Raabe J. Hydro power - the design, use, and function of hydromechanical, hydraulic, and electrical еquipment
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scncc.
I!
is an imaginary cost term, depending on thc loss of a component co~lsidered during its useful
life. Thereby
it
is a function ofthe coinponcnts's six. This facilitates
an
optimizntion of this sizc also
depending
011
the
investment,
the income by salcs of eriergy, and other Dcncfits of
the
plant.
The above capitalized cash value of loss is a part-of what is here called
resulting
cost. It is here
that
thc latcr docs not contain the
OMR
cost (due to operation, mainternnce,
;~nti
The
OMR
cost is dcemcd to be independent of the compcrncnt's size. and hence
irrelevant for optimization. Strictly speaking,
it
depends also on this size. Thercforc the firs! cost
(investment for planning, construction and erection of the plant) could be imagined to i~cludc the
oMR
cost. Here the cash value is identical with the present value at starting power production.
In the following text a "set" denotes
a
generating unit consisting of
a
turbine and its generator.
Moreover the cost of electric energy is referred to
.as electricity rate. Finally the initial investr~xxt
cost of a plant is also referred to as first cost.
4.2.
Optimization of rated discharge of
a
run-of-river plant
4.2.1.
General remarks
and
assumptions
An
exploitable reach of a river with an available head, that results from the river's drop
along this reach, is considered
[4.2;
4.31.
This available head
All
is assumed to equal the
available drop of maximum permissible water levels in the reservoirs
cxisting after the
cornpletion of the development.
The flow
duration curve of the average hydraulic year may be known for this reach. The
development is based on a plan according to which the river, without
any
flow
regulation.
is
developcd in subsequent reaches having available head. From the topograplly of the
river bed, ihe back water effect as a function
of
the known hydrograph for the subsequent
plant is also known and accounted for (Fig.
4.2.1
a).
For this scheme the rated discharge as a function of turbine arrangement is to be
optimized with respect
10
a maxiinum surplus of earnings from the sale of energy over
the
initial cost of plant for the economic life of the facilities. For simplicity the net earnings
-
Fig.
4.2.1.
Lay-out of a run-of-river plant. a) Available
head
Ah,
and backwater effcct
Ah,,
yielding the net head
H,,
=
H
=
Ah
-
Ah,
along the river section exploited. b)
-
\'
Flow duration curve
Q
(t),
head duration curve
H
(r),
and
power duration curve hence derived for the average year
to
t
of
a
certain scheme.
considcrcd are obtaincd
by
s~~btracling the
OklR
cost f~o~n thc gross earnlllg. The
clzcirici(y
satc
is';lssumcd constairt during thc period of dt.prcciation.
-
The
live
storagc voluiiie of thc reservoir is ;tss~r~nctl so sln:~ll, that
it
c;in bc
neglected.
klcnce thc plant consiclt:rcr.l always operntcs under its rnr~xi~num pcr~:iissible hc:~ci water
Icvcl. TO 0ht;ii11 the net hcad
11
froni this,
All
has to bc diminislicd by thc flow-induced
level drop
di:c to the back watcr effect
All
(0).
tlcrice the net liead
I1
=
All
-
~111((2).
For
practical
purposcs and in accordance with cxperiencc
LIII(Q)
is assumed to equal the head
computed by a
wcir formula
dl1
(Q)
=
k
Q'!"
Q
being the discharge and
k
an cinpirical
cons!ant parameter
[4.4].
As
the live storase volume is negligible, any river flow that passcs over the rated flow
Q,
of tht: turbines
at
rated gate has to be spilled to limit cavitation pitting.
It
is
assumed that all the machines
installed
are available for operation, when the river
offers this discharge. Moreover the electric grid is assumed to be large enough to consume
the energy
offered by the plant. The number of sets is assumed to
be
of
a
size so as to allow
(by switching on and off an adequate number of sets) the sets to operate at
thcir bep under
the available river flow. Hence
q
=
q,,
depends only on the net head
H.
