p ocesses
A icle
Flow o Oil and Wa e h ough he Nozzle and Ca i a ion †
Jana Jablonská*, Milada Kozubko áand Ma ian Bojko


Ci a ion: Jablonská, J.; Kozubko á,
M.; Bojko, M. Flow o Oil and Wa e
h ough he Nozzle and Ca i a ion.
P ocesses 2021,9, 1936. h ps://
doi.o g/10.3390/p 9111936
Academic Edi o : Se gio Bobbo
Recei ed: 30 Augus 2021
Accep ed: 25 Oc obe 2021
Published: 28 Oc obe 2021
Publishe ’s No e: MDPI s ays neu al
wi h ega d o ju isdic ional claims in
published maps and ins i u ional a il-
ia ions.
Copy igh : © 2021 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
Depa men o Hyd omechanics and Hyd aulic Equipmen , Facul y o Mechanical Enginee ing,
VSB—Technical Uni e si y o Os a a, 17. lis opadu 2172/15, 708 00 Os a a-Po uba, Czech Republic;
[email p o ec ed] (M.K.); [email p o ec ed] (M.B.)
*Co espondence: [email p o ec ed]
† This pape is an ex ended e sion o a pape published in he in e na ional con e ence: “XXII. In e na ional
Scien i ic Con e ence—The Applica ion o Expe imen al and Nume ical Me hods in Fluid Mechanics and
Ene gy 2020 (AEaNMiFMaE-2020), Pieš ’any, Slo akia, 7–9 Oc obe 2020.
Abs ac :
Today, he co ec unde s anding o he issue o oil and wa e ca i a ion is impo an
due o he g owing demands on wo king condi ions in hyd aulic sys ems (p essu e and low a e).
This a icle deals wi h he measu emen and subsequen ma hema ical modeling o ca i a ion
in
a con e gen -di e gen
nozzle o ci cula c oss-sec ion. Ca i a ion depends on he physical
p ope ies o he lowing medium as a unc ion o empe a u e. Usually, ca i a ion in wa e is de ined
by a wo-phase low o wa e and apo , bu he ai con ained in he wa e signi ican ly a ec s
ca i a ion. The e is usually no apo ca i a ion in he oil. Fa mo e o en, ca i a ion in oil is caused by
he ai i con ains. Fo compa ison, ca i a ion in wa e and oil was gene a ed in expe imen s wi h an
iden ical nozzle. The measu emen was used o de ine bounda y condi ions in ma hema ical models
and o e i y simula ions. The p oblem o ca i a ion was sol ed by h ee a ian s o mul iphase
low, single-phase low (wa e , oil), wo-phase low (wa e – apo , oil–ai ) and h ee-phase low
(wa e – apo –ai , oil– apo –ai ). A u bulen model wi h ca i a ion was used o all a ian s. The
e i ica ion o simula ions shows ha o wa e ca i a ion i is necessa y o use a h ee-phase model
(wa e , apo , ai ) and o oil ca i a ion a wo-phase model (oil, ai ) is su icien . The measu emen
esul s con i m he impo ance o he ai phase in modeling ca i a ion in bo h wa e and oil.
Keywo ds: oil ca i a ion; wa e ca i a ion; ma hema ical modeling; CFD; mul iphase low
1. In oduc ion
Liquid a a cons an empe a u e can be exposed o a dec easing p essu e, which can
each up o he p essu e o sa u a ed apo . each liquid con ains mic obubbles o non-
condensable (dissol ed) gases o nuclei, which can g ow and o m ca i ies as he p essu e
dec eases. This p ocess is called ca i a ion. Ca i a ion occu s in low-p essu e egions
and causes a subs an ial change in densi y. Due o ca i a ion shock, he su ace o he
componen s (pump blades, al es, e c.) is locally damaged and so-called ca i a ion e osion
occu s. I s consequences can mani es hemsel es a e only a ew hou s o ope a ion.
Ca i a ion is suppo ed by la ge empe a u e changes and chemical and elec oly ic
in luences. The solubili y o oxygen in liquids inc eases wi h p essu e [
1
], which a ec s
he highe in ensi y o chemical co osion. In he li e a u e [
1
], i is s a ed ha up o 50
◦
C,
he solubili y o oxygen in wa e dec eases as a unc ion o empe a u e. The in ensi y o
ca i a ion is suppo ed by impu i y pa icles in he liquid.
Cu en ly, he Ven u i ube is widely used o s udy ca i a ion. The li e a u e [
2
]
s a es ha ca i a ion acili a es he syn hesis o bio uels, helps o deg ade con aminan s,
and disin ec ha m ul mic oo ganisms (was ewa e emedia ion), e c. Ven u i ubes a e
widely used o gene a e mic obubbles by hyd odynamic ca i a ion [
3
]. In much o he
li e a u e [
3
,
4
] dealing wi h he low in a Ven u i ube, wa e is conside ed. The au ho s
ha e al eady deal wi h he low o oil h ough cons ic ions in he li e a u e [
5
] and now
his issue is en iched wi h modeling.
P ocesses 2021,9, 1936. h ps://doi.o g/10.3390/p 9111936 h ps://www.mdpi.com/jou nal/p ocesses
P ocesses 2021,9, 1936 2 o 14
In es iga ion o he de elopmen o ca i a ion in oil low is desc ibed in he li e a u e
mainly conce ning low h ough na ow gaps such as he oil ilm seal wi h spi al g oo es [
6
],
ca i a ion in bea ings [
7
] and lub ica ing ilm [
8
]. The li e a u e [
7
] desc ibes he de ec ion o
apo ca i a ion in bea ings using ib a ional and acous ic me hods. The dependence o he
heigh o he lub ica ing ilm on he o ma ion o ca i a ion is desc ibed in he li e a u e [
8
].
Oil ca i a ion, in p ac ice, is impo an in p oduc ion and he li e a u e [
9
] desc ibes he
p oduc ion o biodiesel om used oil using hyd odynamic ca i a ion, and equipmen
using hyd odynamic ca i a ion o p oduce a highly s able oil-in-wa e emulsion [
10
].
Howe e , in con en ional hyd aulic sys ems, ca i a ion can also occu when lowing
h ough
a hyd aulic
elemen . The e o e, o simpli y he p oblem o oil low wi h ca i a ion,
modeling o his low and compa ison o he low in e ms o lowing luid wi h he same
geome y was also pe o med.
