the reservoir and the
RESERVOIR
I >- V »~L .
IV• . •
S
-s:
a
c
yilHHMBP
A
X
> 6O X
x x
.9
eX
(b)(a) (C)
a
7
<TPI3EROVER
PLOW s=rar-7T
o FORTSuoBrtkuHaa aam«JKaiiJ
c
*•>
(a)
ZYA
A
r/77777777?7777777777777777. V
zzz•v z
//////s/£r
%7*
/ ///////////// //// /////// / /YY/77
^^^Fig. 21* 13 Different types of surge tanks
The retarding head so built up in the surge tank reduces the velocity
the reduced discharge
of flow in the pipline corresponding to
required by the turbine.
When the load on the generator increases, the governor opens
the rate of flow entering the runner.
the turbine gates to increase
The increased demand of water by the turbine is partly met by the
water stored between levels a and c in the surge tank. As suca the
tank falls and a falling pressure gradient
water level in the surge
,lirce tank develops an accce^occ,is developed. In other words, the mge
^^^^ tQ a
lerating head which
incases
ten
required by the
value corresponding to tne i
turbine.

FLUID MECHANICS
c i i r e-e tanks a r e also show** .T h e various other types
e tank> type (fi)
n U»
Fig. 21'13. Type (a) is a cornea W
^overflow
SSSCSKT ;Z r$
T
«* .,P.» .h,«
stabilising effect its capacity may be less than that of a sln,ple
cylindrical surge tank. This is so because in a differential
retarding and accelerating heads are develope or® Prornpby than
in a simple surge tank in which the heads only in t up gradually
as the tank fills. Moreover no water is spilled to waste from the
differential tank. Type (d) is also similar in performance to the
differential tank, but it is suitable waen appropriate earth or rock
excavation can be carried out.
HYDRAULICS AND996
same
ta.uk
Illustrative Example 21‘
1 Prove that the maximum efficiency
of Pelton wheel occurs when the ratio of bucket velocity u to the jet veto-city V is given by the experssion
1— cos 0-f- &i
P~
V = 2[l-cosd )+k1+ki
u
where the loss due to bucket friction and shock is expressed askj (V — u)2
and that due to bearing friction and windage lossesvg as
(k2 ~
k ); kx and k2 are
Ut. Neglecting volumetric losses.
Solution :
t
Gonslde g the losses given in the problem thework done per unit weight of water
constants and 0 being the bucket angle at out-
net amount of
1
~
T t(r-«)(i-cos 0)] u— k
(F~M)a «2
1
2g k
%Thus
l
(V -U )2
k —2g
F2
2g
For V t0 be maximum f drLrespect to ut we get du J °» thus by differentiating with
22
_2(F— 2tt)(i#
— Ly
(V ~ 2u)( ( __
du — u) ~~ 2k9u
— 1
— =:0
cos
^(F— u)~ k2u~0
or

22
Performance of Turbines
22 1 INTRODUCTION
Turbines are often required to work under varying conditions
As such, in order to
!
of head, speed, output and gate opening
predict their behaviour, it is essem-;*i *
, ,
the turbines under the •
0 StUCty the performance of
tne turDines under the varying conditions
conditions for working may be as follows :
i
The variations in the
| (i) The head and hence the output of the turbine may change,
1 the speed being correspondingly adjusted so that no appreciable
-change in efficiency occurs, the gate opening remaining constant.
(n) The output may be varied by the movements of the gates
or the spear (or needle) ; the head and speed remaining constant.
J These are the normal operating conditions for most of the turbines.
(Hi) The head and speed may vary. Such variations are common
It may, however, be stated that al-
narrow limits, the
particularly in low head units,
though the speed is permitted to vary within very
head may vary by even 50% or
(iv) The speed may be allowed to vary by adjusting the load
'the turbine ; the head and gate opening remaining constant. These
conditions can be developed only for laboratory turbines or those in
the test plant and are otherwise uncommon.
i4 % N wimore-
on
turbine working under
between the perfor-
to facilitate the comparison
recmired to be constcertain specific quautiii&s a so
these quantities are explained in the next p
i

UNDER UNIT HEAD— UNIT
1016
222 PERFORMANCE
QUANTITIES
From the outputs of a turbine corresponding to different working
heads it is possible to compute the output which would be developed
if the head was reduced to unity (say 1 metre) ; the speed being
adjusted so that the efficiency remains unaffected. The e ciency of
a given turbine will remain unchanged if the velocity triangle under
working head H and under unit head are geometrically similar so-
that shock at entry is avoided.
for Francis turbine-Fig. 22*
1 shows two inlet velocity triangles
under its working head H and under unit head, the subscr ipt
being used to denote unit conditions. But these triangles are similar
It then follows that
w
if their corresponding sides are parallel.
n uu , u __
VW
~
(VWU )
an
Vf (Vfu)
Further it may be shown that
{YmUjg) (Viuu}('Uu )I(j (VUIMIQ) ( VMU) i^'u )Iff
(F/Af ) ~
(r f J/2g)
'
’ (F/12/2?)“ (Vn„2
l2g)
The above expression indicates that the ratio of the useful
energy to wasted velocity energy is same under the unit head and
(22T)*•••
I
v*
w.
1 -v, * bQ
A'
T
’ve
'T
vru t vfu
©
» Vf i±cV
FOR HEAD H
FOR UNIT HEADC
Fig. 22-1 inlet velocity triangles of head H and
the working head conditions. In other words theo unchanged efficiency is fulfilled. Thus
unit head
required condition*
QJ- f
^ut/ u'U'u,
Ph
^hu j
m £txi (22-2}<•••
Eliminating Vw and V
uu2
=
from equations 22Tmu and 22‘2
fc2
„ u
K — " ~
T or
(22-3)V H •••
V/and VfF,2
= u or Vfu= (22-4}.v H t #
•

