Raabe J. Hydro power - the design, use, and function of hydromechanical, hydraulic, and electrical еquipment
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On
tllc otllcr
hnnd
;in nltcrnpt
is
1n3de
here
to
find
which
calcul3(inn ~~rcdicrz~i
rhczc
Icat rcsul!s most
,ccura tcly
ivitli
rcspcct
to
tkc solutions nv;~il;lhlc Irolii Sclr/ic.hrit~g [9.1
171.
Korthc. (4.1
I].
Frtlkv ['I.
1 1
S].
~,,fi~lrj,
19.1
191. Joc.1rt.111
(9.1201,
Ht~hr~to
(5).
12
I],
St~~.sc.hcl~~r:l;y
['I.
122).
C;llirrc~r.l
(9.
I??].
Bir.r~l,o~rr,r 16.61.
s~/I~~/I~III.s/
[6.7], Prrt~rt~ll
(4.10;
9.1241,
Urrz
(0.125], In~hcr~.lr
[9.126].
Kiilrr~c.l
[!I.
1271.
TO
(hc present such tcsts Il;~ve
been
carried
out
ollly
\\lit11
varying
pitcl~ to chord ratio
r
'L
by
Iflihtl
[9.128] at
corresponding
cylindrical sectioils
of
thc
rotor.
The tcsts described here wcrc made on different turbine runners with the same hub to tip
diameter ratio 0,5 and three, four, six and twclve rbnncr vanes as for esample on
a
Kaplan
As shown
(Fig.
9.6.1
a) the vanes are thin, have a sn~all camber and are geomer-
,jcally similar to each other in a certain cylindrical section.
~hc parameters of the straight cascades, originated from unrolling the cylindrical sections
inlo
a
plane; were retgined for sections on a distinct radius. Thus the pitch
to
chord ratic
H~~d angle
13,.
the zero
lift
direction makes with thc circumference, are rctained ivhereas
the vane number varies. For simplified test conditions tile runner operates
\\.it11
a \vhirl-
free absol~ltc flow at inlet. This was obtained by
a
flow straightener upstream of the
rnnner. For the same reason air has been used for the tests.
A
whirl-free admission facilitates also
a
good examination of the secondary flow resulting
from
the relativc whirl (Cap.
5.5.5.4).
Downstream of the runner cunsidsred, there is an
absol~~te whirl opposite in sense to the speecl of the runner (Cap.
9.6.3.3).
Therefore the
runner
vane needs
a
snxtiler twist thail that due to a runner with inlet ~vhirl. I-Icnce thc?
face of vane skeletons can be approximnted fairly well
by
spatially arranged bound vor-
tex stars on a conical
face, see Cap.
9.6.4.2.
An essential difierence exists between runners with no pre\vhirl
and
those ~vith ;vhirljn_r
inlet flow. Contrary to the first case in the second case the whi1.1 and hence the rrLdial
pressure gradient decrease with increasing boundary layer thickness in the streaniivise
direction. Nolv from this, two flow laminze within the boundary layer originate
\\.it11
oppositely
oriented secondary flow (Cap. 5.5.5.3).
9.6.2.
Test
devices
and
instrumentation
used
9.6.2.1.
Air
test
rig
The measurements were carried out in an open air test rig (Fig.
9.6.1
b). Doivnstream of
the
diffuser
7
the flow passes an axial
flow
straightener with four screens in series distri-
buting the flow uniformly over the inlet cross section
0.f
the turbine. The transition to the
test
sectio11 is formed by a nozzle with a contraction of
5
to
1.
An electric torque rcaction dynamometer was used to load the turbine (Fig.
9.6.2
a).
By
extension of the runner shaft supported at the stations
A
and
B,
the dimensions of the
dynamometer could be adapted to the form of the
hub.
Moreover this allowed ior an
overhung design, facilitating dismantling. Besides
this the extension of the runner hub
facilitates the inserting of the rotating probe without the hindrance of bearings. The
wakes
from the stay vanes generated in the generator region have nearly faded out in the
test section. Negligibly
s~nall stall effects occurring in the zone of intersection of stay vanes
and
the casing wcre climinatcd by
a
trip wire.
