Effect of cavitating hydraulic elements on pump characteristics

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