mKruKMANCE OF TURBINES
JxDN
1017
andu
—Since it follows that60 60
N
Nu= V E
where Nu represents the speed of the turbine
•
t js known as speed.
Further it can be shown that both the
flow ratio remain canstant under the head E
The rate of flow and Qa=(kxDB)Vu, hence
Vfu 1
Q v/ V
~
E
~
’
where Qu represents the discharge flowing through the turbine under
a unit head and is known as unit discharge.
The power developed
P^ iwQH )
^P=(wQuX 1)YJ0, hence
Pu Qu
P ~
Qx E
~
’
•••(225)
under unit head and
speed ratio $ and the
and the unit ead.
Ql
•• Qu
— (22'6)•••
VE
and
1 P* p
___u
E3/2
where Pu represents the output of the turbine under unit head and
is known as unit power.T
— —Although the above expressions have been deduced from the
velocity triangles of a Francis turbine, the same are applicable to all
types of turbines. The advantage of obtaining the performance at
unit head is that from the known values of discharge Q, speed N and
power P of a turbine working under head E, the values Qv Nx and P
corresponding to some other head Hx may be computed as indicated
below :
(227)» * •
1
Since Nu and Pu will be same for both the heads, as such
% Q
^iQyfSfth/IT
P1 MPHI3 I2
)I^12
The above derived expressions are based on the assumption that
the efficiencies remain constant at all the heads, which is however
not correct. In practice it has been found that the efficiency vary ’
with head which will cause the
~
scale effect.-
The expressions for the various quantities <as
can be used for comparing the performance of any one turbine only
Under different conditions of operation. HowewF; accoFdmgto
another definition for the unit quantities for the unit quantities as
Q«=(QlJH )={QJV BJj
Nu=( N/VH )=(N1ls/ H1) :
P„=(P/53'2)=(P1/if3
'2) ;
derived above
r

FLUID MECHANICS
; HYDRAULICS AND
1018
indicated below more general[ expressions foi• tbe
perf(J.
be obtained whic* theTametype.
? ° hfunbW unit discharge and unit power
definition the unit speed, ~"
d ~;er 0f a turbine having
rsrszssi
-JU. *
-f »r
- -head)
According to this
are defined
Thus from equation 22*
3
nXlxNuJxJ^!v“
60
1
60
ND (22*
5 a)
Nu—
Similarly from equations 22 6 and 22*
7 the following expressions
for Qu and Pu may be obtained
•* *
or / H
Q (22*
6 a)
Qu= •«
DW H
P (22* 7 a)
Pu—and
•* *
JJZ
H6 ,Z
By using these expressions the values of the unit quantities may
be determined for any one turbine and the same applies to all the
h.similar turbines of the same type. It may however be staled-that the
yalues of the nnlt quantities for similar turbines wLU-Jbee__
^ii
^l-Oiily_if
their efficiencies are ecpial. Since the efficiencies of even the similar
qual the values ofjtKeUnit quantities willldsodiffer.
( ft
tjurbines are not e
22 3 PERFORMANCE UNDER SPECIFIC CONDITIONS
The performance of turbines under unit head facilitates the
‘ comparison of turbines of the same type. However, the turbines of
different types may be compared by considering an imaginary
turbine called specific turbine. The specific turbine is an imaginary
turbine which is identical in shape, geometrical proportions, blade
angles, gate setting etc., with the actual turbine but reduced to such
a size that it will develop one metric horse power under unit head.
The suffix s is used to denote the symbols expressing the performance
* of the specific turbine
Fig. 22 2 shows the actual and the specific runners for two types
of turbines. Since the actual turbine under unit conditions and the
•
specific turbine both work under a head of 1
the velocity triangle abc in Fig.22T will hold
turbine as well.
metre it is evident that
good for the specific
Vf 8=Vfu etc. NowIn other words us— uu,
Qu~ nD(nD)Vfu and Qs=nDs(nDt)V fs
v/Qu
( ~7Q. V D.J•• (22 3)* ••
k

