Raabe J. Hydro power - the design, use, and function of hydromechanical, hydraulic, and electrical еquipment
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10.2.3.
The tubular turbine (TT)
.................
406
10.2.4.
Gencral linlitations and reasons for' Kaplan turbines
.....
418
10.2.5.
The design of an axial turbine
..............
423
10.2.5.1.
The optimization of runner diameter
D
..........
423
10.2.5.2.
The "best possible" type number (specific speed)
......
430
10.2.5.3.
The velocity triangles
..................
431
10.2.5.4.
Design features of axial turbines
.............
431
10.2.6.
Rapids turbines for using kinetic energy only
........
431
10.2.6.1.
Fundamentals, design, head, discharge
...........
431
10.2.6.2.
The optimum diameter
..................
433
10.2.6.6.
The installed power
...................
433
10.2.7.
Some remarks about runner chamber and distributor
....
434
10.2.8.
Flow prediction in the vaneless space and distributor
.....
434
10.2.9.
Tidal power turbines, layout
...............
437
10.2.10.
Runner design, simple procedure
.............
443
10.3.
The project and construction of Francis turbines (FT) with hints at Pelton
...........................
turbines
(PT)
447
10.3.1.
General remarks
....................
447
10.3.2.
Comparison of Francis (FT) and Pelton (PT) turbines
....
449
10.3.3.
The limits of a FT in the lower head range
.........
466
10.3.4.
Efficiency of a
FT
as
a
function of specific speed
......
467
10.3.5.
The design of
a
Francis turbine
..............
467
10.3.6.
Design of runner vane, simplified method
.........
473
.........
10.3.7.
Simple stress calculation of a runner vane
478
10.3.8.
Simple stress calculation of the hub
............
479
10.3.9.
Derivation of the relation
(10.3-23)
............
480
10.4.
Optimization of pump-turbines in terms of efficiency and cavitation also
.....................
applicable to impeller pumps
481
......................
10.4.1.
Introduction
481
10.4.2.
Optimum outside diameter of impeller on pump-turbines in terms
......................
of efficiency
488
10.4.3.
New formula for the type number of a semi-axial centrifugal
.....................
pump impeller
494
10.4.3.1.
Optimum diameter of the impeller with respect to internal losses
494
10.4.3.1
.
1.
Impeller loss
......................
494
......................
10.4.3.1.2.
Diffuser loss
496
10.4.3.1.3.
Optimum diameter of the impeller eye in respect to inernal loss
496
10.4.3.2.
Optimum diameter of the impeller eye with respect to cavitation
497
10.4.3.3.
Optimum type number with respect to efficiency and cavitation
497
.
10.4.4.
The discharge ratio as a function of the speed ratio
.....
498
10.4.5.
Example for the con~puting optimum values of impeller diameter,
type number and discharge ratio
iis compared with correspond-
ing quantities of an actual pump-turbine
..........
499
.........................
10.4.5.1.
Data
499
........................
10.4.5.2.
Results
500
........
10.4.6.
Special operational features of pump-turbines
500
.............
10.4.7.
Sources of troubles and remedies
502
....................
.
10.4.7.1.
Normal pumping
502
..............
10.4.7.2.
Abnormal operating conditions
503
..................
10.4.7.3.
Beginning of pumping
504
XXIII
10.4.8.
Expcrin~cntal rrsearch
of
Fi-~~ncis pl~mp-t~~rl~i~lcs
by
%
.
i\lc~i
.
.
505
............
10.5. Shaft
.
bc~~rings.
accessories
01'
hydro po\\lcr scts
510
10.5.1.
Lilyout of the shaft
...................
510
....................
10.5.1.1. Gcnerr~l remarks 510
...................
10.5.1.3. Flexural vibrations
511
....................
10.5.1.3. Torsional vibrations 513
........
10.5.1.4. Excitation of shaft vibrations by runner
Xi11
514
..........
10.5.2. Intluence of bearings
011
shaft vibration 515
....................
10.5.2.1. General remarks
515
...........
