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

Kc,,,,

follo~vs from

:I

rel.ltion si111il;ir

to

(9.2- IS), in which

Q,,

is subs~itutcd by

a

ilo\u

Q.

lllserting

(9.2-

-

19). (9.2-

20)

ill

(9.2-

12) ii11d (9.2-21), (9.2-22) in (9.2

-

15)

then

(9.2-

16), sives

the

different type numbers as functiolls of the velocity cocficients

X

c,, and

the

design features N, and D,/D

,;

=

2314~112

(1

-

N~)~/~(D,/D)

K

11

KC^,'^^^,

(9.2

-

23)

n,

=

158(1

-

Nf)'I"Dl/~)

K

11

K

CAI:,

(9.2

-

24)

ns

=

578(1

-

N,')l/'rl:;:(~,/~)

KUKC;,':.

(9.2-25)

\vit]l

the obvious relation

Klc,

=

(D,/D)

KII,

(9.2-23) reads

=

23/47[1/2

9

(1

-

A',')'~~

K

u

,

I(

c;,':~~.

(9.2

-

26)

For the bep, characterized

by

c,,

=

0, thc flow coefficient is

-

50iop

-

~,,i~~/tll=

K~~1~~/K1~1

=

fanb1,

(9.2-

27)

in

\vhich

/Il,

the angle between rotor vane and periphery, usually is optimized for

NPSH

or

11,

see Cap.

4.1

1. I-Ience

n'

=

23/4x1/2(1

-

N,')~I~~;;;:,~I(:I~;

4

(9.2

-

28)

,I'

9

=

23/4n1/2(1

-

N:)~/~~~~Z,(D~/D)~/~K~~~/~.

(9.2-29)

Again both the last relations show the type number as functions of the

blade speed

coeficietlt

Kul

nt throat diameter and Kt1 at lotor tip diarr,eter, the hub to

tip

dia~netes

ratio

N,,

the throat to tip diameter ratio v,/D9),

and

the flow coefficient.

111.

How to increase the type number and its limits: According

to

(9.2-

14)

the type

number

n&

is an expression proportional to

the

rotational

speed

(01

or

n)

and

licnce

in\lessely proportional to the size of

a

machine. Rut

nb

increases also with

Q

or

P

anc!

hcnce with the "power density" under a certain head.

n

and

P

are often desired to be as

high as

possible.

This raises the question about measures to increase

11;.

As

the figures

cp,

and

N,

are

usually given, it is seen from both the last relations that

nb

rises with Ku,

Ku,

and

Dl/D.

Thus a machine of higher

nb

operates with a larger blade speed coefficient

K

11.

hloreuver

il

has a larger draft tube inlet to tip diameter ratio. In Kaplan turbines, having Dl

=

D,

only an increase of Ku helps

to

increase the type number. Table 9.2.1 and 10.3.1 reflect

these tendencies.

8sunlly

,I$

is given by the working data

gH

and

Q

of the set, since they, at least in the case of a

rtactio~i machine, limit

)I;

in consequence of suction requirements. This raises the problem of thc

lowest

limits of

11;

for a reaction machine

(FT).

In higli head FTs, with lowest

n$

the angle

/?,,

the

runner vanc makes with the pcriphcry at the inlet, should not exceed

90,

to limit the curvature of

'he

rotor vane. Hence

c,,

=

u.

Thus in this casc the lowest

Ku

value

is obtained from

Eulcr's

relation

as

K

trmi,

=

(q,/2)lt2,

(9.2

-

30)

7,

is

thc

peripheral cficiency. Usually the breadth of the rotor

b

follows the rule

Db

=

D,

b,,

b

holds for the runner inlet and

6,

for the runner exit. Since too small a value of

hlb,

deforms

'he

cross

sectiotlal area of runner channel into a flat rectangle with its uneconomical small hydraulic

"dius

2b,

also too small

a

valuc of

the

diameter ratio

D,/D

is

bad. I-lence

Dl/l)

>'

0,5.

I11

an example:

\

?

Strictly

speaking,

lhront

diameter

D,,

at

a

turbine.

dclincs

thc

outmost

diaineter

of

the

runner

vane's

exit

edge.

and,

at

a

pun?p,

thc

same

diamctcr

of

tl~z

impclier

vane's

illlet

edge.

W

t4

n(rprn)Jp(kGj

I

O0

Table

9.2.1.

