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

10.2.9.

Tidal

power

turbines,

layol~t

1.

~eneral rcnl~i~rks:

The

low hc~d dile to thc lidss prcdcter-mi~~cs the axial tl~rbine

in

its

desigll as a

BT,

ST

0:.

a

?'T

(scc

Fig.

10.2.15)

to

harness tidal po~vcr. For the case

nf

double

acti~~l

plants with one basin (Fig.

10.2.30,

1

b),

this is done by a rnncliine cap;tblc

of

direct

and reversed turbining, and

-

if

necessary

-

also of direct atid reversed pi~tnpi~~g, see

F~s

1.6.2.

~]~crt'Kare the special features of such an axial tubular pump-turbine for direct and

reversed flow

in

both the modes of operation are considered.

Imagine a turbine of the bulb type (Fig.

10.2.15).

The flow enters the runner from the side

of

the bulb with a whirl produced in the stay and mainly in the griidc \.anes of

il

co~lical

distributor.

In

this case of

n

so-called direct

flo\v

whilst turbinillg. the draft tube operates

as

a

diffi~ser. Ob\liously this works at a higlier efficiency as

a

turbine t!lan under reversed

flow,

when

the gate operates

as

a diffuser.

This direct

turbining usually is combined with

a

flow from the reservoir to the ocean

[10.149].

If the basin is situated

at

the mouth of

a

river, then

ill

consequence of the latter's

energy, larger work is done, combined with

a

higher efficiency.

When

reversedly turbining, the in!et whirl of the rotor vanishes, as the draft tube is not

equipped \vith vanes. Then the channels between the guide vanes operate, evcn if undc~

modest pressure recovery, as a diffuser. For the design of gates, it is important thai in

enlergency they close tight.

Fig.

10.2.30.

Various tidal schemes with constant geometry ducts at intake and

flow

off. Notations:

S

-,

spillage (sll.licing) in ccrtain direction

+;

S

e

spillage (sluicing) in both clireotions;

T

-

till-bining

in certain direction

4;

P

4

pumping in ccrtain dircction

-+;

PT

%

pumping and t~lrbiniilg

In

bat11

(jirections. (In c;~cl~ case, pimping can bc omitted.)

I

Schemes with

1

basin:

12

single

effcc! (action);

if

the basill has an inflow, T: basin-sea;

P:

sca-basin;

S:

basin-scu.

If

thc b::sin with or ~virhorlt an

inflow:

P,

T,

S

in arbitrary dircction;

I

h

double effect (action): T,

P,

S

subsequently in direction sea

basin and vice versa.

2

schemes with

2

basins; 2a single effect (action): No

P;

S,

T

in the same

direction

(A

head water pond,

D

tailwatcr pond).

2b

between the sea

and

the hasin

A

double effect

with

S;

P,

2'

subsequently

in the dircction sea b3sin and vice versa. R.lorcover bet\vcen the sea and

the basin

B

single effect

T

and

P.

Pool

U

cmptied

by

P.

Then for

peak

load

T

at rclativcly high head.

2~

between the sca and the pool

A

double effect with

S,

P,

T

subsequectly in the dircction sea-basin

and

vice versa. Upper pool

C

(requiring mountains near coast) filled by surplus energy through

P.

For

pcak load

T

pool

C

to

A

(or sea).

O

power house.

.T'tic:rci;~rc nearly untwistt-cl v;;ncs arc nectlcd. For tlie diffcrc~it operations :lnd

tile

,-orre.

spol:ding g:rtc positions. tl~c stroke of t:ic g;rtc

servomotor

~niist

IIC

ratlicr loiig. I-lowever

*,

this c:ln I)c oh[,linccl e,lsily on

a

stationary part.

3

*

r*

C:ontr:iry to the above. the rotor with its overhung design nceds

a

b1:ldc ildjustment

di

mech;lni\ni with tlic s~iiollest possible stroke. Thcrefore

it

scelns ;ldvis;lble to operate the

pump-turbinc under all its modes of operation with ncarly one and the same position

of

3

the rotor blades.

