Ci a ion: Jablonská, J.; Kozubko á,
M.; D ábko á, S.; Blejchaˇ , T. E ec o
Ca i a ing Hyd aulic Elemen s on
Pump Cha ac e is ics. P ocesses 2023,
11, 2592. h ps://doi.o g/10.3390/
p 11092592
Academic Edi o s: Wenjie Wang,
Gio gio Pa esi, Jin-Hyuk Kim, Ji Pei
and Lijian Shi
Recei ed: 31 July 2023
Re ised: 24 Augus 2023
Accep ed: 28 Augus 2023
Published: 30 Augus 2023
Copy igh : © 2023 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/).
p ocesses
A icle
E ec o Ca i a ing Hyd aulic Elemen s on
Pump Cha ac e is ics
Jana Jablonská* , Milada Kozubko á, Syl a D ábko áand Tomáš Blejchaˇ
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, 70800 Os a a-Po uba, Czech Republic;
[email p o ec ed] (M.K.); [email p o ec ed] (S.D.); [email p o ec ed] (T.B.)
*Co espondence: [email p o ec ed]
Abs ac :
The e ec i e and eliable pe o mance o any pump can be signi ican ly impac ed by
he piping sys em design. One o he essen ial poin s is he pump suc ion pipe. Poo design and
dimension o he suc ion piping can lead o ca i a ion in he pump which a ec s i s head and
e iciency. The p ima y objec i e is o educe he hyd aulic losses o he suc ion piping in o de
o main ain a Ne Posi i e Suc ion Head equi ed by he pump. Suc ion piping is ecommended
o be sho and s aigh , and b anch connec ions, al es and elbows should be a oided, which is
no always possible. In addi ion, ca i a ion can occu in he ac ual hyd aulic elemen s ins alled on
he pump suc ion. This wo k is ocused on he in es iga ion o ca i a ing hyd aulic elemen in he
suc ion pipe on he pump pe o mance. A con e ging-di e ging nozzle wi h a ci cula c oss-sec ion
was used o his pu pose. A s aigh pipe segmen o cons an diame e and he same leng h was
used o compa ison. Bo h elemen s we e cha ac e ized by he loss coe icien and he ca i a ion
numbe . Thei in luence on he pump head and he Ne Posi i e Suc ion Head A ailable (NPSHA)
was in es iga ed.
Keywo ds: ca i a ion; pump; suc ion piping; con e ging-di e ging nozzle; NPSH
1. In oduc ion
The p oblem o ca i a ion in he impelle and on he pump suc ion has been add essed
by many publica ions [
1
]. The undamen al mechanism o ca i y dynamics is desc ibed
as he p ocess o he g ow h and apid collapse o apo bubbles in a luid low ha has
damaging and pe o mance-deg ading e ec s on he pump [
1
–
6
]. A dis inc ion should
be made be ween apo and gas ca i a ion. Vapo ca i a ion a ises i he s a ic p essu e
d ops unde he sa u a ed apo p essu e. Gas ca i a ion in ol es he o ma ion o bubbles
because o he elease o dissol ed gases om he liquid in connec ion wi h di usion [
2
,
3
].
An impo an ac o in e alua ing he suc ion cha ac e is ics o a cen i ugal pump is
he Ne Posi i e Suc ion Head (NPSH). The NPSH can be de ined as he di e ence be ween
he o al head in he inle nozzle and he apo head [7],
NPSH =ps+pb−pw
ρg+c2
s
2g+zs(1)
whe e
ps+pb
is absolu e s a ic p essu e in a suc ion nozzle,
pw
is he apo p essu e o he
luid co esponding o he empe a u e,
cs
is eloci y in a suc ion nozzle,
zs
is he geode ic
heigh be ween he suc ion nozzle cen e and he e e ence le el,
ρ
is he densi y o a luid
and gis g a i y accele a ion.
We dis inguish wo NPSH alues: he Ne Posi i e Suc ion Head Requi ed by he
pump (NPSHR) and he Ne Posi i e Suc ion Head A ailable in he sys em (NPSHA). The
NPSHR is a unc ion o he pump design, i.e., he impelle inle diame e , numbe o blades,
blade inle angle and low inle condi ions. In case o NPSHR, he sum o he e ms in he
P ocesses 2023,11, 2592. h ps://doi.o g/10.3390/p 11092592 h ps://www.mdpi.com/jou nal/p ocesses
P ocesses 2023,11, 2592 2 o 12
Equa ion (1) mus no all below a minimum alue speci ied o a gi en pump. De e mining
NPSHR is pe o med by pump es ing. Mos o en, he cons an low me hod is used o
his pu pose. In his me hod, he pump discha ge al e is kep a a ixed open posi ion so
ha he pump-de eloped head and low a e emain cons an . The pump suc ion al e
is h o led o educe he pump inle p essu e un il a poin whe e he o al pump head
dec eases by a leas 3 pe cen . The ne posi i e suc ion head co esponding o he 3% d op
in he pump head is known as NPSH3 and is commonly used as he ele an ca i a ion
c i e ion; howe e , a ious ca i a ion c i e ia can be de e mined [8–11].
The a ailable suc ion ene gy NPSHA is a unc ion o he design, a angemen and con-
di ion o he suc ion pipe and i ings. NPSHA is he ene gy head de ined by he hyd aulic
loss o he pump suc ion line be ween he liquid le el in he suc ion ank and he pump
suc ion nozzle o 1 kg o liquid (suc ion head). I is dependen on he ope a ing condi ion
o he sys em ( low a e) and p o ides in o ma ion abou he isk o ca i a ion [6,12,13].
Subs i u ing he alues o he sys em in o Equa ion (1), we ob ain:
NPSHA =pe+pb−pw
ρg+c2
e
2g±ze−H s (2)
whe e
pe
is he measu ed ela i e p essu e,
pb
is he ba ome ic p essu e and
pe+pb
is he
absolu e p essu e in a suc ion ank,
pw
is he apo p essu e o he luid co esponding
o he empe a u e,
ce
is he eloci y de ined a he liquid su ace in he suc ion ank,
ze
is he geode ic heigh be ween he liquid su ace in he suc ion ank and he e e ence
le el (suc ion pipe axis in case o ho izon al pipe), and
H s
is he loss head on he sys em
suc ion side.
