Raabe J. Hydro power - the design, use, and function of hydromechanical, hydraulic, and electrical еquipment

$th a pronounced

q

peak at bep, whereas a

KT

especially the double

regulated

one,

b-

afl

atter curve around the bep. Thereby the KT

can

be

adapted better

to

a

changing

&rnand of power. This item must be put aside in

a

plant with a large nurnbcr

of

se~

~~~thermore the

KT

operating with constant speed

can

be

adapted much

better

than

a

FT

to head variations at nearly constant eficiency. Further the runner of

a

FT

witb

n,

'

350 is axially too elastic and is mechanically not suitable for elevated heads,

whereas

a

KT

runner of this n,-range is no problem. Finally the level of bep efidency

of

a

FT

Nnner with highest specific speed n,

2

350 of this type is rather low.

Examples of large and medium FTs in this range of

n,

are shown in

Fig.

10.3.8

and

~jg.

10.3.17.

10.3.4.

Efficiency of

a

F'T

as

a

function of

specific

speed

After a systematic survey carried out on 340 FTs, Zanobetti found the efficiency

at

bep

to

depend as follows on the specific speed n, in the range 145

<

n,

<

340 [10.34]:

For

qopm

see (10.3- 1). Hence for a given head the designer should tend to set

n

and

P,,,

x,

as to obtain n,

=

145. At full load with an efficiency drop

Aq

of about 2% the efficiency

according to Zanobetti [10.34] becomes

Fhlally the same author found the efliciency at 50% of full-load output

Po,,

,,,

to

be

expressed by

'loss

=

op

-

0,0521 (~1,1145

-

(10.3-4)

In

the above

qopm

=

0,935 (0,925) and

70,s

,,

=

qopm

-

0,04. The relations demon-

strate clearly that the efficiencies

qop;

ll,,l;

qOs5

decrease when n, increases above

145.

Note

that these values apply to very large units. Hence for

a

FT with

n,

=

420:

qo,

=

0,92;

ql,l

=

0,87;

=

0,72. This figure makes clear the limits of

n,

due to a

FT.

A

rather bad feature of a FT, also occurring on a

KT

with fixed runner vanes, is the

pressure surge in the draft tube, when the machine operates away from the bep (Cap. 5.5,

8.2

and 9.8) and

if

its runner is of the overhung design.

Contrary to a KT the FT has a more moderate submergence in consequence of its

smal!er

cavitation index (see Table 9.2.1).

Also the runaway speed the set has to withstand, which increases with rising n, from

1,4

to 2 times the normal speed, is considerably smal!er for a FT than for a KT. This appears

in

the dangerous case of off-cam operation at low heads up to 3,5 times the normal speed

[8.132].

10.3.5.

The

design of

a

Francis turbine

The design of a

FT

starts from H,

Q

given the topography and the hydrograph. The speed

fl

must be compatible with the grid frequency and the integer number of pole pairs of the

alternator but set,

if

possible, so as to get a specific speed n,

=

145, where

n,

=

"('prn)

(Pill

(kW))'12/H

(m)5'4.

As outlined in Cap.

9.2,

for reasons of limited submergence

and runner strength, a certain n, is assigned to

H,

a relation which corresponds to the

Fig.

10.3.17. Elevation of the

FT

at Hartwell, Ohio,

USA,

built by Voest Alpine, Linz, Austria.

H

.=

51,s

m;

n

=

112,5

rpm;

P

=

SO

MW.

Runner

D

=

4,877

m. Simplified design of the runner hub.

Bell-shaped guide bearing, machined from shaft, submerged in the oil

pan

to facilitate lubrication

whilst starting up. Parallel plate spiral casing, head cover and bottom ring split into halves for

transportation, bolted together on the site. Gate lever with

a

shear pin as breaking device, if a foreign

body is squeezed between adjacent gates. (Drawing courtesy Voest-Alpine,

Lin~ Austria.)

te

of the art, which in the future will move to assign higher

n,(ns)

values to a certain

sf=

wd-

gctdng

ns

above the actual limit for

H,

means firstly higher submergcllcc and hence

aovation cost. Then the runner is more highly stressed, so as to require

a

stronger

wtcrial or to need

a

more careful stress calculntion

(10.521,

e.g., by finite element method

1391.

