Raabe J. Hydro power - the design, use, and function of hydromechanical, hydraulic, and electrical еquipment
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7.3.7.
The
flow
about
the
inlet
edge
~~,,;,~ly
a
stall or cavitation at or near the inlct edgc
of
a
vane occurs.
if
the
flow
sho~.s
3
n
angl~
of
attack
or
incidence on thc vanc, whence
it
streams
about
the
usually roundcd
no
,,
of
thc inlet.
As
sr;lll
results from the vclocity distributiotl
in
tlie boundary layer.
3
St,,dy ofsiich a
flow
rcquires knowledge
of
~lndisturbcd potetitial
flow
about such
a
rol111d
,,ntour. In
the
following
a
practical approach
is
tried
on
the base
of
a
plane potential
n,-,\v
tllrough
a
straight cascade with
a
nose having
the
form
of
a
parabola
of
tlie
second
mls
is treated approximately by considering sepnratclp the flow in the region
st
close to the
,I;,g~;~tion point
Sf
(Fig.
7.3.5a) and thc section
c
around the point of largest cur\,alure
C
(SCC
kle
7.3.5b) which coincides with the vcrrex of the parabola, and then combining
both
the types of
,lo,v
by
a
simple function.
1:ig
7.3.5. Flow around the leading
,nlcl ccigc of a vanc. a) Stagnation
point
St
distant from the vane's
Illlet
C.
b)
Equilateral hypcrbola as
;I
strcamline close to
Sr.
c) Approxi-
,nation of the
profile's
head
by
a
p;lriibola of the 2nd degrcc.
d)
Point
l',
by
11
level with the centre of prrs-
rurc
M,
where the velocity, induced
h)
circulation approximately
ccluals the value r/21.
-
?'he
flow
in
sf
close to the stagnation point: This flow can be described by the complex potential
a,
=
u,,z2/2 (5.31.
z
=
x
+
iy
bcing the con~plex coordilrate in a
Gairssiun
plane. Splitting this illto
real and imaginary parts yields thc streamlines
YJ
=
.\-y
=
const as the imaginary part. For vanishing
Y
this degcnerates to the axes
of
coordinates.
The complex velocity
\G
=
dm,,/dz
=
n,,z
yiclds the velocity cornponcnts on the
streamline
to the
stngnatioil point and then from
it
along the contour
by
w,
=
-
a,,y and
w,
-
u,,s.
Assitniing now the varlc contour past thc stagnation point
St
as the x-axis, the relatiire velocity along
Ihc contour past the stagnation point reads approximately as function of the distance
x
from the
slagnation point
-
'he flow in thc scction arourld the vertex
C
of a parabola: Now the complex potential reads
m,
=
2dcz112.
In polar coordinates (Fig. 7.3.5~) the contour strcamline as the imaginary part, squarcd,
leads
Yj2
=
ZaJr(1
-
cos
rp)
=
const. where
r
is thc radius from the focus
(Fig.
7.3.5~) to
a
point on
'hc contour,
cp
the angle r makcs with thc 1'-axis.
Pulling r coscp
=
x',
r
=
+
y'2)1/2 and introducing the parameter
p
=
'P2/(2aj),
yields the \\~cll-
linon.n
equation
of
a
parabola,
y'2
=
2p(x8
+
p/2), whose vcrtcx is shifted from
the
origin
of
thc
&)ordinate system by ,112. Now thc cornplrx velocity is
E
=
c~,/z'/~, whence the velocity components:
-.
-
uc
cos
(p/r112,
\vy
;=
a, sin cp/r112 and
-
Ccrnibining tllc vclocitics
Us,
;llld
Uc
of both tllc sc~:tioris
SI
and
c:
Thc contour vcloclty bctwccn
st and
c
is
u
=f,
Y,
+.f*
vc
-11
n,,x
+f2a,lr'12
(7.3-95)
whcrc
f,
and
f2
arc arbitr:~ry stccldy functions along thc contour with thc property of Converging
to
f,
-
1
and
/;
=
0
at tlie point St ancl convcrgirig to
f,
-
0
and
f,
--
1
at thc point
C.
Flcnce
u
rcprcscnts an exact solution only at the points
St
and
C.
The following functio!is satisfy thcse con-
ditions.
n r-r,
.
nr,,-r
f,
=
sin
-
-
,
f2-s~n--.
