Non-linear mathematical models for the jets penetrating liquid pool of different density under diverse physical conditions and their simulation

Transcription

1 Ian V Kazahko Olxandr V Konoal Non-linar mahmaial modls for h js pnraing liquid pool of diffrn dnsiy undr dirs physial ondiions and hir simulaion IVAN V KAZACHKOV ( ( and OLEXANDER V KONOVAL ( ( Dparmn of Enrgy Thnology Diision of Ha and Powr Royal Insiu of Thnology Brinllägn 68 Sokholm 44 - SWEDEN IanKazahko@nrgykhs hp://wwwkhs/im/ins?ln_uk ( Fauly of Physis and Mahmais Diision of Applid Mahmais and Informais Nizhyn Mykola Gogol Sa Unirsiy Kropy yans koho Nizhyn Chrnigi rgion 66 UKRAINE suiidrash@gmailom hp://wwwnduduua/ Absra: - Th problm of modlling and simulaion for h js pnraing pool of ohr liquids is onsidrd undr diffrn physial ondiions and siuaions I may happn for xampl in sr aidns a h nular powr plans in ouh wih dlopmn and opraion of h passi proion sysms agains sr aidns as wll as in many ohr problms Th spifi puliariis of h pnraing js ar disussd and mahmaial modlling of h problm is onsidrd Th non-linar sond-ordr diffrnial quaion and h Cauhy problm is analyzd and sold analyially using h simulanous ransformaion for boh dpndn and indpndn ariabls Th rsul obaind may b usful for horial and praial appliaions whr h liquid j or solid rod is pnraing h pool of ohr liquid Ky-Words: - Non-Linar Sond-Ordr Diffrnial Equaion Analyial Soluion J Pnraion Pool Mahmaial Modlling Transformaion Inroduion o a problm Th puliariis of pnraion dynamis for a liquid j ino h ohr liquid pool of diffrn dnsiis ha bn sudid in a numbr of paprs in h mal nular and ohr indusris g [-] An obji for h prsn papr is rmining h modl for pnraion bhaiours of a hik j ino a fluid pool analyial soluion of h Cauhy problm and analysis of h soluion obaind Phnomnon of a j pnraion ino a pool of ohr liquid is onsidrd as follows Aording o h phnomnon sudid in [6 9 ] j is pnraing a pool wih narly onsan radius up o som poin in a pool hn a h poin of bifuraion i is abruply hanging is radius subsanially (j swihs is on onsan radius o h anohr on as illusrad by xprimnal daa borrowd from [6] shown in Fig Boh daa xprimnal and numrial for h suppor of h abo-mniond puliariis of a j pnraion ino a pool ar prsnd in h Fig for h orrsponding momns of im (in ms This phnomnon onradis o h lassi j shm whn j is onsidring as gradually hanging is radius du o h losss of h loiy [ 7 8] Widning of a j is linar wih a disan and all ross-j loiy profils xp hos ry nar h orifi ar similar o on anohr afr suiabl araging or urbuln fluuaions Similar shmai rprsnaions wr onsidrd for h laminar js as wll This orrspondd ry wll o a hug numbr of xprimnal daa bu no for h ass onsidrd in [6 9 ] and hr whn for h hik pnraing j h main fors ar buoyany hydrodynami drag and inria Non-linar mahmaial modl for a j pnraing a pool of ohr liquid Phnomna of a j pnraion ino pool of ohr liquid a dirs physial ondiions In shor h gnral bhaiours of h plunging j onsiss of surfa aiy of a pool liquid by h iniial impa of h j air pok formaion during h pnraion radial boom sprading of h j and nraind air and inrfaial insabiliy bwn h pool liquid and nraind air I mus b undrlind ha analyial soluions for oninuous and fini js ha rasonably dsribd h hararisis of h pnraion bhaiours E-ISSN: Issu Volum 8 April