2. Physical P ope ies o Fluids
As o he low o liquid h ough a con e gen -di e gen nozzle, i is necessa y o know
he physical p ope ies o he lowing medium. The iscosi y o liquid depends mainly
on he empe a u e. In gene al, wi h inc easing empe a u e, he change in momen um o
pa icles in he mo ing laye s o liquids is mo e in ense and he iscosi y and angen ial
s ess dec ease (see Figu e 1a). The iscosi y o he liquid is also a unc ion o p essu e. Wi h
inc easing p essu e, he iscosi y inc eases signi ican ly, bu only a highe p essu es (up o
100 MPa) [
11
,
12
]. The densi y o he liquid is also signi ican ly a ec ed by empe a u e and
dec eases wi h inc easing empe a u e (see Figu e 1b).
P ocesses 2021, 9, x FOR PEER REVIEW 2 o 14
ha e al eady deal wi h he low o oil h ough cons ic ions in he li e a u e [5] and now
his issue is en iched wi h modeling.
In es iga ion o he de elopmen o ca i a ion in oil low is desc ibed in he li e a u e
mainly conce ning low h ough na ow gaps such as he oil ilm seal wi h spi al g oo es
[6], ca i a ion in bea ings [7] and lub ica ing ilm [8]. The li e a u e [7] desc ibes he de-
ec ion o apo ca i a ion in bea ings using ib a ional and acous ic me hods. The de-
pendence o he heigh o he lub ica ing ilm on he o ma ion o ca i a ion is desc ibed
in he li e a u e [8]. Oil ca i a ion, in p ac ice, is impo an in p oduc ion and he li e a u e
[9] desc ibes he p oduc ion o biodiesel om used oil using hyd odynamic ca i a ion,
and equipmen using hyd odynamic ca i a ion o p oduce a highly s able oil-in-wa e
emulsion [10]. Howe e , in con en ional hyd aulic sys ems, ca i a ion can also occu
when lowing h ough a hyd aulic elemen . The e o e, o simpli y he p oblem o oil low
wi h ca i a ion, modeling o his low and compa ison o he low in e ms o lowing
luid wi h he same geome y was also pe o med.
2. Physical P ope ies o Fluids
As o he low o liquid h ough a con e gen -di e gen nozzle, i is necessa y o
know he physical p ope ies o he lowing medium. The iscosi y o liquid depends
mainly on he empe a u e. In gene al, wi h inc easing empe a u e, he change in mo-
men um o pa icles in he mo ing laye s o liquids is mo e in ense and he iscosi y and
angen ial s ess dec ease (see Figu e 1a). The iscosi y o he liquid is also a unc ion o
p essu e. Wi h inc easing p essu e, he iscosi y inc eases signi ican ly, bu only a highe
p essu es (up o 100 MPa) [11,12]. The densi y o he liquid is also signi ican ly a ec ed
by empe a u e and dec eases wi h inc easing empe a u e (see Figu e 1b).
(a) (b)
Figu e 1. Physical p ope ies o luids. (a) Dependence o dynamic iscosi y 𝜂 s. empe a u e 𝑡;
(b) Dependence o densi y 𝜚 s. empe a u e 𝑡.
I is gene ally assumed ha he ca i a ion p essu e 𝑝 (Pa) is he same as he sa u-
a ed apo p essu e o he liquid 𝑝 (Pa) a a gi en empe a u e 𝑝 =𝑝. Sa u a ed
apo p essu e is he highes p essu e a which a subs ance exis s in an equilib ium gase-
ous s a e a a gi en empe a u e. I is also he lowes p essu e a which a subs ance can
exis in a liquid o solid s a e a a gi en empe a u e. The apo p essu e o subs ances
inc eases exponen ially wi h empe a u e. [13] The sa u a ed apo p essu e o oil is e y
low, in con as o he wa e (see Figu es 2a,b).
Figu e 1.
Physical p ope ies o luids. (
a
) Dependence o dynamic iscosi y
η
s. empe a u e
;
(b) Dependence o densi y $ s. empe a u e .
I is gene ally assumed ha he ca i a ion p essu e
pca
(Pa) is he same as he sa u-
a ed apo p essu e o he liquid
p ap
(Pa) a a gi en empe a u e
pca =p ap
. Sa u a ed
apo p essu e is he highes p essu e a which a subs ance exis s in an equilib ium gaseous
s a e a a gi en empe a u e. I is also he lowes p essu e a which a subs ance can exis in
a liquid o solid s a e a a gi en empe a u e. The apo p essu e o subs ances inc eases
exponen ially wi h empe a u e. [
13
] The sa u a ed apo p essu e o oil is e y low, in
con as o he wa e (see Figu e 2a,b).
P ocesses 2021, 9, x FOR PEER REVIEW 3 o 14
(a) (b)
Figu e 2. (a) Dependence o sa u a ion apo p essu e 𝑝 (Pa) s. empe a u e 𝑡 (°C). (b) De-
pendence o p essu e a he na owes poin 𝑝 (Pa) s. olume low a e 𝑄 (m3·s−1).
As al eady men ioned, he p esence o ai in liquid (wa e , oil) has ad e se e ec s on
he quali y o he liquid, especially oil—oxida ion, oaming, and educing he modulus o
elas ici y. Ai suppo s he o ma ion o ca i a ion. [11] Dissol ed ai is exc e ed om he
liquid, he amoun o which depends on he p ope ies o he liquid, gas, p essu e and
empe a u e. Dissol ed ai can begin o elease and o m bubbles, especially in he a ea
whe e he p essu e dec eases. The amoun o dissol ed ai can be exp essed by he ela-
ionship:
𝑉=𝑉
𝛼
 (m3) (1)
whe e 𝑉 (m3) is he olume o dissol ed gas, 𝑉 (m3) is he olume o liquid, 𝛼 (1) is
he coe icien o solubili y o a gas in a liquid, 𝑝 (Pa) is he absolu e p essu e a he su -
ace and 𝑝 (Pa) is he a mosphe ic p essu e [11].
The alue o he solubili y coe icien is gi en o wa e [1], mine al oil and o he
liquids [14]. The publica ion [15] s a es ha he concen a ion o non-condensable gas in
wa e a 20 °C is om 2–3% and he concen a ion o in asi e oil is up o 8%. All publica-
ions [1,11,14–16] ag ee on he amoun o ai con ained in wa e and oil.