2
ru
-
rc
2
r.,t
-
rc
To express r in (7.3-95) in tcrnls of the contour coordinate
x
(takcn from tlie stagnation point
st),
*
rjx)
has to
bc
known. Assurning
a
point on thc contour, distancc
5
from
the
vertex (Fig. 7.3.5), then
3
by Pyih'rgoros's thcorcm,
r(y)
follows from tlie local thickness 2y of thc profile. Hence
r(s)
can
be
f
reduced to y(.r).
Approximating the vane between the p3ints
C
(leading inlct edge) and
St
(st:lgnation point)
by
a
parabola, the ir~vcrsion
x(y)
can be expressed as an arc length of a parabola between an arbitrary
'
poi31 with profile thickness
2
y
and the stagnation point with profile thickncss
2y.,,.
For a parabola
of equation y2
=
?PC.
~(y)
follows by
x
=
WZ) U~<[Y,.IP
+
JGGZFII[YIP
+
,/l+(yi;;jiij
ilcncc also
y(x)
and r(x)
=
,/-
=
Jm
for an arbitrary polnt a distance
C
from the
leading edge, taken in the direction of. the vane skcleton.
-
Determinntion of the parameter
p:
Since the profile has at the stagnation point at
c,,
the local
thick~i~ss
t,,
=
2y,, the parameter
p
of the parabola bccomcs
P
=
t:,l(S;,,).
-
Determination of the parameters a, and
a,
in (7.3-95): The flow in the section
st
appsoxinrltcd velocity
U,,
has also a vclocity that rises linearly with the distar~ce from the st
point
dong that stagnation streamline which is orientated upstream of
St.
This fcatur
extended als~ to thecurved part ofthis streamline. At a known distancc
yo
from
St,
the vclocity
a
the stagnation streamline is assurncd to have attained the rclative vclocity
U,
before the entrance
the rotor.
Hcnce
a,
=
U,/Y,.
To
obtain an idea how lnrgc a
yo
to choose, the rotor cascade is approximated by a latt
(Fig.
7.3.5d) whose plates are inclined by an angle
P,
to the circumference and have a chord
L.
is well known, such a la!ticc can be approximated by
a
row of straight vortices each locate
chord of
a
plate at a distance
L/4
from the inlet edge [5.3].
The vclociry field of such a vcrtex row is known: Hence the ratio. of the velocity
Q(q)
induced
distance
q
froin the circumference of the vortex row, level with
a
vortex, to the velocity Ui, at infinit
befcre the cascade,
L((l1j/Ui,
=
coth(nq/t).
Approximating now coth(nq/t)
2
t/(nq), the quotient U-JU,, tends to unity, if
r,~
tends to
q,
giv
by
qm
=
t/n.
That means: In a lateral distance: t/n from the vortex row, which represents the cascade, and
I
with vortex. the relative velocity attains nearly- the value uniformly distributed along tlie circ
ference.
As will
be
shown below, the distance of the stagnation point
from the profile's head (vertex o
parabola) dcpends
as
b:lows on the physical angle of attack
b,,
the undisturbed flo*:~, velocity
makes with the plate
ol the lattice
268
&,
=
s;
I
+
(5;).
(7.3- loo)
A,,"lllC
lhal the stre;~mlinc towards the stagnation point starts tnngcntirll to
(inclined
at
pz
to
tllc
~~~~crirnfercnce)) at a distancc
q
from thc vortcx row, which models the c:lrc;ldc. then m;tkes a
q,,ilrtcr of
:i
circlc
(011
bcliiilf of the srnallncs\ of
(5,
relative to
/i,)
;t~id ends normal to the plate
;it
xt,
the
:ire
lcngth of this circle yiclds
yo
=
(t/2)
[tiin P21(l
+
tan P2)] [Ilcos
P2
-
(~14)
(L/I)
+
n(<,,/t)].
lnscr!ing this with
respect
to (7.3-99) and (7.3- 100) in (7.3-95) givcs the dcsircd parameter
(2
I12/L)
(I
+
tan /I,)/tan
p,
asr
=
-
--
l/sin
/12
-
(LIt) [(n/4)
-
niii/(l
+
831'
1-he p~aneter
a,
in (7.3--95) rcst~lts from the known velocity
Uc
(in relation to the undisturbed veloc-
itv)
;it
the leading inlct edge as
tIcre
r,
is thc distancc of 11ic focus of the parabola from its vertex. Usually the radius of curvature
R.
is known for the lending cdgc of the vanc. According to a theoren1 of geometry for a parabolz-like
In
Cap.