Sirice the spillway is designed for the maximum flood discharge, it absorbs a constant
amount of the
initial cost and therefore does not affect the surplus. The same holds for
the cost of high voltage transmission
-
if required
-
with its outdoor switchyard.
The cost term for the barrage (if any at all) may be slightly affected by that of the power
house, if
thf latter forms
a
part of the damming device. However in
;1
run-of-river plant
!ike the one considered, the main cost of the damming dcvice i.5 clut. to t!ie embankment
along
the developed reach of the river. Assuming this, the cost of
clam
may be considered
as
independent of rated discharge. In the following chapter
4.3
an attempt is made to
account for the influence of capacity and hence of the power house
on the cost of a dam
that crosses
a
river of given width.
The cost term for the power house is split here into one term,
K,,
for its
superstructure
and another term for its substructure, mainly owing to excavation for the draft tube and
the spiral casing,
K,,.
The remaining first cost of t3e sets consisting of turbines, alternators and a portion of the
transformer
K,
are assumed to vary with the rated discharge and hence with the size
of
machine which is assumed to determine the cost of fabrication.
As
opposed to Cap.
4.3.,
in
this section the cost
of
the accessories
per
set
arc
assumed to be
pro-
portional to that
of
the set,
and
consequently can be included
in
the nforemcntioned cost
term
K,.
The net head is computed from the known available head
Ah
arid rhe backwater effect
kQ2'3 as
H
=
Ah
-
A-Q'I3, where clearly
11
is a function of the flowr
(2.
Since !he latter is
given by the flow duration curve as a function of time (here taken
in
hours ovcr a year
with
to
hours). a net head duration curve is also given
H
=
H
(Q)
=
1-1
(Q
(t)j
=
M
(t)
and
hence
il
=
q(H)
=
q
(t)
[.1.5]
(Fig.
4.2.1
b),
where
q
denotes the efficiency of the set.
For a comparison with the
initial cost, the present value of earnirigs during the
,I
years
of depreciation under a constant electricity rate
k, and interest rate
p,
denoted by
K,
is
computed for the time, whcn energy production starts.
The optimization of rated discharge
Q,
relative
to
a surplus of earninrs
K,
over the initial
cost terms
K,,
K,,
and
K,
due to the power house, exc=lvation and fabrication of sets,
depending on
Q,,
is made vnder the following alternative cases.
In the first case the nominal runner dianieter II might be given. Naturally this is doliz with
consideration of the number of sets
i,
assumed large, and, of coursc, an inteser.
111
the second case the number of sets,
i,
is given. This must be done, however, to conform
\vitl~
the feasibility imposed by the resulting runner diameter
D.
Turbines with vertical
shafts are assumed.
4.2.2.
Specialization of
the
problem
4.2.2.1.
The case
of
a given runner diameter
D
and
desired
i
Frorn the assumptions made, the flow duration curve Q(t), the net head duration curve
H
(t) and the efficiency duration curve
11
(t) are given (see Fig. 4.2.1
b).
Hence the "power
duration" curve is
P(t)
=
gg
H(t) Q
(t)
q(t) kW. During the
t
=
t, days
of
a year
(measured in
hours from now on) the rated discharge required to make
a
~naximum
surplus is Q,
=
Q
(t,).
Hence and from the assumption, the rated head
H,
due to
Q,
is
j4.41
The unknown quantity is
t,.
According to the assumptions, during the period
t,
the
discharge exceeding
Q,
under H, must be spilled. During t, the turbines consume, aecord-
illg to the similarity laws (Cap. 9.2) at an arbitrary instant, the flow
I
The work done in the period
t
is W(t)
=
JP(r)dr and the associated earnings
0
K,
=
k, W(t). Then the capitalized cash value of earnings at a constant electricity rate
k,
and interest rate
p
during the
z
years of depreciation at the instant of starting energy
production is obtained as
(4.2
-
2)
with to
=
8760
annual hours and the factor
a
due
to
interest
lim
u
=
z.