Characteristic features

of

turbine

vs

specific speed

n,

=

--

H

(rn)''"

*)

referred

to

qi

=

0,88

TY

~e

n,*)

QII

"1

I

Hnnx

"1

1

rr "~~romnr

'

rprn

m3/s

rpm rn

rpm

rl'm

(7)9-

1

0,007

-

0,O

l

I

-39,s

-

39,4 1800- 1650 7 1,6- 70.9

-

1

nozzle, 11-17 0,O

1

-0,024 39,4- 38,9 1650- 700 70,9

-

70,O

- -

17-(26) 0,024-.O,Q55 38.9

-

37,b 700- 350 70,O-67,7

- -

S~OW

51-107 0,l -0,35 60,s

-

63,6 700- 410 102 -112

-

0,03 -0,05

medium

107- 150 0,35 -0,59 63,6- 67,5 410- 240

112 -124

-

0,06 -3.085

150- 190

0.59

-0,83 67,5-72,6 240- 150 124 -138

-

0,085-0.12

TY

PC

e

o

Vf

0

G

5

F

c

0

.

-

".a

0

m

2

6

.s

c

E

190-250 0,83

-

1,13

72,6-81 150- 90 138-15s

-

0,12 -0.185

250

-

300 1.13 -1,2S 81 -92,2 90- 64 158-180

-

0,185-0,27

a

cj

8 vanes 240 340 450 0,930- 1,220 85-145 50 205

-

250 300 0.3 -0,55

6 vanes 330450 560 1,290- 1,600 100- 155 3 5 260-280 330 0,65 -0,85

5 vanes 390 540 690 1,600-2,000 1 10- 17G 20

245

-

300 330-360 0,s

-

1,2

4

Sranes

490 610 750

2,000-2,350 120- 180

15

300- 355 300 1,2

-1,6

3

vanes 570 630 920 2,350-2,450

135

-

200 6 355-400

430

1,s -33

=

tan

13,

=

tan

I8

=

0,377,

I],,,

=

038,

11,

=

0,92,

K

11

=

0,677,

D,ID

=

0.5:

11;

=

0.31

S

or

"'?

57.

111 gcncr;,l

11.

in FTs should not fall short of

!I,

=

60

cnrrespondinp to

11;

=

0.33.

Eh;lrnplc

ns

-

m-

Rossliaag. Zcmnigrund, Austria, with

H

=

672

ni,

rr

=

750

rpm,

I:,,

=

79

500

hp, liclds

*,(lip)

'

6273.

lnstcad of the con~idcration in Cap.

6.2.2

concerning thc limit of

11,

duc to ground excavation

b!

S,blllergcnce required, the following

consideration

on the

rt,(l-1)

relation is bnsed on an estimate of

rotor strength and optimum valuc for thc

flow

numbcr at rilnner cxit

qI0,.

lnlrodu~ing

D,lD

=

h/O,

(often realized) and

Ku

from

(9.2-17)

into

(9.2-29)

results in

11;

=

IL"~(I

-

N:)'I2 (p~l~(b/b1)3/2~3/2

/(B

H)31"