8.2.3, the velocity
Uc
(there denotcd by
iv,,,)
is reduced to the blade velocity
u
(Kaplan tur-
bine)
by
mcans of
k
.=
L',llr
(8.2.3-
19).
(8.2.3-21) shows
k
as a function of the cascade parameters
r!L,
z
and thc curvature 1/X the vanc contour has at its leading edge.
*she distance
<,,
of thc stagnation point from the inlet: As a simple approach to the effects under a
strong angle of attack. thc profilc is motlcllcd by
a
plate of chord
L.
After Bir-nbarrtt~
[6.6]
the density
;,(<)
of bound vortices uridcr an angle of attack
h,
bctwcen undistarbcd velocity
U,
and thc plate
reads
at
stations clow to the leading edge (sec (6.2-30))
y(<j
=
2b,
U,
[(I,
-
(7.3- 103)
As the contvur vclocity
givcn by
U
=
U,
-
?(;)I2
vanishes at thc stagnation point
st(c
=
<,,)
according to
0
=
(1%
t
1
-
80
[(I-
--
~s,)/~.,,]1'2
1,
(7.3
--
104)
this gives
t1,e
clist:~ncc
(:,,
of the stxgnation point from the Icading cdgc of the vane according to
(7.3
100).
Eflects
of
clustic
walls:
Hitherto the wall has been assumed to be inelastic. \Vith machine
size
increased, any wall bccomes elastic. Then the flow along the wall depends mainly on
its
lateral vibrations, and not so much on the boundary layer development. These vibra-
tions
are either excited by unstcady flow or by flutter.
E.g.,
at
a
trailing edge, the latter
may
originate from the formation of
a
Knrntarz
vortex street.
8.
Cavitation
armd
water
hammer
as
detrimental
effects
8.1.
Introduction
For given site conditions, cavitation and water hammer i~lfluence the design and opera-
tion of a hydro po\ver plant. They a!so affect the economy and reliability, because wear,
efficiency drop, possiblz loss of control, and wall thicknesses are to be kcpt within
reasonable limits. 90th are unsteady flow phenomena.
Generally both effects are related to each other, since water hammer may cause
water
column separation, the worst form of cavitation, which in turn may lead to reversed water
"
lianln~cr as its worst
consequence,
whilst any cavitation also causes watcr hammer.
Cavitation results from the fact that
real water with its impurities evaporritcs, when the
3
pressure frtI!s short
of
a
critical value. Si~icc the density of cold water is
10'
timzs that
of
i
stcam
at the
same
przssure and temperature, the sudden condensatio~~ of a vapour cavity
r
f
o:l
a
solid \\all causes
a
severe droplet impact there with the consequence of pitting and
+
rndtel
i,~l
dcstruct~on.
9
T11erefm-e the fur,d;:nientals for the onset of cavity for~nation and collapse and its final
impact
OII
matcrial, hrive to be studied with respect to different effccts on various
materials.
Cavity t'ormrction and pittings, their i~fluence on characteristics in reaction machines, as
ivel!
as
remedies to encounter them, have to be considered with respect to
operational
limits.
\lTater hanl~ncr rcsults from the elasticity of pipe walls and the compressibility of real
Lvater as
a
consequence of speed control at load regulation, under transients like starting
or
sudden load rejection of
a
set. The real phenomena with sets on different pipe systems,
their prediction by
the
method of characteristics and rernedies are given in this chapter.
Thcs
various aspccts of cavitation and water hammcr are reflccted by numerous refercnccs. Concern-
ing
cavi;ation. general sur\;eys are
given
in
[8.1
to
8.71,
from which the lirst four refecs:lces deal
with
turbornacl~ines. The follo\\,ins references treat special problems
in
~nachinery, and
(8.8
to
8.121
~iutncl:
in
pillnps,
[S.13: S.141
in Francis turbines,
[8.15]
in pump-turbines,
[8.16: 3.171
in
Kaplan
turbines, and
[8.1S]
in
Pclton turbines.
In
the latter
it
is
mainly
a
kind of droplet erosion.
The
pu;>lications
fS.19:
S.'rS]
deal
with
the problem of the existrncc of nuclei.
[8.21
to
8.301
treat
the
problcms
of
their
rncn~i~rt.mcnts and their influence, e.g., by ndditiws. The ~011tribuii0il.s
[8.31
to
S.461
dznl
\\-it11
gro\rth
and
coll;lpse of bubbles. The following rcfcrences treat scale effects:
(8.471
i~1tlucnc.c. of turbtllence.