2 Ian V Kazahko Olxandr V Konoal And numrial modl simulad hs gnral bhaiours of h plunging j and proidd rasonabl mah on h pnraion loiis [6 9-] Th mulipl xprimnal rsuls ha larly shown ha pnraing j is going firs wih approximaly sabl radius and hn is abruply hanging is radius o h ohr on biggr This bifuraion poin may b xplaind from h analyial soluion obaind Thus h j pnraing h pool is supposd as a body of a ariabl mass assuming ha h j is moing undr an inria for aing agains h drag and buoyany fors Hr h surfa fors ar supposd o b ngligibl omparing o h hr abo-mniond fors Radius of a j is assumd o b approximaly onsan during a j pnraion or a las during som par of h lngh of pnraion This allows alulaing h j pnraion pross sp-by-sp in a gnral as aking for h bginning som onsan radius of a j and hn hang i o h ohr on onsan j radius and so on Wih aoun of h abo-mniond h quaion of a j momnum onsidring a j as a body of ariabl mass is wrin as follows []: whr h apour prssur aion (h las rm in an quaion on a j is akn maximal possibl is a apour dnsiy h las rm in q ( rprsns prssur du o aporizaion of h oolan in a pool πr d( h πr h ( g πr ( for h as of h isohrmal j wih a radius r Hr and ar dnsiis of h pool and j rspily dh/ - loiy of a j im h- lngh of a j pnraion ino a pool (from is fr surfa g is alraion du o graiy is a j ross-sion ara πr 5 πr is a drag for aing on a had of h j from h pool whih was in a rs bfor j pnraion Th muliplayr 5 is akn in h abo quaion as a maximal possibl alu In a raliy som par of h kini nrgy of h disurbd pool is wasd Thn h quaion of a j momnum for h non-isohrmal ondiions may b wrin for h uni ross-sion in a following form []: d( h h ( ( g RT Fig J pnraion xprimnal daa: iniial loiy of a j - 4m/s 6m/s 9m/s rspily E-ISSN: Issu Volum 8 April

3 Ian V Kazahko Olxandr V Konoal R is h unirsal gas onsan and T is a j mpraur Th nrgy onsraion quaions may b prsnd in h following simplifid form T T T κ x x dt κ T ( Hr κ is a hrmal diffusiiy offiin of h mdium (j or pool Th orrsponding iniial and boundary ondiions mus b sad for h quaion array ( Mahmaial modl in a dimnsionlss form Th non-linar quaion ( may b ransformd o h dimnsionlss form: d h dh h (4 whr / u /(gr is h oud numbr whih hararizs h raio of h inria and buoyany fors Hr h following sals wr implmnd for h loiy and for h im rspily: u and r /u Hr h and ar dimnsionlss alus u is h iniial loiy of a j bfor is pnraion ino h pool Th iniial ondiions for h j momnum quaion (4 ar sad as: h d h / (5 So ha i yilds o h Cauhy problm (4 (5 Th non-linar sond-ordr Cauhy problm (4 (5 hus obaind is now sold analyially using h simulanous ransformaions for h boh dpndn as wll as indpndn ariabls aording o h mhodology prsnd in [] Pnraion of h solid rod ino a pool If a solid rod is pnraing a pool of liquid h quaion of momnum ( hangs o h following on πr d H πr ( H h g πr (6 for h pnraion dph lngh of h rod And πr d H πr h H whr H is h H( g πr -(7 for h pnraion dph h > H whn h solid rod is moing oally insid h pool Th quaions hus obaind for wo ass of h solid rod pnraion ino a pool ar ransformd o h mor gnral dimnsionlss forms Th quaion (6 for h firs as whih orrsponds o an iniial sag of pnraion is wrin as h following dimnsionlss array: d h H h H dh dh (8 Equaion (7 in a dimnsionlss form is as follows d H( H (9 Boh quaions (8 and (9 show ha solid rod pnraion subsanially diffr from pnraion of h j haing h sam radius as solid rod Th las quaion (9 dsribs momn of h rod whn i is oally in a pool I is asily ingrad wih h iniial ondiion h ( whr mus b rplad by in as of quaion (9 as far as loiy of h rod drasd by is oal immrs ino a pool For h hr diffrn ass dpnding on h dnsiy raio (solid rod o a pool h soluion of (9 is go as shown blow: > ( > rod is dnsr han a pool: H H H H H ( E-ISSN: Issu Volum 8 April