10.5.2.2. Thc spring rate of the film of lubricant 515
.......
10.5.2.3. The damping coefficient of the lubricant's lilm 515
...................
10.5.2.4. Control of bearings 516
.............
10.5.3. Bearing design and arrangement 516
..................
10.5.3.1. General remarks
.
;
516
.....................
10.5.3.2. Guidc bearings 517
...............
10.5.3.3. Thrust bearing design, brake 517
...............
10.5.4. Lubricants and their cooling 521
................
10.5.5. Runaway and its problems 522
....................
20.5.5.1. General remarks 522
...........
10.5.5.2. Protection of the set against runaway 523
.............
10.6 The coi~putation of flow in
a
Francis runner 524
.....
10.6.1. Flow predicition for given runner and working data 524
......................
10.6.1.1. Introduction 524
......................
10.6.1.2. Assuinptions 524
...........
10.6.2.
List of symbols used: (Fig
.
10.6.4 to
6)
526
........
10.6.3. Computation or relative velocity distribution 526
...............
13.6.3.1. In the peripheral 9-direction 526
............
10.6.3.2. In the shroud to hub
(=
cr)
direction 528
.......
10.6.3.3. Method of solution with exact
~v(n)
distribution 529
........
10.6.3.4. Relations needed for the coefficients
A
and
B
530
......
10.6.4. Simplified relation for shroud to hub distribution 531
......
10.6.5.
The twist
of
the stream face in the runner channel
532
....
10.6.6.
The state of knowledge in predicting the relative flow 534
...............
10.7.
Computer aided
desigil of hydro turbines
537
.......................
10.7.1. Introduction 537
......................
10.7.2. General goals 537
...............
10.7.3. Hardware for
a
CAD
system 537
...............
10.7.4. Software for the
CAD
system 537
10.7.5.
ProliIe of a comercial computer aided design program for Francis
........................
turbines
538
......................
10.7.6. Conclusions 539
.............
.
11
Regr~lation
of
hydro
power
sets.
the
generator
540
...........................
11.1. Introduction 540
.........
11
.2
.
Regulation of hydro power sets. governors. accessories
542
...........
11.2.1.
The
governor. its purposes. its design 542
11
.2.1.1.
Survey of control
.
different kinds.
why
speed control
.....
542
.........
11.2.1.2. Control loop. governor. controlled system 543
......
11.2.1.3.
Working together of turbine and grid. self control
543
............
...
11.2.1.4.
The tasks of speed control
;
545
.....
11.2.2.
Design of speed and acceleration metering members
548
.........
11 22.1.
Mechanical and hydromechanical members
548
...................
11.2.2.2.
Electronic governor
548
...
11.2.3.
Simple treatment of the dynamic behaviour of governors
558
..............
11.2.3.1.
Nomenclature and assumptions
558
11.2.3.2.
Proportional governor, servomotor with stiff feedback (speed
.......................
governor)
558
11.2.3.3.
Proportional-integral governor with acceleration feedback (ac-
.............
celeration governor,
PD
governor)
560
11.2.3.4.
Proportional integral governor, servomotor with elastic retarding
..................
feedback (PI governor)
560
..............
11.2.4.
Other time parameters, tuning
560
................
11.2.5.
Special regulating devices
562
11.3.
Stability of control with respect to water hammer. autoregulation of grid. and
..................
turbine characteristics
562
...............
11.3.1.
Introduction to the problem
562
......................
11.3.2.
Assumptions
563
.........
11.3.3.
Dynamically equivalent types of governors
563
.......
11.3.4.
The controlled system (turbine, grid, penstock)
564
............
11.3.4.1.
General remarks on the electric grid
564
.........
11.3.4.2.
Linkage of turbine characteristic and grid
564
.................
11.3.4.3.
Intervention of penstock
565
...
11.3.4.4.
Boundary conditions of water hammer in a simple pipe
565
.............
11.3.4.5.
Relation for the controlled system
566
....
11.3.5.
Relation for closed control loop, stability parameters
566
.......