[S.IS
to
8.501
rou~hncss effects,
[8.23]
influence of bubble history,
[S.3
I]
added
mass.
[S.35;
S.363
ens
diffusion,
[S.51]
pressure pulsntion,
[S.52]
c:~vitation type,
[S.53]
second
v~rticity,
[Q.15; S.541
tc'::silc stress,
[S.Sj]
boundary layer turbclencc,
(8.42; 8-56]
viscosity,
[8.34;
8.
S.??:
8.531
hc.;it
transfer
iS.591
other effccts observed,
(8.601
hysteresis,
[S.13]
head.
[8.40]
bu
conlcscence,
[S.61]
cavity sizc proportional to that of apparatus.
270
F,,rtll,:r
~?ublicntions are dcdic:ctcd to cvvitntinn erosion and cnvi!y coll;lpsc. n:imely
;I)
prediction
gb2
1,'
8.641.
h)
c:ivitation rcsi.;t:~ncc [8.h5]; c) tcsting of the lattcr
[S.60
to 8.691; d)
jc't
forn~atio~~
C)
~mcdirs :~g;iinst crosinn
[X.Z5;
8.71; 8.721:
f)
scale effcrts due to eraion
IS.73:
S.?J]:
;l,cchanism of erosion (8.751;
h)
its connccrion with hydrody11:lmic c;~vit;~tion
[X.7(;];
il
detection
;rcritic;~l poil:t by isotopes [8.77];
j)
prediction ofcclcriry ~vith respect to partial conrlcnsntion u~?dcr
prcrsi~r~
~LIISCS
[S.64] (Notc LIlat
1111:
spced of sound
ill
u
fluid, which rurrouadr
:i
coll~psiog bubble,
is
llceJ~d for the picdiction of the pressure pulse duc to implosion using Joukovsky's water hammer
h,rmul:l);
k)
cavitation in spool valvcs [8.78];
1)
collective
bubblc collapse [8.79].
Fin;,lly
solne authors Ilave considered the phenomena on large, quasi fixed, partly vantcd cavities
ly.80 to 8.831.
Concerning
transients and Mrntcr hammer, a survey is given
in
thc books of
C.
Ja~gcr
[5.84;
8.851.
nlc
~:;~rious characteristics of controlled and cavitating fluid mrichines with their surging draft tube
,,,,rt~x
and thc
different
modes as the origin of transients or pulsations arc trcatcd in [8.56 to 5.97j.
,rncdics for
preventing
or optimizing certain sources of excitztion in [8.93 to 8.961, the predictior~
Jnd
mcasurcrncnt of speed of sound in different types of flow
ill
[S.64;
8.97; 8.981. prcdictio~l
and
mcl~urements of penstock rnsponses in [S.99 to 8.1051. The influence of air is given in jS.1061.
dnlnpillg nlechanism in [8.107], water colum1.1 separation in [8.108 to 8.1 101, accidents by reversed
,+;,ter hammer in [8.109], and needs for
improvements
in [S.lli].
8.2.
Cavitation
8.2.1.
Sur~ey
of
the
fundarlientals
of
cavitation
and
resulting erosion
8.2.1.1.
I~ltroduction
to
the
various
p!~cnornena
and
aspec!s
C;l\litation
is
evaporation of a liquid, caused by lowering
its
pressure.
It
occurs
in
liquids
with
free and dissolved 23s at stations, where the pressure falls short
of
a
certain critical
v;llue, roughly corresponding to the vapour pressure. Before the proper cavitation,
the
liquid diffuses dissolved gas in the gaseous nuclei, when under lowered pressure,
e.g.
when
passing the critical zone.
The following phenomena can be divided into two parts:
a)
Formati011
of
bubble-, patch-, hose-, bubble ring- or sheet-like maitily vapourous
cavities, that rest on walls or float within the liquid,
in
areas
of
lowest pressure. These are
located on high points
of
closcd ducts, on the centre
of
streamline curvature,
on
low
pressure (suction) faces, on convex or vibrating wetted walls, and slightly downstream
of
waks
of
wall roughnesscs.
b)
Implosive, irreversible collapse
of
the vapourous part of the cavity in areas of rising
Pressure
by
sudden condensation, which is favoured in thc
case
of
"cold
water" by
a
liquid
lo steam dcnsity ra!io
of
lo'.
Simultaneously, this leaves the residual gas
in
smaller
cavities, slowly reabsorbed aftel-wards.