For a lack of sufficient

tinic between the different operations, due t9 speeds, reversed

to

each' btiier, the rotor vanes should also be capable of operating as

a

hydraulic brake. rhus

a

pitch to chord ratio below

1

should enable a turnover of the blades through the closed

'

position.

11.

Some peculiarities of tlie velocity triangles during dlrect and reversed turbining and

pumping: Contrary to thc use made so far in this book, the rotor edge towards the

reservoir is dcr,oted by subscript 2 and the opposite edge with

1.

I

1)

Direct turbining: Flow from the distributor to thc: draft tube. Vane arlgle at overgate

about

70-.

On the rotor inlet,

a

whirl in the sense of speed

ti.

Relative flow at rotor inlet

more inclined to the circunlfcrence than at the exit. Zero whirl at the rotor exit

(c,,

=

0).

Hence (Fig. 10.2.3

1.1)

n

forward curved rotor vane is required. Flow from basin to ocean.

2) Direct pumping: Flow from basin to ocean. Gate angle of 90" frrcilitates

a

vanishing

whirl at the rotor inlet. At the rotor exit, whirl is in

the setlse of ratation. Sirice this gives

a bad prcssille recovery, :dso

a

gate angle of more than

90"

may be applied, which causes

a whirl

at

the rotor Inlet contrary to its speed but avoids the whirl at the outlet. Rotation

in

tlic scilst. of casc

1).

Compared with c;ise 1) the rotor requires backward curved vanes

(Fis.

10.2.31.2).

3)

licver;t.d turhining: Flow from ocean to basin and from draft tube to gates. Rotation

reverses its sense compared with 2). Therefore fast

brakin2 is required at the end of the

foregoing operation

2)

by

turning the blades through its zero position in a plane normal

to the

axis.

The

gates

now opened by about 90" operate as a part of diffuser. For the

generation of torque, thc whirl at the rotor exit must be opposite to the latter's speed,

since

c,,,

=

0.

Now

back:vard curved vanes required as in case

2)

(Fig. 10.2.31.3).

4)

Reversed pumping: Flow from ocean to basin. Zero whirl at rotor inlet. Hcnce at the

rotor

exit, whirl is in the di~ection of rotation. Gates, now operating as a part ofa diffuser,

make

an

angle about 90' with the circumference. Forward curved rotor vanes are re-

quired

(Fiz. 10.2.31.4), as in case 1).

To reverse the

speed for transition frcm

4)

to l),

a

turnover of rotor vanes for braking

is required.

Fig.

1.6.2 shows the timing of the operations 1 to

4

needed for a one pool

double effect tidal power plant,

e.g. the French "Rance".

Conclutling, the following requirements can be stated for the design of a tubular pump

turbine, for direct and reversed flow, in both the modes of operation, for such a tidal

power plant:

a) The stroke of the gate must enable a gate-position of about

90"

with the circumference.

The

sate vane must be rounded at both its ends, since they may both serve as entrance

(,Fig. 10.2.3 1.5 a).

b) The rotor vane ad;ustn~ent must enable

a

vane position of about 20" with the circum-

ference for normal operation and a position of

-

10"

for brakirg (Fig. 10.2.31.5

b).

C)

The required camber of the roior blade is the same for direct turhining and reversed

pumping,

namely that due to forward curved blades. For the other two operations

Fig

10

2.3

1.

Kccll~ired posiiion, camber

Direr

t.

Turbining basin ocenn

311d

roundctl irllct cdgcs for the

gntcs

and

,hc

rotor bladcs at rhe following

4

modes

ofoPcrating:

1

direct turbirling(basin-sea);

jircct pumping (basin-sea);

3

rcversed

7)

;urbining ocean-basin;

4

rcvcrscd pump-

ing

ocertn-basin.