-

(9.2-31)

~~~:aIly

N,

is negligibly small. In general

cp,.,

=

tan j?!.,.is prescribed either

by

suction requircmcnt

NPSH)

or

by efficiency

requirement

(optlmlzing

11).

The pcriphernl blade velocity

u

at

tip

dian~eter is limited

by

the strength of the rotor

(especially

with respcct to runaway). For

a

type

blb,

is also limitcd

by

the rotor strength. Hcnce for reaction machines the

11;

limit

dcpcllds on

H

by

This dependence of

H

on

ni

is

reflected

by existing dcsigns of reaction machines (Fig.

8.2.6~)

in the

of

a

strip approximating a function of this type.

For an impulse turbine the throat diametcr, introduced in (9.2-

18)

must be replaced

by

thejet diameter

do.

Obviously

N,

=

0. Since then c,, corresponds to the spouting veloc-

ity

Kc,,,

=

Kc,,

=

1

(when loss in the nozzle is neglected), hence

n;

=

2314 illi2(do/~)

Ku,

nq

=

158(do/D)

K

u;

(9.2

-

33)

n,

=

578

9:;:

(d,/D)

K

11,

where

D

is now the jet circle diameter (Fig. 9.2.1) as the reference diameter

in

an impulse

turbine. From this it is seen, that the type number (specific speed) of

a

single impulse wheel

with one nozzle iilcreases by the

jet

diameter ratio d,/D. Usual values of D/d, are

(6)8

10

lS(17)

at

H

of(200) 300

m

to 1200 (1800) m, where high D/do values are related

to

~nulti

nozzle wheels.

For an impulse turbine with

i,,

being the number of wheels and

in

the number of nozzles

pcr wheel, the resulting type number follows from the fact that

QnOzzle

=

iwinQwSheel as

111

the casc of

PT

with

a vertical shaft there is a maximum iw

=

1

and

in

=

6.

In the case

of

a

PT

with

a

horizontal shaft

i,

=

2 (in single case 3), in

=

2. Both

the

last formulas can

be

applied also to reaction machines. In rare cases there are more stages in series (for

Pump-turbines up to

5

stages) in impeller pumps the number of stages can be even more.

liere with is, as the number of stages

-314

ni

res

=

wheell1st

.

(9.2

-

35)

&sides the characteristic factors

nil,

H,,

,

Ku,

Q;,,

Q,,,

q1,

Kc.,

Y

=

Kc,,,/K1[, also the

Pressure coeflicient

Y

is used

Y

=

2g~/(o~/2)2

=

~[Ku'.

(9.2-36)

The

following similarity relations are needed for

a

force

F

and its moment

M:

F

-

PA

--

L?

gHD2

'

@It2

D4,

M

-

FD.

In

thc

LISII;I~

t/l(,;~)

pl;~~ic' tllc ~Ii;~r;~ctcristi~ or

;I

rc;~ctio~i

iypc

fluid milcllinc

may

be

ap

proximarccl

1,)

ivhere

11

is

a

dimcnslonlc~s spcecl. pT'lle upl,cr sign IioIJs for

:I

pllnip, the lower for

a

turbine

Hol!~

thc

fint

tcrnir

arc

~LIC

to

tl~c chnr\gc

of

whirl

within thc rotor in consequence

of

l~rrl~~r.'s

~clatlon.

[I,;

~tricl

ti,

lic~iig ~Icsig~i-li~ikcd p;~ri1111cters, 1'1-orn which

(1,

in

a

turbine

depcndh alsc or:

the

git~c po\ir~o~i.

Thc

thil~i Lcrnl is duc

Lo

thc anzle

of

incidence

at

the

rotor's inlet

edge

\isllcn tllc niaclline opcr.ltes

away

from its bcp due to (pop. Here the

coefficient

tr,

vi11ics \tro~l~lj \vhcii pasing

the

bep so as to

bc

smallest either in the over-

load range

of

:I

pitnlp

01.

ill

thc part load ranze

of

a

turbine. The fourth term

is

due

to

the

loss

in tlic

/lo\\.

p;is\;~gcs

of

the

admlcsion duct (suidc appitratlls, suction line), rotor

,

and dlff1.1ser.

Tl~c

lifth

[el-m

I,

ciuc

to

disk frictio~i loss

of

tlie rotor and the last term due

*

j

to

leakage

flow

loss tlierc.

In

nn inipillse turbinc, both thc

last

losses

and

the third term

,

due to

shock

loss

at rotor ir~let, are vanishing.

4

Adding thcse lolscs. ih~ n~~l-d~mcnsic~~li~l

ioss

11:

of

a

hydro turbonlachine becomes

4

I

4

Hence lincs of ccnhtant lo>>

11:

or constant efficiency

11

=

1

-

hi,

are rcprescntcd by the graph

t

Naturally such

a

SIITIDIC

:Ipi)ro\lIii;lt1on dcscribcs only the range of constant efficiency for

'P

value

not too

hr

:I\\~IJ

from thc bcp. [-he real line>

of

conslant cfficicncy arc

closed

(Fig. 9.2.2), hencc the

name hill di;~~rn!n.