P-0
The cost of the superstructure of the power house with
i
units of diameter
i)
is approxi-
mately
Kp
=
ikpk,
~,D'(H,
+
k6D)~,
(4.2
-
5)
where
k,
is the cost per unit volurne of the power house superstructure, k, the streamwise
extension of the superstructure related to
D,
Ho the height of the pit (usually of the order
of(2
to
3).
D),
k,
a
design parameter for the power house, being about
0
for an open air
plant
and reacliing about
3
for
a
tall hall design, containing the main crane in its interior,
k,
a
parameter depending on the lateral size of the block of the set related to
D
and being
k,
=
2
for a TT, k,
=
3
for a semi-spiral casing, k,
=
4
for a spiral casing,
x
the ratio of
'he
average height of power house in relation to Ho
+
k,D (height due to the crane
rullway above the turbine pit), where
<
1.
The
cost due to excavation
K,
now considered requires the volume excavated of one set
K,
==
ti,Lhs
(sec
Fig.
4.3.1
b).
7'11is
is composed of the depth
dl.,
whicl~ follows from the
figure above by
where
Dl is the runlier throat diametcr [D,
-
(D,/D2)D and D,/D, is roughly determined
by the
head and licnce thc type numberj,
k,
=
1,O to
1,5
or
2,6
to
3,2
for
a
straight or
elbow
drart tube,
11,
is the suctiotl head (Fig. 4.3.1 b),
(1,
the original depth of river bed
below the tailwiiter levcl (scc Fig. 4.3.1
G),
$
<
1 accounts for the varying depth of
excavation.
The streamwisc length of the excavation is (Fig. 4.3.1 b)
where
k,
=
5
to
6,
or
7
to
I3
for an elbow, or horizontal straight draft tube. Finally the
exctlv3tt.d volume consists of the width per set (see Fig. 4.3.l.c)
where
k,
is from (4.2-5).
The suction head in (4.2-6) depends on the
NPSH
required (Cap.
8.2)
by
h,
=
B
-
NPSH,
uhere
E
depeilds on the altitude
h
of the plant and its critical pressure
According to cap. 8.2,
NPSH
as function of the pressure number
1.
of the critical point
in the rotor
aild the draft iube efficiency
17,
for the bep (with
c,,
=
0) is
where
c,,,, is the meridiorlal speed and
u,
the peripheral rotor blade speed at station 1
(rotor throat diameter 3).
Tllc
above relation can also be used
for
the general case, where the rated poirlt
has
a
whirl
c,, past the ruzner. Then under the assumptions leading
to
(4.2- lo),
NPSH
is given by
2g;JPSH
=
i\v:
+
F,I,C;,
\I.,
and
c,
being the relative
and
absolute speed at point
1.
From the velocity triangle there, the following relations can bc obtained:
c:
=
c:,,
+
ci,,
with the whirl c,,
=
11,
-
c
,,,,
cotfl,
=
z1,(1
-
cp,cotP,),
41,
=
c,,/u,,c,,
=
Q,/(A,i)due
to the rated flow
Q,,
Dl
being the runner vane angle at station
1
and
A,
the rotor's cross
section there. Hence
2 g
iVPSH
=
(]./sin2
P,
+
r,~~)
clSll
+
qs(l
-
(P,
cot
u:
.
(4.2
-
1 1)
It can be seen that the structure of relation
(4.2- 10) corresponds to the more general case
of (4.2-
1
I),
if
(;/sin2
Dl
+
qs)
is substituted for
I.
+
qs,
and qs(l
-
41,
cot is substi-
tuted for
I..
Obviol~sly.
i,
in (4.2- 11) and (4.2-
10)
are different.
The peripheral blade speed
14,
=
(o/2)(Dl/D,)D where D2/D1 is a design-conditioned
value following from the type number
n, as a non dimensional specific speed defined
according to cap.