5

compromises to

bc

Gz

needed

In3de

for all

thc

4

modes of operating

runner blade

rncntion~d previously:

a

large stroke of

and

gate

blade

rounded on

both

cnds;

Direr

f

Pumping basin ocean

t,

rotor hladc doublc cambered and

l,u~~ded on the

both

cnds.

For hydraulic

braking: pitchlchord ratio smaller

1.

11

*)

c2-f

-

needed

LuI

.

runner

blade

Ocean

Reversed Turbining

ocean

basin

/7/

3)

*/

CUZ

needed

tcl

runner

blade

Reversed

Fumplng

ocean

basin

Y\\

Cr

needed

irnpelle,-

blade

namely

reversed turbining and direct pumping, the opposite camber is required. To

overcome this discrepancy, double-cambered blades are realized as a compromise with

both ends rounded, in the Rance tidal power plant [10.149],

Fig.

10.2.31.5.

Another

colnpromise would be the use of non-cambered skeletons. Tests made by

Rayin-

dran

[10.150] showed clearly the superiority of this double-cambered solution compared

with

non-cambered solution with one blade end sharpened.

Since the suction face of the rotor bladc is oriented towards the draft tube during the

operiltio~i-;

I!

ancl

4)

and oriented tow;~rds tile g.ltes during the opcr:ltions

2)

and

31,

hot

the f;~ccs

ol

thc ro!or bl:ldc ~nurt be li~iisbed s~lioollily so as to R~~~ction

as

a

suction

raw

cI)

Tlie guide hc:lrings of the sct havc to enable lubric:ition in hot11

SCIISCS

of rotatiOK

8

C)

Silicc tllc

il~iill

thrust chiltiges its direction whc'11 the head wntcr clinnges

from

the

.+

Imsin to tllc oceiln, ttie sct nceds two thrust berlrings for both axial flow directions.

,,

f)

Tlle distributor with

its

adjustable gales must operate as

a

ili!Tilscr in the case of the

-$

"b

revccsed turbine

:lrliI

direct pump flow. Therefore both ends of the distributor vanes mus<

%

be'proliled carefully. l~ccause of thc rather poor pressure recovery of a distributor

,,

a

clilTuscr the cflic~cncy

of

:I

machine opcrating that way is lower than that of a machine

operating the other

W,IJ

with a draft tube as

a

dinuser.

iJence best efficiency car1 be achieved only, when the flow direction is retained. This

can

be done either

by

complicntcd bifi~rcations and

valvcs

at both encls of the machine (also

draft tubes which

may

bc turned on hinges have been proposed, surely not reliable) or

bq'

a single actio~; working cyclc (Fig.

10.2.33.).

In the largest tidal power unit of Annapolis

Canada (Fig.

10.2.22)

designed by Sulzer Escher Wyss the singit. action was preferred

of

using a Straflo turbine wit11 rim generator,

110.321.

Fig.

10.2.32. Thc sequence of operating modes in a tidal

powcr plaat with

1

basin

and single uffcct (action):

1

filling;

2 waiting plus eventual pumping;

3

turbining.

0

level

sea;

1

2

3

b level

basin;

p.levcl

basin

with pumping. Utility lifted

by

l

l,,,,

,,,,I-

some sets suitcd

to

return

flow. (Drawing courtesy

EdF).

111.

The energy _gain

by

additional pumping: Imagine the head during pumping changes

p:~rabolically with

ti~nc

according to

Hp

=

k

r2.

When

[I,

has rcaclicd a value

H,,,

after

1

period

t,,

tu1-bii~ing starts. Whence

k

=

flp,/tb.

Assumin:. nn electricity rate

ke,

efficiency

q,,

and accounting for unit flow

ell,

whilst pumping, the expenditure during

pumping Secon~es

%'hen the plant turbines without pumping during the period

t,.,

the following profit is

11iadc

by the s;ile of energy at

a

constant turbir.:: efficiericy

11,

and

electricity rate

k,

with

a

unit

flow

Q,

,,

under tilrbining at a head

HTo

Then generating the same time

t,

with a head incrzased by

AH,

as

a

consequellce of

prtmping would increase the profit

by

Ah',

=

(312)

k,qT

Q1 lT@(~pog)3'2

tT(l

-

dN!HPU)

D2.