This holds especially for larger and hence more elastic units as seen in

[lo'

10.3.17

and

10.3.8.

Fig

rab1e

10.3.1.

Empirical values for a Francis turbine runner as a function of the specific

n

-~Q~/~H-~/~ n

Ipeed

9-

,

rpm,

Q

m3/s,

H

m.

1.

rpm

17

29 4 3 57 70 85 100

-

mxH

m

700

Qll

111

m3/s

0,123

81

1

rpm

61

maxn,,

rpm

105

QII

JQII

111

0,8

1

li

L

0,88

0,845

2

y2

0,024

#

0,045

D,

ilD

1

,o

DJD

1,04

DS1D

0,6

*dD

0,055

Ii'D

0,36

yo

0,27

qil

efficiencies

for model runners of

0,3

to

0,6

m

0

under

a

head

of

5

to 60

m

For

;I

gibsn spe~ific spced

11,

or

:I,,.

tl~c r,ited unit dischilrgt' Q,,

:lnd

~iilit

speed n,,

nr

obtilincd fl-orn Tablc

10.3.1.

Ilcncc,

rhc

runncr tlirln~ctcr altcrnntivcly (for control)

!\s

;I

third control thc

blade

tip speed coefficient Ka may also bc @ptirnizctl with respect

to

il

n~axiniuin surplus during the useful life of the turbine, following the lincs of Cap.

4

-

.

Oiicc ri(rpm) and

=

7t

11/30 arc known, thcn

Y

0nc'c'~ is known aild the figures D2JD, D,/D. bJD, IJD,

li/D,

D,/D are taken

from

Table 10.3.1 as a function of

n,,

the elevation of the runner, inclusive of the gate's outlet

."

edee

C

and draft tube inlet may be plotted, aided

by

the contours of thc runner given in

the

above table as

it

function of

r?,.

(For qlI1

=

0,9:

11,

=

2,97 n,.)

It

must

be

emphasized that the contour is more or less set at will and reflects the

experience of

a

special maker.

In the following some design features

(as

a function of H or n,) may be mentioned, found

in the survey of modern trends in the design of FTs by

de Siervo

and

cle

Leva

[10.33]

n,

=

3470 H-~.~~~,

KU

=

0,323

+

2,2

rt,.

(10.3-7)

*Thc

outer diamtter of the runner outlet edge Dl, follows from the rated output

P,,,

and

a

coefficient usu;~lly referred to as the coefficient of the largest runner outlet diameter

C,,

a

figure Ivhich

for

machines above

i-\

r~,

about

200

corresponds

to

what

is

called the runner

throat

coeficient

Dla

=

Cl (P,~~)~~~. (10.3-8)

This

rule folloivs t!le experience that the flow through a device or machine under

a

certain

head is l;sually dictated by the size of its narrowest passage. C, may

be

!raced to the

similarit::

laws

(Cap.

9.2).

according to which, wit11

D

=

D,,

Dl,

=

Dl,

the speed may be

expressed by

t~

=

C

Ku(D,,'D,)

II'/~/D,

with

C

as

n

dimension-l~nked constant. Elimi-

nating

tz

from this by means of the definition of

n,

=.

n(P,ll)112/H51" gives

C,

=

C

Ku(D,/D,)/(n, H3I4).

(10.3-9)

Since the values

Kli,

n,,

!i

and Dl jD, depend on n,, C, also is

a

function of

n,.

According

ta

Znnob?tti's

review [10.34]

C,

=

0,075

+

O,S55

.

n,. (10.3- 10)

For the output, according to Cap.

9.2:

P

=

PI,

D2 H3I2. Hence by logarithmic differen-

tiation and putting

tlP

=

P

-

PI/,

and

dH

=

H

-

IIlI,:

PIP,,

=

-

0,5

+

1,5 H/HIIl.

Af-

ter

ZanoSrtti

for the machines reviewed

P,'Plll

=

-

0,433

+

1,44 H/Hlil.

(10.3- 11)

The span

of

the gate depends according to

de

Sierro

[10.33] on

11,

as

b,iD,

=

0,188

+

0,5

-

10-3n,.

(10.3-12)

The axial

depth

'a'

of runner (Fig. 10.3.18) as the distance

of

the outmost point on the

runner outlet edge from the distributor's centre line, according to

de Siervo

and

de

Leva,

is given by [10.33]

a/D,

=

-

0,05

+

42/n,,

(50

<

n,

<

110)

a/Dl=1/(3,16-0,0013nS), (110<nS<350).

.

(10.3-

13)

10.3.18.

Elevation of set with the

figimeterr to calculate earth excavation,

and

pa'

cbaracteristi~ relations for the lay-out of a

FT.

fie

distance of the centre lines due to adjacent sets is after

[10.34]

in relation to

Dl

Hence

AID,

=

5,06

and

3,75,

at

n,

=

100

and

300.

The submergence (or the suction head

h,)

of the runner outlet follows by the well-known

nlation of Cap.

8.3

by means of a cavitation index from Table

10.3.

From

de

Siervo

[10.33]

the cavitation index

a

reads as a function of

ns

This approximates the values recommended by