PERFORMANCE OF TURBINES
%
10J 9
P.Again
Assuming 7]0 to be same for both
* wQuX 1 Xvj0 ; and pa^==wQs x i xi)o
P„ Qu f D 2
p
(22*
9)3
* ••
f-no
—j
~
^nDsh-
0 0S
ACTUAL
RUNNER
SPECIFIC
RUNNER
(a) FRANCIS TURBINE
/ A
,0*
* EEzd=( Ds /m )
SPECIFIC
WHEEL
C$= (D/m) ACTUAL
' WHEEL (b) PELTON WHEEL
Fig. 22*
2 Actual and specific runners
Equation 22 9 shows that the power of geometrically similar turbines
working under the same head varies as the square of the runner
diameters.
Since
P
Ps=l h.p. (by definition) and Pu= jpT2
D DH31
*
nDNU
Dt=
Further uu 60
7TDSNa
and
60
but Uu Us 5
7TDSN(,7iDNnbence 6060

N,=NU V Pu
1020
(22 10>•••
JL D
N
Nu=but V H
P
and £3/2
N */
~
P
Bbl*
(22*
11)Na= •» •
ecific runner is known asThis value of N8 , the speed of_s .
specific speed. For any other turbine also exactly same relationship,
for Ns may be derived.
the above noted expression for theA significant point about
specific speed is that it is independent of the imensions oi size,
both of the actual turbine and of the specific turbine. It therefore
means that all turbines of the same geometrical shape, working
under the same values of <f> and 4*» an(
^ thus having the same
efficiency, will have the same specific speed, no matter what their
sizes be and what powers they develop under what heads. As
such it may be stated that Ns represents the specific speed of the
actual turbine, as well as of the specific turbine. T. herefore, the
following general definition for the specific speed may be given.
The. specific speed of any turbine is the speed in r.p.m. of a tur-bine geometrically similar to the actual turbine but of such a size that
under corresponding conditions it will develop 1 metric horse power when
working under unit head (i.e., 1 metre).
The specific speed is usually computed for the operating condi-tions corresponding to the maximum efficiency.
It may be seen from equation 2211, which is dimensionally
^
I'homogeneous, that the specific speed is not a pure number. By
considering the dimensions of N , P and H it can be shown that the
value o f N s in metric units is equal to about 4'
44 times its value in
F.P.S. units.
non
There exists a general trend to select a turbine of higher
specific speed because higher specific speed of a turbine results in the
reduction of the runner diameter as well as the overall size of the
due to which the weight and the cost of runner are reduced.This may be explained from
runner,
equation 22T1 which shows that for
a given power output and head, an increase in the specific speedpermits a higher speed of rotation for the turbineing to which even for a runner of smaller diameter
runner, correspon-
a higher peri-

PERFORMANCE OF TURBINES 1021
pheral velocity may he devel
speed the runnerdiameter asTn As s
^h hv
Furtheras shown in Illustret; !38 its overall'?Crea*hgthe
turbinemay be expressed
^^ple Ss /** he r
*"* •**»»%i
.p,??-
^,=3-65 JVU VQur,0
jYom the above expression it is seen that the specific speed may be
eased by increasing the unit speed Nu or unit discharge Qu. It is,incr
ph
Ns =50
m-—i
NS=I00
Ns=2f0
mj
Ns=300 Ns=500 Ns=700
Fig. 22’ 3 Relative sizes of turbine runners developing
same power under same head for different
specific speeds
given discharge Q and head H th
tionally to the r0M
*-*-fjstr»« inc«®d ,
-^SSCia
Z
^Z.
discharge may be increase
wing through t ie
a low head
reduced and the discharge ,
turbine operating . charge to flow
As indicated in Chapter
allow larger
passage,
and
consequently required to
& larger area ofough the runner must
square
thr

1FLUID MECHANICS
HYDRAULICS AND
1022
— , bv designing the runner with axial fl.
which may be obtame y
^hus axiaJ flow turbines are h6 1
r r s^(or discharge,and head, he d.ameter a
^ ^^ ^of an axial flow tnrb.ne.s h
speedS. This fact £ j
other types of turbines having lowet r act nas
been illustrated in Fig. 22-3 which shows relat.ve s.zes of
dlffereat
types of runners designed to develop same power and operating u„det
the same head, for different specific speeds. It may be seen froni
Fig. 22 3 that for low specific speed radial flow runner with
diameter and relatively narrow fluid passage, is required to be
vided, and for increased specific speeds tne 1 minus of smaller
diameters and wider flow passages are required to !>< provided,
ov
a large
pro.
It may thus be stated that an increase in the specific speed of
a turbine results in the reduction of the diameter as well as the
all sizes of the turbine runner,
bine runner will lead to a corresponding reduction in the cost, it is
evident that for economic reasons a turbine runner with highest
SDecific speed possible should be selected.
22 4 EXPRESSIONS FOR SPECIFIC SPEEDS IN TERMS OF
KNOWN CO-EFFICIENTS FOR DIFFERENT TURBINES
From the general expression for the specific speed represented
by equation 2211 it is possible to obtain the
speed in terms of known
cated below :
over*
Since this reduction in size of a tur-
expressions for specific
co-eflicients for different turbines as indi-
(a, Specific Speed of Pelton Wheel. An expression for the
specific speed of Pelton wheel may be obtained ? ,?,
Kv), ,,and „as indicated below • “* tCrmS
°f (ot
u=<f, V 2gB as
Y-f60
^
6U
A
KD
p wQH Cv V 2gH X #7)0
75 Xylo=
75
P — 46*
36 (CvdQV2
)
Since «>=1000 kg/ms and
or
0=9-81 m/sec2
.