The concen a ion o dissol ed gas 𝐶 (mg.L−1) in wa e is di ec ly p opo ional o he
pa ial p essu e o he gas 𝑃 (Pa) abo e i s le el. This dependence is explained by Hen-
y's law:
𝐶=𝐾𝑃 (2)
whe e he alue o 𝐾 is he abso p ion coe icien cha ac e izing he solubili y o gas as
a unc ion o empe a u e.
When modeling, he p ope ies o liquid, i.e., densi y and dynamic iscosi y, had o
be chosen. Fo wa e and oil, he dynamic iscosi y and densi y as a unc ion o
empe a u e we e de ined using an app oxima ion equa ion, see Figu e 1a. The densi y
o wa e apo , oil apo and ai we e de ined as a unc ion o empe a u e and p essu e
using he equa ion o s a e. The dynamic iscosi y o gases was de ined in [17].
3. Desc ip ion o Measu ing Equipmen
The basic hyd aulic pa ame e s we e measu ed on he es ci cui . Since he in es i-
ga ed elemen , he con e gen -di e gen nozzle, was anspa en he eme gence and de-
elopmen o ca i a ion could be obse ed. The same nozzle was used on he wa e and
oil ci cui pipes; he di e ences we e due o echnical easons only in he loca ion o sen-
so s (see Table 1). The liquid was pumped by a pump/hyd ogene a o in o he ci cui
whe e he ball al e and he lowme e we e connec ed. The liquid hen lowed h ough
he anspa en nozzle and he ball al e back in o he ank.
The oo h lowme e GFM-70 used had a ange om 0.7–70 dm3·min−1 on he oil ci -
cui wi h an accu acy o 0.5% and he induc ion low me e FLONET FN20XX.1 had a
ange om 0.72–72 m3·s−1 on he wa e ci cui wi h an accu acy o 0.2%. In on o and
Figu e 2.
(
a
) Dependence o sa u a ion apo p essu e
p ap
(Pa) s. empe a u e
(
◦
C).
(b) Dependence o p essu e a he na owes poin pnozzle (Pa) s. olume low a e Q(m3·s−1).
P ocesses 2021,9, 1936 3 o 14
As al eady men ioned, he p esence o ai in liquid (wa e , oil) has ad e se e ec s on
he quali y o he liquid, especially oil—oxida ion, oaming, and educing he modulus
o elas ici y. Ai suppo s he o ma ion o ca i a ion. [
11
] Dissol ed ai is exc e ed om
he liquid, he amoun o which depends on he p ope ies o he liquid, gas, p essu e
and empe a u e. Dissol ed ai can begin o elease and o m bubbles, especially in he
a ea whe e he p essu e dec eases. The amoun o dissol ed ai can be exp essed by
he ela ionship:
Vp=Vkαpp
p0(m3)(1)
whe e
Vp
(m
3
) is he olume o dissol ed gas,
Vk
(m
3
) is he olume o liquid,
αp
(1) is he
coe icien o solubili y o a gas in a liquid,
p
(Pa) is he absolu e p essu e a he su ace and
p0(Pa) is he a mosphe ic p essu e [11].
The alue o he solubili y coe icien is gi en o wa e [
1
], mine al oil and o he
liquids [
14
]. The publica ion [
15
] s a es ha he concen a ion o non-condensable gas
in wa e a 20
◦
C is om 2–3% and he concen a ion o in asi e oil is up o 8%. All
publica ions [1,11,14–16] ag ee on he amoun o ai con ained in wa e and oil.
The concen a ion o dissol ed gas
Ci
(mg
·
L
−1
) in wa e is di ec ly p opo ional o
he pa ial p essu e o he gas
Pi
(Pa) abo e i s le el. This dependence is explained by
Hen y’s law:
Ci=KiPi(2)
whe e he alue o
Ki
is he abso p ion coe icien cha ac e izing he solubili y o gas as
a unc ion o empe a u e.
When modeling, he p ope ies o liquid, i.e., densi y and dynamic iscosi y, had o be
chosen. Fo wa e and oil, he dynamic iscosi y and densi y as a unc ion o empe a u e
we e de ined using an app oxima ion equa ion, see Figu e 1a. The densi y o wa e apo ,
oil apo and ai we e de ined as a unc ion o empe a u e and p essu e using he equa ion
o s a e. The dynamic iscosi y o gases was de ined in [17].
3. Desc ip ion o Measu ing Equipmen
The basic hyd aulic pa ame e s we e measu ed on he es ci cui . Since he in es-
iga ed elemen , he con e gen -di e gen nozzle, was anspa en he eme gence and
de elopmen o ca i a ion could be obse ed. The same nozzle was used on he wa e
and oil ci cui pipes; he di e ences we e due o echnical easons only in he loca ion o
senso s (see Table 1). The liquid was pumped by a pump/hyd ogene a o in o he ci cui
whe e he ball al e and he lowme e we e connec ed. The liquid hen lowed h ough
he anspa en nozzle and he ball al e back in o he ank.
Table 1. Basic dimensions (mm), see Figu e 3.
Dimensions Wa e Oil
∅D20 20
∅d6 6
a300 110
b25 25
c70 70
d385 195
(L)780 400
posi ion ho izon al e ical
The oo h lowme e GFM-70 used had a ange om 0.7–70 dm
3·
min
−1
on he oil
ci cui wi h an accu acy o 0.5% and he induc ion low me e FLONET FN20XX.1 had
a ange om 0.72–72 m
3·
s
−1
on he wa e ci cui wi h an accu acy o 0.2%. In on o
and behind he nozzle we used senso s om he manu ac u e Hyd o echnik wi h anges
om (
−
1–1) ba and (
−
1–5) ba , he p essu es we e measu ed, w i en in o he e alua ion
equipmen and hen copied in o he compu e (all p essu e gauges had an accu acy o 0.5%).
P ocesses 2021,9, 1936 4 o 14
When ca i a ion occu ed, a high-speed came a eco d was made and sen o a compu e .
The eco ding was cap u ed by a high-speed Pho on Mini UX50 came a. The ame a e
was 8000 ames pe second wi h a esolu ion o 1280
×
296 pixels. A he mome e was
placed in he ank. The ci cui diag am and nozzle dimensions a e shown in Figu e 3.