The following investigations have been carried out at Lehrstuhl und Laboratorium fur
Hydraulische Maschinen und Anlagen of The Technical University Munich, West
Germany.
Fig.
9.6.1.
Runner and tcst rig for investigations. a) Bladc design of the
3
vancd runner in elevation (above) and plan (below).
-
pressure face
ofthe
vane
---
suctio~r face of the vane. b) Schematic representation
of
the horizontal air test rig.
1
gauze;
2
intake nozzle;
3
settling chamber;
4
Venturi
meter;
5
diffuser;
6
blowcr;
7
diffuser;
8
rectifier;
9
gauze section;
10
settling scction;
11
casing for brake generator;
12
stay ribs;
13
nozzle;
13
turbine
-T
-
iuw
i
.-.
Fig.
9.6.2.
Dctails of the tcst section.
a)
Pcrspcctive sectional drawing of the turbine with an electric reaction torque dynamometer and a weighing
machine.
S
trip wirc;
A,
B
stations of support for gcncrator rotor.
b)
Dctail of thc leaf spring
suspension
of thc
generator
stator;
1
uppcr half ring;
2
lowcr Iialf ring;
3
bridging ring;
4
Icaf springs. c) Location
of
thc
measuring
cross scctions for mcasurcmcnt of the absoli~tc Ilow. Circular projcction
of runner vane:
--
.-z
3;
---z 4;
---z
6;
-z
12. Chord Icngth
L,
of a nlciln cylindrical scction, radius
r,;
L!:
=
chord lcngth
of
z-vancd
runncr.
Bl;~dc
numbcr
12,6,4,3
in
rcspcctivc ~ncasuring scctio~ls
211,
3/1,4/1, 5jl. d) Location of thc ~ncasuring scctions for thc mcasurcmcnt
of
rclativc flow
within a
runner. Circular projcction of runner vane:
-.
.-z
2;
---2
4;
---z
6;
-
z
12.
Chord length in
a
mean cylindrical section
for
z-vancd runner:
$
L!:,!.
Blade numbcr 12161413 in thc measuring scctiolls in thc runncr
12/13,
11/14.
10/15,
and
past thc runncr
13,
15,
16,
17. Aftcr
Kiilrnel
[9.127].
Thc greatest proirlc 111ic.kncsx clccrc;~\cs Iilic;~rIy ;ilons the r:ldius.
being
2.1
inm at hub
arid
13 nim at tip. Tllc pitch to clior-d ra~io
t
'L
wits :ls~ulncd col~stilnl
:IIIJ
C~LI~IS
to one
for
;!I!
tlic ritdii arid l>l:1i!i112.
For the clcbign tl~c
flo\v
\!it3
i~\surnccl
to
be ofcon.stiinl angilli~r momc~tlilni and hence
of
conbtant ericrgy
\\ill1
a
constant
axial vclocity
long
c;!cll rncllus ups1r::lrn the rotor. ~h~
perform;~nce daia arc
ns
Ihllows: Specific Iictld
Y-
900
m'/s2,
l1o\v
Q
=
5
m3/s,
Speed
11
=
1700
rpm. Tl~c
ticbign
\\.;is made unclel- thc ;lssii~nptioii of
it
p1:111c, in~onlpressib]~
no\\ without i11tcr11;11 fl-iction [9.124: 9.129: 4.101.
9.6.2.3.
.In.jtruincnts
and
probes
uscrl
1.
Probes for
:h\:
ntcasurcmcn
t
of absolute ve1ocit;l:
--
Six-l~ole probe: -Phe vclncil;! vector, llie to~al a~ld the
static
pressl.lrc
of
the assumed
stcady
rlbsolute
fio~s-
Lverc n~casurcd ilpstrcam an11 tlownstrci~m rhe rotor. To do this,
a
six-11olt: COIII-~~~! probe
[9.109]
was
liscd
iF'12.
9.4.3
g).