(10.2-33)

For profitable purnying the ratio

f

=

AKJK,

must be larger than one. Hence

Ascuming

Ql,,!Q,

,,

=

1,3;

11~

=

0,9;

q,

=

0,s; then the condition

f

>

1

leads to

(tp/rr)(l

-

AH/HpG)

>

472.

This gives for

AH/llp,

=

0,2

the time ratio

t,/.rp

>

0,9.

This

proyes the rule that to

make

a

profit

by

pumping, the duration of turl~~ning should

be

equal

or larger than that

cf

pumping.

I\'.

Problems with two hasins:

A

typc of plant not made to dntc may bc bet~veen tu.0

rescr~~ir~ (Fig. 10.2.30.2 a). 'The one is kept at high Icvel by connectins

it

with the occ;ln

nu~.illg

a

time period from somewhat before high tide up to high tide. The other is

kept

at

low level by co~lnecting it with the ocean during a period from sorncwhat before lo\v

tide up to low tide. This is eflected by sluice gates.

Even

if

the flow is now more continuous, this arrangement also shows the general

disadvantages of tidal

power. The first is the linkage of power production to the lunar

position when there is no storage capacity, and hence the in~possibility to cover peak load

demand.

The

second is the rather low head and hence large and expensive sets.

This situation can

bc improved, when the topography permits it. by building a second

basin

C

above the basin of a one basin plant in the neighbourhood of the Irltter

(Fig. 10.2.30.2~). Imagine this second reservoir, whose time-averaged level elevation

above

thc lirst is Hp to be filled during a period t,, when the electricity rate

kc.

is lour,

by a flow Qp

zt

a pump elliciency

11,.

Hence the expenditure of this filling

Kp

=

kep

_O

Qp

~Hpfpl~,.

This stored energy may

be

used to cover peak load demand, electricity rate k,,, d~lring

the period

r,,

under

a

time-averaged head

H,,

flow QT, efficiency

q,.

l-lence the earning

by the sale of energy

KT

=

keTeQ,gHTtrqT. Imagine the live storage volume of the

upper basin

AV

=

Q,

t,

=

(2,

t,.

'Then the surplus made during both thc last operations

In the utmost favourable case, pumping is during high tide and turbining during low tide.

Then

11,-

=

H,,

+

AH,

,1Ij

being thc tidal stroke. Then the maximum surplus becomes

The base load produced by this plant with live storage volume

AVl

brings under the tidal

stroke

All

at least

Hence the ratio of maximum surplus by means of

st1

addition&] basin for pumped storage

to minimum earning

with a tidal cycle \vithout storing capacity becomes

sn,oA/Emin

=

(A

v/AJ'l)

(J3f'/A1a

(keT/kel>)

-

(kel>/lieT)/(r~T

+

A1l!ljPl

-

(10.2--

37)

Examplc: k,,/k,,

=

2;

H,,/AII

=

10; AV/AV,

=

10;

11,

=

0,85;

11,

=

0,9;

S

,,,,

/El,,,

=

89,3. However, this procedure suffers from a lack of availability of the plant.

A

special form of the multi-pool schen~e is the two-pool scheme proposed by

Shaw

in his

paper on: "Some considerations of a possible new role for tidal power", presented at the

international conference on the

utili~ation of tidal power in Halifax, 1970 (see

Fig.

10.2.30).

Here the smaller pool, basin

B,

is pumped empty with surplus energy, so that for peak

load production, water can be passed through the turbines from sea to basin

zt a

relalively high head.

A

large pool, basin

A,

is operated in single or double effect in the

same way as described above.

V.

Plant wit11 a ccntr~l generator: The usually small tida! range requires even for the

highest tidal range,

c.g., Rance, a runner tip diameter

of

5

nl for

10

MM1

output. There-

fore,

11.