To dcmonstr;~tc the nlachine's hcliuvio\

(Fig.

9.2.31.

This

is

sl>o

cal!c.d

a

hill

di;

rflicicncy to the coritour lines of

w

hill.

If

the

!<)SF

\i

ithin

a

mi~chinc is of

the

constant ef!-icie~;c.> must be parabolus

Q,

1(~~1

,)

pli111;.

Th:

drlkiitiion of ihe real lines

due

to

)I

=

const from these parabolas may hint

al

shock

Io~s,

Icakngc. tlisk friction :ind scale effects, due to a partial linkagc of thc

cri

to

Re.

1

with

fixed

runner b:ades in

Y,

cp-plane.

1

lines

O~CO

d

stunt cflicicncy

rl:

2

idealized col?tinuation of

11

co

.*

lincs as parabolas;

3

runaway-lint: (approxima!

parnbolil):

4

zcro spccd line (rlpprosimately

pd

bola]:

5

::

:

band of whirl-frcc and surge-frec operat'

A

in draft ttilbc;

6

part load range with precessing

scrc:v vortex:

7

ovcr!oad range with more

cenl

axial vortex in thc draft tubc;

8

-

line of adrnissi

.rr. also the

Q,,

(11,

,I

planc is used instead of the

P(cp)

plan

agramm bzcaus: of the similarity of its curves of consta

simple

form

h:.

-

cf/(gH)

-

Q2/(gfI), then the graphs

1)

=

kq'

ill

thc

LY((p)

plalic arid lincs Q,,

=

const in th

Fig.

9.2.2.

Hill (shcll) diagram 0f.a reaction turb

pressure

surgc;

9

limit by maximum output of

g

tor;

10

limit due to opening

of

wicket gates;

11

/

u

-

/

,z3

-

limit of

efficiency

rl;

13

uppcr limit of cavitation

im

cp

o,,

given by plant.

rig,

9

2.3.

Efficiency hill dia-

gri,,,,

of

3

Fr;incis t~~rbine

fiq

=

60

in

thc

G),

,

,I,,-plnnc.

F

o,,u';~t~ng range frcc

of

surcgc

,,lid

whirl in draft ttrhc.

T

part

jo;ld lnngc with cork

screw

vor-

,,,:

ill

tlic draft tubc,

u

overload

r;,,lgc. with straight vortex in the

dr;lft

I

ube.

---

lines of constant gatc open-

~ng;

-*-

lines of constant

mvi-

tnlion index

U.

"11

nrf nri n11

Fig.

9.2.4.

Efficiency hill diagram with lines of constant opening

a,.

a)

Impulse turbine,

n,

.:

15.

b)

Francis turbine,

11,

=

26.

c) Francis turbine,

n,

=

90.

d) Kaplan turbine,

11,

=

160

double regulated

with cam-on operation.

However, in two

cascs the simple similarity law fits also to loss-linked real flow. The first case is

runaway.

Here the conversion of the specific head

gN

into heat through dissipation results in

a

run-

away

parabola

'P

-

rp2.

A

similar relation holds also for a set at rest, which functions as a throttle.

This indeed results also from tests (Fig.

9.2.2).

-

Different flow vs speed behaviour of individual turbines: For impulse turbines at con-

stant needle position, the flow

Q,,

is independent of the speed

n,,

(Fig. 9.2.4a).

For

Francis turbines of low specific speed

at

constant gate position, the flow

Q,,

drops vs

speed

n,

,,

whereas for Kaplan turbines and Francis turbines of high specific speed

it

increases (see Fig.

9.2.4

b

to

d).

The first meritioned is

a

conscqucnce of thc large radial extension of such a Francis runner. It orig-

inatcs from centrifitgal forces acting opposite to the flow direction. The effect secondly mentioned

follows from

the approximate proportionality of thc flow-linked meridional velocity and the speed-

linked peripheral blade vclocity for the case that the difference of absolute whirl velocities at the

runner's iniet and outlet becomes negligibly small

compared with the blade velocity.

-

The limits for operation: The operation of a turbine is

first

limited by the maximum

gate (needle) opening. The next limit follows from the maximum power capacity

of

the

electric machine. In

a

hill diagram of the

q.

$-plane this limit is described by a line

c~yrl*'

=

const where

11

is the shaft efficiency of the machine

(+

for turbine). In the

Q,,

,

"1

l-plane this line is

described

by

Q,,

v*'

n;:

=

const.

A

thirtl li~nit

011

thc hill di3gr;lin

is

drlc

lo

tllc c;~vil:ition i1lt1c.u tcquil-cti rising with

the

cli\tancc

of'

thc opcralir~g

point

from lhc bel), whcrc

o,,,,

is lo~vcsi bp

111~

then vn~lislli~~

0

anglc