9.2
by
n,
=
o(Q,/i)112/(g HJ31J. Hence
w
=
n,(i/Q,)'I2
(g
H,)3"4,
11,
=
D (D,/D2)
11,
(g
HJ3I4
[i/(4
Qr)] 'I2. Inserting this,
and
c,,
from continuity
c,,
=
(4
Q/[ni(
1
-
N"])
(D,/D,)~/D~ in (4.2-
lo),
putting NPSH in
It,
=
B
-
NPSH,
and this in
d,
(4.2-
6),
thz cost due to excavation of
i
sets wit11 rotor diameter
D
becomes
with respect to submergence required and the cost per unit
volume
k,,
')
Strictiy speaking. rotor throat diameter (here and rlsewhcre) defines the largest diameter
of
the rotor exit edge
in a turbine.
Kp,
=
$
i
k,, k,
k3
DD1
{k2
D,
--
B
-.A,
+
8 (1.
+
4,)
Qf/(fj
[i(l
-
Ar2) n(Dl/D,)2]2 D4)
+
1.
TI:
9"i2
11;7i2
02.
('1
1'2)'
i/('
Qr)}
.
(4.2
-
12)
The cost of fabrication, transport and i~ssembly of the
i
units with the runner diameter
D
is assumed as
[4.6]
K,
=
ik,Dn,
(4.2-
1
3)
where
k,
is an empirical cost factor,
11
=
2.3
to
2,5.
For
i
sets with
Q,,
as unit discharge
(Cap.
9.2),
the rated discl~nrge of the station reads
Qr
=
i
Q,, D2
(g
Hr)'I2.
Hence
i
becomes
the followi~lg function
i(t,)
of the unknown time
t,
by
which also
Q,= Q(t,)
slid
H,
.=
H
(t,)
are determined
i
=
(Qr/1IfI2)
(Q,
,
D2
g112)
-
=
i
(t,)
(4.2- 14)
~nserting this in
(4.2-
5),
(4.2- 12), (4.2- 13)
the i~ldividual terms are obtained as functions
of
the unknown time
t,,
namely
K,,
=
$
k,,
kl
k3
Dl
D
{k2 Dl
-
B
-
dR
+
(8(L
+
11s)
Q:ll[n(]
-
N2)(Dl/D2)212
+'.11:
(DlID2)21(S Q11)) HrJ i(tr),
Expressing the suction head
1.1,
in
(4.2-6)
according to cap.
8.2.
by the cavitation indcx
a
as
h,
=
B
-
G
If,
K,,
(4.2-
15)
yields
K,,
=
$
k,,
kl
k3
(Dl/Dz)02 [k2 (D,/D~)D
-
B
-
dR
+
o
H,]
i(tr).
(4.2- 18)
The earnings given in
(4.2- 2)
are already expressed as functions of
t,
taking into account
that
H,
and
Q,
are also such fur~ctions according to
Q,
=
Q (t,)
and
(4.2--
1).
The surplus
of
profit reads
S
=
K,
-
K,-
Kp-
Kc,..
(4.2-
19)
Its desired maximum folIows by putting its differential coefficient with respect
to
time
t,,
correlated to the rated discharge
Q,,
to zero. Accordingly
dS(t,)/dt,
=
0.
With the simplifications:
dH,/dt,
=
li,,
dQ,/dt,
=
Q,,
then from the extremum condition,
from
(4.2-
19),
(4.2-2), (4.2- 16)
through
(4.2- 18),
the following equatioil is obtained
-
A,
(o
+
H, da/dN)
~-j,
H,
Q,,
(4.2
-
20)
where the parameters
A,
read as follows
A3
=
kpkl
k5
D2(Ho
+
k6
D)):,
-P!;r:
c;~\;ir:~tioli indcx
rr
:111tl Ilc~icc also its clcrivati:.~
tlrrjrli-i
arc kliown
;IS
functio~is
of head
fror11 T:~blc
0.2.1.
T.'or
(,4.2
-.:(I)
this
IIC;I~I
is Soi111d
by
trial :i11d
error.
7'hc ~.cl;ttion
(4.2
20)
c;111 I-c c\;~l\~a~c~l ullly
if
(lie Cunc~it)ns
I1
(1)
;11it1
Q
(1)
a~ld subscqut.ntly
I!,,
Q,,
. .