P ocesses 2021, 9, x FOR PEER REVIEW 4 o 14
behind he nozzle we used senso s om he manu ac u e Hyd o echnik wi h anges om
(−1–1) ba and (−1–5) ba , he p essu es we e measu ed, w i en in o he e alua ion equip-
men and hen copied in o he compu e (all p essu e gauges had an accu acy o 0.5%).
When ca i a ion occu ed, a high-speed came a eco d was made and sen o a compu e .
The eco ding was cap u ed by a high-speed Pho on Mini UX50 came a. The ame a e
was 8000 ames pe second wi h a esolu ion o 1280 × 296 pixels. A he mome e was
placed in he ank. The ci cui diag am and nozzle dimensions a e shown in Figu e 3.
Figu e 3. Measu ing equipmen and basic dimensions (mm): (1) ank; (2) hyd o gene a o /pump;
(3) ball al e; (4) lowme e ; (5) con e gen -di e gen nozzle; (6) p essu e senso s; (7) e alua ing
and eco ding equipmen ; (8) compu e ; (9) ligh ; (10) high speed came a; (11) he mome e .
The ball al es we e included in he ci cui o con ol low. In he case o a wa e line,
he low was con olled by a equency con e e loca ed on he pump. The ci cui was
composed o DN20 plas ic pipes. In he case o an oil line, he low was con olled by a
con ol hyd ogene a o wi h a il ing pla e and he ci cui was composed o a DN20 s eel
pipe. I was possible o hea he oil on his ci cui .
Table 1. Basic dimensions (mm), see Figu e 3.
Dimensions Wa e Oil
∅𝐷 20 20
∅𝑑 6 6
𝑎 300 110
𝑏 25 25
𝑐 70 70
𝑑 385 195
(𝐿) 780 400
posi ion ho izon al e ical
4. Ma hema ical Model
Flow is de ined as mul iphase low (wa e –ai – apo , oil–ai ), so he mul iphase mix-
u e model was used. The model can sol e homogeneous mul iphase low, whe e he in-
di idual phases mo e a di e en eloci ies. The mix u e model consis ed o he con inu-
i y equa ion o he mix u e [18]:

 +∇∙(𝜌𝑣

)=0 (kg·s−1·m−3) (3)
Figu e 3.
Measu ing equipmen and basic dimensions (mm): (1) ank; (2) hyd o gene a o /pump;
(3) ball al e;
(4) lowme e ; (5) con e gen -di e gen nozzle; (6) p essu e senso s; (7) e alua ing and
eco ding equipmen ; (8) compu e ; (9) ligh ; (10) high speed came a; (11) he mome e .
The ball al es we e included in he ci cui o con ol low. In he case o a wa e
line, he low was con olled by a equency con e e loca ed on he pump. The ci cui
was composed o DN20 plas ic pipes. In he case o an oil line, he low was con olled by
a con ol
hyd ogene a o wi h a il ing pla e and he ci cui was composed o a DN20 s eel
pipe. I was possible o hea he oil on his ci cui .
4. Ma hema ical Model
Flow is de ined as mul iphase low (wa e –ai – apo , oil–ai ), so he mul iphase
mix u e model was used. The model can sol e homogeneous mul iphase low, whe e
he indi idual phases mo e a di e en eloci ies. The mix u e model consis ed o he
con inui y equa ion o he mix u e [18]:
∂ρm
∂ +∇· ρm→
m=0(kg·s−1·m−3)(3)
whe e
ρm
(kg
·
m
−3
) is je mix u e densi y,
→
m
(m
·
s
−1
) is mass a e aged eloci y.
A e subs i u ing:
ρm=
n
∑
k=1
(αkρk)(kg·m−3)(4)
→
m=
∑n
k=1αkρk→
k
ρm
(m·s−1)(5)
αk
(1) is he olume ac ion o phase
k
,
n=
3 (1 is liquid, 2 is ai and 3 is apo ). The index
msigni ies he mix u e.
The momen um equa ion o he mix u e is [18]:
∂
∂ ρm→
m+∇·ρm→
m→
m=−∇p+∇·µm∇→
m+∇→
T
m+ρm→
g+→
F−∇· n
∑
k=1αkρk→
d ,k→
d ,k!(N)(6)
P ocesses 2021,9, 1936 5 o 14
whe e
n
is he numbe o phases,
→
F
is a body o ce,
→
d ,k=→
k−→
m
is he d i eloci y
o he seconda y phase k,µm(Pa·s) is he iscosi y o he mix u e:
µm=
n
∑
k=1
(αkµk) (Pa ·s)(7)
The gas densi y ollows he equa ion o s a e wi h cons an empe a u e, he liquid
densi y is assumed o be incomp essible. The e o e, he p ocess is assumed o be iso he mal.
Ca i a ion is cha ac e ized by a phase change be ween liquid and apo , he equa ion
o he olume ac ion o apo is de ined as ollows [18]:
∂
∂ α apρ ap+∇· α apρ ap→
m=−∇· α apρ ap→
d , ap+
n
∑
q=1.
ml ap −.
m ap l(kg·s−1·m−3)(8)
whe e he indexes indica e he indi idual phases: ap— apo , l—liquid, ai —ai .
The Schne and Saue ca i a ion model [
18
] used is compa ible wi h his mix u e
model. When en e ing he model, i is assumed ha he mix u e consis s o a liquid and
a apo phase. The equa ion o he olume ac ion o apo has he o m:
∂
∂ α apρ ap+∇· α apρ ap→
m=R(kg·s−1·m−3)(9)
whe e
α ap
(1) is he apo olume ac ion,
ρ ap
(kg
·
m
−3
) is he apo densi y,
→
m
(m
·
s
−1
)
is he mix u e eloci y and R(kg·s−1·m−3) is he mass ans e a e:
R=ρ apρl
ρm
α apαl
3
<Bs2
3
p ap −p
ρl
(kg·s−1·m−3)(10)
whe e
ρl
(kg
·
m
−3
) is liquid densi y,
αl
(1) is he liquid olume ac ion,
p ap
(Pa) is sa u a-
ion apo p essu e and <B(m) is he bubble adius:
<B=α ap
αl
3
4π
1
n1
3(m)(11)
whe e
n
is bubble numbe densi y. The Schne and Saue models use apo o connec a
olume ac ion o apo o he numbe o bubbles pe olume o liquid:
α ap =nb4
3π<3
B
1+nb4
3π<3
B
(–)(12)
The ca i a ion model o Schne and Saue is based on he Rayleigh–Plesse
equa ion [13]
.