of

incidence. licncc

ci

I-cquircd may

bc

clcscribed approxiunatcly by (Fig.

9.2.2)

Morcovcr the hill diagram

of

a reaction turbine with

fixed

runner vanes shows only

a

small

hatched

strip of quiet operation crossing thc bep and usually connected with

a

nearly vanishing whirl

component

in the draft tubc.

Outside this zone there are areas

of

draft tubc surge. Thc one with a flow larger than

in

the qi~icl zone corresponds

to

a

straight vortex ollly surging at larger distance from the

quiet zone. The other area has a flow below the quiet one. It is surging in consequence

of a precessing cork screw formed vortex in the draft tube

[9.45;

9.461.

To calculi~tc transien~s, the characteristic (9.2-36) must be plottc~i for diffcre~nt speeds )I, but also

(in

the case of turbincsl for diffcrcnt opening angles of thc gatcs

a,

and (in the case of adjustr~blc runner

vanes) for

different

position anglc;

/I,

of

tilt

runr;er vancs. Both the nnglcs appear

with

the parameter

u,

in the fcrm of

ii

factor cot r3

+

(b3/b,) cot

/I,.

Hencc for s~~iall angles

2,

and

/I,,

the gate vanc and the runner vanc make with the periphery, the

-

term

a,

in

(9.2-37)

becomes inversely proportional to each of the angles by

tr,

-

1/r,

+

(b3/b,)//jl,

u

here

b,ib,

1s

01c ~nteirotor oi~tlet breadth ratio. Transicnis also htlvc to account for ihe constant

torque iincs.

In

the

$,

cp-sysrenl they read const

=

rp

'P(1

-

11:)'

',

-

(

+)

sign for pump (turbine).

As

an intcrestins feature in case of high head imptller pulnps. the torque at zero flow

(azainst closed valve) may reach

up

to

50%

the rated value. In any turbine the torque

drops vs speed.

I-Iencc any machine must resist the starting torque. TIiis attains

1,4

of the

ra:ed one in KTs and up to twice the rated one in PTs.

Vaii!sl~ing

!orque

appears undcr runaway speed. This attains in FTs of low specific specd 1,5 times

%

the

rated specd and in KTs under off-cam operation (when the usual linkage betwcen gate and

rurlncr vane position is intcrruptcd by failure), as the other extreme, up to

3

times the rated speed,

~alucs, for which the set must bc eventually ~nechanically suitablc.

To

study the behaviour

of

reaction turbomachines under various modes of operation, e.

as

a

putnp-turbine or under transients (occurring at start up and slowing dcwn the

se

the 4-quadrant characteristic in the

n,

Q-plane is used. See

Fig.

9.2.5.

Here lines either

constant

gnte

pusition or constant torque or head, unequivocally assign

a

certai

operating pcint. \vith

a

certain speed and discharge, to a certain mode of operation oft

Fig.

9.2.5. Four quaduadrant characteristics

radial reaction

turbon~r~chine in speed

(n),

(Q)-plane.

1

pump regime with normal sense of

tion given by its volutc;

2

dissipating (brake) r

3

centripetal (Francis) turbine regime:

4

dissipa

(biske) regime; 5 reversed pumping regime;

6

d

pating (brakc) regime;

7

centrifugal (Fourneyron)

bine regime;

8

dissipating (brake) regime.

1Ll

cons

torque line

---.

Y

const specific head line

-.

of constant gate opening have a hump in

2

si

that of constant torque lines. Hcnce a possiblz lac

coordinating

flow

to spced with troubles in opera

the machine.

m,lcl~i~ic such

as

normill pi~~~~ping, rcversed pi~~nping. ccntripctnl turbining (Fr;:ncis

l,,rt-,i~ic), cc~ltrifug:~l ii~rbini~lp (Fourncyron turbine)

:i114

11ic foitr br:lLin~ i-illlacs

~IISC~IC~

in

bctwccii the :~forcmc~~lioned nlodcs

of

proper m:lcIiine opcr;ltiori

nith

thcir houi~d;~~)

collditio~l~, a~u~la\~ay-(zuo torque) or zero-ht.;id-lines.

irlJtllr;~lly tlic~e diflcrcnt motlcs ofopcr;~tion may :~lso be reprcscntccl in

a

I],

Q-plane. tio\\cvcr. hcrc

the

diflercllt modes of

operation

with thcir lirres of constiint spcctl or constiint gate opc~:ing may

o,,rl;lp c;~cIi othcr in certain rcgions. Then the uneqt~ivocal assignment of a ccrt:iin point

(H,

Q)

to

,

,,-,tail1 mudc of opcr;ltion of the machine docs not exist any niorc.