11.3.6.
Special
h
(q)
relations at the lower penstock end
568
........................
11.3.7.
Example
568
.....
.
11.3.8.
Theory of motion and resonance in the pipe system
569
..........
11.4.
The electric machine (generator. alternator. motor)
572
.........................
11.4.1.
Survey
572
....................
.
11.4.1 .1
General remarks
572
...............
11.4.1.2.
Output as a function of speed
573
.....................
11.4.1.3.
Water cooling
573
.............
11.4.1.4.
Pumped storage motor-generators
574
..............
11.4.1.5.
Special machines for low head
575
....................
1 1.4.1.6.
Operating regimes
575
..................
11.4.1.7.
Computerized design
575
................
11.4.2.
Alternator, electric feztures
575
........
11.4.3.
Design features with respect to critical speed
577
.............
11.4.4.
Dimensioning the alternator rotor
578
.......
11.4.5.
Alternator output as a function of working data
579
...................
11.4.6.
Cooling in practice
581
...................
11.4.7.
Rotor construction
584
11.4.8.
Stator construction
...................
589
.........................
12
.
List
of
References
593
xxv
.......
13
.
List
of
further
Litcrnturc
in
olpl~abctic
ortlcr'of
the
auil~ors
639
...
13.1.
Suyplctncnt
of
ti~rtlic
r.
Literature
in
alp11;lbctic
order
of
the
authors
635
...........................
14
.
Subject
index
641
..........................
15
.
Author
indcx
675
Nomenclature
a)
Romnn
letters
celerity
(=
velocity of sound); thermal diffusivity; distance, distance between
adjacent towers; direction from shroud to hub along the rotor vane. and along
lines of approximately constant moment of momentum.
projection of the a-line
in the meridian
revenue factor.
capital recovery factor
=
l/a.
constant, e.g., due to the bound vortex distribution
y,.,
surface area, cross' sectional area.
real wetted cross sectional area of channel.
cross sectional area of a rotor vane's intersection with an axisymmetric
floiir
plane.
projection of
A,
onto a plane normal to the rotor axis.
alternating current.
cost term due to dam section
(4.3
-
16).
b
width (i.e. of river valley); span (width) of a blade (vane); depth of
flow
layer
or elementary turbine; minimum distance of conductor from
pound; accel-
eration; rotor breadth.
Ab
axial thickness of shroud and crown (hub) at external diameter.
b~
length of power house normal to flow direction.
b,
width of excavated ground volume per set.
B
barometric head [IA-critical head
h,,
(due to cavitation);
constant,
i.e. due to the bound vortex distribution
y,.
c
absolute velocity; specific heat.
C,,
meridional component of
c.
Cu
whirl component of
c.
Ca
axial component of
c.
C
constant, due to the bound vortex distribution
;,,;
gas concentration of solved
gas (kg/m3), iron utilization factor of alternator (Esson number).
d
damping coefficient; diameter of pipe; diameter of conductor
diameter.
do
jet diameter (impulse turbine).
d,
depth of excavated ground volume.
4
original depth of river bed below minimum tailwater level.
D
Nominal rotor diameter (reaction machines: outmost diameter
of
rotor
passage;
i~npulse turbines: jet circle diameter); mass diffusivity; diameter of
vortex tube.
Do~q
optimum diameter
D
with respect to efficiency of axial turbine.
DopNpsH
optimun~ diameter
D
with respect to
NPSH
of axial turbine.
D,,,,,,,,
opti~num dian~ctcr of rotor eyc (throat diamctcs)
D,,,
with rcspcct to cflici~~,cy
;II
scn1iasi;il or
rilciirll
pump-turbine.
D,,,,,,
,,,,,y,,
optinitlni tli;~nictur of rotor cyc (throat dianlctcr)
L),,
with rcspcct to
NPSti
at
scrni;l.ui:rl or ratiial pi~nip-turbinc.
DC
direct cvrrcnt.
e
internal energy; distance froni neutr~tl axis in
a
certain cross
sectional
area.