Fiscltcr. Ilas proposed one central generator should

be

driven

by

a large number

of

(may hc

ui>

to !CO) turbincs by means of hydrostatic gear to savc first cost. Here each

turbine

gcnt'ratt's

a pressur.izcd oil flow by

:i

rnultipiston bcvcl clisk pump.

The flow

resulting

from all the turbincs of one group propcls the rotor of thc ccntral generator,

This can

bc

clone, c.g., by a niultipiston bcvcl disk positivc displaccmcnt motor.

By

adjustment

or

the bcvcl disk in thc ccntral gcncrator, thc gcar ratio bctween turbinc

anci

ahcrnator

can

be

changed

under constant altcrnator spccd.

~htk'the ;aijustable bcvcl disk allows an adaptation of the turbine's optimum

speed

to the head,

whicll varics continuously in

a

titl;tl power plant. Thc improvcd cflicicncy has to compensate the

cost

and

loss

of

cncrgy duc to thc gcar. Moreover such n gcar enables an economically high speed

for

thc altcrnator, which would bc cheaper than a slowly rotating direct couplcd one.

VI.

Plant with turbines for sluicing operation:

A

further increase in energy production

can be achieved if the turbines may run also in the sluicing operation to fill or to empty

the basin, when the water levels of the sea and the basin are nearly the same.

VII.

Economic consideratiotls: In his paper "Choice of hydroelectric equipment for tidal

energy" presented at the Korean tidal power symposium, Seoul in 1975,

H.

lMiller

reported that the direct costs of different Fundy Bay schemes are distributed as follows:

a) Electromechanical equipment 46,7%,

b) Power i.louse

21,6%,

c) Sluice\vays 13,7

%,

d) Dikes 16,3

%,

e) Fixed site cost 1,7

%.

Hence the hydroelectric equipment, directly influences about 50% of the costs, and

indirectly

about 35%, since the power house and sluiceways depend largely on the

machine's sluicing capacity.

For the single pool single effect scheme, an optimally designed turbine can be used.

However, within the usual range of operation

HIH,,

=

0,2 the pump efficiency of the

turbine goes down to 50% of the turbine's bep efficiency. If therefore there is no pumping

a

fixzd b!ade propeller would suffice.

This reduces the machine price by about

12%, together with lower maintenance costs.

Si~ce sufficient discharge is always available, the turbine can always operate at its bep.

Whzn compared with the single effect scheme, the double effect scheme, as

a

compromise,

results

in a lower mean efficiency and in a lower energy output per tidal cycle. Accord-

ingly a larger number of units is required. This decreases the benefit-cost ratio. Hence, the

latest results of studies on Fundy Bay have reconfirmed the single effect schemes to

be

the most economical.

All

rittemp:~ to eliminate, e.g., by added pumped storage, the intermittent energy produc-

tion, are unsatisfactory, since they require:

1)

Increase of investment,

2)

more complicated hydroelectric equipment,

3)

higher maintenance costs,

4) shorter working life.

Therefore tidal power can be economic when it solely converts the available tidal

energy

directly into electric energy.

VIII. The superiority of straflo design compared with bulb design: Again

following

124illur,

simplicity

and

compactness are requirements to make use of caisson constructio*

which has proved so successful

in

the building of dikes. These requirements are

not

fulfilled by the vertical Kaplan turbine or the tubular S-turbine.

file

ilqe of bulb tiirbines for tidal power to d:ltc originated from the fact. th;u

the

1:irgc

~urbinc tlccded was

not

available, and

tli;it

the bulb turbinc

can

fulfil

sc>n;c

of

tllc

Str~

rcquil.cmsnts of

i~

tidal power machine. I-lowever bulb tl~rhirles have some lilnitations.