4 Ian V Kazahko Olxandr V Konoal whr from for < H on an g formula for loiy of h rod pnraion: H H ( H H H < ( < pool is dnsr han a solid rod: H for h iniial ondiions: arg H g ( H - ( ( H H and h h (4 wih h modifid (loiy insad of iniial ondiion ( Rmarkably i is h sam rlaion in as of H For h quaion abo gis for low >> dnsiy pool and long rod ( / H << h following simplifiaion wih aoun of H by << : H H H [ ( H ] H H ( H H whih shows slow dras of loiy ] ( h rod and h pool ar of h sam dnsiy: ( H In as of h sam dnsiis as shown by ( loiy of h rod pnraion is asympoially approahing zro And h longr is rod h slowr is his pross Soluion ( shows slow hyprboli drasing h loiy of h rod wih im ( H g - (5 H for h iniial ondiions: h whih ar pariular as of (4 whn rod (j sops a som dph of pnraion (loiy is zro and afrwards furhr momn is going bakwards o h pool surfa du o buoyany for Th soluions ( (4 and (5 (6 hus obaind dsrib h as whn j or rod is pnraing a pool and h ohr as whn j/rod is moing bak o a pool surfa du o h buoyany fors orrspondingly Analysis of h formula ( shows ha by pnraion of a gas j ino dns pool ( << h loiy of pnraion is approximaly H g arg H H whr from by >> (high-spd j on an g approximaion h H g H so ha loiy drass fas wih h im E-ISSN: Issu Volum 8 April

5 Ian V Kazahko Olxandr V Konoal Analyial soluion of h non-linar sond-ordr diffrnial quaion Th quaion (4 is subsanially non-linar du o h sond rm whih onains produ of h funion / h and ( d h / In his sion h ransformaion for analyial soluion of his quaion is found and hn h quaion is sold Transformaion of h quaion o a sandard form Firs h quaion (4 is prsnd in h following form wih h nw funion y aording o []: a b y x x r s y'' y' y' dx g (6 Hr ar: r s b rs ( a d ψ Thn for y h sandard quaion (6 is ransformd o b x x r ( s ( ψ' ψ' dx ψ''a ( ψ (7 wih h nx nw funion ψ Th indpndn ariabl hr is x Simulanous ransformaion of h dpndn and indpndn ariabls Now h funion ψ (dpndn ariabl and indpndn ariabl x ar ransformd as follows dx x f df df y dξ x f ( r( s whr from h quaion (7 yilds o h hird-ordr diffrnial quaion of h form f a b r s f ff f df f f f f Hr ar: n d f f n n n dξ (8 Th an b sn h quaion (8 is sill subsanially non-linar and nds furhr ransformaion Afrwards h quaion (8 may b rplad by h following quaion array dy ( r /( s a r x z w( z z dx s (9 a a az wz w' d y ( r /( s x r w( z a dx z s ( r ( ( r r / s a x z w( z z s s Now subsiuing h quaions (9 and ( ino h ougoing sandard quaion (6 on an g ( r /( s afr diision on h alu x w: dw G ( ( z F z dz w ( whr ar G dz F r s ( z a b( a z ( z b ( a as r s r s z a Transformaion o h firs-ordr diffrnial quaion Now h quaion ( an b ransformd o h following form wih h assignmns r A b( a and α : s Wd dw dz A Z d Z ( a ( s a W A / ( s z ( a ( s w a as ( KZ Z ( ( E-ISSN: Issu Volum 8 April

6 Ian V Kazahko Olxandr V Konoal whr ar: K ( br ( s( r ( a ( r ( s( r ( s [{ s( a ( r } ( s b] { ( s( b 4( a ( s } {( s( b 4( a } r ( a r [( s( b 4( a ( a r] Th firs-ordr diffrnial quaion ( is of dz h yp as w( z a whih is in his as dx x z as follows: a a ( KZ Z dw a Z dz W (4 wih h assignmns and 4( a [ 4( a ] ( a [ 4( a ] ( ( K Z KV ( Z s KV for as K or ( 4as ν kwv V kwv V ( as /( s ( /( a ( a /( s k ( 4as K a ( K ( ( ( b Subsiuing all hs abo quaions ino (4 yilds h following firs-ordr quaion or dv dν dv dν Vν ν ν V ν ν (5 Gnral soluion of h quaion (5 hus obaind is wrin as follows δ θ ( ν V ( ν dν whr ar: (6 δ ; ons ; δ ( θ ; ; / θ 4 Analysis of h analyial soluion obaind Th following ingral is onsidrd ( ν dν lnν lnν ln ν ν Th ingral ( ν θ dν xiss for h ingr and raional alus of h paramr θ Assuming ha θ 5/ 4; < < θ / on an g ( / ν ν dν ( ν / Thn for h as of < < h alus of h paramrs ar: and i yilds o: δ k K 9 d a A E-ISSN: Issu Volum 8 April