To
niiliilnize tllc error, measure-
In?nts
have
been repeated
by
means
of
a
two tingilr ~>i.obc (Fig.
9.4.3;1).
The head of the
probc has
a
diameter- of
4
mix.
1
I.
Rotating probcs for relari~x flow: The measurement of relative Ilotv I)e!~veen
and
past
ti?c rotor
has
been car-ricci oilt by mc;\ils of
~1
probc I-otr:tins tc~cthcr
\vitli
t!le
runner.
The
probe stem reaclics itito the
flow
passing
,r1
slot within the hub \\.all.
--
.-4cljustmertt ds\.il:t.s for rotating p~.obts:
Ttic
clc..;ip~
of
thc
:~~!jtlstrncnt device for
a
rl.?t;ltiiic~ probe
is
shdi\
n
in
~~rincipli: in Fig.
9.4.8
a.
--
Each of
[he
tappit~gj of
tlic
prol~
is
~Ol~ii~~t~cl
b~
1it)sc
~IIICI
brass
pipcs
\r.itllin
thc
liollow
shaft
v:iih
on?
of
tlic ch;imbcrs
of
~hc
prcssul-c
~1-ali5niittcr
1T.'i?.
9.4.3
h).
?'hi. 1:robt: srcm could also
be
:r;ustcd
Jul-ins
rotaiinn
ro
cariccl tlie
>;,;\.
angle.
1-his
\\.;I.;
effcctcd
by
~.IS~:II
~lispl;~ccnicnt
or
an
;1~!,:1s:ri:~:ri!
rod
similar
it)
Fig.
0.4.8
a.
co:~\i;ill:\. 1oc:trcd
ivithin
tlic shaft. This
rod
!i;ts on one
of
its
ends
a
bldtred
link-iikc
lcver
for
turning rlic probc about
its
racli:~lly oriented stcln by lneans
of
a
hlock-link
~nechallisnl.
-
Ilesigns
of
r0t3tii:g i3robes: Tc-o dcsigns of rotating probes Lvsrc LISCJ.
111
thc
11su:il
?)tie
(Fig.
9.4.4)
thc sttm
is
turned
thr~>ugh
n
short U-slial>e ;tnd
in
the other
fIrig.
9.4.5')
throl.lg!i
a
wide
U-turn. This
\\.us
~nndc to ob\iaic
an!
iniluencc
of
tlic total pressure c;tirsed
by
the radial stein on tlie head
of
probe.
-
Cnlibrtliion: The calibration of the indication of the relative flow ang!e
p'
was made
{
4
n.ithin the Ineasurlns section with thc rotating probe, but with the stltiner removed.
;
111.
kleasurement
of
flow: To measure the flow,
a
st:~ndardizcd Venturi meter
with
m
Q
iusa ratio
HI
=
0,j-I
(Fig.
9.6.1
b)
was used. The rhroughllow cocflicient was evaluated
by
c;~libratio:l test.
Thc velocity profilc tv:rs mcasurcd
by
3
J'rcirrclrl
probe
in
the narrou cross scction. Hence the
flow
\vas cbtrlincd
by
in~csration. Tlic dcviaticw of the results given
13~
the Venturi rlictcr
was
in the
range
of
2
0.240.
Angal:ir. deviations from the axial flow
direction
amounted about
+0,4'.
IV.
.Measuremznr of torque and
speed:
'fhe torque
ivas
measurcd by mcans of an electric
reaction torclue J~n;tmo:neter
(Fig.
9.6.3;1!. The fulcra of its stator colisist of two semi-
circulrtr rinss conn~ctccl nith e;~ch,other
by
a
pair of leaf springs norlniil to each other
r
Fig.
9.6.2
b).
[9.13
11.
7Phc uppt'r half ~-itlg is con~lected with the stator. 7'he lower
is
fixed
370
llc
~[;i~ion'lry n11d solid srlpport. With t11c aid of
a
c;ilil)rntion
it
is ensurcd
t11:it
thc
by'Spr~~lgs
do
lloi exert any to~.qilc about the turbine axis in the 7cr0 positioll
01
11111
Id
,,ighi]lg 111:1cl1inc.