I desc ibes he g ow h o one apo bubble in a liquid. Equa ion (13) desc ibes he mass
ans e be ween he liquid and apo phase in ca i a ion:
<b
D2<b
D 2+3
2D<b
D 2
=pb−p
ρl−4νl
<b
.
<b−2σ
ρl<b
(m2·s−2)(13)
whe e
<b
(m) is he bubble adius,
σ
(Pa
·
m
−1
) is he liquid su ace ension coe icien ,
νl(m2·s−1)
is he liquid kinema ic iscosi y,
p
(Pa) is local a - ield p essu e. Neglec ing
he second-o de e ms and he su ace ension o ce, Equa ion (13) is simpli ied o:
D<b
D =s2
3
pb−p
ρl
(m·s−1)(14)
Due o he Reynolds numbe de e mined om he expe imen , i is a low ansien
be ween he lamina and u bulen modes. The e o e, he p oblem will be sol ed by

P ocesses 2021,9, 1936 6 o 14
a ime-a e aged u bulen model sui able o low Reynolds numbe s. The goal o he
Reynolds-A e aged Na ie –S okes (RANS) app oach is o simula e a ime-a e aged low
ield. A la ge numbe o RANS models a e based on he Boussinesque hypo hesis. This
assumes ha he s ess enso de ined by New on’s ela ion can be used o a u bulen
s ess enso . The molecula iscosi y is hen eplaced by he so-called u bulen iscosi y.
The con inui y Equa ion (3), he equa ion o mo ion (6) and o he s in he mix u e model (8)
will be de ined o ime-a e aged quan i ies.
The wo-equa ion model
k−ω
adds wo di e en ial equa ions [
18
,
19
]. The eloci y
scale o u bulence is speci ied by means o he u bulen kine ic ene gy k(m
−2·
s
−2
). The
longi udinal scale o u bulence is sol ed by means o he o ici y ω(s−1):
∂k
∂ +∇· →
mk=∇· (Γk∇k)(m2·s−3)(15)
∂ω
∂ +∇· →
mω=∇· (Γω∇ω)(s−2)(16)
In hese equa ions
Γk
(m
2·
s
−1
) and
Γω
(m
2·
s
−1
) a e he e ec i e di usi i y o
k(m−2·s−2)
and ω(s−1), espec i ely.
The Shea S ess T anspo SST k-
ω
model used in solu ion [
4
,
18
–
24
] is a modi ica ion
o he basic k-
ω
model. I has a modi ied di e en ial equa ion o o ici y. The model is
sui able whe e i is necessa y o p ecisely de e mine he ea -o poin and he size o he
closing a ea.
The SST model akes in o accoun he ans e o he shea s ess o u bulence in he
de ini ion o u bulen iscosi y:
µ =ρk
ω
1
maxh1
α∗,SF2
a1ωi(Pa·s)(17)
whe e
S
(Pa
·
s
−1
). is he magni ude o he s ain a e,
α∗
(1) is de ined in Equa ion (19) and
F2(1) is gi en by:
F2= anh max"2√k
0.09ωy,500µ
ρy2ω#!2
(–)(18)
whe e y(m). is he dis ance o he nex a ea.
α∗=α∗
∞

0.072
3+ρk
6µω
1+ρk
6µω 
(–)(19)
Ca i a ion is cha ac e ized by ime changes o sol ed hyd aulic quan i ies wi h high
equency. To achie e con e gence, i is necessa y o sol e he p oblem as ime-dependen
wi h a small ime s ep and a e aged alues used o e alua ion.
The simula ion was pe o med using Ansys Fluen 2020 R2 so wa e. F om he au ho s’
expe ience, i was decided o model he geome y as 2D axially symme ic (see
Figu e 4
).
Howe e , i was necessa y o c ea e a e y ine mesh. Se e al a ian s o mesh wi h
e inemen a he walls we e c ea ed. G ids we e es ed using g id con e gence analysis.
Due o he di e en dis ances om he inle and ou le , he e was a di e ence in he numbe
o cells— he mesh c ea ed o he wa e low had 242,450 cells, and he mesh c ea ed o
he oil low had 166,090 cells.
A he inle , he a ious low a e alues o indi idual phases we e se . The measu ed
olume low o wa e anged om Q = 1.74
×
10
−4
m
3·
s
−1
o Q = 7.23
×
10
−4
m
3·
s
−1
and o oil om Q = 1.70
×
10
−4
m
3·
s
−1
o Q = 5.86
×
10
−4
m
3·
s
−1
. The ou le bounda y
condi ion was gi en by se ing he measu ed p essu e alues a he ou le . Fo wa e , he
absolu e p essu e ange was om 93,500 Pa o 99,000 Pa. Fo oil, his ange was om
91,300 Pa
o 104,100 Pa. A hyd aulic diame e o 0.02 m and a u bulence in ensi y alue o
P ocesses 2021,9, 1936 7 o 14
1% we e se o he inle and ou le . In he case o ca i a ion, he p oblems we e sol ed as
ime-dependen .
P ocesses 2021, 9, x FOR PEER REVIEW 7 o 14
in ensi y alue o 1% we e se o he inle and ou le . In he case o ca i a ion, he
p oblems we e sol ed as ime-dependen .
Figu e 4. Bounda y condi ions and mesh o he modeled a ea o he con e gen -di e gen nozzle.
5. Resul s o Measu emen s and Modeling
5.1. Measu emen Resul s
Figu e 5 p esen s he measu ed alues o inle and ou le p essu e s. olume low
a e. Wi h an inc easing low a e, he p essu e a he inle and ou le inc eased, which
co esponds o Weissbach's heo y o luid mechanics. The p essu e alues o wa e ag ee
wi h he Weissbach heo y up o a olume low alue o 4.51·10−4 m3.s−1. Fo highe low
alues, he inle p essu e alues 𝑝 (Pa) had a s eepe inc ease due o ca i a ion. The hy-
d aulic esis ance o he elemen inc eased du ing low wi h ca i a ion. In he case o wa-
e , a b eak in cha ac e is ics co esponding o he ca i a ion o ma ion is e iden . In he
case o oil, he p essu es o h ee a ian s o empe a u e we e e alua ed. No ca i a ion
occu ed a 25 °C. A 40 °C and 50 °C, ca i a ion occu ed a highe low a es. The mini-
mum low alue o ca i a ion o 40 °C is g ea e han 5.5·10−4 m3.s−1and he minimum
low alue o 50 °C is g ea e han 5.02·10−4 m3.s−1. The inle p essu e inc eased bu less
han acco ding o Weissbach’s heo y. The e is no appa en b eak in oil cha ac e is ics
wi h ca i a ion.