D.

Thoma.

-

Draft tube: Depth of draft tube bottom below the runner outlet (Fig.

10.3.18),

from

de

Siervo

[10.33]

The same author recommends for the horizontal length

h

of an elbow draft tube

(Fig.

10.3.18)

taken from its centre line

hlD,

=

ns/(-

9,28

+

0,25

n,).

(10.3- 17)

Depth

of

draft tube bottom below the deepest point of the runner outlet edge

2,5

to

3

Dl.

Length of draft tube cone

0,8 Dl

.

Horizontal distance of bend outlet from the turbine axis

about

1,35

Dl

(Fig.

10.3.18).

IIeight of bend outlet section

0,6

D,.

Radius of curvature of

the bend's centre line

1,4 Dl.

-

Spiral casing: Mean velocity in the spiral intake at full load flow

Q,,,

for

n,

=

60

to

350

[8.132]:

c,,

,,,

=

(0,15

to

0,25)(2 gH)1'2.

Hence inlet diameter

D,,

of spiral casing by continuity. Internal diameter of spiral casing

at

stay vane inlet for a non-parallel plate stayring:

D,,

=

(1,4

to

2,4)

D,,,

at

n,

=

77

to

240.

The

last value is subjected to large changes. This follows from the ratio of stay vane

number to gate vane number and from the gate vane

lumber

itself. For large machines

both vane numbers are equal. Moreover the gate number increases from say

8

for very

small units to

28

for very large ones.

-

3

1:inally thc dii~metcr is inflncnccd by

the

hct wliet!~cr the spiral

ciising

is

of

tllc

pnralld

pl;ltc

dcsign nficr 1'ig:lct

(Fig.

10.3.4)

13.1281

or the

sl;~ssic:ll

dcsigo (Fig.

10.3.5).

The

latter

is

rno1.e

sp;lce

consuminp.

and

strcsscs the stay va~ics, its crown

;lad

bottom

cover

more

than

in

the first design but

if

offers

a

more smooth intake for the stay vanes.

As

a

gcncral rulc

attention

has to be paid to the strcngth of the spiral, stay vane rings and stay van

CQ,

hcad and bottom cover and distributor vanes under the following rclcv;ult lond9 especially devcl-

oped at closed ptes: Water load ur.dcr test pressure, which is ust~aliy 1,5 times the highest working

prcssu'rc composed of thc static prcssurc plus the highcst adn~issiblc prcssurc surge due to

hammer.

b

Load of the thrust bearing, whcn this is supported by a truncated cone on the head cover. Membrane

stress from the shell of the spiral casing usually calculated so as to assume the top most

quasi

cylindrical section through thc spiral casing wall as free of shear parallel tc the machine axis. Besides

the junction bctwcen spiral casing wall and thc stay vane rings (usually casings) also transfers

bending moments The strength of the stay vane ring is the core of a FT design.

By adopting

Piguet's contracted spiral casing with parallcl platcs of the stay vane ring, the ca1cula.

tion bccomcs more reliable and transparent, since the stay vanes are then of constant span over their

streamwise length, and are attacked more centrally by the membrane pu!l from the spiral casing wall.

'The bending moment in the junction of a spiral casing is a conscquence of t'nc closure of an imaginary

gap that would cxist if the load-induccd deformations of the stay ring and spiral casing were

consitiercd indepcndcnt of each other. This gap would exist alotlg the circular junction of stay ring

and spiral casing. In rcality it is closed mainly by the action of an internal bending moment around

a tangent of this circle. This causes a bending stress in the meridional direction, which has to

be

superimposed on the meridional melnbran stress due to internal pressure of the spiral casing. In a

wclded design, for reasons of fr~brication, this highly btrcssed junction is

implemented

by

a

weld seam.

Manhole and pedestal of the spiral casing, or

the concretc propping it, must not overstress it.

Moreover

thcre are fatigue problems in the spiral casing. the stay rings, and the stay vanes of sets

w!lich often change their mode of operation [10.124 to 10.1311.

The stress calculation of the

gate and its drive in closed position under squeezing adjacent gates

necds ths same care tv~th respect to strength [10.59; 10.89; 10.1221. Here thc "submergence" of the

gate