The NPSHA can be plo ed agains he low a e in he same way as he NPSHR. To
ensu e he con inuous and eliable ope a ion o a pump, he NPSHA should always be
highe han he NPSHR in all du y poin s, and a ce ain sa e y NPSHA ma gin o e he
NPSHR mus be kep . Mos o en, NPSHA = (1.1
÷
1.5) NPSHR. Inadequa e NPSHA
can lead o liquid e apo a ion in he low-p essu e zone a he eye o he impelle . The
p e ailing low p essu e o he liquid ises as he apo pocke s a e se he impelle blades.
The ising p essu e causes he apo pocke s o collapse agains he impelle and pump
housing, p oducing noise and damage o ca i a ion [
14
]. We can imp o e he NPSHA by
ei he inc easing he suc ion s a ic head abo e he pump cen eline and he p essu e abo e
he liquid in he suc ion ank o by educing he losses on he sys em suc ion side, which
sub ac s ene gy om he liquid.
The suc ion pipe design should ensu e an undis u bed app oach o he low o he
pump impelle . Elbows, ees and al es should no be ins alled close o he pump’s suc ion
nozzle o p e en an i egula low pa e n inside he suc ion line. Na ow pipes and
cons ic ions p oducing u bulence and p essu e losses should be a oided as hey can wo k
as a sou ce o ca i a ion.
Many p esen ed pape s deal wi h he in es iga ion o p essu e losses in hyd aulic
elemen s in a ious applica ions. The au ho s o [
15
] in es iga ed he in luence o angu-
la i ing connec ions in pipeline sys ems o anspo machines’ hyd aulic d i es. The
pe o med esea ch showed ha each ype o angula connec ion equi ed an accu a e
de ini ion o he low coe icien based on he low a e ha could be ob ained bo h expe i-
men ally and by means o nume ical simula ion. The s anda d equi alen leng h me hod
was no capable o aking complex low phenomena in o accoun . Nume ical modelling can
be used as a ool, which p o ides de ailed desc ip ion o eloci y and p essu e ield in he
hyd aulic componen s in a ious condi ions. Simila esea ch was p esen ed in [
16
]. The
analysis o luid low and ene gy e iciency inside non- epai ed and epai ed high-p essu e
hoses was p esen ed in his wo k. The p essu e d op, powe losses and low coe icien s
we e in es iga ed. Nume ical simula ion esul s we e e i ied by he physical expe imen .
I was concluded ha he epai o he hose wi h a junc ion i ing led o changes in he
low cha ac e is ic and he inc ease o powe losses.
P ocesses 2023,11, 2592 3 o 12
As can be seen om [
15
,
16
], i is possible o analyse he cha ac e is ics o hyd aulic
elemen s by nume ical modelling, bu in he case o mo e complex elemen geome y and
unde complica ed ope a ing condi ions (high empe a u e, sa u a ed apo p essu e, phys-
ical p ope ies o liquids, ai con en ), i is ad isable o e i y he me hod expe imen ally,
as p esen ed in [3,8].
The au ho s o [
8
] ocused hei a en ion on he CFD (Compu a ional Fluid Dy-
namics) in es iga ion o he low ield in he con e ging-di e ging nozzle. The wide
ange o he ca i a ion egimes we e in es iga ed nume ically o complemen he p e ious
expe imen al esul s.
The au ho s o [
3
] ocused on he measu emen and e alua ion o he hyd aulic
pa ame e s o a con e gen -di e gen nozzle o ec angula and ci cula c oss-sec ion
du ing he low o wa e ha was sa u a ed wi h ai . The dependence o he size o he
ca i a ion cloud and he change o hyd aulic pa ame e s (p essu e and low) on he amoun
o abso bed ai we e moni o ed. In conclusion, i is necessa y o model he ai phase in he
low wi h ca i a ion.
The p essu e losses in hyd aulic elemen s can be in es iga ed in connec ion wi h he
pumping sys ems. Bo h he pump cons uc ion and suc ion pipe design ha e been subjec ed
o esea ch. This issue has also been he subjec o analysis in he la es publica ions [
17
].
In [
17
], he au ho s compa ed wo me hods o he axial low wa e je pump design wi h
espec o ca i a ion. Nume ical simula ions, including he equa ions o ca i a ion low
and physical expe imen s, we e conduc ed o in es iga e he low cha ac e is ics. A en ion
was paid o he changes o in e nal low ields wi h he a ia ion o he ca i a ion numbe .
The design o he suc ion pipeline can signi ican ly a ec he pump ope a ion. The
o ma ion o ca i a ion in hyd aulic elemen s can con ibu e o dec ease o he pump
pe o mance and he pa ame e s a which he pump head s a s o all.
Ca i a ion can be iden i ied acous ically [
4
,
18
], by ib a ions [
19
], isually and by
measu ing hyd aulic cha ac e is ics [
4
,
8
,
14
,
15
]. An o e iew o ca i a ion de ec ion in
pumps is gi en in [20,21].
The au ho s o [
4
] ocused on he a ailable me hods o iden i ying ca i a ion in
a hyd aulic elemen . Hyd aulic quan i ies we e measu ed he e, om which he loss
coe icien was e alua ed. F om he dependence o he loss coe icien on he Reynolds
numbe (expe imen ), he o ma ion o ca i a ion was clea ly isible, which was also
suppo ed by he isualiza ion on he anspa en elemen . The main con ibu ion o he
publica ion [
4
] was he de ec ion and dynamic beha iou o he ca i a ion cloud eco ded
by a high-speed came a, while he a ailable quan i ies we e measu ed simul aneously
(p essu e, low a e, noise, ib a ion and he amoun o he ai dissol ed in he liquid).
The e alua ion o all quan i ies con i med he change in he low ield, which he au ho s
a ibu ed o he occu ence o ca i a ion.
The au ho s o [
19
] ocused on he de ec ion and moni o ing o he ca i a ion phe-
nomenon wi hin a cen i ugal pump using he ib a ion echnique. Vib a ion signals we e
analysed in bo h ime and equency domains unde di e en ope a ing condi ions and
p o ed o be a good ool o ca i a ion indica ion.