-
Dctrimen~al effects: Change
of
machine characteristics, efficiency drop, water column
separation; generation of sonic and ultrasonic noise, water hammer, flow dissipation.
quick sponge-like erosion of walls
by
pitting. For the engineer the most essential effects
ofcavitation, which he has
to
control are, material erosion, efficiency and head drop, and
'
loss
of
colltrol over the machinc or device.
-
A~pearance in
practice
and
consequences:
Excepting highest heads. any hydr;iuIic apparatus or
'""hinu,
which has
a
longer
uscful life, cspcciillly when
it
works away from its bcp, due to economic
cOnlpulsions, usually opcr;~tcs under limited cavitation. As extreme examples are ~nentioned: Super-
c;lvit;itin_c, ship propcllcrs ;tnd
inducer
pumps, cnvitating.v;~lvcs or
britrrri
norzlcs. Due to
this
liq~iicl
is
cc)nvcrtctl luc:~lly into n two-ph;~sc mixturc with predominant vapourous cavities.
subscclucnt "rcvc~.scd wnlcr
hammer".
Thus thc water colunin in
a
condcnscr loop,
c.g.
m
scpar;~tcd undcr sliut~lowl~
of
pump drive. Ql~ickly closing gatcs may cause water column scparatio
past them, follo\i.cd by tlangcrous collapsc oftliis cavity u~idcr rcvcrscd
W;I~CT
hamnicr. Dcstro
head covcrs, broken runners, power drop, power swi:ig, loss of control or at Icast noise, vibra
and material erosion
on
cavitatillg hydro turbines make evident the ~ffc'ct of cavitation.
-
Cavitation
a
high speed phenomenon: Cavitation
is
always combinccl with press"
and
vclocity fluctuations of high and highest frequency. Any observation of elementa
effects like bubble collapsc requires light exposure time intervals down to
10-
%.
Cavit
tion in the words of
Ile~-lnanrt
Fiittinger
is
a
phenomenon, which limits the transmissio
of any observaiion m:de on gaseous
fluids
to liquid fluids
[8.114].
8.2.1.2.
Nuciei
as
origin
I.
General remarks: Cavitation has its origin in the existence of gaseous nuclei, especial
in cold water. Such nuclei may be also within crevices of floating particles or station
~valls (Fig.
8.2.1
a).
They constitute the so-called "free
gas
content", decisively responsi
for the onset of cavitation. In cold water
(290K)
this amounts to only about
voi
fraction of the total gas content.
Fig.
8.2.1.
Nuclei. a) Ilarvey's model of the gas
remaining in a
liquiclo-phobe crevice of
a
suspended
solid body.
b)
Forces on a bubble wall. c) External
excess pressure
p,
-
p,
vs
bubble radius
R.
The lin
of the critical bubble radius
R,(----)
divides th
stable range (left) from the
illlstable range (righ
d) Mass conservation for the
motion of
a
spheri
bubble in
an
inconipressible fluid,
if
accounting
evaporation
(R
>
rr
(R)).
As
a
model consider
a
stationary spherical bubble, radius
R,
resting within the surround-
ins
liquid.
uniformly and rectilinearily moved, (Fig.
8.3.1
b).
The equilibrium
between
surface tension
G,
internal pressure
pi
and external pressure
p,
requires:
p,
=
pi
+
2
o/R
11.
Vapolrr
bubbles: In a vapour bubble, the pressure
pi
equals the constant steam pressure
p,
of th
liquid. An
accidental
growth of bubble radius R diminishes the
term
2
a/R.
Equilibrium
now
needs
a
smaller pressure
pi
than the retained stcarn pressure. Hence an unlimite
growth of bubble. Similarily an accidental dccrcase of bubble radius
R
leads to a bubble collaps
Thus a vapour bubSle is unstable.
Ill.
Gi~~t~olis blrbhle
within
a
~trs-yomeable bubble wall: Here the pressure
pi
of gas within the bubbl
increases with accidental decrease of bubble radius
H.
Consequently all the gas diffuses into
th
liquid. Any accidental increase of radius
R
would be continued by diffusion of
gas
into the bubbl
272
In
Illis contcut
a
bubblc model aftcr
Horvcv
may be mcntioncd
(8.201.
Hcrc the
ga5
is ;issun~cd to
bc
jnclu~icd in crcviccs
of
suspcntlcd bodics with liquido-phobc witlls
(t
ig.
S.2.1
a).