%
c
ThC
of the generator rotor is supposted so as to make the friction:iI torque
in
these
warings
a11
iiittcrnal load
011
the rotor (see Fig. 9.3.1 b). The sensitiiity of the nlessurilig
&"ice amounts
10
0,05
%
of the smallest torque.
y.
Mcasiircnlents of the pressurc: For the measurement of the pressure, an inclincd tube
,,,,n~metcr a11J
n
projtclion manometer of the
Bet:
type were used (Cap. 9.4.1.1).
9.6.3.
Measurement
of
flow,
discussion
of
results
9.6.3.1.
Illvcstiga tions of
the
absolute
flo,~
1.
pteasuri~ig
programme:
Distribution of absolute velocity
c,
static pressure
p
and total
pressure
g
was measured upstream and past
the
runner. Tests carried out at
I5
stntio~~s
along
the radius by 6-hole and two-finger probes (Fig. 9.4.2) after axisymmetry of
floiv
was
tested. Each blading was assigned to
a
measuring section, at
a
distance of
0,4
times
the
chord length of the mean cylilldrical vane section from the plane of runncr vane
ases
(Fig. 9.6.2
c).
~.)ICV~I~CS
werc fixed on thc
hub.
Therefore
a
varying angle of attack of tlic rel;iti\.e flo\v \villi respect
10
the cylindrical scclion of \lane was obtained by the variation of speed. The range of
thc
Reynold.;
"umber varied bet\vccn
Re
=
2
.
lo5
ant1
3
. lo6
in which
Kc
=
lv,
L~Y,
with
1\.,
as
the undisturbed
velocity,
dclined by
~c,
=
Or,
+
n,)/2,
w,
and
IV,
being
the
reldtive \~elocitics at runner ii~let
2nd
ouilct,
and \vith
1,
as the chord. Hence any disturbance by hub clcarancc flow was rerno~fcd.
11.
Evaluation of measure~nents:
-
Computation of the velocity components: The absolute velocity
c
i5
obtained as
ill
Cap. 9.4.1.2
fl-0111
the dynamic pressure, sensed by the six-port probe
as
the differential
pressure between the total
prcssure
y
and the static pressure
p,
and
in the casc of
t1:e
two-finger probc as the difkrential pressure
pi
between the
two
ports of tl~c probe.
In
the case of
a
six-hole probc, thc spatial flow direction is fixcd
by
the two angles
x'
and
7
=
y'
(Fig. 9.4.4). Thus the three components of velocity are obtained from
C,
=
-
csincc'cosy
=
ccosacosy,
(9.6-
1)
c,
=
c
siny,
(9.6-
3)
r'
being defined by
a'
=
cr
-
n/2. The time-averaged measurements past the rotor show
that
7
is sinall, hence cosy
=
1.
In
the following text the subscript
1
is used for the section upstream of the runner and
2
for the section
2,
3,4, 5 past the rotor. The bar over any value characterizes the mean
value of a certain cross section.
Hence
Go
=
e
yo
=
(e/2)
5;
+
Do,
(9.6
-
4)
which was kept constant as
a
reference value during the measurement by speed control
of
the blower.
111. Analysis of measured results:
-
The volume flow: From the distribution of the axial velocity obtained by
(9.6-
1
to
4)
the
flow can be calculated as
The solution of this relation agrces with the flow mcasurcd si~nultaneously by
the
Ventu"
meter with
an
crror below
1
%.
--
Torque: Application of the theorem of moment
of
momentum gives the torque for
the
existins case of vanishing inlet whirl
as
The
torque from the weighing machine equalled that from the velocities
Kc,,,,
Kc,,
nleasured with an error up to 3,5
%.
-
Critical analysis of results measurcd: The valucs obtained by both thc probcs aforcmcntioned,
for
spczds of
1400,
1700.2000
rpm showcd good agreement.