The au ho s o [
18
] p esen ed he simula ion esul s o he acous ic ield a ound an
unde wa e supe ca i a ion ehicle. They p o ided a comp ehensi e nume ical s udy o
he in luence o a ious ope a ing condi ions on he en ila ed ca i a ion phenomenon
and he noise cha ac e is ic spec um in a wide- equency domain. The ob ained esul s
p o ide heo e ical suppo o s udying he supe ca i a ion ehicles’ noise and applying
he en ila ed supe ca i a ion echnology.
The au ho s o [
20
] ecommended he de ec ion o ca i a ion in p oblema ic cases by
combining wo me hods. Ca i a ion de ec ion by he acous ic me hod is non-des uc i e
and has a high accu acy o ea ly ca i a ion de ec ion, bu he p ice o senso s is high, and
his me hod is di icul o use in p ac ice. Vib a ion measu emen is widely used in p ac ice,
bu his me hod is unsui able o ca i a ion de ec ion due o he a enua ion o he signal
ha p opaga es h ough he s uc u e and i s low accu acy o he p ima y de ec ion o
P ocesses 2023,11, 2592 4 o 12
ca i a ion. The isual me hod is applicable p ima ily in labo a o ies whe e anspa en
elemen s can be used.
When de ec ing ca i a ion using he measu emen o hyd aulic quan i ies o p essu e
pulsa ions, pipe des uc ion may occu when p essu e senso s a e connec ed; he ins alla-
ion o senso s is complex and no always desi able in p ac ice, and he low accu acy o
de ec ion du ing ini ial ca i a ion is also a disad an age. Howe e , he me hod is esis an
o in e e ence om su ounding in luences [17].
Hyd aulic sys ems allow o anspo he luid h ough he se o in e connec ed
disc e e elemen s. The pe o mance o such sys em can be desc ibed by he sys em cha -
ac e is ic cu e, which is a esponse in he head o p essu e o a gi en low a e. The
sys em cha ac e is ic is calcula ed based on majo ( ic ion) losses in he pipelines and
mino losses in he hyd aulic elemen s. The p esence o ca i a ion is o en only disco e ed
du ing he implemen a ion and ope a ion o he sys em. The main mo i a ion o his pape
was he e o e he e alua ion o he cha ac e is ics a ec ed by ca i a ion o indi idual
elemen s (pipe, con e gen -di e gen nozzle) and, subsequen ly, he cha ac e is ics a ec ed
by ca i a ion o he pump.
The ca i a ing elemen s loca ed in on o he pump signi ican ly a ec he pump
cha ac e is ics, while he elemen s behind he pump do no ha e a signi ican e ec on
he cha ac e is ics o he pump supplied by he manu ac u e . By de aul , p essu e losses
in he ins alled elemen s a e de ined du ing he design o he suc ion pipe using a loss
coe icien depending on he low a e (o speed, Reynolds numbe ). The loss coe icien s
a e de ined on he basis o he p essu e d op on he hyd aulic elemen in he ca i a ion- ee
mode. Howe e , he speci ic alue o he absolu e p essu e a he en ance o he pump
is essen ial, which can change signi ican ly in connec ion wi h he o ma ion o ca i a ion
in he hyd aulic elemen s on he suc ion, as demons a ed by he expe imen al esea ch
ca ied ou . This conclusion can be s a ed as he main con ibu ion o he a icle.
2. Objec i e and P ocedu e o he Expe imen
The main objec i e o his wo k was o in es iga e he in luence o ca i a ing hyd aulic
elemen s ins alled in he pump suc ion pipe on he pump pe o mance. We chose a
con e ging-di e ging nozzle wi h a ci cula c oss-sec ion o his pu pose. A s aigh pipe
segmen o cons an diame e and he same leng h was used o compa ison. The design
and main dimensions o bo h a e p esen ed in Figu e 1.
P ocesses 2023, 11, x FOR PEER REVIEW 4 o 12
signal ha p opaga es h ough he s uc u e and i s low accu acy o he p ima y de ec-
ion o ca i a ion. The isual me hod is applicable p ima ily in labo a o ies whe e ans-
pa en elemen s can be used.
When de ec ing ca i a ion using he measu emen o hyd aulic quan i ies o p essu e
pulsa ions, pipe des uc ion may occu when p essu e senso s a e connec ed; he ins alla-
ion o senso s is complex and no always desi able in p ac ice, and he low accu acy o
de ec ion du ing ini ial ca i a ion is also a disad an age. Howe e , he me hod is esis an
o in e e ence om su ounding in luences [17].
Hyd aulic sys ems allow o anspo he luid h ough he se o in e connec ed dis-
c e e elemen s. The pe o mance o such sys em can be desc ibed by he sys em cha ac-
e is ic cu e, which is a esponse in he head o p essu e o a gi en low a e. The sys em
cha ac e is ic is calcula ed based on majo ( ic ion) losses in he pipelines and mino
losses in he hyd aulic elemen s. The p esence o ca i a ion is o en only disco e ed du -
ing he implemen a ion and ope a ion o he sys em. The main mo i a ion o his pape
was he e o e he e alua ion o he cha ac e is ics affec ed by ca i a ion o indi idual el-
emen s (pipe, con e gen -di e gen nozzle) and, subsequen ly, he cha ac e is ics affec ed
by ca i a ion o he pump.
The ca i a ing elemen s loca ed in on o he pump signi ican ly affec he pump
cha ac e is ics, while he elemen s behind he pump do no ha e a signi ican effec on he
cha ac e is ics o he pump supplied by he manu ac u e . By de aul , p essu e losses in
he ins alled elemen s a e de ined du ing he design o he suc ion pipe using a loss coe -
icien depending on he low a e (o speed, Reynolds numbe ). The loss coefficien s a e
de ined on he basis o he p essu e d op on he hyd aulic elemen in he ca i a ion- ee
mode. Howe e , he speci ic alue o he absolu e p essu e a he en ance o he pump is
essen ial, which can change signi ican ly in connec ion wi h he o ma ion o ca i a ion in
he hyd aulic elemen s on he suc ion, as demons a ed by he expe imen al esea ch ca -
ied ou . This conclusion can be s a ed as he main con ibu ion o he a icle.