7 Ian V Kazahko Olxandr V Konoal ν κwv ν 9 WV 6 y whr from follows as far as W x y Z 9 hn hr is x V x KZ y Finally ( ν / V ν V 4V y ν V x Now i is go h following: whih rsuls in V ν V 4V x y x y 4x y ν V x x y ( / V νν Exluding from h quaions alu ν and subsiuing h alu V insad of x / y yilds: x y 4 ( whr from follows x y y / (7 x ± 4( In h soluion (7 squar roo is always posii for all alus of h onsan с hrfor wih aoun of < < on an g h physially righ onlusion ha graiaional fors alras h j whil h drag for dlras i om his poin on of h soluions is physially wrong and finally h soluion in h ariabls h is as follows h 4 ( (8 Th soluion obaind (8 is riial and has rsrid appliaions for h as whn j is pnraing a pool saring wih zro loiy a h pool fr surfa 5 Analyial soluion for gnral as Th Cauhy problm (4 (5 onains singulariy du o a ona of moing j wih a pool in a rs baus a h ona ara of a j and pool loiy will hang abruply (shok J is loosing som loiy whil liquid in a pool is ging moing To aoid singulariy of h iniial ondiions (5 h following iniial ondiions migh b onsidrd insad of h abo-mniond iniial ondiions []: h h p d h / u (9 whr h and u p ar h iniial lngh and loiy of a j pnraion (afr a firs ona of a j wih a pool Th quaion (4 is sold wih h iniial ondiions (9 whr h and u p ar ompud in a dimnsion form using h Brnoulli quaion and h j momnum quaion as follows: 5u 5u p ( gh / u 5up Hu Hup hu p ( whr H is h iniial lngh for h fini j falling ino h pool In as of a j sprading from a nozzl (no a j of h fini lngh his alu is rmind by a prssur a h nozzl E-ISSN: Issu Volum 8 April

8 Ian V Kazahko Olxandr V Konoal Soluion of h quaion array ( is prsnd in a dimnsionlss form as follows h H ( h / h ( / u p ( Th Cauhy problm (4 (9 was sold wih h spial simulanous ransformaions of h dpndn and indpndn ariabls wih aoun of (: A A A X h A A X dτ A ( whr A / Thn using h quaion ( wih a fw furhr simpl ransformaions yilds h following linar sond-ordr quaion in h nw ariabls: d y dτ Hr y is a nw ariabl by h quaion y X Finally h soluion is y whr ar h onsans ompud from h iniial ondiions (9 Th ign alus k ar ompud as [ 5( ] k / ( In as a pool is dnsr han a j ( > k is an imaginary alu and h soluion is ґ ґ y os sin Th xa analyial soluion hus obaind was basd on h assumpion abou h onsan j radius; hrfor i is sri for a solid rod pnraion ino h pool and for som iniial par of a j pnraion bfor h rmarkabl growing of is radius I migh b usd as approxima sp-bysp soluion for a j pnraion ino a pool for small mporal inrals orring a j radius from sp o sp 6 Bifuraion of h j radius during is pnraion ino a pool On ould sima an oluion of h j s radius o g suppor of h assumpions mad or obain an ida on how o orr soluion in a good orrspondn o h xising xprimnal daa For his h Brnoulli quaion and h mass onsraion quaion wr onsidrd [9-] for a j pnraing a pool In a dimnsion form hy ar: S(( hg 5 5uS ( S u S S is a ross-sion ara of h j Indxs and dno h iniial sa and h urrn sa of h j rspily In a dimnsionlss form h quaion array ( yilds: [ h( / ] S S Equaion array (4 has h following soluion: [ ± 8h( ] ( (4 S / 4h (5 S / Th soluion (5 larly shows ha hr ar wo possibl alus for h radius of h j: on is h iniial radius of h j whil h ohr alu mans ha h j may loos is sabiliy and hang abruply is radius o a biggr on This orrsponds ry wll o h xprimnal daa som of ha shown in Fig abo Thus w ha wo aailabl soluions for a j radius wih h poin of bifuraion whih dpnds on h Fourir numbr and dnsiy raio as follows: h (6 ( 8 Th quaion (6 was go from rquirmn of posii alu undr squar roo in (5 A his E-ISSN: Issu Volum 8 April