dribe (see Fips. 10.3.1 or 10.4.17) into the hcad covcr

(if

possible) avoids jamming of the gate stem

by

defor~nation of head cover. enable5 statically dctcrrnined support of thc gates (two bearing

arrangement) and

eliminates

the fatigue problem at the tramition between stem and blade of the

gate.

Thc latter follows from the fact that in a two bcaring arrangement, this section with its strongly

varying

sectiona! modulus is not subjected to axial bendins stresses (acting parallcl to the stem),

ivl~ose flux would there experience a strong contraction connected with notch effects i2.231. However

such a design is restricted to heads below 450

nl (the FTs of Grand Coulee

111

built by Allis Chalmers

can be

considered

to be an exemplary constrt~ction of this type). (See Cap. 2.3.)

The

head cover of huge and high head machines must be stiff so as to avoid deformation causing

fouling of

thc runner within the labyrinths and exciting runner vibrations by variation of the

labyrinth's gap clearance

[8.132].

The circumferential development of the

cross

sectional

area

of

the

spiral

casing tube

follows

from

the intention to distribute the flow to the

runner

uniformly over the circum-

ference, with

an

angular momentum near to that required

by

the runner.

To reduce the

loss in the case of highest heads, usually the angular monlentum within the spiral tube

is smaller

than required

by

the runner. To save space in the case of low-he:~d plants with scmi-spiral

casings, the angular

nlomentum in the duct of the latter is larger than required. Because of the

relatively

smal! kinetic energy in the ducts of the guide apparatus there, this does not influence the

eficiency. At any rate, the change to angular momenturn required is effected by the stay and guide

vanes.

Dimensioning of the

cross

sections

is

based

on

the assumption that at

least

in

the first

three quarters

of

the circumference

a

velocity distribution

is

generated

that

corresponds

to

that of a free vortex. Hence the distance

e

of the outmost spiral wall from the machine

depends as follows on the azimuthal distance

cp

from the radius due to the inlet

section of the spiral case near its tongue: cp(e/D,,) 90"

(13..

.2,5);

180" (1,7.. .2,3);

270"

(1,5

.

.1,9); 360" (1,3..

.

1,4).

For

heads below about 30 m also semi-spiral casings are applied. Inlet width:

2,9

-

D.

Eccentricity: related to the axis of the turbine: 0,2

D.

Maximum height: 1,4

-

D.

Maxi-

mum

lateral distance of the spiral chamber wall from the axis: 1,4

-

D,

[10.147].

10.3.6.

Design

of

the

runner

vane, simplified

method

1)

~ivision of the runner vane channel into

i

elementary turbines

(i

=

2

for small

n,,

i

=

5

for

high n,, according to Fig. 10.3.19), having the same flow

AQ

=

Q,,,!i.

QSp

follows from

the

ratio

Qll

.,/Ql1

,,,

in Table 10.3.1. Continuity for an elementary turbine, local span

~b,

local distance from axis

r,

local meridional velocity

c,,

local contraction coefficient

@

(due

to vane and boundary layer), requires

AQ

G=

2nAbrc,@

=

const.

(10.3- 18)

The

procedure is best satisfied along a normal to the preliminarily set meridional

streamlines in the n-direction. Strictly speaking introduction

of

c,

requires knowledge

of

the

c,

distribution, Cap. 10.6 or 6.4.

A

simplified mean value of

c,,

follows from

Fig.

10.3.19.

Elevation

and

plan

of

an

arbitrary

FT

(hcre

with

n,

=

100)

with ch-nentary turbines,

mdial

and pattern sections and angles.

473

c,,

-

Q,,p

!?

ii:

I.,,

h

+),

\Y~L'I-C

I)

is the local brc:ldth of rdtor,

is,,

tile mean 11ist;rnce of this

t~rc:rdt!l Croln

:I\;,.

Tlic~!

(.,,

-

<

c,,,,,f.

hlore rigorously

is

ob~itined from thc potelltial

How

p;i!

tcrn

in

I

he meridt;~n, !:unccl by

thc

~raphical method after Cap.

5.2;

ro~g'illy with

11,

=

60.

.3

10.

on

the 41roucl

<

=

1

.

1,S

ancl on the hub

5:

-

1

. .

.0,5.

The contraction

cocfiicicn

t

~1)

1nay

bc

prclilr1;narily

estimated

between

0,9

and

0,7.