2. Objec i e and P ocedu e o he Expe imen
The main objec i e o his wo k was o in es iga e he in luence o ca i a ing hyd au-
lic elemen s ins alled in he pump suc ion pipe on he pump pe o mance. We chose a
con e ging-di e ging nozzle wi h a ci cula c oss-sec ion o his pu pose. A s aigh pipe
segmen o cons an diame e and he same leng h was used o compa ison. The design
and main dimensions o bo h a e p esen ed in Figu e 1.
(A)
(B)
Figu e 1. Geome y o he con e ging-di e ging nozzle (A) and pipe segmen (B). The main dimen-
sions o he nozzle and pipe a e gi en in millime es.
The nozzle was made o anspa en ma e ial, which made i possible o obse e he
o ma ion o ca i a ion bubbles, as illus a ed in Figu e 2 [5]. The s udy o he one-dimen-
sional low wi h bubble ca i a ion effec can help in unde s anding he mechanism o
Figu e 1. Geome y o he con e ging-di e ging nozzle (A) and pipe segmen (B). The main dimen-
sions o he nozzle and pipe a e gi en in millime es.
The nozzle was made o anspa en ma e ial, which made i possible o obse e
he o ma ion o ca i a ion bubbles, as illus a ed in Figu e 2[
5
]. The s udy o he one-
dimensional low wi h bubble ca i a ion e ec can help in unde s anding he mechanism o
ca i a ion o igin in di e en condi ions o low and o e alua e i s con ibu ion o NPSHA
and pump head dec eases.
P ocesses 2023,11, 2592 5 o 12
P ocesses 2023, 11, x FOR PEER REVIEW 5 o 12
ca i a ion o igin in diffe en condi ions o low and o e alua e i s con ibu ion o NPSHA
and pump head dec eases.
Re = 19,000
ini ial
ca i a ion
Re = 23,500
de eloped
ca i a ion
Figu e 2. Ini ial and de eloped ca i a ion in he con e ging-di e ging nozzle o he ci cula c oss-
sec ion. The Reynolds numbe is de e mined a he inle (illus a i e pho o).
Basic cha ac e is ics ∆𝑝 = 𝑓󰇛𝑄󰇜 we e e alua ed o bo h hyd aulic elemen s (nozzle
and s aigh pipe). Fu he , he loss coefficien 𝜁 and ca i a ion numbe Ca we e calcu-
la ed and plo ed agains he Reynolds numbe Re.
Re = 𝑐 𝑑 𝜌
𝜇
whe e 𝑐 is he eloci y, 𝑑 is he diame e , 𝜌 is he densi y o a luid and 𝜇 is he dy-
namics iscosi y. The loss coefficien 𝜁 can be de e mined om he Be noulli equa ion:
∆𝑝 = 𝜌 𝜁𝑐
2⇒𝜁=2 ∆𝑝
𝜌 𝑐 (3)
whe e ∆𝑝 is he p essu e loss, 𝜌 is he densi y o a luid and 𝑐 is he eloci y. The ca i-
a ion numbe Ca ep esen s he a io o he s a ic p essu e ( he diffe ence be ween he
inle absolu e p essu e and he sa u a ed apo p essu e) o he dynamic p essu e as gi en
by Equa ion (4). The c i ical Ca co esponds o he o igin o ca i a ion in he con e ging-
di e ging nozzle.
Ca = 2 󰇛𝑝−𝑝
󰇜
𝜚 𝑐 (4)
whe e 𝑝 is he p essu e a he inle o he measu ed elemen , 𝑝 is he apo p essu e
o he luid co esponding o he empe a u e, 𝜌 is he densi y o a luid and 𝑐 is he
eloci y in na owing. The pump was es ed in he labo a o y unde a ious discha ge
and head condi ions. The cha ac e is ic 𝐻−𝑄 cu e (head— low a e) o he pump was
p o ided by he manu ac u e . The pump cu e was hen measu ed wi h he con e ging-
di e ging nozzle and s aigh pipe ins alled in he suc ion pipe.
Applying Be noulli’s equa ion be ween poin s P3 and P2 in Figu e 3 esul s in:
𝐻=𝑝−𝑝

𝜚 𝑔 +𝑐
−𝑐


𝜚 𝑔 +𝑧 (5)
whe e 𝑝 is p essu e poin P3 (see Figu e 3), 𝑝 is p essu e poin P2 (see Figu e 3), 𝑐 is
eloci y poin P3 (see Figu e 3), 𝑣 is eloci y poin P2 (see Figu e 3),
𝜌 is he densi y o
a luid and 𝑧 is he geode ic heigh be ween he liquid su ace in he suc ion ank and he
e e ence le el. Since he diffe ence be ween he ele a ions 𝑧 and eloci ies a poin s P2
and P3 a e negligible, he equa ion becomes:
𝐻=𝑝−𝑝

𝜚 𝑔 =∆𝑝
𝜚 𝑔 (6)
The NPSHA was e alua ed om Equa ion (2). Since he ank is opened, 𝑝=0,𝑐
=
0 and he equa ion becomes:
Figu e 2.
Ini ial and de eloped ca i a ion in he con e ging-di e ging nozzle o he ci cula c oss-
sec ion. The Reynolds numbe is de e mined a he inle (illus a i e pho o).
Basic cha ac e is ics
∆p= (Q)
we e e alua ed o bo h hyd aulic elemen s (nozzle
and s aigh pipe). Fu he , he loss coe icien
ζ
and ca i a ion numbe
Ca
we e calcula ed
and plo ed agains he Reynolds numbe Re.
Re =c d ρ
µ
whe e
c
is he eloci y,
d
is he diame e ,
ρ
is he densi y o a luid and
µ
is he dynamics
iscosi y. The loss coe icien ζcan be de e mined om he Be noulli equa ion:
∆p=ρ ζ c2
2⇒ζ=2∆p
ρc2(3)
whe e
∆p
is he p essu e loss,
ρ
is he densi y o a luid and
c
is he eloci y. The ca i a ion
numbe
Ca
ep esen s he a io o he s a ic p essu e ( he di e ence be ween he inle
absolu e p essu e and he sa u a ed apo p essu e) o he dynamic p essu e as gi en by
Equa ion (4). The c i ical
Ca
co esponds o he o igin o ca i a ion in he con e ging-
di e ging nozzle.