9 Ian V Kazahko Olxandr V Konoal poin h alu undr squar roo quals zro L us all i h poin of bifuraion [9-] Afr his poin h soluion (5 dos no xis in ral numbrs hrfor h j swihs is radius abruply bwn hs wo aailabl sabl sas Saring pnraion ino h pool wih h iniial ross-sions g S afr som small pnraion dph or mor gnrally in as of 8h ( / << h soluion (5 gis h following pair of h aailabl j radiuss o swih bwn hm: S S h( >> (7 Obiously hr ar no physial rasons for a j o grow abruply from a sion ara o a biggr on a h bginning of h j pnraion ino a pool baus h j momnum dirs mainly along is axis Bu hn wih a j furhr pnraion ino a pool du o insabiliy of a j ausing by is fr surfa prurbaions and by a loss of momnum h j ara may hang a any momn 7 Exampls illusraing h puliariis of h j bifuraion phnomnon J is saring pnraion ino a pool from S and hn i grows o a ross-sion S a h poin h 8( 8Ri whn furhr xisn of h wo possibl j s radiuss is impossibl Hr Ri ( / is h Rihardson s numbr (h raio of h momnum o h buoyany fors Th j pnras ino a pool a h disan h h rmind by h iniial lngh of a j oud numbr and dnsiy raio In as of a long j as wll as h j whih is prmannly sprading from a nozzl h iniial pnraion lngh is rmind by h oud numbr and h dnsiy raio Th j is going wih inras of is radius unill h bifuraion poin h h Afr bifuraion h j is abruply swihing o anohr narly onsan radius Applying h soluion obaind o hos pars wih hir own iniial daa h whol j migh b ompud basd on h analyial soluion obaind orrspondn o h abo xprimnal piurs 8 Dimnsionlss im for h non-linar analyial soluion Dimnsionlss im in h ransformaion ( is rmind hrough h ariabl τ as follows: J is dnsr han pool ( <- Pool is dnsr han j ( >- ' ' os sin dτ ; (8 d τ (9 whr h onsans ar alulad lar on For τ << h following linar approximaions by k τ ± τ ar saisfid: k ± os sin Thrfor h quaions (8 and (9 yild: for < for > τ / k( [( k( ] ' ' ' k Th onsans ar go from hs quaions rquiring whih lads o τ Consqunly h ral dimnsionlss im is ompud hrough h inrodud ariabl τ as follows: < k( [( k( τ] }; { > (4 E-ISSN: Issu Volum 8 April

10 Ian V Kazahko Olxandr V Konoal ' k ' ' ' ( (4 u p ln h h Th quaions (4 (4 ar saisfid in a iiniy of τ In gnral as on nds ompuing h ingrals in h xprssions (8 (9 numrially Bu for / ~ and >> h muliplayr of τ mus b small alu whih is aailabl for ompuing wih h approximaions (4 (4 in a widr rgion of τ and n if τ is no small bu h ondiion k τ << is sill saisfid Furhr ransformaion of h xprssion (4 is prsnd in h form: k( (4 k( (4 Th orrsponding xprssions for > ar obaind from (4 similarly 9 Calulaion of h onsans for soluion Using h iniial ondiions (9 and orrlaions ( subsiuing h quaion (4 ino (9 on an g h onsans For < (j is dnsr han pool h onsans ar ompur from h quaions: ln h u p ( h whr from yilds: u p ln h h (44 ( ( For > (pool is dnsr han j from (4 (9 aouning (7 yild h onsans : ' ln h u ' p ( h (45 4 Transformaion of h non-linar soluion o an xplii form 4 Explii form of h soluion obaind Th soluion (4 an b ransformd o an xplii form as h funion of (xlud h arifiial im τ For his purpos from ( (4 yilds and furhr i gos o or k( [( k( τ] [ ( k( ( up h α (46 (47 E-ISSN: Issu Volum 8 April