Strictly speaking

(b

depends on the real vane thickness

It,

the anglc

/I'

between the pilttcrn

making vane sectior, and tl?e periphery, thc angle

O

between the radial vane section and

radius, the r:ttiius

r

and thc vane numbcr

z

:IS

follows

[8.132]

L

(b

=

1

-

[It

2/(2

n

r)]

(1

+

cot /I'/sin2 @)'I2

.

(10.3-

19)

~trictl~ spcilking

It

contains also the displacement thickness of

the

boundary layers

on

both the faces. Ro~~ghly

h

corresponds to the vane thickness resulting from the stress

calculatior! given later on.

In general

the vane nurnber

z

is odd to avoid excitation of torsionai vibratioils at

the

instant the rotor vane cuts

a

wake from the gate, whose number is in general even.

The

numbcr of runner vanes

may

vary from

z

=

7

to 17 for hcads of

20

to 700 m.

,

2)

Ass~~rnption of ttic strca~nwise cl~ange of aiigular mornenturn: After tests on Francis

turbines

by

Uiir

[5.17]

a!ld

Schler)~nzer

[5.18]

(Figs. 10.3.20 and 10.3.21),

earl-ied

out at

Lehrstuhl und Lab~r fiir Flydraulische Maschinen und Anlagen

TU

Miinchen, West

Germany, in a streamwise directio~~ at

a

working point near the bep. the angular momen-

Fig.

10.3.20.

Test results, show-

ing

thc

graph of the'absolute

flow

cnergy

Y,

and rothalpy

Y,,

both related to specific head

gH

vs

unrolled lllcridional stream

i

f

liiw length

s

for various elemen-

tary

tur'oincs of

a

FT

with

n,

=

70

at bep according to

the

hill

diagram. of

Fig.

10.3.21.

B

elementary

turbine close to

the shroud;

F

ditto close to the

hub.

Plccordingly lines

C,

D,

E

belonging to the mean meridi-

onal

streamlines

of

the

elemen-

tary

turbincs from thz shroud lo

the hub. Frorn

E.

Bar:

Messung

des relativen Strijn~ungsfeldes

in

Wasser an der ~aufschaufel

0

eincr schnelliiufigcn Francistut-

o

bine. Forschung Ing. Wesen

36

coordinate

s

(I

970),

no

2.

p.

54/61.

.

1;o

1.2

nll/"llopt

Fig.

10.3.21.

Hill

diagram of a

FT

with

n,

=

70

in the

bep,

obtained

by

tests on a model with

D

=

0,455

m

runner diameter.

With

relative velocity

w

and angle

j3

with

circumference

at runner inlet

at

different operating points according to the measurements of

Schlemmer,

in a mean flow plane

(5.181.

Beginning and increase of surging in the runner indicated by arrows

A:

suction face, shroud;

B:

suction face, hub;

C:

suction' face, shroud unsteady flow, pressure face shroud, and hub;

E:

suction face, shroud.

turn drops nearly linearly from the leading inlet edge (there having a magnit~de

cU2

r2

=

(gH qu

-

c,

rl

o)/o)

towards the value

c,,

r, at the outlet edge of the rotor. The

above tests also show the derivative of angular momentum along a normal

n

to the

meridional streamlines

(d(c,r)/an) to be vanishing small at the bep and away from the

leading edge of the runner vane, see below.

For the case of

the bep

c,,

=

0.

Hence the whirl at a certain station with the radius r,

where AY,(S,~) is an arbitrary steadily falling function vs the meridional stream path s,,

measured from the inlet edge and being gH quo, at the outlet.

Such an arbitrary drop of the flow

energy

Yo

within the runner in the streamwise direction

an be set only along one elementary turbine. The relation (6.5-4) for the distribution of

he

meiidional velocity as a function of the distribution of angular momentum normal

0

the streamlines shows that only in the case of

a

loss free flow and

a

(c,

r)lan

=

0,

the

iistribution of the meridional velocity in the n-direction (normal to the meridional

Jreamlines) becomes independent of the cur distribution normal to the streamlines.

\ssuming heace this sinlple case, then two things have to be satisfied:

1)

The cur distribution along the other elementary turbines has now to account for the

)ne prescribed

c,

r-course such as to have always

d

(c, r)/dil

=

0.

Raabe J. Hydro power - the design, use, and function of hydromechanical, hydraulic, and electrical еquipment

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