Ca =2(p1−pw)
$c2(4)
whe e
p1
is he p essu e a he inle o he measu ed elemen ,
pw
is he apo p essu e o
he luid co esponding o he empe a u e,
ρ
is he densi y o a luid and
c
is he eloci y
in na owing. The pump was es ed in he labo a o y unde a ious discha ge and head
condi ions. The cha ac e is ic
H−Q
cu e (head— low a e) o he pump was p o ided
by he manu ac u e . The pump cu e was hen measu ed wi h he con e ging-di e ging
nozzle and s aigh pipe ins alled in he suc ion pipe.
Applying Be noulli’s equa ion be ween poin s P3 and P2 in Figu e 3 esul s in:
H=p3−p2
$g+c2
3−c2
2
$g+z(5)
whe e
p3
is p essu e poin P3 (see Figu e 3),
p2
is p essu e poin P2 (see Figu e 3),
c3
is
eloci y poin P3 (see Figu e 3),
2
is eloci y poin P2 (see Figu e 3),
ρ
is he densi y o a
luid and
z
is he geode ic heigh be ween he liquid su ace in he suc ion ank and he
e e ence le el. Since he di e ence be ween he ele a ions
z
and eloci ies a poin s P2
and P3 a e negligible, he equa ion becomes:
H=p3−p2
$g=∆p
$g(6)

P ocesses 2023,11, 2592 6 o 12
The NPSHA was e alua ed om Equa ion (2). Since he ank is opened,
pe=
0,
ce=
0
and he equa ion becomes:
NPSHA =pb−pw
ρg±ze−H s (7)
whe e
pb
is ba ome ic p essu e,
pw
is he apo p essu e o he luid co esponding o he
empe a u e,
ρ
is he densi y o a luid,
ze
is he geode ic heigh be ween he liquid su ace
in he suc ion ank and he e e ence le el (suc ion pipe axis in case o ho izon al pipe) and
H s
is loss head on he sys em suc ion side. Fo compa ison, he nozzle was also si ua ed
behind he pump on he discha ge pipe. The pump cu e
∆p= (Q)
and
NPSHA = (Q)
we e e alua ed.
3. Expe imen al Se up
The hyd aulics bench was i ed wi h a single cen i ugal pump G und os MAGNA1.
The liquid (wa e ) was pumped om he opened ank (T) wi h a wa e le el o 500 mm
u he in o he sys em, which was designed as o e p essu e (o e p essu e was induced by
a column o liquid in he ank). Tempe a u e (TM) and oxygen (O2) senso s we e loca ed in
he ank. A low me e (FM) behind he ank was used o measu e he olume low a e.
Pump (P) suc ion and discha ge p essu es we e measu ed o e alua e he pump head. The
low a e was con olled by he al e (V) on he discha ge pipe.
The e we e h ee a ian s o he loca ion o he measu ed hyd aulic elemen (HE):
HE_A—a anspa en con e ging-di e ging nozzle o ci cula c oss-sec ion wi h an inle
diame e o 20 mm and minimum diame e o 6 mm, loca ed ups eam o he pump. The
p essu e d op was e alua ed om he p essu e di e ences
∆p=p2−p1
, and he low a e
was measu ed by an FM low me e .
HE_B—a anspa en ube wi h a cons an inne diame e o 20 mm, loca ed ups eam o
he pump. The p essu e d op was e alua ed om he p essu e di e ences
∆p=p2−p1
,
and he low a e was measu ed by an FM low me e .
HE_C—a anspa en con e ging-di e ging nozzle o ci cula c oss-sec ion wi h an inle
diame e o 20 mm and minimum diame e o 6 mm, loca ed downs eam o he pump.
The p essu e d op was e alua ed om he p essu e di e ences
∆p=p3−p4
, and he low
a e was measu ed by an FM low me e .
P ocesses 2023, 11, x FOR PEER REVIEW 6 o 12
NPSHA = 𝑝−𝑝

𝜌 𝑔 ±𝑧
−𝐻 (7)
whe e 𝑝 is ba ome ic p essu e, 𝑝 is he apo p essu e o he luid co esponding o
he empe a u e, 𝜌 is he densi y o a luid, 𝑧 is he geode ic heigh be ween he liquid
su ace in he suc ion ank and he e e ence le el (suc ion pipe axis in case o ho izon al
pipe) and 𝐻 is loss head on he sys em suc ion side. Fo compa ison, he nozzle was
also si ua ed behind he pump on he discha ge pipe. The pump cu e ∆𝑝 = 𝑓󰇛𝑄󰇜 and
NPSHA = 𝑓󰇛𝑄󰇜 we e e alua ed.
3. Expe imen al Se up
The hyd aulics bench was i ed wi h a single cen i ugal pump G und os MAGNA1.
The liquid (wa e ) was pumped om he opened ank (T) wi h a wa e le el o 500 mm
u he in o he sys em, which was designed as o e p essu e (o e p essu e was induced
by a column o liquid in he ank). Tempe a u e (TM) and oxygen (O2) senso s we e lo-
ca ed in he ank. A low me e (FM) behind he ank was used o measu e he olume
low a e. Pump (P) suc ion and discha ge p essu es we e measu ed o e alua e he pump
head. The low a e was con olled by he al e (V) on he discha ge pipe.
The e we e h ee a ian s o he loca ion o he measu ed hyd aulic elemen (HE):
HE_A—a anspa en con e ging-di e ging nozzle o ci cula c oss-sec ion wi h an inle
diame e o 20 mm and minimum diame e o 6 mm, loca ed ups eam o he pump. The
p essu e d op was e alua ed om he p essu e diffe ences ∆𝑝 = 𝑝−𝑝
, and he low
a e was measu ed by an FM low me e .
HE_B—a anspa en ube wi h a cons an inne diame e o 20 mm, loca ed ups eam o
he pump. The p essu e d op was e alua ed om he p essu e diffe ences ∆𝑝 = 𝑝−𝑝
,
and he low a e was measu ed by an FM low me e .