Considerable
diffcrcnces exist between the
rc;ults of thc six-hole probc and two-fingcr probc for working points
with
large deflection of
the
:?bsolutc flow, e.g.
in
the cases of 800
and
1100
rpm, espccially in
the
hub region and for the smallest
:.ar,e numbcr
z
=
3.
The two-finger probe
was
preferred as the more accuratc one.
9.6.3.2.
Investigation
of
the
relative
flow
a
I.
~~lcasuring program: The measurement of thc relative flow was carried out at spccds of 800 and
1400
rpm since the bcp was
in
this range. The measuring stations were distributed for the runnen
n.ith
3.
3,
6
vancs
in
three planes normal to
the
machinc axis (Fig.
9.6.2d),
the first scctions 10, 11,
11
being close bcllind the rotor's inlet edge,
the
second stations
13,
14,
15
upstream of the runnet
ssit
and the third
past
the latter. Due to
a
lack
of
space, thc mcitsurcments of
the
runner
with
12
..-anes
were
lirnitctl
to
thc flow past thc rotor, scction
13.
Thc location of the mcasuring s:aticns rela-
ri\c
to the cascades of the unrollcd cylindrical sections collsidcrcd can be seen
in
Fig.
9.6.2d
and the
ioilowing corresponding figures.
11.
Evaluation of measurements:
-
Corrcctior,h clue to centrifugal force: The effect of centrifugal load on
thc
air
columns encl
cet\\.ecn
rhc
probc head and manometer as well as thc Mach numbcr effect under isothe
bci;s~iou: of the gas wsre accountetl for.
-
Criticcll examination of the measured results: For an estimation of the disturbances
by
thz stem of the rotating probe on the relative
Flow,
the measurements at
800
rpm
have
been compared with
a
specially shaped six-hole probe (with wide U-turn in the ste
Fig.
9.4.5).
The results with both probes coincide also in the wakes. Only at very sm
distances
between the probe tapping and the runner vane face, the usual form of t
six-hole probe indicates
up
to
two
degree larger or smhller
flow
angle than that indicate
by the probe with
an
LI-shaped stem.
This
is
due
to the fact that the stem of the usual probe displaces the flow from the vane..
111.
Pt-lotographic observations of the boundary layer flow: The flow direction within th
boundary layer adjacent to the rotating wall has been visualized under stroboscopic
illun~ination using eiderdowns fsmall feathers) fixed at one end to the vane face. Due
irs small mass in relation
to
its large flow resistance the ratio of centrifugal force
rssistance force under
1400
rpm, for instance
is
only
1
in
20.
Therefore a photosrap11 of
a
rotating vane carrying eitierdowns illuminated strobosco
cally facilitates qualitative analysis of the flow field close to and within the boundary la
of the rotating vane (Fig. 9.6.3a) relative flow
is
from left to right.
3
72
1700
rpm
yr"TI---uc..-
--.I---
1
ZOO0
rpm
--..-
n.--
n
---.-r--..
-+--''q
Fig.
9.6.3.
Visunlizcd flow closc to
thc
riinncr vanc f;~cc.
a)
Flow
direction
in
thc boi~ndary laycr on the suction facc of the runncr with
3
bladcs: fabovc)
and
I?
blndcs (bclow)
;it
spccds of 800, 1100, and
2000
rpm \~isu:~lizcd by Eidcr down tufts pnstcd
at
onc
of
tllcir cnds
on
thc rotating runncr vanc
facc,
b)
Comparativc ccntrirugnl
tcsts
with
a
woolcn tlircnd
and
Eidcr down fixcd
on
a
rotating rod. Abovc: down and woolcn thrcad
at
2000
rpm,
down nc;lrly trlngcntinl in flo\\l tlircction; micldlc: do\\.n
:~t
1400
rpm
:~g:~in r?carly tnngcntial; bclow: Down
and
woolcn tlirc;~d at 800 rpm.
down
ag;~in
nearly
tangential
in
flo\v
direction.
After
Kiilr~tc~l
[0.1271.
b]adl: speed
Ku.
This means thc
flow
was ~cnrly undeflccted
(Fig.