11 Ian V Kazahko Olxandr V Konoal Wih aoun of (46 yilds: α ( h u p ( (48 Th aouning and using h quaions (47 (48 (4-(4 on an om o a soluion (4 for h pnraion dph h as a funion of h ral mporal ariabl (for k τ <<: < > h ( / - k( ; h ( ' k 4 ' ' ' ( / ' ' ' ( / (5 (5 (h sh h α h h (49 whr h sh dno h hyprboli osin and sin rspily is xprssd hrough by (47 Th loiy of a j pnraion ino a pool is rmind from (4 or (49 using h rlaions dh / ( dh / dτ dτ / In h ass of /< and / > i rsuls in and dh ( ( k dh ' ' ( k( os sin rspily ' ' os sink τ (5 (5 4 Basi paramrs of j pnraion ino pool Th quaions obaind for h non-linar modl of a j g (49-(5 allow ompuing h paramrs of a j pnraing a pool For xampl h pnraion dph h is rmind by ondiion dh/ hrfor h and h orrspondn pnraion im for < and > ar ompud orrspondingly as Th soluion (5 and (5 for wo diffrn ass (for h pool whih is dnsr han a j and for h inrs siuaion ar idnly absoluly diffrn 4 Analysis of h soluion for limi ass Analysis of h analyial soluion obaind is asir prformd for h limi ass whn h soluion is subsanially simplifid For xampl if << (- h << hn ( (4 (46 (48 rsul in: H h H H α 85 α τ k H whih an b asily analysd Hr should b nod ha his approximaion saisfis a wid rang of paramrs baus many praial siuaions orrspond o h larg oud numbrs Aouning (6 (44 from (49 (5 yilds h following approxima soluion for h dph of a j pnraion as wll as for is loiy and alraion: h / (h H h sh 4 H dh h H 85 H H / H {ln sh h} 4 H or wih xplii xprssion for dh/ (55 is (54 (55 E-ISSN: Issu Volum 8 April

12 Ian V Kazahko Olxandr V Konoal / / h H shk τ 4 H H 85 H H H whr ar: / H {ln sh h} 4 H 85 H/ 85 H/ h H H 85 H / 85 H/ sh H H (56 Analysis of h xprssions (55 (56 hus obaind shows dpndn of h j hararisis on h paramrs H/ / H A ky faur of a j pnraion is rmind by h oud numbr and iniial j lngh g for H/ <<: H 85 H H 85 ln H H sh 85 ln hk τ H up o a limi / H~ and n highr For xampl hrfor h approximaions usd hr saisfy a wid rang of h arying paramrs By suh assumpions linarizaion of h soluion (5 by h paramr H/ yilds H h H (57 / H H {46 ln ln } H H Wih an ordr of h rm / ln[ H / (/ ] ln( / H rsrid by furhr simplifiaion is as follows / d h H ln H H H Thus in his as loiy of a j is linar funion of im and hyprboli by h j s lngh Hr H~ or H>> wr onsidrd baus by H<< hr is aually no j (a lngh of a j supposd o b a las largr han is diamr Bu his as migh b also onsidrd using h soluion obaind 44 Paramrs of h j in a gnral as Wih aoun of h abo mniond subsiuion of (4 ino (46-(49 rsuls for << in h following: dh whr ar: h sh / / h h / ln / / h h τ k Th quaions (59 yild for following approximaions: (58 h sh / (59 << h / sh /( / hk τ hrfor soluion of h problm in a form (58 gos o h following simplifid xprssions: h h (6 Analysis of h simpl limi soluion (6 shows ha a h bginning of h j pnraion h dph E-ISSN: Issu Volum 8 April