.it

cert;iin nlodcs of operalion, loops in the curves of constant speed or glttc position \\.Iiich may

,,sign

t\vo diffcrent flows to

a

given licad or

a

given gate opening.. are supposccl to causc

a

d!.-

r;3mically unstnble bchit~iollr of the set. Such

a

conclusion has to

bc

based

011

thc scl's \\,orking

IL,getIicr \i,ilh its piping, its grid. its governor

in

connection

with

the incrti;) of the set and the w;lti.r

hammer of the piping.

9.2.5.

The

cam

curves

of

double

rcgulafed

turbines

Fis.

9.2.6

~llo\\~s

a

set of so-called "cam curves" as a p!ot

of

efficiency

1,s

flo~v

iunder con-

stant

riil;ner vane position

(p

(proportional

to

/?,),

the whole

at

conslant head. Sin~ul!a~le-

ously

thc dingram contains also lines

of

_gate openins

(1,

(or

Y,)

vs

flcw

under constant

runner vane position, all at constant head.

Fig

9.2.6.

Cam

curves

(11

((I,

,)-

curves.

and

gilt~

o,

vs

flow

6'

(@,I)-~i~~\~s). hot11

at

cvI1st;int

c

posltlon

(,oL

of runncr bladc

lor double rcpuliited turbines

(Kaplan tilrb~!ics, tubular

t

ur-

bines, bulb turbine5, Str;tflo tur-

bines. diagonal

(Dcria~) tur-

bines).

The

m:~xixum of the cfficicncy, is located on t!ic c:lvelope ci~r\~e for tlic incli\,idt~;tl c;trn c.t~r\~i.s.

A

perpendicular

from the point of contact of tlie crivclopc curve

tvitli

;I

dcfinite cam clrr\~ in~ersccrs

the line of tlic

sate

openins bclongins to thc same

cp

;it

;I

point \vliicli dctcrmincs thc most effective

€ate

opening for

:i

given runner vane angle (Fig.

9.2.6).

It is reco~nizcd that

a

double regulated turbine (of ihe

k"ipl(rtr

or

I)cr.icl:

typc)

has

in the

whole

range. between

0,2

Q,,

and

1,3

Q,,

3

suycrior efficicnc) than

;i

single regiilated

turbine with adjustable runner vanes. Othernrisc such

n

single regill;\ ted tarbinc

has

also

a

superior efficiency over the wholc operating rang tllari

:i

turbinc rcgul:ited only

by

\vicket gates.

The

ilssignment of gate and runner vane position

is

by

a

cam connecting

(1-72.

11.7.8)

tlie

control

loops

of gate and runner vane scrvornotors in parallel or in series.

9.3.

Head

ant1

eflicicllcy

lneas~lrenient

by

the

t~iern~otlyn;rrnic mcthod

and

the

rlsual nlctliod

Strictly speaking the spccific head of

:I

fluid machine, the work per unit mass of the fllid,

converted

withill the machine from fluid energy into shaft work and loss implied, or vice

versa, is given by the difference of total enthalpy

i,

betwecn the prcssure

(11)

and suction

(1)

llange of the machine (Fig.

9.3.1

a)

Y=

di,

=

d(p/~

+

e+

c2/2

+

gh):'

(9.3

-

1)

where the terms are defined in Cap. 1.6.1.

A

strict treat~nent of the last relation should also

account for the change of density

g

within the machine. Under atmospheric pressure

of

water is lo3 k_g/m3. This ~aluc call

be

used approximately for the suction flange

I.

To

obtain the density

Q,,

in the pressure connection, the simple equation of state

(5.3-5)

is

used. In Cap.

5.1.1

it

was shown that the ;rtual Lighest head of

H

=

1760 m would result

in

a

compression

of

,,I

e/,o

=

0,86

-

10

-

2.

.g

e

Ho~vever, accoonting also for the change of internal eilergy

A

ril, with its sign, adverse

1

to

that due to the error of the d(p/Q)~l-term, then the error due to lleglectirlg the corn-

$

pressibility of water under the actual highest head approximately amounts to one half

of

the above figure.

4

Thus

a

short recapitulation of basic

thermodynamic

principles necessary for

fluid

nl;~chines is now made. Neglecting heat flow

(rlq

=

0) into ?he machine, the first law

of

thermodynamics

liq

=

tle

+

pclv,

u

specific volume, gives the change of internal energy

as

%1

t

The efficiency

ili

of

a

high head machine is proportional to

a

More concisely the influence of compressibility results from the so-called control surfa

11

work

1'

ctlp

as

the surplus of displacement work per unit fluid mass due to admissio

I