Now
thc tc1.1~1 due
surf;~cc lcnsion reverscs its
sign.
I
Icnce
p,
=
p,
-
2
a/K.
Now
gas
diff~~ses from thc liquid inlo the
11nti1 thc crcvicz is filled
\h8itll
gas. Thcn such
a
cavity n:ay supply
free
g3S.
\vhcu
the prcssure
in the liquid is lowered
at
tllc cntrancc into thc cavitation zone.
PI
1~
,:~oseolrs
on0 voporolrs
~~ttclei
with
\ualls inll~errneahleJor
gcrs:
Now
if
the gas follo\vs
prect gas law:
pi
=
NT/II'
+
p,.
Hence
~vl~cn the bubble radius
R
grows accidentally within the branch, fnlli~lg vs
R
(Fig.
8.2.1
c),
the external pressure required p, for
a
vapour.pressure
p,,
to be retained, falls short of
the actually existing pressure
11,.
This shrinks the bubble into its ori~innl size. \Vhen the
bubble accidentally shrinks under retained vapour pressure, the external pl.essul-e re-
quired
grobfs above the actl~ally existingp,. Thus the bubble expands at last to its initial
r;ldiu~
R.
Hence this falling branch of the p,(li) graph is stable.
On
the branch (Fig. 8.2.1
c)
of the
p,
(R)
graph, that rises vs
R,
the bubble: accidentally
disturbed, grows
either to infinity or shrinks to the radius
R*.
at the minimum of the
pph. Hence the largest stable bubble radius, possible, the so-called "critical bubble
follo\~s from
d(p,
-'p,)/dR
=
0
as
Thus the critical pressure due to
R*
[S.6]
I!
The
critical
pressure
as ajtnctiori ofjlorv velocity:
After tests
of
Pnrkir~
et
crl
[8.112], the
critical pressure for the onset of cavitation past the station of
minimum pressurg i~wr?te_r
dcpends strongly on th~flm Telociiyclv, (Fig.8.22)- ~hefoloiiily~ relation can be
cxtracted from these tests
where
p
=
3,3 and
C,
=
0,0137 kg
rn-"-'
sp-*
Negative values ofp,, hint
at
tensile stress in the water between the nuclei such
as
was
found by tests
of
Knrrpp
in real water, when this has been pressurized before, or also ill
pure water
(8.61.
Fig.
8.2.2.
P~lrkin's
tcst [8.112] showing the critical
Pressure on
a
hcmisplicrical body vs velocity.
P
probe
station.
8.2.1.3.
3Iqustion
of
n~otiori
of
a
bubble
rcbting
i~
tl:c
Ilt:itl
The dvn;~mics
of
sphcr-ical bt~bblc, radi~~s
K,
rcsting
I\
it
liiri
;III
inli:ii
tc..\i~~iall~
incompressiblt: liquid is bnscd on the niotlcl of
Rrr~.l(,i~l~
(S:lO].
~t~p~~l~11111 ]atcr
ol
Scri~vri
[8.43],
and
Poritzky
(8.441.
Continuity rciluircs,
that
lllc vc'locl~y
11
of
tllc
1.
due to sphcrosymmctric wall n~otio~l of a bubblc. raclit~s
I\',
ilcpcncls as follows
on
velocity
u
(R), the liquid has at the bubble wnll (I.'ig.
5.2.1
(1)
u
=
li
(I<)
~'lr'.
-
Relation betweeii the velocity
ti
(R)
and growth rate
I<
of bubble d11c
LO
its cvaporati
The mass growth rate of the vapour bubble, density
Q,
.
lit,
=
4
nc,.
1~'
R
mus
with the mass flow of liquid, density
Q,,
passing the bubble wall
lil,
=
4nR'
-
u(R)].
(By evaporation, the bubble is "eating tlirough" the licl~lid.) Introdacing
t
figure
E
=
1
-
gv/QL
(being nearly "cne" for cold water with
e,/eL
=
1C'-5,
and bcing "~ero" at the criti
point), the above mass flow balance gives
u(R)
=
ER.
Inserting this in (5.2-5) results in the velocity of the licluid due
to
bubble growth at
arbitrary station
r
as
u
=
R~
dc/r2.
For
E
=
constant, the flow is ir-rotational. Hence
t
equatio;l
of
motion of the spherosymnletric field
d~/2t
+
du/dr
=
-
(lie,-)
C7plJr.