(a) (b)
Figu e 5. (a) Dependence o ela i e inle p essu e 𝑝 (Pa) s. olume low a e 𝑄 (m3·s−1); (b)
Dependence o ela i e ou le p essu e 𝑝 (Pa) s. olume low a e 𝑄 (m3·s−1).
The loss coe icien 𝜉 (1) was de e mined using Be noulli's equa ion:
∆𝑝=𝜉𝜌 
𝑄→𝜉=∆𝑝
 (Pa) (20)
and discha ge coe icien 𝜇 (1): 𝜇=


 (–) (21)
Figu e 4. Bounda y condi ions and mesh o he modeled a ea o he con e gen -di e gen nozzle.
5. Resul s o Measu emen s and Modeling
5.1. Measu emen Resul s
Figu e 5p esen s he measu ed alues o inle and ou le p essu e s. olume low
a e. Wi h an inc easing low a e, he p essu e a he inle and ou le inc eased, which
co esponds o Weissbach’s heo y o luid mechanics. The p essu e alues o wa e ag ee
wi h he Weissbach heo y up o a olume low alue o 4.51
×
10
−4
m
3·
s
−1
. Fo highe
low alues, he inle p essu e alues
p1
(Pa) had a s eepe inc ease due o ca i a ion. The
hyd aulic esis ance o he elemen inc eased du ing low wi h ca i a ion. In he case
o wa e , a b eak in cha ac e is ics co esponding o he ca i a ion o ma ion is e iden .
In he case o oil, he p essu es o h ee a ian s o empe a u e we e e alua ed. No
ca i a ion occu ed a 25
◦
C. A 40
◦
C and 50
◦
C, ca i a ion occu ed a highe low a es.
The minimum low alue o ca i a ion o 40
◦
C is g ea e han 5.5
×
10
−4
m
3·
s
−1
and
he minimum low alue o 50
◦
C is g ea e han 5.02
×
10
−4
m
3·
s
−1
. The inle p essu e
inc eased bu less han acco ding o Weissbach’s heo y. The e is no appa en b eak in oil
cha ac e is ics wi h ca i a ion.
P ocesses 2021, 9, x FOR PEER REVIEW 7 o 14
in ensi y alue o 1% we e se o he inle and ou le . In he case o ca i a ion, he
p oblems we e sol ed as ime-dependen .
Figu e 4. Bounda y condi ions and mesh o he modeled a ea o he con e gen -di e gen nozzle.
5. Resul s o Measu emen s and Modeling
5.1. Measu emen Resul s
Figu e 5 p esen s he measu ed alues o inle and ou le p essu e s. olume low
a e. Wi h an inc easing low a e, he p essu e a he inle and ou le inc eased, which
co esponds o Weissbach's heo y o luid mechanics. The p essu e alues o wa e ag ee
wi h he Weissbach heo y up o a olume low alue o 4.51·10−4 m3.s−1. Fo highe low
alues, he inle p essu e alues 𝑝 (Pa) had a s eepe inc ease due o ca i a ion. The hy-
d aulic esis ance o he elemen inc eased du ing low wi h ca i a ion. In he case o wa-
e , a b eak in cha ac e is ics co esponding o he ca i a ion o ma ion is e iden . In he
case o oil, he p essu es o h ee a ian s o empe a u e we e e alua ed. No ca i a ion
occu ed a 25 °C. A 40 °C and 50 °C, ca i a ion occu ed a highe low a es. The mini-
mum low alue o ca i a ion o 40 °C is g ea e han 5.5·10−4 m3.s−1and he minimum
low alue o 50 °C is g ea e han 5.02·10−4 m3.s−1. The inle p essu e inc eased bu less
han acco ding o Weissbach’s heo y. The e is no appa en b eak in oil cha ac e is ics
wi h ca i a ion.
(a) (b)
Figu e 5. (a) Dependence o ela i e inle p essu e 𝑝 (Pa) s. olume low a e 𝑄 (m3·s−1); (b)
Dependence o ela i e ou le p essu e 𝑝 (Pa) s. olume low a e 𝑄 (m3·s−1).
The loss coe icien 𝜉 (1) was de e mined using Be noulli's equa ion:
∆𝑝=𝜉𝜌 
𝑄→𝜉=∆𝑝
 (Pa) (20)
and discha ge coe icien 𝜇 (1): 𝜇=


 (–) (21)
Figu e 5.
(
a
) Dependence o ela i e inle p essu e
p1
(Pa) s. olume low a e
Q
(m
3·
s
−1
);
(b) Dependence o ela i e ou le p essu e p2(Pa) s. olume low a e Q(m3·s−1).
The loss coe icien ξ(1) was de e mined using Be noulli’s equa ion:
∆p=ξρ 8
π2D4Q2→ξ=∆pπ2D4
8ρQ2(Pa)(20)
and discha ge coe icien µ(1):
µ=s1
1−ξ(–)(21)
P ocesses 2021,9, 1936 8 o 14
The low coe icien exp esses he loss caused by he low h ough a na owed c oss-
sec ion (nozzles, o i ices, al e). I is commonly used in hyd aulics. The ype o low was
speci ied by he Reynolds numbe using pa ame e s o inle c oss-sec ion and iscosi y:
Reinle =ρ inle D
η(–)(22)
The c i ical Reynolds numbe was conside ed 2320. The wa e low was u bulen in
he ange o measu emen , see Figu e 6a. The oil low was lamina , bu wi h inc easing
empe a u e and dec easing oil iscosi y, u bulen low could be achie ed. The lamina
low was also a he na owes poin o he nozzle du ing oil low. F om he g aphs, i is
e iden ha he loss coe icien changes signi ican ly wi h ca i a ion.