expansion and exhaust of fluid and is represented

by

the term A(?/@

+

e)il in

(9.3-1

which

at

vanishing heat input and small compressibility of the fluid is converted into

11

A(~l,o

+

4:'

=

J

L)~P

,--

h1

-

PI) [VII

+

(1/2)

(0,

-

v11)1,

1

Able

+

4:'

=

[bll

-

~~)le111[1

+

(PII

-

PI)I(~EL)I-

No\\: neglecting compressibility in the example of Reisseck,

H

=

1760

m, would yield only an err

of

+

0,42%

in the calculation of eficiency. In practice the error made by neglecting the expansio

work

due to compressibility of the fluid is superirnposcd by other effects. When the dissipation

9

I

the mnchine heats the fluid, the change of its internal energy becomes el,

-

el

=

@

+

I

pdv.

I1

The non-dimcnsional loss

hi

of a machine heatsits fluid of specific heat

c.

The coeflicient of

cubi

expansion of the fluid results in an addition:~l expansion work

ph;(g~)~/!2c)

(Cap.

7.3).

In relati

to the expansion work

due

to compressibility, the latter becomes

f

=

2Ph;

E

J(Qc).

For water at

20

with

c

=

4187

m2/(s2

K),

EL

=

2,06. 109Pa,

=

1,8

.

10-'K-',

the usual loss of

11;

=

0,l

results

j'

=

1,765

-

lo-'.

Below

4

'C

f

becomes negative.

334

Thccl;~sticity

of

tl~e wettcd ducts cxposcd to thc prcssurc causcs another error

in

the cxp:lnsion work.

Asj,:lni~ig \v;ltcr hnrnnlcr in

il

pcnstock

as

a

model (Cap. 8.3), the bulk modulus I:,

!ins

to

bc

suh-

jlilutcd

by

thc value 1i,/[1

+

E,di(a

Eh)].

whcre

d

is

a

n1c;ln "di;inictcr" of the d~lct. 11 thc

thickness

of

its

willl,

E

thc Yol~ng mo~lulus of thc

wail

~natcrial and

n

a dcsign-linked par;lmctcr. In larce

,,,achincs, the sccond tcrm in the square

bracket

is always lnrgcr (!Ian

I.

Hcricc tl~c influcncc

of

the

error

duc to cspansion with

EL

in the dcliominator becomes larger than shown by (9.3-2).

The

mensuretncnt of the static pressure (simply pressure) call be made only by tappings,

uniformly

over

the periphery of the flow passage of interest. thus resultinrr

-

in

,

cross sectionnl-avcraged value. Thc method is only applicable, when the streamline

,,,\/ature

in

this cross scction is negligibly sn~all so as to make any incrcase of thc

pressure across the passagc due to centrifugal force irrelevant.

~hc

flo~-a\~ernged square of the velocity at section

1I

and

I

of arbitrary cross section

,4

can be found by measurement of the iocal velocity

c

under the assumption of steady flo\v

with

Q

as the flow

Q

=

5

c

d~.

A

In thc simple case of

an

axial

velocity profile, e.g. of the type

c,(l

-

r/R)"'",

r-

distancc from pipc axis,

R

pipe radius,

c,

vclocily on pipc axis (for

m,

Cap.

5.4),

the time consuming measurement of the

within the passagc is reduced to that of the vclocity on thc pipc axis

c,.

Hence from (9.3-3)

Thus for

trr

=

7,

c2

=

0,7058

ci.

The measurerncnt of

c,

can

be

madc with

a

flow meter. An

cxact

mcasurcmen! of the velocity c.g., by means of

a

PranJtl

tube can bc made only under

a

stcady flo\v

regimc.

Owing to turbulence, the real flow is unsteady. The temporal distribution of the velocity

is

assumed to be

c

=

c1

+

c2

sin

cot,

(9.3

-

5)

with

c,

and

c2

as

constants. Since the

Prar~tlll

probe measures

--

the time-averaged dynamic

pressure

as

a

basis for the ~neasurement of velocity

c2

=

c:

[I

+

(c2!c,)'/2],

an

en-91-

of

(c,/c,)'/~

is

made. This error may be negligible if the level of turbulence

is

small

and

if

the kinetic energy term is small as colnpared with

gh

+

pie.

For high head machines the term

cf

usually is not ncgligible as compared with the hcad

gII

-

p,,;~),

on the other hand in such a machine the levcl of turbulence

at

the inlct (11) given

by

(c,/c,)~

falls

short of

10

".

This docs not hold true for the o~ltlet of the machine (I) especially when

it

operates

under off-design load.

Here the degree

of

turbulence can easily reach

20%,

which includes an error

d2%.