9.6.4)
Under
:I
spceci
of
j
400
rprn cvcn a pos~tive absolute whirl of
2
to
4%
of
Klr
occut-s.
An
incl-ernent
of
h'
\r.,,
(llle 'ic1;itivc whirl) lownr(ls the suction face indicates tllc bcpinning of
flow
dcficction
,,,itllin the cascade.
The
cul-ves of the static pressure, only depending on radius and speed,
sl~o\\~
a
pronounced
of
low pressure caused
by
the flow around the head
of
the ~>rofilcs
in
the
,,ighbourhood
of
the suction face. In connection with this, steep gradients
of
the flow
fl
and the axial velocity
Klv,
are observed.
~lle pressure rise following the reduced pressure causcs
a
stall in the speed range between
800
and
1100
rprn on the suction face past the inlet edge of runner vane. This can easily
bk ;cognized from the orientation
of
the eiderdowns pasled to the suction face
(Fig
9.6.3).
The rctation
of
the stalling region is seen
to
be opposed
to
the sense
of
tlie
runner rotation for the case of the 3-vane runner
at
speeds
of
800
rprn (Fig.
9.6.3).
Thc
staIlcd region is subjected
to
the influellce
of
centrifugal force and grows therefore from
the
hub towards the casing wall. The length
and
thickness
of
the stalled zone grows
\\lit11
&creasing vane number.
-
Flow effects within thc boundary layer: The course of the flow bctwecn the vanes and past them
is essentially inflr~enced by
secondary
flow
in
the boundary layer region (Cap.
5.5.5.4).
Rec;~use of the
pressure
trou_el~ i~nmediatcly past tile leading cdge a turbulelit boundary layer on the
whole
suction
face may
be
expected.
AS
it
has been found that the \+.hid
Kc,
increases along thc runner vanc in the streamwise direction,
the radial pressure grcdicnt of the main flow also increases towards the casing. Vcry rol~ghly
Since in an axial runner thc
Coriolis
force and centrifi~gal force due to rotation :Ire nearly radial in
direction, the pressure
gradient
dyldr
outside the boundary layer acts through it.
A
dccrcase of thc relati\-c \clocity within the boundary layer froin
\c
to
w'
(Fig. 9.6.5a) causes a
dccreasc of
the whirl from
c,,
to
ck
with
angle
fi
retained.
This leads by rclntion (9.6-7) to
a
disturbance of thc radial equilibrium, which exists in the main flow at thc outer edgc
of
the bounda:~
Fig.
9.6.5.
Volocity trianglcs of
a
runncr with zcro prewhirl.
a)
Vclocity triangles for an axial runner
with
zcro preullirl
in
the rcgion of boundary layer under the assumption of a plane flow parallel to
[he v:tnc f:~ce.
b)
Velocity triangles in three dirncnsional flov! (flow in bctwcen irnd downstrcan~ of
'he
runncr vancs).
c)
Inlet
and outlet trianglns (assuming
ii
floiv on coaxial cylindriitl stream faces,
Cv
=
0).
After
Kiiht~cl
[9.127].
I;~>c-r.
l11c Illilrl c'l~t~:ct?t~
111
tI11:
cblttcr rcgio~i
oi
thl:
bo111iJ;iry Iaycr liiovc tow.irds tlic
;\XIS.
hi^
.I
~;OIISC,JIICIICC
ot'
111~
I\
1111
1
h
(.,
itlcrc,isilig
iri
I
Iic >trc;lm\vlx tl~rccl Ion
and
t1111s c.iuSil~g ;in
incrcnse
of2t;itlc prz.,\iIrc :o\i,110,
ill(:
1111
(1-ig.
9.6.5a).
I<;iril;rl
prcssurc gradlcnt and boilnd;iry laycr
thick
11icr1:;i~c
ill
1112
SI
rc';~,llwisc dit c--lion.
very
cloxc
I(,
11ic
V,IIIC
f;tcc tlic fluicl c1cnic1-its in tlic boundary 1;iycr arc !Iling tow:irds the casing
wall
In this rcpio~i tlic rc'l,itttc vclocity
\V
dccreascs ~owards the v:iluc
w".