PERFORMANCE OF TURBINES 10 23:
N*f P
Nt=Thus, //r,M
_ (84 67) 0 (V H )(46'
36)l/2(O1,/i2y]0fir3/2)l /2
~
D(l p/4 )
_ (576-6
^( / C
^) (576‘6)
^(Ot,Ylo)1/a
( D/d )N,= (22*
12)••*
or m
Now taking <£=0*46, C„=0-98 and v)0=0’85, which are the
al values, equation 22'
12 gives
242-1
11SU
Ns=^ (22-13)•••m
Equation 22*
13 gives a relationship between the specific speed
jf and jet ratio m for a single jet Pelton Wheel.
m is more or less constant and in most of the cases it is equal to 12,
almost all the Pelton wheels have the same specific speed,
in some exceptional cases an abnormally low value of m equal to 7
has also been used. As such from equation 22‘13 it is found that for
in varying from 7 to 12 the value of Ns varies from about 35 to 20,
which is quite a narrow range.
For a multiple jet Pelton wheel having number of jets n, the ex-
pression for specific speed may be obtained in a similar manner. The
discharge will be (nQ) and hence the power will be (nP). Therefore,
the specific speed in this case will be (/ n ) times the specific speed
for a single jet Pekon wheel.
(b) Specific Speed of Francis Turbine. For Francis turbines
an expression for the specific speed may be obtained in terms of
V aiJd n as indicated below :
Since the value of
However,
uDN
u— (f) V 2gU =
84-67 <f, y/ B
60 ’
N = D
w^n(nD2 )
^42gH (HX'f]0wQH
XY)0 = 75/5
or P= l 85’ 5{knD2
'>H3 l2
ri0 )
NVT• »
tf5/*
_ (84-67)
^V H (185-5)1/2(&tt#‘-fyffa/ 2/)0)1/2
DE5li
or (22*
14)N'^UMMVJcn
^o)] •••

HYDRAULICS AND FLUID MECHANICS"2024
» 7 A HP constant, equation 22'14 indicates
that specific
*
s p e e d ,for Francis turbines depends on speed ratio
*,
flow ratio iji and breadth ratio n. Generally
* ranges from 0 6 to
0-9, 4,ranges from 0'15 to 0'
30 and » ranges from 0 10 to 0 45. The
variation of any or all of these will alter N., and hence a much
greater range of N,is available for Francis turbines as compared to
that for Pelton wheel.
classified as slow runners {Ns=60Francis turbine runners are
medium runners [Ns— 120 to 180) ana fast runnersto 120), normal or
( JSf9 —180 to 300). It has been observed that the increase in A, from
60 to 180 is obtained by increasing n and tj/, but to attain the higher
value, it would be necessary to increase simultaneously 0 and n.
It may, however, be pointed out that in order to increase Na the
values of and t|; cannot be increased to any value, because unduly
higher values will result in increased hydraulic losses and consequently
lower efficiency. It has been explained in Chapter 21 that as the
value of n increases the runner shapes are also altered. For slow
runners the flow is predominantly radial and exit is axial. For fast
runners the flow is more or less axial both at entry and exit.
(c) Specific Speed for Kaplan and Propeller Turbines.
Since these axial flow turbines work under low heads, it is evident
from equation 22*
11 for N8 that these turbines have very high specific
speeds ranging from 300 to 1000. As for the Francis turbine in this
case also it can be shown that
(VCy[#V <Kl-»s)]
_ 60V Vg
L l ~ z
(22*15)••«
(V2gr)Y)ownwhere and C2 — 4x 75TC
Now if yj0 =90% =0 35 then
iV'
s =512'4 (4& v/ ).
22 5 PERFORMANCE CHARACTERISTIC CURVES
The turbines are generally designed to work at particular
values of H,Q,P, N and v)0 which are known
ditions. But often the turbines
as the designed con-
r
aie re(
luired to work at conditions
different from those for which they have been
it is essential to determine the exact behaviour of
the varying conditions by carrying
turbines or on their small scale models,
usually graphically represented and the
characteristic curves.
designed. Therefore,
the turbines under
out tests either on the actual
The results of these tests
resulting curves are known as
For the sake of convenience the characteristic
are