P ocesses 2021, 9, x FOR PEER REVIEW 8 o 14
The low coe icien exp esses he loss caused by he low h ough a na owed c oss-
sec ion (nozzles, o i ices, al e). I is commonly used in hyd aulics. The ype o low was
speci ied by he Reynolds numbe using pa ame e s o inle c oss-sec ion and iscosi y:
Re =

(–) (22)
The c i ical Reynolds numbe was conside ed 2320. The wa e low was u bulen in
he ange o measu emen , see Figu e 6a. The oil low was lamina , bu wi h inc easing
empe a u e and dec easing oil iscosi y, u bulen low could be achie ed. The lamina
low was also a he na owes poin o he nozzle du ing oil low. F om he g aphs, i is
e iden ha he loss coe icien changes signi ican ly wi h ca i a ion.
In Figu e 6a, when oil lows in he lamina egion, he loss coe icien is con inuously
dec easing excep o lows whe e ca i a ion occu s. The loss coe icien o wa e a ies
signi ican ly. Fo lows wi hou ca i a ion, a dec ease in he loss coe icien is no iceable,
while he loss coe icien inc eases sha ply wi h he inc easing low in he ca i a ion a ea.
Fo low 4.51·10−4 m3.s−1. he loss coe icien a he in e ace be ween he ca i a ion and
ca i a ion modes is minimal and has a alue o 17.02.
Figu e 6b shows he in e se cou se o he low coe icien wi h espec o he loss co-
e icien . Fo low 4.51·10−4 m3.s−1. he low coe icien a he in e ace be ween ca i a ion
and wi hou ca i a ion mode is maximal and has a alue o 0.25.
(a) (b)
Figu e 6. (a) Dependence o loss coe icien 𝜉 (1) s. Reynolds numbe a he inle 𝑅𝑒 (1); (b)
Dependence o discha ge coe icien 𝜇 (1) s. Reynolds numbe a he inle 𝑅𝑒 (1).
The po en ial o ca i a ion o ma ion is gi en by he ca i a ion numbe and i was
used o analyze he esul s. The ca i a ion numbe is de ined as he a io o he p essu e
d op a he nozzle inle o he kine ic ene gy [19]:
𝜎=

 (–) (23)
The ela ionship be ween bo h dimensionless pa ame e s (ca i a ion numbe , Reyn-
olds numbe in he nozzle) is displayed in Figu e 7. The c i ical ca i a ion numbe de ined
om he measu emen (by obse ing he ca i a ion o ma ion) was he same o bo h liq-
uids and ook he alues 𝜎=0,747(1) o he gi en nozzle geome y. I is e iden ha he
o ma ion o ca i a ion does no depend on he ype o luid, bu on he hyd aulic pa am-
e e s, i.e., p essu e and low a e. The e a e ma ked a ian s, which will be nume ically
analyzed.
Figu e 6.
(
a
) Dependence o loss coe icien
ξ
(1) s. Reynolds numbe a he inle
Reinle
(1);
(b) Dependence o discha ge coe icien µ(1) s. Reynolds numbe a he inle Reinle (1).
In Figu e 6a, when oil lows in he lamina egion, he loss coe icien is con inuously
dec easing excep o lows whe e ca i a ion occu s. The loss coe icien o wa e a ies
signi ican ly. Fo lows wi hou ca i a ion, a dec ease in he loss coe icien is no iceable,
while he loss coe icien inc eases sha ply wi h he inc easing low in he ca i a ion a ea.
Fo low 4.51
×
10
−4
m
3·
s
−1
. he loss coe icien a he in e ace be ween he ca i a ion and
ca i a ion modes is minimal and has a alue o 17.02.
Figu e 6b shows he in e se cou se o he low coe icien wi h espec o he loss
coe icien . Fo low 4.51
×
10
−4
m
3·
s
−1
. he low coe icien a he in e ace be ween
ca i a ion and wi hou ca i a ion mode is maximal and has a alue o 0.25.
The po en ial o ca i a ion o ma ion is gi en by he ca i a ion numbe and i was
used o analyze he esul s. The ca i a ion numbe is de ined as he a io o he p essu e
d op a he nozzle inle o he kine ic ene gy [19]:
σ=2pin −p ap
ρ 2
nozzle
(–)(23)
The ela ionship be ween bo h dimensionless pa ame e s (ca i a ion numbe , Reynolds
numbe in he nozzle) is displayed in Figu e 7. The c i ical ca i a ion numbe de ined
om he measu emen (by obse ing he ca i a ion o ma ion) was he same o bo h
liquids and ook he alues
σ=
0.747 (1) o he gi en nozzle geome y. I is e iden
ha he o ma ion o ca i a ion does no depend on he ype o luid, bu on he hy-
d aulic pa ame e s,
i.e., p essu e
and low a e. The e a e ma ked a ian s, which will be
nume ically analyzed.
P ocesses 2021,9, 1936 9 o 14
P ocesses 2021, 9, x FOR PEER REVIEW 9 o 14
Figu e 7. Dependence o ca i a ion numbe 𝜎 (1) s. Reynolds numbe in he na owes c oss-
sec ion Renozzle (1); de ail o he ca i a ion a ea is o he igh .
5.2. Resul s o Modeling
The single-phase model can be used o es ima e he ype o low, simple ask en y
and sho compu a ional ime. In he case o wa e ca i a ion, a wo-phase model (wa e –
apo ) is commonly used. Due o he inaccu acies o he esul s, he model was ex ended
o a h ee-phase model (wa e – apo –ai ). The accu acy o he calcula ion depends on he
co ec de e mina ion o he amoun o ai in he wa e depending on he empe a u e and
p essu e.
The e we e h ee simula ing a ian s o he wa e low h ough he nozzle:
(a) single-phase low: wa e low;
(b) wo-phase low: wa e and apo low;
(c) h ee-phase low: wa e , apo and ai low.
Due o he inpu bounda y condi ions, absolu e p essu e 𝑝 was simula ed and sub-
sequen ly e i ied wi h he measu emen . Figu e 8 p esen s he a e age alues o he ab-
solu e inle p essu e depending on he olume low a e when wa e and oil lowed
h ough he nozzle. Figu e 8a shows he di e ences be ween he simula ed and measu ed
inle p essu e in he wa e low as a unc ion o he numbe o phases. In he case o low
wi hou ca i a ion, he di e ences o he indi idual a ian s we e sligh . In he case o
low wi hou ca i a ion, he de ia ions o he measu ed p essu e 𝑝 (Pa) om he simu-
la ed alues 𝑝 (Pa) o he single-phase, wo-phase and h ee-phase models we e mini-
mal. In he case o low wi h ca i a ion, he one-phase and wo-phase models di e ed
signi ican ly om he expe imen and he h ee-phase model shows minimal de ia ions.