E
Youn~'s mod~llus (modulus of elasticity), internal
effective
vo1t;ige of alter-
nator; cost V~ctor duc to
excavation
(4.3-
12).
E(ni2,
k)
complete elliptic integral of the first order
E!-
bulk modulus.
E
energy flux.
E
11
Euler (pressure) number.
f
deflection of the shaft centre line; 1ir.e frequency; spacing of opposite rotor
shroud (hub) and casing wall.
fxyf,
influence functions of velocity induced by a whirl row.
F(ni2,
k)
co~npletc elliptic integral of the second order.
Fr
Froude number.
Y
acceleration due to gravity; total pressure.
G
cnst term due to darn section (4.3-17).
h
altitude; vane thickness; elementary length along the contour of
a
profile;
ord~r of harmonics; number of days per year;
=
A
NIH
nondimensiofial small
fluctuations of pressure head.
11'
nor~diniensional loss.
lls
suction head.
hi
insulator !ength.
3
It
drop in river level; backwater effect.
H
head; delivery head.
i
imaginary unit; enthalpy; number of sets; number of wheels; number of stages;
runaway speed
to rated speed ratio.
I
investment; electric (effective) current; cost due to fabrication and assembly of
sets (4.3
-
18);
elliptic integrals.
i
imaginary unit.
J
nioment of inertia due to cross sectional area; cost factor due to cash value of
loss
during useful life (4.3-22).
design bound stress concentration factor
(1
1.4-
12).
critical to undisturbed speed ratio; design bound parameter: due to start up
time of set (1 1.4- 15); excavated width per set to diameter
D
ratio.
streamwise excavated length to diameter
D
ratio.
cost factor due to accessories of one set.
cost per unit volume of power house superstructure.
cost per unit volume of excavation.
electricity rate.
roughness.
cost factor; gas constant.
=
iic,i(dM)
penstock parameter due to wter hammer, cost term.
cost of
dam.
=
(yV/Q
,
,)
(2
Q
,
,/an),
for turbine.
nb
"s
n4
nb
q
NFSIl
"1
1
N, N,
NPSH
=
(mh,/Q
I)
(ZQ,
I,53m)n for turbine.
=
(n,/rl) (aq,lan),, for turbine.
=
(m,/tl)
(atl/c?n~), for turbine.
=
1
+
u
+
K,,
+
KQn self regulation parameter of grid and turbine.
=
ci/
J2
g
H
velocity coefficient, by which velocity
ci
is referred to the spouting
velocity under the head.
lever of force; distance of critical point from stagnation point.
chord length; pipe length; conductor length; streamwise length of excavation;
loss parameter
(4.3
-
2
1).
mass; dimensionless deviation of servomotor piston Amlm,,; Poisson's ratio
(0,3).
servomotor-piston stroke.
torque; moment.
,
.
scale of length and velocity under conformal mapping.
rotational speed (rpm) or (rps); integer number; exponent of cost term:
direction normal to meridional streamline in the meridian; direction normal
to arbitrary surface, polytropic exponent.
nondimensional specific speed
(type number)
=
COQ'/~(~H)-~"~.
specific speed
=
n(rpm) P(kW)'12
H(m)-514.
specific speed
=
n(rpm) Q(m3/s)'I2
H
(m)-
314.
best type number of axial turbine or semiaxial pump-turbine NPSN.
unit speed n(rpm) D(m)
H
(m)-
'I2.
hub to tip ratio in axial turbines or in the throat of semi axial machine.
net positive suction head.
.
c
pressure; interest rate.
.
+
atmospheric pressure.
impact pressure due to bubble implosion.
number of pole pairs.
power (input, output).
.
.
internal output (input).
.
.
peripheral output (input).
shaft output (input).
gross output, net input
QQ~
H.
interest factor
1
+
p;
dimensionless flow fluctuation AQ/QtV; flow ratio of
pump turbine
=
Q,/Q,; Weinig factor (n/2) (Llt)
x
sin
Q,.
flow (discharge).
flow at
bep.