13 Ian V Kazahko Olxandr V Konoal of pnraion is a linar funion of im and h loiy of pnraion is narly onsan bing inrsly proporional o h dnsiy raio and o h oud numbr 45 Esimaion of h soluion for long im Similar approximaion for h xndd im >> is h following: / / h (6 h lngh of a j pnraion is growing wih im A full pnraion lngh is rmind by h ondiion of so ha sh ln / h wih a im and a dph of pnraion τ h rspily Soling his quaion wih (6 yilds h / / h / ln h / γ ln / / ( / γ and furhr gos for h pnraion im: (6 5/ / γ (6 γ whr If h 5 / ln( / 5 / ln ( / 5 5 / ( ln / 5 / ln( / 5 ( / / 5 << ln( / hk τ [ ] / 5 >> hn from (6 yilds: [ ] ln ( / / 5 (64 h 5 Th formula (64 is lar onrning h oud numbr (h highr loiy h dpr j s pnraion Bu onrning h dnsiy raion influn is mor omplx funion 5 Corrspondn of h modl o xprimnal daa To alida h modl dlopd and h analyial soluion obaind h ompud pnraion lph of a j had bn ompard o xprimnal daa from h liraur and h orrspondn was go good for a numbr of diffrn ass [9-] 6 Conlusion Non-linar analyial modls for h oninuous and fini js o prdi h maximum pnraion of h plunging j wr dlopd and rasonably dsribd h hararisis of h pnraion bhaiours Th non-linar sond-ordr diffrnial quaion and h Cauhy problm was analyzd and sold analyially using h simulanous ransformaions for h boh dpndn as wll as indpndn ariabls in h diffrnial quaion An analyial soluion obaind is aura for h solid rod pnraion ino a liquid pool and is approxima for h j pnraion ino a pool of ohr liquid Th j pnraing a pool of ohr liquid was insigad for diffrn ondiions Problm is of inrs for modlling and simulaion of dirs praial appliaions g sr NPP aidns in ouh wih dlopmn of h passi proion sysms E-ISSN: Issu Volum 8 April

14 Ian V Kazahko Olxandr V Konoal Analyss on h pnraion phnomna of a j ino anohr liquid a h isohrmal and nonisohrmal ondiions wr prformd Th nonlinar analyial modls for h j o prdi h maximum pnraion ino a pool wr dlopd and rasonably dsribd h hararisis of h pnraion bhaiours Th analyial soluion for h non-linar sondordr diffrnial quaion may b of inrs as a mahmaial rsul oo Mhanis- - Vol 7- Issu 4- P [] IV Kazahko Th Mahmaial Modls for Pnraion of a Liquid Js ino a Pool// WSEAS Тransaions on fluid mhanis- Issu Volum 6 April - P 7-9 [] PL Sahd Non-linar ordinary diffrnial quaions and hir appliaions Marl Dkkr In 99 Rfrns: [] MI Gurih Thory of js in idal fluids Translad from h Russian diion by: Robr l Sr and Konsanin Zagusin Aadmi prss Nw York/London 965 [] JS Turnr Js and plums wih ngai or rrsing buoyany Journal of Fluid Mhanis Vol6 No4 966 pp [] MA Larn' and BV Shaba Th problms of hydrodynamis and hir mahmaial modls Mosow Nauka 97 (In Russian [4] ML Corradini BJ Kim MD Oh Vapor xplosions in ligh war raors: A riw of hory and modling Progrss in Nular Enrgy Vol No 988 pp -7 [5] M Saio al Exprimnal sudy on pnraion bhaiors of war j ino on- and Liquid Nirogn ANS Prodings Nal Ha Transfr Confrn Houson Txas USA July [6] HS Park IV Kazahko BR Shgal Y Maruyama and J Sugimoo Analysis of Plunging J Pnraion ino Liquid Pool in Isohrmal Condiions ICMF : Fourh Inrnaional Confrn on Muliphas Flow Nw Orlans Louisiana USA May 7 - Jun [7] GN Abramoih L Shindl Th Thory of Turbuln Js MIT Prss 96 [8] F Bono D Drw and RT Lahy Jr Th analysis of a plunging liquid j-air nrainmn pross Chmial Enginring Communiaions Vol 994 pp -9 [9] IV Kazahko AH Moghaddam Modling of hrmal hydrauli prosss during sr aidns a nular powr plans Naional Thnial Unirsiy of Ukrain KPI Kyi 8 (in Russian [] IV Kazahko and VH Moghaddam Spifi puliariis of h js pnraing h liquid pool of diffrn dnsiy undr sr aidns a h NPP ondiions and hir modling and simulaion// WSEAS Transaions on Applid and Thorial E-ISSN: Issu Volum 8 April