TIILI.; wirh
>
c,
the cclltrifug.
force r,';r dc!l~inntcs ovcr
tha
pressure gradicnt dp/?r from outside thc bound:iry layer, resullin&
from
ci/r
3s
in
(9.6-7).
If
thc bl,~tlc
:\pexi
I,
oil the innzrniost radius cqilals
the
whirl vclocity (.,, as niay happen with spccb
of
800
and
I
I00 rpm. thcn
the
layer with thc dominating cclltrifugal forcc completely disappears
and
the
boi!ndary I:i)cr n?ateri;~l is :lccumulated at thc hub. In othcr parts of tlic runner the boundary
laqe: licvt to thc w,tll is ccn~riiugcd and the external boundary laycr is sliiitcd towards the
hub,
The tip clcarr~nce vortes: I?ctween the tip of the vane and the casing
n
gap exists
with
;l'
cltilrancc
of
0.8
min.
T!IC
lcakagc through this gap is wound up together with the
ten.
trifuged boundary laxer fluid on the suctioli face into a vortex, transported streamwise
bq
tl~c siiiroundin:
flew
ancl rotating in the seqse of the runner.
The hydrodynamic paranietcrs i~ifluenced by this tip cleararlce vortex are shown
in
Fis.
9.6.4.
Its centrc
line
coincides with a minimum of total pressure. The static pressure
rezchil?g
i~s minimum at the same station increases parabolically on both the sides of the
vortex i:xis. The vortex causes slsn
15.31
an
aclciitional radi'll flow component
K
w,
with
rc~ erscd sign on both the sides from the vortex axis along a normal to the blade.
Bi
t\
convex plot of flow angle
vs
circumference hints at
a
vortex axis within the measuring
sec~inn
il-
ig.
9.6.6
fsr
3
vanes).
A
concave shape
(Fig.
9.6.6
for
4
and
6
vanes) indicates
a
vortel centrc line outside this section. Becausr: of the
!arge
loss of total pressure due
to
tf.2
tip
vor1c.x
the relative velocity also dccreases in the nc~ghbourhood of this vortex.
-
Relative
flo\v
before the outlet edge of runner vane: Outsid2 the boi~l~dary layer and
outside the tip clearance vortex the total
pressure
y,,,,'ij,
within the relative flow remain(
cor~stant past the inlet edge of the rotor. Since at
14CO
rpm the static pressure
p/g,
grows
approximately
parabolicaily vs the circumft.rel:ce from the suction to the pressure face,
the relative velocity drops linearly along the unobstriictcd pitch. The curves of the
axial
component
Kit.,
are nearly constant, with
a
slight growth tov:ards the prebsurc face.
Therefore the variation
of
the relative velocity is cacsed mainly by the whirl component
I<
\v,,
whose linear variation along the circumference is sigrlilicarlt over the whole measur-
ing range.
Mcnce tile relative flow does not fo1lc;w the dirzctio!i of the vane.
At
800
rpm
the growth
of
the static pressure is restricted (more than that at
1400
rprnj to the region
close to
the pressure face with
a
smaller differential pressure between adjacent vane faces.
The
lirsc difference the component
KW,
has past the inlet edge betweell suction an
pressure fxe. is nearly conlpensated. The fluid elements ~vithin
a
plane perpendicular
the turbine
asis and moving with
a
nearly constant axid ~clocity rernain in rhis pla
Relative to thc annular sector formed by two adjacent vanes, the casing and hub,
flu
rotates with an an~ular velocity increasin~ together with deflection from zero at inlet
t
a mrlximum at the outlet in the same sense
3s
tile runner
[5.62].
As a consequence of
th
a
secondary flow due to the relative eddy is generated. This causes a positive
radi
velocity at the outer edge of the boundary layer on the suctio~i
Fact
and
oli the pressu
face
a
radial negative velocity (Fig.
9.5.6).
This flow can bc predicted according
Cap.
5.5.5
but with
a
C,
now negative because of the negative whirl
c,,
oriented
~gai
the speed
u.
376