PERFORMANCE OF TURBINES
lotted in terms of unit quantities,
of the following three types :
Constant head characteristic curves, (or Main characteristic
1025
These characteristicare p
curves
curves are
W
curves)•
(fl) Constant speed characteristic curves, (or Operating charac-
ic curves).(eristic
( Hi) Constant efficiency curves.
(i) Constant Head Characteristic Curves,
obtain these curves the tests are performed on the turbine by main-
taining a constant head and a constant gate opening and the speed
is varied by changing the load on the turbine. A series of values of
# are thus obtained and corresponding to each value of N, discharge
$ and the output power P are measured. A series of such tests are
performed by varying the gate opening, the head being maintained
constant at the previous value. From the data of the tests the
values of Qu> Pu, Nu and yj0 are computed for each gate opening.
Then with Nu as abscissa the values of QUi Pu and TQ0 for each gate
opening are plotted. The curves thus obtained for Pelton wheel and
the reaction turbines for four different gate openings are shown in
Fig. 22*
4.
In order to
For Pelton wheels since depends only on the gate opening
and is independent of Na, the Nu v/s Qu plots are horizontal straight
lines. However, for low specific speed, Francis turbines Nu vjs Qu
are drooping curves, thereby indicating that as the speed increases
the discharge through the turbine decreases. This is so because in
these turbines a centrifugal head is developed which retards the
flow. Since the centrifugal head increases with the speed, the flow
through the turbine is reduced as the speed increases. On the other
hand, for high specific speed, Francis turbines as well as Kaplan
turbines since the flow is axial there is no such centrifugal head
developed which may cause the retardation of the flow.
The curves of Nu v/s Pu and Nu v/s Y]0
for the different turbines as shown in
that for a Pelton wheel, for each gate opening the maximum value
°f >jo is attained at almost the same value ot Nu which corresponds
to 0=0*
46 [4>=( NunD)/80{ V 2g) ]- However, in case of reaction
turbines for each gate opening the maximum value of % is attained at
different values of Nu.
parabolic in shapeare
Fig. 22'4. It will be observed
In order to
(it) Constant Speed Characteristic Carves.
formed on the turbines at cons-
attained by regulating the gate°dtain these
tent
curves the tests are per
speed. The constant speed is

1026
, •
*1,- rifcrhar^e flowing through the turbine-opening thereby varying the discnarge &
. °
as the load varies. The head may or may not reman constant. The-
power developed corresponding to each setting of the gate openmg
© FULL GATE OPENING
I
3!
^3/4 GATE OPENINGCi
LkJ
e>
^l /2 GATE OPENING
1/4 GATE OPENING
<
{CO FOR PELTON WHEELx
uifi
Q
Z
UNIT SPEED Nu
©
t FULL
Z5
a. FULL
3/4
o:
LU
1/2§
o
CL
1/4z
O
UNIT SPEED Nu
FULT-
GATE OPENING '
UNIT SPEED Nu
©jj 2 FULL GATE OPENING3/4 GATE
OPENING 3/4 GATE OPENING*31
<3
1/2 GATE
OPENING
LU
iD
1/2 GATE **
-^OPENING
cr
<x
-1/4 GATE
OPENING
oCO
1/4 GATE"'3?
‘'V
OPENING
(FOR FRANCIS TURBINE)
i
atMWaBOTW l.JITUBP
—E
UNIT SPEED Nu
Q
|j (FOR KAPLAN TURBINE)
UNIT SPEED Nu
@
®„t|
I >•
o
2
o: LU
u
PULL LLcc ILill
FULL(JJ
3/4oa X <
cr V2 ?/4'1/42 Ul
Z3 1/4>
o
UNIT SPEED Nu
UNIT SPEED Nu(b) FOR REACTION
Fig. 22-4.
iwewie
Constant head character*wheel and
ristics for. Peltonreaction turbines

PERFORMANCE OF TURBINES 1027
measured and the corresponding values of „0 are computed.
Further knowing the total load capacity of the turbine the percent-
2<Te ohfoil load niay be computed from the measured power, and a
plot of percentage of full load v/s
^ is prepared. The curves thus
obtain^f
°r I0UJ different types of turbines working at constant
ed under constant head at varying gate openings are shown in
Fig 22’5. From Fig. 22*
5, it will be seen that as the % full load
spe
too
80
z
1
UJ
0- 60
w (1) KAPLAN TURBINE - Ns»700
(2) FELTON WHEEL - Ns«=15
(3) FRANCIS TURBINE- W£=350 _
14) PROPELLER TURBINE- 1
^=600
4LU
100
PERCENTAGE OF FULL LOAD
Fig. 22-5. Percentage of full load v/s v)„curves for different
types of turbines
In other words at reduced loads 7}0 is
At 100% full load v]0 is near about the maximum efficiency
It will be observed that the Kaplan turbine and the
increases vj0 also increases.
also less,
in all cases.
Felton wheel maintain a high efficiency over a longer range of the
part load as compared with either the Francis or the fixed blade
important factor in theTherefore this is anpropeller turbine,
selection of turbines.
above noted graph, discharge (Q) v/s BHP
also plotted which are shown in
from Fig. 22‘6 that BHP is directly
if the head is constant, and hence Q,
However a minimum discharge Q0
In addition to the
aRd discharge v/s rj0 graphs
Fl? 22-6. It will be
are
seen
proportional to the discharge
vls BBP graph is a straight line. ,
Will be required to run the turbine at no load. The overall efficency
remains more or less constant beyond%
^creases with discharge and
a Particular value of Q.