11,.
ti,.
Q,.
;IIILI
tIle11 1l1c
~I~IC.~I~;II
in
(4.2--20)
arc
k110\v1i
i111~l
:1pproxi111;11cd by polyno~ni;lls
in
1,.
111
[his casc
(-I.?--
10)
i!;
an cql~;l~io~l
I,,
tlic snli~ii~lls of \vhicIi arc (he v;llucs
making
tlic surplus
n
ri~;!.ii~~li~~il.
I'llcy c;ln
Ix
I;>uncl
by
Nc\\tton's ;~l)proxirnation. 'I'lic sc~liitio~i of in:crcst is that Jlic to
largest s~rrplus
S.
7'hc ti11;c
;,
Ic;~ds
to
thc ~.;ltcd Jisch;il.ge
Q,
=
(I([,)
of
the pl;lnt's
110~
Juriltion line
and
hcncc
by
(4.2--
I)
;tlso 11ic ~.atcJ lie;ld
li,.
The rccli~ircd numhcr of scls
i
follows from
(4.2-
14),
and ins to l1c ;~d;~ptctI
to
;in
i~itcgcr
[4.7].
For ~hc nd~~ptation of the gcncrntor's spccd
and
output
scc ch:~ptcr 11.4. Obviously the time
1,
has :o
b~:
smaller than
I,
cluc to one year.
4.2.2.2.
7'11~
cilsc
of
3
gi~cn
liumbcr
i
of
sets
and
runner
dii~rneter
D
to
be estimated
Equation
(3.3-
14) givcs
:hc diamctcr
D
as
a
function of
t,,
namely
Inserting
this
irito
(4.2
--
S),
(4.2
-
13), (4.2-
J
8),
the optimum condition
dSjtlr,
=
0
for the
surplus
after 14.2
-
19)
gives
0
=
k,,L,g'1(~rt1r-112
Q,~~,H,-'~~)~I~(c)
(.i.(~)~/~dr
0
-
i.[k,~lD"-'+ ~~(3k,,k,/<~k,%+ 3kL,,k,k2k3(D,/~2)2~I/)
+
Z1)[kl,klk5%I-Io
--
k,,
k1 k311/(E1/1D2)(B
+
dR)
+~~,r~i1kj~I/(D,/D2)al~r])
D
-
i
k,, k,
k,
+
(D,/D,)
Il2
k1,(!1, tlaldff
+
o),
nllete
d
=
LID
(tr)jdtr
and
D
=
D (t,)
depend on
t,.
4.2.3.
Example
of
the
estimation
of
runner
diameter
D
Tlie
flow
durativll
curve is
assurncd
to
be
linear,
Q
(t)
=
Q,,
--
(Q,,,
-
Qo)(t~t,)
ivith
Q,
and
(2,)
as
~nnxi!li~.im
~nd
n~ii~ii~urn
flow
rates. Hcnce with
x
=
!,/to
as thc ur~krlown figure
Q,
=
Q,,,
--
con,
-
Qo)x
9
fir
=
dl!
-
k
[Q,,
-
(Q,
-
Qo)
xI2I3.
1,:
h,lort.over
the
efficiency is assumed as constant. Hence the integral in
(4.2-20)
J
H3I2 dr
0
can
be
strictly solveci by the substitution
k
[Q,,,
-
(Q,,
-
Q,)
.rite]
2!3
=
Ah
sin2
11.
This
brings
(1s
=
--
(3
Ah3/'
t,;'[(Q,,
-
Q,) k3I2]}
sin2
u
cos
ri
du
and
1.
1,
H3I2 dr
=
-
3
dh3 to/[k312
(Q,,
-
Q,)]
j
cos4
11
sin2
11
du.