:
-(
unit flow Q(m3/s) D(m)-* H(m)-
'I2.
radius, distance from axis.
bubble radius; distance between space points; ohmic resistance
radius of curvature of meridional streamline.
radius of curvature of streamline in an axisymmetric stream plane.
Reynolds number.
streamwise length; pipe wall thickness; entropy.
surplus; cost factor due to fabrication and erection of set
(4.3-
19).
Strouhal number.
I
,
,
1.
tinic elapsed; time pcriod; pi:ch;
dcptli
of
flow
I;~ycr
(7.3-71).
peak
ldud
pcriotl
(4.5-
1
I):
iihsolutc tctnper;lture.
travel time or prcssurc pu:se along penstock.
reflection
time of prcssurc pulse in penstock.
resct (isodrome) time of governor.
rate (accclernlion) timc of
governor.
start up time of sets.
start up
time of penstock.
opening timc of distributor.
closing
tilnc of distributor.
-
ro
blade speed; voltage drop along a line
A
U/U;
exponent of generator
torque
duc to certain kinds of grid load
(11.3-7);
blade specd at rotor
diameter
D.
voltage; voltage at generator terminals (eflkctive
v.).
undisturbed velocity outside of wake.
wetted circumference of real channel.
specific volume; velocity
conlponent normal to wall.
particie velocity normal to fringes due to laser measurement.
volume.
volume due to seepage, evaporation, precipitation.
live storage volume.
relative velocity (in the rotating frame of reference); velocity component
ill
secondary flow direction; weight of conductor.
velocity induced
by the deficit of relative eddy within the interior of the vane
of a
mixed llow rotor.
internal friction; annual work; sectional modulus.
coordinate in
main flow direction; unknown variable: coordinatc in the
direction of
the undisturbed velocity from
the
~nidcliord; dummy variable;
nondirntnsional speed deviation
Anln,,;
rnericlional coordinate.
coordinate of point considered.
mesidional coordinate in axisymmetric flow plane.
coordinate normal to wall; coordinate in peripheral direction of cascade;
coordinate
normal to undisturbed .flow; sagging coordinate of high voltage
transmission
line.
lensth of critical zone referred to
I.
specific head (delivery head)
gti.
mechanical flow energy
per
unit mass
=
p,'~
+
c2/2
+
gh.
mecharliial rothalpy per unit mas5
=
p/p
+
\v2/2
-
u2/2
+
gh.
coordinate in the
direction
of secondary flow; coordinate in the direction of
rotor axis; number of years of
useful life (depreciation), Gaussian colnplex
coordinate of point considered on the contour of profile, at which the
kinematic boundary condition is satisfied;
vatle number; number of stages in
a labyrinth.
h)
Greek
letters
-
-
'l
'li
'l"
'lsh
'ls
9
9i
911
0
acute angle between absolute velocity and periphery; interest kictor during
useful life
[(I
+
p)'
-
l]/[p(l
+
p)']; stagger angle of cilscrtde
=
n/2
-
P;
heat
transfer coefficient.
guide vane (gate) position angle.
acute angle between relative velocity and circumference.
acute angle between pattern making vane section and circumference.
acute angle between undisturbed flow and circumference.
acute angle between zero lift direction and circumference.
longitudinal dihedral angle of vane; strength of bound vortex, continuously
distributed along the contour of the profile, and referred to unit length of
contoor.
circulation around a profile or a vortex tube; cost ratio
AK,/K,
due to
pumping of tidal power plant (10.2-34).
angle of attack between chord and undisturbed flow; boundary layer
thickness; parameter of rotor diameter (4.4-3).
physical angle of attack between zero lift direction and undisturbed velocity.
zero angle of attack between chord and zero lift direction.
boundary layer parameter.
permanent speed droop, proportional band.
temporary speed droop, transient proportional band.
difference;
Laplace operator.