FLurO MECHANICS
HYDRAULICS AND
1023
I i A
a
X
! /
w
-9.-*
DISCHARGE Q
Fig. 22’6 Discharge v/s BHP and YJ0 curves
Efficiency (or Iso-efficiency)
Fig. 22'
7 shows the constant efficiency or iso-efficiency curves.
These curves show the efficiencies of the turbine for ail conditions
(Hi) Constant Curves.
of running and hence these are also known as universal characteris-tic curves of the turbine. In order to draw these curves the follow-mg procedure is adopted. By operating the turbine at about 8 to
10 gate openings, the corresponding number of Nu v/a rJo and N„v/s
Qu (or Pu) curves are plotted as shown in Fig. 22 4. On the Nu »/«,)
(LCUrVeS a Set horizontal lines, (each line representing the same
efficency) are drawn which will cut the curves corresponding to
each gate opening at different points. These points are projectedthe corresponding N,v/s Qu (or P„) curves for each gate
^ening
and the points of the same efficiency are joined by smooth curveswhich are the iso-efficiency curves Tf it. i, r
^titof • •
urves‘ u ls> however, clear from the
graph that the innermost iso-efficiencv rnr i
efficiency of the turbine and the Z rePre^nts the highest
efficiencies. It will be seen from Fig. 22'7 that for a given unit
discharge (or unit power) if a vertical line is drawn it will cut some
iso-efficiency curve at two points and it will touch some other inner
iso-efficiency curve of higher efficiency, which may not have been
drawn, just at one point only. Thus for a given unit discharge the
on

1 UZTfA V Ai
-;i 1SI -.il
verticalline touches the curve
point. It, therefore, mean'?
°. max
^urri •
iso-efficiency curves are j0
;
^if the peak n -Ciency at only on
obtain the best perfor
^f
of each iso-efficiency cu7^the tUri>ine
°
be
lhen we
7 CUrve are the
’ becau-'« the peat
various
points of maximum
for some value of the unit discharge (or unit power).
points
efficiency
Vith the help of these characteristic curves it is possible to predict
best performance of the turbine. Thus if the gate opening is
then the point of intersection of the Nu v/s Qu (or PM) curve
the
known
for this gate opening and the best performance curve will represent
the conditions for the best performance for that particular gate
ISO-EFFICIENCY CURVES
BEST PERFORMANCE CUtVE
60
70
j 60
Nu50
<J0
30
mm
260 ' 300IGO 140 180 220
Fig. 22'
7 Constant efficiency curve or Universal
characteristics ul a Francis turbine
opening. From this point unit discharge (or unit power) can be
known by drawing a vertical line and unit speed by drawing a
horizontal line. Now if the head on the turbine is known the
discharge Q, the power output P and the speed N for maximum
efficiency at this gate opening can
hand if the head H and the speed N are known, then unit speed Na
can be computed. Then by drawing a horizontal line for this value
°f Nu, to cut the best performance line, the point of maximum
efficiency is known corresponding to which unit discharge or unit
power can be obtained, and as before the discharge Q and power
^Can be calculated at which the turbine efficiency will be maximum
for the given head E and the speed N.
be computed. On the other

n1 n
J U-iU
OF TURBINESMODEL TESTING22'
geometrically similar to the actual turbine is first prepared.
various linear dimensions of the model turbme bear the s a t n *
portion to their corresponding dimensions ot the actual turhj
The model turbine is then tested under a known head, speed
well as the efficiency are dete
a
^tUal
rhichk
and
flow rate and its output as
From these test results it will be possible to predict the perforrnanCe
of the actual turbine. Moreover on the basis of the test results if
the design is to be modified it may be so done in the model turbine
without incurring much expenditure. Thus model testing of turbines
a perfect design for the actual turbines,
rmined,
assists in obtaining as Well
as in the development of the new types oi turbines with higher specific
speed and better efficiency.
The various variables involved in this case are dischargeQ
head H, speed of rotation of runner N , runner diameter D, output
power Pt mass density p and viscosity /A of the flowing fluid. It may
however be stated that in the problems of turbomachines generally
shaft work glJ is used as one of the variables instead of head H
These variables may thus be grouped into fallowing dimensionless
parameters :
Q gE P A
ND3
’ N*D*J' * pgHND3 /
’
9ND*
GVT)3)*S known as discharge number
The parameter
or flow
number,
®y Combini
“g these parameters!'
Their
"
alternativeexpressions may be obtained.
also be
Pis known as head number, and is known
pgHND
i. bus the discharge number may
Qexpressed as and the power number asZ)2 %/ gJJ
dividing the
(pgH3/ 2D2 )' Furtker by square root of the power
to power (3/4) the following
number by the head
dimension!
number raised6SS Parameter n,is obtained
p!/2
The parameter ».is knownnumber of the turbine.
out t
1 ma
^’
how
-1
', be noted that iPUt Power P has been considered
as dimensionless shapspecific speed or
theln the above expressions
in terms of (kg-m/sec) But ^