0
0
h'laking use of the abbreviations
the integral in
(4.2-20)
has
the
solution
From (4.2-28), (4.2- 29), (4.2- 30), (4.2- 31) the tr-dependent parameters in (4.2-
27)
can
be reduced to s, with
s
=
t,/t,, which is also a t,-dependent parameter, as follows
Hr
=
Ah(1
-
sq);
H,
=
-
2 dkq/(3 t,.~"~),
'
Q~
=
-
(Qn,
-
Qo)ito,
5
114
'=(2~r~r-~r"r)l[4('~llQr)"~(gHr)
I.
(4.2
-
33)
Inserting (4.2-32) and (4.2-33) in the governing relation (4.2-27) yields an equation for
s
and hence also for
t,,
whose numerical solution makes the profit
S
a maximum.
Ob\liously here the absolute maximum is desired under the side condition
tr
<
to,
accord-
ing to the assumptions. Inserting this in (4.2-26) gives the required runner
diameter
D,
which naturally has to be suitable for the facilities of machining and transpol-tation.
4.2.4.
Reasons
for
the
r~umber
of
sets relatir~g to
the
turbines
Disregarding the transportation and fabrication facilities and contrary to the introduc-
tory assumptions, now attention
is
directed to a plant with a lour number of sets
i.
With
respect to eventual repair of the turbines, the lowest number of sets is
i
=
2.
This also fits
the usual run-of-river plant with its strongly varying flow and head, in consequence of the
flow duration curve
[4.5].
More generally, in a run-of-river plant with strongly varying head and flow,
i
must lie
above a certain minimum. This results
froin the similarity laws and the type of turbine.
In the following it is assumed that all the turbines are of the same size
I).
Under a head Hop corresponding to the bep, lying between the maximum and minimum
head
Ii,,,,
and H,,, of the flow duration curve, iop sets are assumed to take a
flow
Q,,.
This lies between and Qmin of this curve. If the machine with its unit discharge
Q,,
op
hep (Cap. 9.2) is to operate with Qop at its
bep,
its runner diameter foilows from
similarity laws (Cap. 9.2.) as
D
=
{Qop/[iop Q,
,
op
(g H,,)~~'])"~
.
The k~lown rated discharge
Q,
at the rated head Hr changes, at maximum flow
Q,,,,
and
corresponding head
H,,
(from the similarity laws), to the following flow: Qr(Hn,in/13r)1'2.
It
is
assumed that the maximum number of sets installed i,,,., is provided for this case.
Contrary to Qop the sets now operate with fully opened gates (corresponding to the
Unit discharge Q,, or
Q,,
,,,). Then the flow Qop/io, passing through a turbine
at
'P
under Qop and Hop now changes into the total flow passing the
i,,,
sets:
~~p~~max/io,)(~m,n/~o,)112
(Q,,
n,a,/Ql, ..). Equating both the flow rates last mentioned
Yelds
A
ccrtnin
Ilow
and hcad dut-ation curvc has
JI,,/ll,.
=
2,
Q,!(Z,,
=
3.
Using
:I
high
specific
spccd Francis ti~rhillc
with
(I,,
,,,,/Q,,
,!,
=
1.1
tlic ];)st rcl;ttion sives
i
,,!,,x
Ii.,
-
3,k6
--
4.
The
bep
((I,,,
If,,,)
may be conncctcd w~th an efficiency curve
ts
Q
such that one set can
cover the rnngc
bctween
Q,,
aid
Qmi,.
In this case only onc sct is ~icedcd for
Q,,.
The
whole plant then rcquircs at least four sets.
For the
atse of Kaplnn turbines with
Q,,
,,,/Q,,
,,
=
2,5
the maximum number of sels
i,,,,
=
1,70
i,,.
~erc
for the same plant at least
i,,,,
=
2
sets are needed
(4.7;
4.81.
4.3.
The
optimum
size
(D)
and
number
of
sets
for
a
river
power
plant
with
given
rated
discharge
and
rated
head
4.3.1.
Introduction
to
the
probleni
and
assumptio~~s
The resulting rated discharge
Q,
and rated head
H
of
a
river power plant are assumed
known,
=.g.,
from a calculation as in the previous section. Also the load factor
cp,
the
propot tion
of
hours per annum under rated load, is known. Now thc problem arises of
fi~ding either the ~iamber of sets
i
or their size, here represented by tiic runner diameter
D.