Inserting
u
=
R"c/I-'
and integrating this from
r
=
R
(bubble
wnll)
to infinity
r
=
(where
ir
vanishes) at
E
=
constant yields
[~/(EQ~)](P~,,.
-
p:)
-
RR
+
d2(2
-
&/2),
at
which
p*,
is tl~e pressure at infinity of the bubble,
y,,,.
=
p(K)
the presbui-e
in
th
at station
r
=
K.
For
a
real Newtonian liquid, now assumed, with kinenlatic vis
by
means of the equilibrium normal
t~
the bubble surfrice, the pressure
y,,,
can
decluczd from the surface terision
a,
the vapour pressure
y,.,
the
prcssurep, of
a
perman
t13s
wi~hin the 'subble and the radial viscous tellsile stress, induced by the radial str
L
rate close to the bubble wall, being according to
Il'clvi~r
and
Sr01;~s
(Eq.
(5.4-6))
[5.2
5.22j,
[8.44],
and
with respect to (8.2-
5)
2
ti
o7u/c7r,=,
=
-
4
v
Q,
c,
RIR:
pbw
=pi
+pv-
2a/R
-~EvQ~~~/R.
Putting this in (5.2-9) results in
[l/(e,,
E)]
(p,
+
pi
-
pz
-
2
a/R)
=
R
R
+
R*
(2
-
E/?)
+
4
YR/R.
For cold water the surface tension term
2
a/R
is relevant only in the lnm range of
R.
F
radii greater than
1
mm
aiso the viscous tertn
3
1,
R~R
vanishes. bloreover in con
of
c,.!,o,
=
lo-',
E
=
1.
.Also
pi
can be neglected for larger radii
R.
tlence, aft
(8.421, the last relation
may
be written
(l/~,)(p,
-
pz)
=
RR
+
3
~~12
=
tl(~~
~')j(2
R2
R
tlt).
The upper limit of growth rate
R
yields from
R
=
0
as
rim
=
[(2/3
e,)
(P,
-
~31
'I2
.
274
FroIll
this
it
is sccn. that thc growth of a bubble is mainly induced
by
tcnsile stress. n:imely
n,g:l~;~~c
r:ilucs of
pf.
This acts
ill
tllc pure liql~id, between thc bubblcs.
8-2.1.4.
Somc cstimalcs on
ca\if:ltion scale
effccls
at
onset
The
onset of cavitation requires
y*,
=
p,,.
Inserting
p,,
from
(8.2--4)
in
(5.2-
13)
brinss
,rnafine the critical zonc to have a length
yol,
with
I
as
a typicill dinlension of tile body
considered and
yo
as
a
figure
due to a certain geometry of the body and its streamline
Pi
,~rcr-n. Thc bubble passes this zone with velocity
\v,,,,
=
k1
w,,
in wllicl~
w,
denotes the
LIIldisturbed
flow
velocity in front of the body. As shown in Cap.
5.2.2.3,
kl
=
\v,,,,,/\\~,
jcpc~lds on the cavitation indcx
rr,,
due to the critical zone considered
(ith
omitted) by
During the time period
t
=
I
y,/(k,~o,),
in which the bubble covcrs the critical zone, the
bubble sadills increases from its initial value
R,
to its final value
R,
=
R,
+
t
R,,,
.
Hence
For cornparins siniilar phenomena at cavitation inception on similar bodies and under
flow (same
yo)
the incrcase of the bubble
Rj
-
R,
in the critical zone is assumed
to be proportional to the length
yol
of this zone. Accordingly
(Rj
-
R,)/(y,
I)
=
x
=
constant.
Putting this into
(8.2-
16)
and squaring the result brings
z2
=
(2 C2/3
Q,)
\vk-
'lkf.
Insert-
ing
k,
from
(8.2-15)
and making
p
=
3,3
after
POI-kilt
[8.112],
gives the real cavitation
index for the onset of cavitation due to a certain body
From
this
it
is seen, that at a certain proportional bubble growth
l,
the cavitation index
rr
increases
\villi
the vclocity
iv,
the fluid flo\\,s against the cavitnting body. This tendency is
reconfirmed
by
tcsts
(I.'&
8.2.3
a). Also the i~lcrcasc of
a
\ritli Io\vered density
p,
of thc fluid around the bubble
i3
rcflccred
by
thc
increasing
of
a
with rising frce air contents, also found in tcsts of incipient cavitation
(Fig.
8.2.8).
6)
a
&a
&
0
0
a
$9
-a
0
0
@b
zBBT
BziZF
4l(o
zzm%
?zmm
Fig.