Howcver, this is also negligible, if we co~lsidcr a high head machine, where thc kizetic energy

at

I1

or

I

fiills short of

6%

of the specific head. Moreover the cross sectional arcas

A,,

and

A,

are

usually of the same ordcr so as to cancel the difference

ci,

-

ci

in

(9.3-

1).

However, for part load operation of the machine, account must bz taken of the fac!. that

by

stall the main flow occupics only

a

portion of the available cross section of the draft

lube outlct. Once the sections

A,,

and

A,

are equal and the

stall

blocks

50%

of the section

A,,

in a machine, where the kinetic energy at

A,

under regular flow is only

1

YO

of the head,

no\\.

the kinetic energy becomes

4%

of the head.

And

this would yield

a

surplus

of

3%

of kinetic cnergy at the exit, which is not used.

.

The latter is a considerable amount, since at continental price level the revenues due to

1%

loss at rated load may be of the order

of

the cost of one set

lo).

-

10

Tllis

statement

for

I

%

loss

;~t

ratcd load

is

equivalent

lo

3%

loss

at

3026

of

rated

load.

7'0

avoid unr~cccss;~ry troi~blcs ant1 ~nisunticrst;~ndings tiul.iug tllc cv,~luatiun

of

hcarl auti cfficicncy

mcasurcm:r;ts.

IIIC

com~nittccs

of

IL:.C

li;~vc ;~gr.ccd sta~~i.!;~rtl proccrlu;cs

of

mc;lsurcmcnts. At this

intcr~ii~tion;ll Icvcl there cxist also st;intl;~rds

~.i(ll

rcspcct to [hr: sclcction

uf

tile Ic1c;ltion oftllc

section

I

and

If

for the indivi~lu;il typcs of ti~rbirlcs (scc Tablc 1.0.1).

Thc

Il:C

code also providcs rulcs whcrc to inst;~ll flow mctcrs for cfficicncy

tests.

For Cxilmplc, in

a

run-of-river plar~t will1 its slow cilrrcnt ,in tlic head pond :tnd through the thrush-rack (to reduce

loss tlicrc) the flow mctcr is to be niountcd in the exit section

I,

wlicrc thc vclocity

c,

should 11ot

bclow that ofthc tail water.

In

pump-turbines, whcre

;I

low thrash-rack loss during pumping require

a

low velocity

c,,

only thc vclocity

c,,

is ncccssilry.

1

lc~iuc the flow mctcr (also used for

Q)

is to

bc

mountcd in cross scction

11.

It must be clarified to the m;inufr~cturer, in what way one intends to control the guar-

antees of the

~naker. On the one hand the customer must follow the advice, that "trust

is good, but control is better".

On

the oll~er hand, lic must bc aware of the fact, that any

kind of guarantcc _given by the modcl test, cannot I;e obtained exactly on the prototype.

This is due to t11e fact ;hat machining, transportation and assembly of the prototype

is

subject to imponderabilities and unfoceseeable events, which cannot bc included in exact

formulas. Moreover full scale measurements on the prototype arc time-consuming,

expensive

and less exact than model tests.

9.3.2.

Fundamcx~tals

of

thernlodynamic

head

~neasurement

Vjith the enthalpy

i

=

p/g

+

e

and its increment

[9.86]

tli

=

(2i/dT),dT+

(Ci/dp),dp

=

c,tlT+

[t;

-

T(du/8T),]

tlp,

the enthalpy change in the machine can be written as

Ai

=

(l,/C)

c,,,

AT+

A,,,

u,dp,

where

C

=

4187

kcal/Nm,

T

is t!le absolute temperature in

K,

2,

the spccific volume

relnted ro

177

K

:tnd

1

bar pressure,

c,

the specific heat under cot1st;int pressure, related

to the state

11,

A,,,

the so-called isothermal coefticient related to the state 11 and defined

The coeflicient

A,

varies for water as shown in Table

9.3.1.1.

The spccilic hent

c,,

for water can

be"

obt.tinsd from ihc same table. These values were taken from

-4.

S.

Tllo~n

(scc Water Power vol. 7,110

3'

11965)).

Alllring

i~nd

Overli

(9.901

have found the

cueJ/icieni

A,,,

to depend slightly

011

additives

of

the uatcr.

It

should be remembercd, that the specific heat contrary to

A,

is known

in

the

low

t

perature range with an accuracy of

otlly

four decimals. This point may cause some

trou

in the application of the thermometric efficiency measurement after

Wil!m

and

Cainpm

[9.56];

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

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