HE_C—a anspa en con e ging-di e ging nozzle o ci cula c oss-sec ion wi h an inle
diame e o 20 mm and minimum diame e o 6 mm, loca ed downs eam o he pump.
The p essu e d op was e alua ed om he p essu e diffe ences ∆𝑝 = 𝑝−𝑝
 , and he
low a e was measu ed by an FM low me e .
Figu e 3. Hyd aulic ci cui . T— ank, FM— low me e , HE—hyd aulic elemen , P—pump, V al e,
DR—da a eco de , Comp—compu e , 𝑝

p essu e a he inle o he hyd aulic elemen A, B, 𝑝

p essu e a he ou le o he hyd aulic elemen A, B and a he inle o he pump, 𝑝

p essu e a he
ou le o he pump and a he inle o he hyd aulic elemen C, 𝑝

p essu e a he ou le o he hy-
d aulic elemen C. TM— he mome e , O2—oxygen me e .
In all a ian s, he p essu e ups eam and downs eam o he elemen HE was meas-
u ed, and he p essu e d op was e alua ed. The con igu a ion o he hyd aulic ci cui is
illus a ed in Figu e 3. The cen i ugal pump was si ua ed unde he liquid le el in he
ank. The discha ge om he e u n pipe was subme ged unde he liquid le el in he
opposi e side o he ank. The e was a pa i ion be ween he suc ion and e u n pipes.
The hyd aulic elemen s we e es ed sepa a ely; only one elemen was inse ed in o
he ci cui , and he hyd aulic cha ac e is ics we e in es iga ed. A e eplacing he ele-
men , he ci cui had o be en ed p ope ly. The HMG 3000 po able da a eco de was
used o eco ding he measu ed da a simul aneously a a a e o 1 ms o 10 s. The a e age
Figu e 3.
Hyd aulic ci cui . T— ank, FM— low me e , HE—hyd aulic elemen , P—pump, V al e,
DR—da a eco de , Comp—compu e ,
p1
p essu e a he inle o he hyd aulic elemen A, B,
p2
p essu e a he ou le o he hyd aulic elemen A, B and a he inle o he pump,
p3
p essu e a
he ou le o he pump and a he inle o he hyd aulic elemen C,
p4
p essu e a he ou le o he
hyd aulic elemen C. TM— he mome e , O2—oxygen me e .
In all a ian s, he p essu e ups eam and downs eam o he elemen HE was mea-
su ed, and he p essu e d op was e alua ed. The con igu a ion o he hyd aulic ci cui is
illus a ed in Figu e 3. The cen i ugal pump was si ua ed unde he liquid le el in he ank.
The discha ge om he e u n pipe was subme ged unde he liquid le el in he opposi e
side o he ank. The e was a pa i ion be ween he suc ion and e u n pipes.
The hyd aulic elemen s we e es ed sepa a ely; only one elemen was inse ed in o he
ci cui , and he hyd aulic cha ac e is ics we e in es iga ed. A e eplacing he elemen ,
P ocesses 2023,11, 2592 7 o 12
he ci cui had o be en ed p ope ly. The HMG 3000 po able da a eco de was used
o eco ding he measu ed da a simul aneously a a a e o 1 ms o 10 s. The a e age
alue was hen calcula ed om eco ded alues. Table 1gi es a summa y o he pump
expe imen al se up de ices.
Table 1. Pump expe imen al se up de ices.
Equipmen Speci ica ion
pump G und os MAGNA1 25–120–180
ul asound low me e
FLOMIC 1014
Range (0; 3) m3/h,
Accu acy ±0.5%
p essu e senso s
Hydac HDA 4346
Range (−1; 1) ba , (−1; 5) ba , (−1; 9) ba
Accu acy ±0.5%
he mome e Range (−10; 60) ◦C
Accu acy ±1%
dissol ed oxygen me e
End ess + Hause Liguisys, M COM253
Range (0; 20) mg L−1
Accu acy ±0.5%
4. Resul s and Discussion
Measu emen was pe o med and epea ed o a a ious ange o pump low a es up
o 0.0008 m
3·
s
−1
, which co esponds o a Reynolds numbe up o 40,000. The ba ome ic
p essu e
pb
was 99,180 Pa. The physical cons an s o wa e du ing he measu emen a e
summa ised in Table 2. Due o he amoun o liquid in he ci cui and he o al leng h o he
measu emen , i was no necessa y o s abilize he empe a u e. The measu ed empe a u e
change was up o 1 ◦C.
Table 2. Physical cons an s o wa e .
Physical Cons an s
Tempe a u e T=21 °C
densi y o wa e ρ=998 kg·m−3
Viscosi y µ=0.001 Pa·s
apo p essu e o he luid pw=2406 Pa
Figu e 4shows he cha ac e is ics o he con e ging-di e ging nozzle ( a ian A) and
he pipe segmen ( a ian B) loca ed in on o he pump and con e ging-di e ging nozzle
( a ian C) loca ed behind he pump. Filled poin s exp ess he low wi hou ca i a ion, and
emp y poin s indica e he low wi h ca i a ion.
In he non-ca i a ion egion, he cha ac e is ics o all h ee a ian s (A, B, C) a e app ox-
ima ely p opo ional o he squa e o he low a e wi h espec o he loss coe icien [5,7].
A sha p inc ease in p essu e loss in he con e ging-di e ging nozzle ( a ian s A and C) was
associa ed wi h he o ma ion o ca i a ion and co esponded o he isual obse a ion o
ca i a ion in he anspa en nozzle (see Figu e 2). Ca i a ion did no occu in he pipe ( a i-
an B), and he p essu e loss was signi ican ly smalle compa ed o con e ging-di e ging
nozzle. I can be obse ed ha ca i a ion in he con e ging-di e ging nozzle si ua ed
behind he pump ( a ian C) occu ed a highe alues o he low a e i compa ed wi h
a ian A.
P ocesses 2023,11, 2592 8 o 12
P ocesses 2023, 11, x FOR PEER REVIEW 7 o 12
alue was hen calcula ed om eco ded alues. Table 1 gi es a summa y o he pump
expe imen al se up de ices.