8.2.3.
a) Scale effects at incipient cavitation. Cavitatio~~ indcx
ai
vs frce stream velocity at
dinerent diameters
d
of an ogival body after
J.
F.
Ripilken
and
J.
M.
Killen:
Gas bubbles, their
mcasurerncnt and influellce
in
cavitation tcsting. Proc.
IAHK
Symp. Scndai 1962
p.
37.
b) 1-0rm.s of
bubblcs.
.To untlcrstrlnd tlic I;lltcr dl-cct, in1:lginc th:ll lhc volume of thc
free
illr
colltunl ccp:~nds ~Onsid~~,~b~
'
Y
uithin the critic;11
LOIIC
of low prcssurc.
'The
grc;~tcr i~~tlii'fcrcncc ofn
vs
risillg
1(,1;11
air
content shows
;I
ccrtain inilcpc~ltlcncc of thc rclcv:int frcc air contcnt from thc toti11
;lir
contcnl. Note
thi~t
tIic
mass
of
the
free
air contcnt usu;~IIy
is
only about one t1iot1s;lntlth of thc tot;ll.
Any
CXC~~;III~~
ofr~llbs~~b~~
and free air by mass diffusion
in
thc critical zone nccds timc, usu;llly not rlv;til:rble during the quick
passage
of
nuclei across the critical zonc.
8.2.1.5.
Transfer of energy under
cavitation
The kinetic energy communicated to the ambient liquid by a sudden expanding bubble,
radius
R,
growth rate
6,
amounts from (8.2-5) with
11
(R)
=
d
according to
R(iyleigh
,
[5.46]
*
rn
a
E
=
(112)
1
1r2
dl71
=
2
n
Q,
R'
R3.
(8.2-
18)
#
9
r=R
I
It originates partly from positive displacement work
jpdc
of the differential pressure
between the ambient fluid and the bubble but is partly due to the lieat transfer from
the
liquid into the bubble. This accompanics the evaporation of thc liquid into the bubble,
The lattcr
originates
from the superheating of the liquid, when crossing the critical zone,
whose pressure falls short of the steam pressure due to the liquid's temperature
T,.
Il.loreover an additional temperature drop of the gaseous content in the bubble, that
favours heat transfer, may be caused by the sudden expansion, when the bubble
pene-
trates quickly into the low pressure zone. Limiting the heat transfer to heat conduction,
thc
gcvernin? differential equation for
a
spherosymmetricril temperature field with
'a'
as
thermal diffusivity reads
11
beins
the
radial velocity according to (8.2-5).
a
For larger bubbler the surface availriblc usually limits the bubble growth by heat transfer. In this
cad
tlic bubble _pro\vth
is
dcscribcd by the simple rclntion
R
=
2/3(ur)'!',
in which
depends,
afte
Scrl~etl
18.431,
on the density ratio of hapour to liquid
e,/pL,
on thc late~~t hcat
11,
the specific heat
of
kapour and liquid
c,,
c,
and thc superheating
To
-
T,.
8.2.1.6.
Gas
diffusion under
cavitation
A
strict computation of cavity growth must also account for gas diffusion from thc liquid into
th
cak ity.
El
cn
if
diffusion in general needs much more time than evaporation,
it
may become
irn
tant in the case. whcn the liquid entering the critical zone has passed before
a
longer low pres
zocc, e-g.
a
suction line in case of an impeller.
The bubble
gronth by diffusion is controlled by Fick's law. Hcnce the governing differential equatio
is of the
t}
pc
(8.2-
19).
But now the concentration of dissolved gas
C(kg!m3)
appears instead oft
temperature
7:
and the mass diffusivity ~(m'ls) instead of the thermal diffusivity n(m2/s). For
relativclq large bubblc this converts to
R
--
(D
t)'I2.
A
special case of diffusion is the rectified diffusion of an oscillating bubble within a surging press
field. During one oscillation the
_eas contcnt ofthe bubble is altcrnatingly compressed and expand
In
the compressed phase gas diffuses from the bubble into the liquid. In the expanded phase
fr
the l~quid into thc bubble. Since the area for diffusing
ib
larger in the
expanded
bubblc than in
conlpresscd bubblc, the gas content grows
by
this so-called "rectified diffusion".
Any
diffusion is favoured, when the pressure falls short of the saturation pressure, which
is
requi
to prevent a certain air quantity absorbed to diffuse. According to Henry's law this saturati
quantity grows with the pressure.
276