1031PERFORMANCE OF TURBINES
the output power P is considered in terms of horse power then in the
above expressions P will have to be replaced by (75 xP). Further
for water since p=102 msl/m3, the dimensionless specific speed na
becomes
iW75 xP _W,“
(102)1,2(9'
8i )5
'4#5/4 20 24
iVa— 2Q‘24 ns
which gives the relationship between the specific speed (equation
22'
11) and the dimensionless specific speed.
For complete similarity to exist between the model and the
actual (or prototype) turbines the above noted parameters must
-have the same values for the model and the prototype turbines.
The parameter f rePresents Reynolds number which should
also be equal in the model and the prototype turbines for complete
similitude. But in actual practice, it is not possible to have the
same value of the Reynolds number in the model and the prototype
turbines, on account of considerable difference in their sizes.
However, since the flow in the prototype turbine is turbulent, it has
been observed that if the flow in the model turbine is also turbulent
then even if the Reynolds numbers are not equal for the model
and the prototype turbines the similarity between them can be
^ ensured.
N,
or
Thus for the complete similarity to exist between the model
and the prototype turbines, the following conditions may be required
to be satisfied :
1r QQ
ND3 ND»
m
y (20'16)••®
QQ
DWgH / v jor
DWgH m
( J &_
N2D2 / m N2D2 )V
)m
=(pjHN& X
(20-17)e e o
1P
pgHND3
(20*
18)# •*
or
).NV PN*f P ) -( (20-19)
( r •c
p1/2 (gU)5
'4

Since in most of the cases gm=(h>
conditions may be simplified as follows.
1032
and also pm Px> the abov©-
). 1f QQ
=! I N D3
ND3
Jm (20*
16 a)
t). J
•••
QQ
DWHor
DWB m
N%D%
Jm ' K
(20 17 a)9
••
1P )( - V J )
KHNDVm
-f
IV END3 ' p
f P
(20*
18 a}>•••
or
( N 4 P N 4 P
H51*
(20'
19 a )f * * *
/V /75/4
m
where the subscripts m and p refer to model and prototype turbines,
respectively. By determining the values of P, H, N and D from the
model tests the values of the above noted parameters are determined*
which will be same for both the model and the prototype turbines.
its diameter D and head H areFor the prototype turbine since
known, its speed N , power P and discharge Q can thus be calculated
with the help of the known values of these parameters.
)
A little consideration will, show that the above noted conditions;
may be achieved if <f>,
^and Ns have the same values for the model
and the prototype turbines. Further it may be seen that the various
terms in the above expressions represent the unit quantities and
hence for similarity between the model and the prototype turbines
the values of the various unit quantities must be equal for the both.
Ihe above noted conditions for establishing the similarity
between the modej. and the prototype turbines are based on the
assumption that tne efficiencv of the model is equal to that of the
prototype. However, tne efficiencies of the model and the prototype
turbines are not equal. This is so because the: energy losses are
proportionately more in the model turbine than that in the prototype
turbine. As such the efficiency of the model turbine is lower than
* . of
.
the Prot<>type turbine. On account of difference in the
efficiencies of the mode! and the prototype turbines the scale effectwould be developed and it would therefore be a possible source oferror m predicting the performance of the prototype turbine on thebasis of the mode! test results. However by modifying the above-

vtKrUKMANCE OF TURBINES 1033
d conditions for the similarity the error due to such
may be eliminated. The modified expressions for the
conditions may be obtained as indicated below.
If and n„are the overall efficiencies of
orototype turbines respectively, then
* TT 1
(rw* )
note a scale effect
similarity
the model and the
H )9
‘‘
Oom
’
and v)0i>m
where H and TIL are respectively the head acting on the turbine and
the loss of head in che turbine and the subscripts m and p refer to
model and prototype respectively. From the above expressions the
net effective heads available for the model and the prototype turbines
may be obtained as
I1 TiL)m vj om[H )m
{ II TIL),p j>
By considering these values of the head in place of II in equations
22*
16 (a) to 22"
19 (a) the following expressions are obtained :
and
(D2
V r,0
ji )m
~
(
Q
(1216 6)•* •
W-%n /„
( yisM. 'i (22-17 b)•« «*
)Pi (22-18 b)m ••
N/ P
( ).-(N/ P (22*
19 b)•9 1
7)05'4£6 /4 /*7)o5
'4#5
'4
the conditions for similarity
These equations thus represent
between the model and the prototype turbines with due allowance for
the difference in their efficiencies.
From equation 22'
19 {b)
model and prototype turbines may be expressed as
the ratio of the specific speeds for the
5/4
f TFL.
r] j>'
(Ns)m
For determining the efficiency of a prototype
efficiency obtained for its model, the following general expression
been given :
turbine from the
has
(22-20)
I]OP
1
•••
the overall rf
*•“£££ •
erical
where of
’lom and r)ov are
Prototype respectively, Dm arK
* ®‘p a
e ,
h-n(T on
1
runners, IIm and Ev are the eacs
^ ^ ^num
Prototype turbines respectively andthe