Table 1. Pump expe imen al se up de ices.
Equipmen Speci ica ion
pump G und os MAGNA1 25–120–180
ul asound low me e
FLOMIC 1014
Range (0; 3) m
3
/h,
Accu acy ± 0.5%
p essu e senso s
Hydac HDA 4346
Range (−1; 1) ba , (−1; 5) ba , (−1; 9) ba
Accu acy ± 0.5%
he mome e Range (−10; 60) °C
Accu acy ± 1%
dissol ed oxygen me e
End ess + Hause Liguisys, M COM253
Range (0; 20) mg L
−1
Accu acy ± 0.5%
4. Resul s and Discussion
Measu emen was pe o med and epea ed o a a ious ange o pump low a es
up o 0.0008 m
3
·s
−1
, which co esponds o a Reynolds numbe up o 40,000. The ba ome ic
p essu e 𝑝 was 99,180 Pa. The physical cons an s o wa e du ing he measu emen a e
summa ised in Table 2. Due o he amoun o liquid in he ci cui and he o al leng h o
he measu emen , i was no necessa y o s abilize he empe a u e. The measu ed em-
pe a u e change was up o 1 °C.
Table 2. Physical cons an s o wa e .
Physical Cons an s
Tempe a u e 𝑇 = 21 ℃
densi y o wa e 𝜌 = 998 kg ∙ m

Viscosi y 𝜇 = 0.001 Pa ∙ s
apo p essu e o he luid 𝑝

= 2406 Pa
Figu e 4 shows he cha ac e is ics o he con e ging-di e ging nozzle ( a ian A) and
he pipe segmen ( a ian B) loca ed in on o he pump and con e ging-di e ging noz-
zle ( a ian C) loca ed behind he pump. Filled poin s exp ess he low wi hou ca i a ion,
and emp y poin s indica e he low wi h ca i a ion.
Figu e 4. P essu e d op s. olume low a e o all a ian s.
Figu e 4. P essu e d op s. olume low a e o all a ian s.
The pump inle p essu e was moni o ed and compa ed o all a ian s (A, B, C), as
p esen ed in Figu e 5. The con e ging-di e ging nozzle in on o he pump ( a ian A)
exhibi ed a s eepe p essu e d op in compa ison wi h he pipe segmen wi h a cons an
diame e ( a ian B). Fu he mo e, i is no iceable ha om a ce ain alue o he low a e,
i dec eased signi ican ly as e , which was due o he o ma ion o ca i a ion. In he case o
he s aigh pipe ( a ian B), he p essu e d op was gi en only by he ic ion loss o he
pipe. The nozzle placed behind he pump did no a ec he p essu e a he pump inle ,
e en i ca i a ion was obse ed.
P ocesses 2023, 11, x FOR PEER REVIEW 8 o 12
In he non-ca i a ion egion, he cha ac e is ics o all h ee a ian s (A, B, C) a e ap-
p oxima ely p opo ional o he squa e o he low a e wi h espec o he loss coefficien
[5,7]. A sha p inc ease in p essu e loss in he con e ging-di e ging nozzle ( a ian s A and
C) was associa ed wi h he o ma ion o ca i a ion and co esponded o he isual obse -
a ion o ca i a ion in he anspa en nozzle (see Figu e 2). Ca i a ion did no occu in
he pipe ( a ian B), and he p essu e loss was signi ican ly smalle compa ed o con e g-
ing-di e ging nozzle. I can be obse ed ha ca i a ion in he con e ging-di e ging noz-
zle si ua ed behind he pump ( a ian C) occu ed a highe alues o he low a e i com-
pa ed wi h a ian A.
The pump inle p essu e was moni o ed and compa ed o all a ian s (A, B, C), as
p esen ed in Figu e 5. The con e ging-di e ging nozzle in on o he pump ( a ian A)
exhibi ed a s eepe p essu e d op in compa ison wi h he pipe segmen wi h a cons an
diame e ( a ian B). Fu he mo e, i is no iceable ha om a ce ain alue o he low
a e, i dec eased signi ican ly as e , which was due o he o ma ion o ca i a ion. In he
case o he s aigh pipe ( a ian B), he p essu e d op was gi en only by he ic ion loss
o he pipe. The nozzle placed behind he pump did no affec he p essu e a he pump
inle , e en i ca i a ion was obse ed.
Figu e 5. Inle pump p essu e s. olume low a e o all a ian s.
The loss coefficien 𝜁 was e alua ed om he p essu e d op on he hyd aulic ele-
men s Equa ion (3) and is p esen ed in Figu e 6. In he non-ca i a ion egion, he loss co-
efficien dec eased wi h he Reynolds numbe in he case o all a ian s (A, B, C). The
o igin o ca i a ion did no impac he loss coefficien signi ican ly, bu wi h he de eloped
ca i a ion, he loss coefficien inc eased apidly. Figu e 7 shows he dependence o he
dimensionless ca i a ion numbe Ca on he Reynolds numbe o a ian s A and C. I is
e iden ha a diffe en c i ical ca i a ion numbe applied o he con e ging-di e ging
nozzle placed in on and behind he pump, so he p oblem could no be simply gene al-
ized.
Figu e 5. Inle pump p essu e s. olume low a e o all a ian s.
The loss coe icien
ζ
was e alua ed om he p essu e d op on he hyd aulic elemen s
Equa ion (3) and is p esen ed in Figu e 6. In he non-ca i a ion egion, he loss coe icien
dec eased wi h he Reynolds numbe in he case o all a ian s (A, B, C). The o igin o
ca i a ion did no impac he loss coe icien signi ican ly, bu wi h he de eloped ca i a ion,
he loss coe icien inc eased apidly. Figu e 7shows he dependence o he dimensionless
ca i a ion numbe Ca on he Reynolds numbe o a ian s A and C. I is e iden ha a
di e en c i ical ca i a ion numbe applied o he con e ging-di e ging nozzle placed in
on and behind he pump, so he p oblem could no be simply gene alized.