For
this parpose
a
certain site with given topography of river bed, namely its width
b
(see
Fig.
4.3.1
a)
is considered. The widtn of spillway required, across the river, is assumed not
to constrain placement of the power house also across the river either as a part of the
dnrnming device
or
situated on the base of the dam. Subsequently the case of a forebay
dam is also de'ilt with (see Fig.
4.3.1
d).
Only reaction turbines are considered, needing some submergence and hence excavation
of
the ground. From the rated head
H,
the specific speed or the nondimensional type
t
Fig.
4.3.1. Run-of-river power
plant
(hcrc
wiih
KTs). a) Plan,
1
power house;
2
spillway;
3
dam
(if
any), b width of
the
river
valley required
for
the
plant.
b)
Elevation,
4
surface of river
bed at
the
instant of planning
(---).
C)
Plan
of
adjacent sets
(here
with semi-spiral casing).
d)
An
angular dam with fore-
bay.
a), b), c), d) to a different
scale.
is known (Cap. 9.2.) as:
n,
=
o
Q"~/(~
where
Q
=
Q,ii
is the flow of one set
and
o
its a1:gular \zelocity. Once
i
is set,
co
follows mainly f1o111
H,
)I,,,
0,.
leaving aside
the adaptation of
co
to the grid freqlrency (which is of minor iniporti~nce. whcn thz
consideration
is limited to large multipole generators) but leaving also aside the depen-
dence of the unit output on the
spccd. For lirniting generator design this dependence
follows from the restrictions on the electric machine due to the space for the conductor
and pole shoes and due to the requirements of fly wheel moment and cooling, see
Cap.
1
1.4.
Now
i
or Dl
alternatively
required are set siich ns to make thc surplus of revenues during
the depreciation period a maximum compared with the initial cost of the plant. Hence
the
initial cost of fabrication and erection of the sets and thcir accessories, due to earth
excavation (substructure of the power house), the superstructure of
potver house, the
spillway, the residual dam plus the monetary equivalent of the losses during the depreci-
ation of plant at the instant of starting power production are
tilade a minimum. The
calculation starts from the runner diameter as
a
fiinction of turbine type and workins
data.
4.3.2.
The
runner diameter
as
a function
of
working data
4.3.2.1.
Kaplan turbines (KTs)
I.
Assumptions: The flow downstream of the runner exit has zero whirl
(c,,,
=
0)
st
bep.
For an arbitrary number of sets working under
varyiilg head and flow this assunlption
requires
a
KT with at least adjustable runner vanes.
11.
Optim~im diameter with respect to efficiency: According to Cap. 10.2 after
C.
Kcllei-
P.91
D
=
Do?,
=
1,155 {(I
-
T~,)/(K~~E~H)}'!'[Q,'(~
-
N~)]~/',
(4.3-
1)
where
Q
is the rated flow,
11
the rated head,
w
the angular velocity,
N
the hub to tip
diameter ratio,
F.
the glide angle of the runner blade, KO
=
(2/3)(1
-
N3)/(1
-
I\"),
r;l,
the
draft tube efficiency.
111.
Optimum diameter with respect to
NPSH
and thereby ground excavation: According
to Cap.
10.2
and reflecting the relation D
-
Qli3
first mentioned by
Alllfors
[4.10;
4.1
I]:
where
is the pressure number of the critical rotor point with respect to cavitation. This
is also valid for FTs at D
=
Dl.
It
(nay be mentioned that contrary to expectation, especially in large low head sets with
vertical shafts
Dop,,,,, requires a larger earth excavation than Do,,(D,p,,sII
>
Do,,).
4.3.2.2.
Francis turtines (FTs)
Here the calculatioll starts from an economical optinlization of the runner tip speed
coeficient KuOp
=
L,,,/(~~H)~~~ shown in Cap.
4.4.