Different values have been suggested for the exponents
» different investigators, but the most commonly adopts
those recommended by L-f . Moody winch are a=0*
20
exponents-
a and {3 b
values are
add p=0. Thus equation 22 20 becomes
0 20
<W P )1 "Cop
1 f]om
#<150 m. For head H>150 m the following relationship has been
recommended :
(22-20* * »
a)
is however applicable for head
o io0* 25 / 77
Dm1 — T)oj> (22 20 h)) *•«
HPDP
It may however be stated that different investigators have recom-
mended different values for the exponent ot varying from
0*
04 to 010 in equation 22‘20 (b).
22-7 CAVITATION IN TURBINES
When the pressure in any part of the turbine reaches the
smallit boils andvapour pressure of the flowing water,
bubbles of vapour form in large numbers. These bubbles (or
vapour-filled pockets or cavities) are earned along by the flow,
and on reaching the high pressure zones these bubbles suddenly
collapse as the vapour condenses to liquid again. Due to sudden
collapsing of the bubbles or cavities the surrounding liquid rushes
in to fill them. The liquid moving from all directions collides at
the centre of the cavity, thus giving rise to very high local pressures,
which may be as high as 7000 kg/cm2. Any solid surface in the
vicinity is also subjected to these intense pressures. The alternate
formation and collapse of vapour bubbles may cause severe damage
to the surface which ultimately fails by fatigue and the surface
becomes badly scored and pitted. This phenomenon is known as
camtcdion.
>
In reaction turbines the cavitation may occur at the runner j
exit or the inlet to the draft tube where the pressure is considerably
reduced. Due to cavitation the metal of the runner vanes
draft tube is gradually eaten away in these zones, which
lowering the efficiency of the turbine. As such the turbine compo- I
nents should be so designed that as far as possible cavitation is
eliminated. In order to determine whether cavitation will occur in
any portion o the tuibine, D. I boma of Germany has developed j
a dimensionless parameter called Thom,a’* cavitation factor a which is
expressed as J
and the
results in
(22-21)a •
= E * •»

1035
here Ba b atmospheric pressure head ; H.is vapour pressure
d ; #» *"pre
-are head (or height of runner outlet above
Wi> race) ; and H is worloug head of turbine. Complete similarity
j« respect of cavitation can be ensured if the value of a is same in
both the model and the prototype Moreover it has been f
a depends on Ns of the turbine, and for a
can be reduced
hea
ound that
turbine of particular Nsupto a certain value upto which its
A iurther decrease in the value of
The value of
the factor cr
efficiency v)0 remains constant.
a results in a sharp fall in rj0 .. . . . . at this turning point
is called critical cavitation factor ae. The value of ac for different
turbines may be determined with the help of the following empirical
relationships .
For Francis turbines
v 444 )ac=0*
625 (22-22)••«
For Propeller turbines
L7 5V444 Jcrc=0'
28 + (22-23)••
For Kaplan turbines, values of a0 obtained by equation 22" 23
should be increased by 10 percent.
(a) Soeiion Specific Speed. In addition to Thoma’s criterion
the consideration of suction specific speed provides another very
useful criterion for establishing similarity in respect of cavitation in
the turbines. The suction specific speed 8 may be defined as the
speed of a geometrically similar turbine such that when it is develop-ing a power of 1 hp the total suction head H8V is equal to 1 m (in
absolute units). According to this definition the expression for
suction specific speed may be obtained by replacing the total head
H in equation 22*
11 for the specific speed by the total suction head
ThusHSV •
. N±l
°~~~
TJ 5/ 4IJ- SV
(22-24)* « a
value of the suction specific speed for the modelBy having the
and the prototype turbines the similarity in respect of cavitation can
Be established.
same
The total suction head Hsv can be expressed
H8V =Ha-Hv -Hs
as
and hence from equation 22*
21
Htv — oH (22*
25)* ••

in equation 22 24 we get
1036
By substituting the value oi Hsv
4 /5
Ns
<22-26)•* •
°~
sor
the relation between the two para,.Equation 22' 26 represents
meters a and S, both of which are useful for establishing a similarity
in respect of cavitation in the model and prototype turbines. How.
ever, the concept of suction specific speed is more commonly used in
the case of pumps.
22*
8 SELECTION OF TURBINES
The selection of a suitable type of turbine is usually governed
by the following factors :
(i) Head and Specific Speed. It has been found that there
is a range of head and specific speed for which each type of turbine
is most suitable which is given in Table 22'1.
TABLE 221
8.N. Head in metres Types of turbine Specific speed
1 300 or more Felton wheel
Single or Multi-ple jet
Pelton or
Francis
10 to 55
v 2 150 to 300 35 to 100
3 60 to 150 Francis or
Deriaz (or Dia-gonal)
Kaplan or
Propeller or
Deriaz or
Tubular
100 to 220
4 Less than 60
220 to 1000
However
turhfrl3 §
re,ral
,rU
'e
’ k may be stated that as far asturbine with highest permissible
chosen, which will
small
possible a
specific speed should be
size and hioh f
°
^^ .6 c
*)
eaPest in itself but its relatively
generator as well as power”house^B
'7'l
^
the size of *®
I wei nouse. But the specific speed cannot

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