In he case o a ian A, he ca i a ion occu ed a a lowe Reynolds numbe and
a ec ed he o ma ion o ca i a ion in he pump, as can be seen in Figu e 8. The e ec
o he ca i a ing elemen on he pump suc ion could be de e mined om he all o he
pump head. I is ob ious ha du ing ca i a ion, he pump cha ac e is ic de ia ed om he
cha ac e is ic gi en by he manu ac u e .
P ocesses 2023,11, 2592 9 o 12
P ocesses 2023, 11, x FOR PEER REVIEW 8 o 12
In he non-ca i a ion egion, he cha ac e is ics o all h ee a ian s (A, B, C) a e ap-
p oxima ely p opo ional o he squa e o he low a e wi h espec o he loss coefficien
[5,7]. A sha p inc ease in p essu e loss in he con e ging-di e ging nozzle ( a ian s A and
C) was associa ed wi h he o ma ion o ca i a ion and co esponded o he isual obse -
a ion o ca i a ion in he anspa en nozzle (see Figu e 2). Ca i a ion did no occu in
he pipe ( a ian B), and he p essu e loss was signi ican ly smalle compa ed o con e g-
ing-di e ging nozzle. I can be obse ed ha ca i a ion in he con e ging-di e ging noz-
zle si ua ed behind he pump ( a ian C) occu ed a highe alues o he low a e i com-
pa ed wi h a ian A.
The pump inle p essu e was moni o ed and compa ed o all a ian s (A, B, C), as
p esen ed in Figu e 5. The con e ging-di e ging nozzle in on o he pump ( a ian A)
exhibi ed a s eepe p essu e d op in compa ison wi h he pipe segmen wi h a cons an
diame e ( a ian B). Fu he mo e, i is no iceable ha om a ce ain alue o he low
a e, i dec eased signi ican ly as e , which was due o he o ma ion o ca i a ion. In he
case o he s aigh pipe ( a ian B), he p essu e d op was gi en only by he ic ion loss
o he pipe. The nozzle placed behind he pump did no affec he p essu e a he pump
inle , e en i ca i a ion was obse ed.
Figu e 5. Inle pump p essu e s. olume low a e o all a ian s.
The loss coefficien 𝜁 was e alua ed om he p essu e d op on he hyd aulic ele-
men s Equa ion (3) and is p esen ed in Figu e 6. In he non-ca i a ion egion, he loss co-
efficien dec eased wi h he Reynolds numbe in he case o all a ian s (A, B, C). The
o igin o ca i a ion did no impac he loss coefficien signi ican ly, bu wi h he de eloped
ca i a ion, he loss coefficien inc eased apidly. Figu e 7 shows he dependence o he
dimensionless ca i a ion numbe Ca on he Reynolds numbe o a ian s A and C. I is
e iden ha a diffe en c i ical ca i a ion numbe applied o he con e ging-di e ging
nozzle placed in on and behind he pump, so he p oblem could no be simply gene al-
ized.
Figu e 6. Loss coe icien s. Reynolds numbe .
P ocesses 2023, 11, x FOR PEER REVIEW 9 o 12
Figu e 6. Loss coefficien s. Reynolds numbe .
Figu e 7. Ca i a ion numbe s. Reynolds numbe .
In he case o a ian A, he ca i a ion occu ed a a lowe Reynolds numbe and
affec ed he o ma ion o ca i a ion in he pump, as can be seen in Figu e 8. The effec o
he ca i a ing elemen on he pump suc ion could be de e mined om he all o he pump
head. I is ob ious ha du ing ca i a ion, he pump cha ac e is ic de ia ed om he cha -
ac e is ic gi en by he manu ac u e .
Figu e 8. Pump cu e impac ed by hyd aulic elemen s in he ci cui .
The cou se o he p essu e a he pump inle co esponded o he NPSHA acco ding
o Equa ion (1). A nozzle in he suc ion pipe led o an inc ease in hyd aulic esis ance and
he o ma ion o ca i a ion, he eby educing NPSHA, as shown in Figu e 9.
Figu e 9. NPSHA-Q cha ac e is ics impac ed by hyd aulic elemen s in he ci cui .
Figu e 7. Ca i a ion numbe s. Reynolds numbe .
P ocesses 2023, 11, x FOR PEER REVIEW 9 o 12
Figu e 6. Loss coefficien s. Reynolds numbe .
Figu e 7. Ca i a ion numbe s. Reynolds numbe .
In he case o a ian A, he ca i a ion occu ed a a lowe Reynolds numbe and
affec ed he o ma ion o ca i a ion in he pump, as can be seen in Figu e 8. The effec o
he ca i a ing elemen on he pump suc ion could be de e mined om he all o he pump
head. I is ob ious ha du ing ca i a ion, he pump cha ac e is ic de ia ed om he cha -
ac e is ic gi en by he manu ac u e .
Figu e 8. Pump cu e impac ed by hyd aulic elemen s in he ci cui .
The cou se o he p essu e a he pump inle co esponded o he NPSHA acco ding
o Equa ion (1). A nozzle in he suc ion pipe led o an inc ease in hyd aulic esis ance and
he o ma ion o ca i a ion, he eby educing NPSHA, as shown in Figu e 9.
Figu e 9. NPSHA-Q cha ac e is ics impac ed by hyd aulic elemen s in he ci cui .
Figu e 8. Pump cu e impac ed by hyd aulic elemen s in he ci cui .
The cou se o he p essu e a he pump inle co esponded o he NPSHA acco ding o
Equa ion (1). A nozzle in he suc ion pipe led o an inc ease in hyd aulic esis ance and he
o ma ion o ca i a ion, he eby educing NPSHA, as shown in Figu e 9.
Ca i a ion is cha ac e ized by he o ma ion o a apo phase and he elease o ai
om wa e . The elease o ai is iden i iable by measu ing he oxygen concen a ion in he
liquid. Simul aneously, ni ogen is also eleased, and he amoun o eleased ai can hen
be de e mined. This con i ms he p esence o he ai ca i a ion besides he apo ca i a ion.
We can see ha he measu ed ela i e ai concen a ion is cons an i ca i a ion does no
occu in he ci cui (Figu e 10). I ca i a ion ully de elops in he ci cui , hen he ela i e ai