Hence the pzripheral blade speed
UI
=
u
(D, ID) follows from the diameter ratio
D,
ID2
=
Dl ID, which is known from
H
via
the
resulting type number
n,,
by the relation
u,
=
(D,/D,) ~u,,12
BH)'".
Continuity in the runner throat with its hub to tip diameter ratio
N,
gives the constant
aQumed mcridional speed by
c,,,
-
4
Q/[Z
D:
(1
-
N:)]. As is vlell known from
C.Pfl~iderrr
[4.12]
the runner vane anglc at the outlet at the tip section
/?,
can be
optimized
with rcspcct to minimum susccplibility to c;:\itation. Flericc tai1
op
=
{
1/[2
(11~
+
i)]
1
I:'. For tlic wl~irl-frce absolli tc
flow
assumcd
:
tan /Il
,,
=
c,
,111,.
ticnce
tlie oprin~iim ruiincr rip di;~inctcr
I)
=r
D (w1icl.c from now o:i t!~e indcu
2
is olnittcd for
YP
cotivcniclice)
Lia
D
=
(D,/D,)
I),
is obtaiiicrl as
4.3.3.
The
r111lner dianietcr
2s
a
fi~nction
of
the number
of
sets
The given rated discharge
Q,
of the plant and of one sct Q (and the number of
sets
i)
are
rel:~ted by
Q
=
Q,/i.
To limit the submergence, the type number
ti,
=
w
Q"2/(gH)31J is
mainly given by the rated head H.
I-lence according to the assumptions, the
angular velocity for
a
plant with given rated flow
Q,
and
i
sets results from
=
,,
j'
1j)3!-'0-
12
i"2
clJ
-
I
.
Inserting this in the relations (4.3-
I),
(4.3-2)
or
inserting
Q
=
Q,/i in (4.3-3) the respective throat diameter Dl (coinciding for
a
KT
with
1))
as a
function of !he
number of sets reads
where
C
reads as follows as
a
function of the turbine type or advances in design:
For a
KT,
designed on lines optimizing
rl:
C
=
C,
=
1,155
{(I
-
q,)/[K,
n,
Z(~H)'I~])
'I7
~i~~
(I
-
N~)-~~~
.
(4.3-5)
For
a
KT
or
FT
(then
N
r-
N,) designed on lines optimizing NPSH:
C
=
C,*
=
2
[2
(tls
+
A)/l,]
'I6
Q?2/[~
tlq
(gH)314 (1
-
N2)]
'I3.
(4.3-
6)
For
a
FT,
but in general also for a
KT,
designed on lines optimizing as v~ell the overall
economy
by
KII,,
and
NPSH
C
=
Cg"
=
2
[(i
+
r1$li]
'I"
{Q,(D2/D ,)3;[n (1
-
Nf)
k'li,,
(2
tlH)'i2])
'IZ
(4.3
--
7
a)
With
the unit flow Q,,, known by H, the similarity law from Cap.
9.2
gives another
C:
C,***
=
(Dl/D2) Q;;l2 Qil'
(IIH)--114.
(4.3
--
7
b)
4.3.4.
The
NPSH
as
a
function
of
the
worlting
data
and
type
Inserting in NPSII according to (4.2-10) the peripheral blade speed at the throat diame-
ter
u1
=
LL)
D1/2, the meridional speed from continuity
c,,
=
{4 Q,/[n(l
-
Ni)])i(i Df) and
using the angular velccity
m
from the type number
n,,
mainly given by H, and the rated
flow Q, of plant with
i
sets as
o
=
t~,(i/Q,)"~ (gH)314, then
NPSH
results as
a
function of
the type number
)I,
(according to turbine type), the rated flow
of
plant Q,, the number
cf
sets
i,
the throat diameter D,, the design feature N1 and the parameters
I.
and
Q,
due to
cavitational
susceptibil~ty in the form
Inserting
Dl from (4.3-4) reconfirms the well known fact that
NPSH
as
a parameter
of
cavitational sensitivity of
a
certain machine cannot depend on the numbzr of sets
i.