Collsid~r the gcnerator out put

P=

KD~L~,

,here

K

is thc compactness factor,

DR

the rotor tip diameter,

L

the rotor length. and

11

the spccd.

mc

slim

bulb, hydraulically rcquircd. makcs thc generator rotor dinnicter

D,

considerably sm:llIcr

lhan

the turbine runncr diameter. Therefore, thc gcncrator length

L,

and the compitctncss factor

li

,,,,,t

bc incrcirsed, which both increases thc temperature. This may require complicated \vatu--

cooling

in

a

badly accessible gcnerator even at smaller outputs (about

10

MW).

1hc small rotor diameter results also in a low inrrtii~ of the set. This contradicts the rcquircments

or

stable regulation, rspeciallv when thc plant supplies a grid by itself. Moreover a Ilirge inertia

nJuceS the consequenccs of power swing,

a

tid;il unit is subjected

by

wave-induced pressure

fl,ctuations.

It

would help also to reduce thc fluctuating loading on thc critical connections of the

and

windings due to this powcr swing. l'hesc problenls yield a lo\vcr limit for tlic runner

diameter, which lies, accordi~ig to recent studies

on

Fundy

Bay,

at

7500

mm.

In

the

compact bulb,

it

is dificult to carry out extensive repairs, especially on the stator. Therefore

generator

sliould be removable through a special hatch on the top of the bulb (Fig.

10.2.1

1).

~~rnoving the gcnzrator through the turbinc pit requires removal of the whole unit, jvhich alolie

lakes about

6

months for units of the size needed.

1"

the Straflo turbilie all the disadvantages mentioned above have been eliminated,

namely

1)

50%

larger generator diameter than for a

BT;

effectively cooled generator,

2)

no limitation

to

size of turbine and generator,

3)

3

to 4 times as much inertia makes the gcnerator more stable and stabilizes the

regulation,

4)

turbine and generator in the same vertical plane require only

1

erection pit,

5)

good accessibility to ge~erator.

A

comparison by

hlillcr

bctwcen

a

straflo turbine plant having

18

units,

25

MW

each, and

a

tulb

turbine plant, having

30

units,

15

MW

each; results

in

the following:

At

a width of

421

ni for the

Straflo plant,

thc bulb plant ha5

a

width that is

1,27

times larger, and a price of hydro electric

equipment,

that

is

1,5

times larger than the rcspcctive valucs of thc Straflo piant. Hence tlie Straflo

turbine seems to be

thc tool to utilize tidal power

at

rcduccd cost.

10.2.10.

Runner

design,

simple

procedure

I.

Runner vane design

by

cascade theory:

For

given velocity triangles with their cocffi-

ciellts

h'u,

Kw,,

K

Ac,,

from 10.2.5.3, the maximum lift coeficient at the hub

CAC

=

1,4, with the vane number

z

from Table 10.2.1, the known hub diameter

D,,

the

pitch chord ratio from relation (10.2-

13)

gives the chord

Li

at the hub.

Runner diameter

D,

vane number

2,

the hence given tip pitch and the pitch

to

chord ratio:

from

Table 10.2.1 yields thc chord

La

at the runner tip. Since the inlet edge and rear edge

of

the runner vane should

be

as

straight as possible for reasons

of

stable flow and vane

Strength,

the.pitch chorct ratio and hence the chord

is

given for an arbitrary cylindrical

Rotion, for which the velocity triangles are

also

known. The relation (10.2-

13)

yields the

lift

coeficient in the cascade

c,,.

'Then

fro111 Table 6.2.1 thc ci~scrldc coc.fiicit.~lls

C,,

C,, C,,

C.,

LIIC

cieternlirlcd

:IS

functiolls

ol't/L

a~ld thc anflc

11,

which may be approxinlated to

/I,,

I'rorn tl~c trii~ngles,

as

the angle

)r,

=Z

Or,

+

w2)/2

n~akcs with the circumference. Then from

(6.2-47)

the physical angle

of attack

(So

=

i,,

(C,/C,)

is

obtained.

Now

the

vane

skeleton

must

be set. This may be done with respect to pressure distribu-

tion tcavit;~tion sensitivity) or the hydrodynan~ic torque abaut the pivot of the blade.