angle of glide; angle
betiveer] resultant velocity close to the wall aria nlain flow
direction in consequence of secondary flow; liquid property during cavitation
1
-
Q,/Q~;
orientation angle of a vortex tube within a duct
(7.2-9).
complex contour coordinate.
loss coefficient due to
i;
lift coefficient
(i
=
A).
efficiency; dynamic viscosity; dummy variable in
y
direction.
internal efficiency.
circumferential (peripheral) efficiency.
shaft (coupling) efficiency.
draft tube (diffuser)
eficiency.
angle of
a'
line and radius in the meridian.
angle between skeleton and chord at Birnbaum point
xi.
momentum thickness of the boundary layer.
angle of radial vane section to radius; rotor's moment of inertia.
isentropic
coeflicient
=
c,/c, (c,
and
c,
being the specific heats at constant
pressure and volume respectively); cascade coefficient
=
1
iaoG
=
boE;
speed ratio:
n,/n,
of pump-turbine with pole changing alternator.
pressure number of critical point.
wave length of laser beam.
latent heat; breaking length.
angle of meridional streamline to radius; exponent of (8.2-4).
kinematic viscosity; angle of a-direction to circumference.
frequency of scattered light of laser beam.
0,
p;
~lc~i\i[y
01.
\\
~IICI-,
clct~sity
of
III:I~CI
ii11
or 111ccli11111
i.
0
I)
spcciljc
I~C\I.;[;II~CC
of co~~<li~~.tor.
(T
surhce tctision
iri
licluicl-gas i~iterfilcc; ci~vitation indux.
Gud
admissible
tensile stress.
ci
sr
rchs in pl;~lic
\i'itli
normnl
in
i-dircctio~l. ;lctillg in j-direction.
T-
u
sum.
T
tin~c pcriod. slicur
st
rcss.
cP
:~zirnur
h:
Ilo~v
cocKificitlnt
=
~.,,,,r.r!):
load
f:~ctor of
plant.
cos
$9
i~i>v.cr
fxt
o
r.
@
coijtr-action cnclTicicnt:
potential,
cscnvritcd cross section of penstock to
un-
oh\iiLrcrtd
scction;
-
I//
Ii,
k,
(scc
excavation).
'I
/.
11e:~ti ratio
If;,
11,
:
angle of
(I
to 11-direction: rotor pararrleter
(3.4-8).
'P
prt-,surl: coelficient
-
2[g11/1r';
stre;~nifunctinn; excavation
dzpth
parameter.
o
nngi11,tr
\
clocily.
0
--
form yar~lnister
of
rotor
(
10.4 13) or
(4.4--5).
Q
,
\.oriici~
ii~
i-direction.
1
iou
prc\bilrC
e(ige
of rotor
vane.
2
high
prtss~~~e
ccl;c
of
rotor vano.
3
lo\l;
P~CSF?I!-~
cc12c
01'
diffuser
(guidc) vane.
1.
1
fllli
Io:1ct.
+
(
-
)
on
posili~
c
(i~c~;~ti\~c) cb;~s;t~tcristic
iil
!hi
x/t)
plane
for
::!I
3bst.l-~er,
moving
with
11
+
c(-
ir
+
(.)
ill
.Y-direction.
A
lift:
d11e
10
bou~id vort:.~ distribution
;:,.
LI
actii
e
c;~>c;~de;
;l~iill
direction; e.uternal: alternator.
r7d
adrnisjib!~. calue.
/)
adrni:siblc i.ctriable depth of flow
layer:
bending.
B
adinisbible
bound
vortcs ile~isity
7,
admissible I?ount! vortex density
7,.
critic;~l ~cai-i tat ion): critical (speed).
diffuser.
dr;~g.
tiiffuscr. ~x~criciinna! vcluci ty.
diKuser.
lvhirl velocity.
disk friction.
rt.latin_r to clcctricity rate.
I-el at in^
to
excavation.
rcl;lting
to
singlc profile.
relaiins to profile in cascade.
.
<;IS.
L
hydraulic. relatins to diffuser, rotor, guide apparatus
and
suction
pipe.
inlet.