Ai ca i a ion occu s mainly in oil and apo ca i a ion was no achie ed a he gi en
p essu e and empe a u e (see Figu e 2). Fo compa ison, he same a ian s we e simu-
la ed.
(a) single-phase low: oil low;
(b) wo-phase low: oil and ai low;
(c) h ee-phase low: oil, apo and ai low.
Figu e 8b again shows no di e ences o indi idual a ian s in modeling low wi hou
ca i a ion. When modeling oil low wi h ca i a ion, a signi ican di e ence a inle p es-
su e is e iden o he one-phase a ian . The de ia ions be ween he wo-phase (oil, ai )
and h ee-phase model (oil, ai , apo ) a e no so impo an , because he apo ca i a ion
was no achie ed. A he na owes poin , he lowes modeled p essu e was 8500 Pa.
Figu e 7.
Dependence o ca i a ion numbe
σ
(1) s. Reynolds numbe in he na owes c oss-sec ion
Renozzle (1); de ail o he ca i a ion a ea is o he igh .
5.2. Resul s o Modeling
The single-phase model can be used o es ima e he ype o low, simple ask en y and
sho compu a ional ime. In he case o wa e ca i a ion, a wo-phase model (wa e – apo )
is commonly used. Due o he inaccu acies o he esul s, he model was ex ended o
a h ee-phase
model (wa e – apo –ai ). The accu acy o he calcula ion depends on he
co ec de e mina ion o he amoun o ai in he wa e depending on he empe a u e
and p essu e.
The e we e h ee simula ing a ian s o he wa e low h ough he nozzle:
(a)
single-phase low: wa e low;
(b)
wo-phase low: wa e and apo low;
(c)
h ee-phase low: wa e , apo and ai low.
Due o he inpu bounda y condi ions, absolu e p essu e
p1
was simula ed and sub-
sequen ly e i ied wi h he measu emen . Figu e 8p esen s he a e age alues o he
absolu e inle p essu e depending on he olume low a e when wa e and oil lowed
h ough he nozzle. Figu e 8a shows he di e ences be ween he simula ed and measu ed
inle p essu e in he wa e low as a unc ion o he numbe o phases. In he case o low
wi hou ca i a ion, he di e ences o he indi idual a ian s we e sligh . In he case o low
wi hou ca i a ion, he de ia ions o he measu ed p essu e
p1
(Pa) om he simula ed
alues
p1
(Pa) o he single-phase, wo-phase and h ee-phase models we e minimal. In
he case o low wi h ca i a ion, he one-phase and wo-phase models di e ed signi ican ly
om he expe imen and he h ee-phase model shows minimal de ia ions.
P ocesses 2021, 9, x FOR PEER REVIEW 10 o 14
(a) (b)
Figu e 8. (a) Dependence o inle p essu e 𝑝 (Pa) s. olume low a e 𝑄 (m3·s−1) o wa e ; (b)
Dependence o inle p essu e 𝑝 (Pa) s. olume low a e 𝑄 (m3·s−1) o Oil.
Figu e 9 e alua es he de ia ions o he simula ed alues o indi idual a ian s and
he measu ed alues depending on he ca i a ion numbe . Figu e 9 (a) p esen s a esul
o wa e . The p essu e de ia ions wi hou ca i a ion we e e y small, hey had alues up
o 2%. Howe e , when modeling low wi h ca i a ion, i is necessa y o de ine apo and
ai componen s in he liquid. In he wo-phase model (wa e , apo ) he de ia ions we e
smalle , bu he model was s ill no accu a e enough. Wi h he ex ended de ini ion o he
concen a ion o non-condensable gas in he wa e , he p essu e de ia ions a he inle
du ing he low wi h ca i a ion we e up o 5%.
(a) (b)
Figu e 9. (a) Dependence o de ia ion measu emen 𝛿 (%) s. ca i a ion numbe 𝜎 (1) o wa e ;
(b) Dependence o de ia ion measu emen δ (%) s. ca i a ion numbe σ (1) o oil.
Figu e 9b shows ha he oil de ia ions o oil we e he smalles o he wo-phase
model (oil, ai ), and had alues up o 2.5%. The h ee-phase model (oil, apo , ai ) di e s
e y li le om he wo-phase model due o a mo e complex ma hema ical model and
ounding e o s. When calcula ing he h ee-phase model, he compu a ional ime was
longe , and he calcula ion was less s able.
The s a is ically a e age alues o he p essu es on he wall in he na owed pa o
he nozzle a e p esen ed in de ail in Figu e 10. A h ee-phase model (wa e , apo , ai ) is
simula ed o wa e , and a wo-phase model o oil (oil, ai ). Fo wa e and oil, a a ian
o he ini ial ca i a ion σ = 0.747 (1) is plo ed, ollowed by wo a ian s o he de eloped
ca i a ion. The low-p essu e ange o he oil is sho e and smoo he . The ange o mini-
mum p essu e in wa e is caused by he o ma ion o apo and he low o ai as a com-
p essible medium. The size o he ca i a ion cloud in wa e low is signi ican ly longe
han in oil. This is a ec ed by he low o ai in he oil.
Volume lows a e gi en o hese ca i a ion numbe s:
(a) wa e compa ison:
𝜎=0.747 (ini ial ca i a ion, Q=4.51∙10 m·𝑠)
𝜎=0.552 (de eloped ca i a ion, Q= 5.53∙10 m·𝑠),
𝜎=0.433 ( ully de eloped ca i a ion, Q=5.98∙10 m·𝑠)
Figu e 8.
(
a
) Dependence o inle p essu e
p1
(Pa) s. olume low a e
Q
(m
3·
s
−1
) o
wa e ; (b) Dependence o inle p essu e p1(Pa) s. olume low a e Q(m3·s−1) o Oil.
Ai ca i a ion occu s mainly in oil and apo ca i a ion was no achie ed a he
gi en p essu e and empe a u e (see Figu e 2). Fo compa ison, he same a ian s we e
simula ed.
(a)
single-phase low: oil low;
(b)
wo-phase low: oil and ai low;
(c)
h ee-phase low: oil, apo and ai low.