Then

~lcicrmination of the angles

9i

the skeleton makes with its chord at Birnbaun~'~

3

stations

(Cap.

6.2-37). Hcnce

K,,

K,

and then the zero lift angle of attack

6(FA

=

01,

are obt'lincd by (6.2-4). Thus the anglc the chord of the cylindrical sectiorl makes with

the circumference

/I,

=

0,

-t

6

(FA

=

O),.

The skeletons then are built up so that the centre of gravity of each cylindrical section

coincides with the blade axis.

For blades with

a

usual standard profile such as those of the basic Giittinger series or the basic

NACA

scries

[10.151]

the centre of gravity is aproximntely a distance

0,4

L

from the profile's leading

edge.

Hcnce there is no bending stress at the blade's pivot due

to

centrifugal load of the blade.

The

real thickness of the blade must be determined at first on the base of the blade on

the hub.

Mere

the

bending moment as the relevant load caused by the hydrodynamic

lift

may be

approximated as

iV

being the hub to tip diameter ratio,

z

the vane number. The root cross section

of

the

blade on the hub

dlsk is nearly

a

rectangle, whose length equals the diameter

D,

of the

blade disk (Fig. 10.2.33). Roughly

D,

=

0,4

Dlz.

This gives the sectional modulus of this

cross sectional area as

hi,,

0,/6,

where

hmi

is the maximum blade thickness at the hub

section.

VJith the carefully set admissible bending s:ress

a,,

(which is strictly speaking only about

fi

399'0

of the stress acting there, as a consequence of centrifugal load-induced torque,

*

torque due to hydrodynamic lift, to mention the most important ones), the maximum

bl~de thickness is

a

According to

Barp

and

Krller

[10.152], in blades of alloyed chromium steel, the assess-

ment

of

a,3,

has to account for fatigue effects. To reduce the latter a fillet is provided a10

the corner between the disk and the blade. Moreover a rounded cut out should

provided at the corner between the disk and blade in the plan view.

This cut out has to be closed by an insert bolted on the blade to prevent leakage

betwe

pressure and suction face of the blade.

The :hickness distr~bution of the real profile of

a

given cylindrical section based on a certain

seri

e.g.,

IVACA

is set so as to strain the chosen rnodel profile normal to its centre line at any sta

proportiolial to the ratio of calculated maximu~n thickness, to maximum thickness of the

m

profile.

The

mnximuni thickness required at an arbitrary cylindrical section

h,,

drops continuously

wi

radius from its maximum

h,,

on the hub towards a value

h,,

at tip diameter, tolerable for le

and tip clearance cavitation erosion. To reduce the latter. an anti-cavitation fin may be welded

a

a part

of

the blade's tip (sec Fig.

10.2.25).

The fin is usually worn off by tip clearance cavitation

therefore in general not

pravided.

444

50

60

70

80

90

rill

U/rpm

08

1.0

1;2

nll/nll

opt

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

/?

with circumference at runner inlet

at different operating points according to the measurements of

Schlemtner,

in a mean flow plane

[5.18].

Beginning and increase of -.urging 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.

tun1 drops nearly linearly from the leading inlet edge (there having a magnit~de

c,, r,

=

(gH

9,

-

c,,

r,

o)/w)

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

strea~nlines

(d(c,r)/dn)

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.

Ijence 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

Zn arbitrary drop of the flow energy

Y,

within the runner in the streamwise direction

can be

sct only along one elementary turbine. The relation

(6.5-3)

for the distribution of

the

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

to the streamlines shows

that only in the case of a loss free flow and

d(c,r)ldn

=

0,

the

distribution of the

meridional velocity in the n-direction (nonxal to the meridionill

streamlines)

becomes independent of the

c, r

distribution normal to the streamlines.

Assuming heilce this sin~ple case, then two things have to bc satisfied:

a)

The

c,r

distributior~ along the other elementary turbines has now to account for the

one prescribed

c,

r-course such as to have a1 ~ays

a

jc, r)jd~z

=

0.

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

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