Transactions of the VŠB Technical University of Ostrava, Mechanical Series No. 1, 2014, vol. LX article No Daniel HIMR *

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1 rscios of he VŠB echicl Uiversiy of Osrv, Mechicl eries No., 04, vol. LX ricle No. 975 Diel HIM * NUMEICAL MODEL OF AI VALVE FO COMPUAION OF ONE-DIMENIONAL FLOW NUMEICKÝ MODEL ZAVZDUŠŇOVACÍHO VENILU PO VÝPOČE JEDNOOZMĚNÉHO POUDĚNÍ Absrc he er is focused o umericl simulio of usedy flow i ielie. he secil eio is id o umericl model of ir vlve, which hs o iclude ll ossible regimes: criicl/subcriicl iflow d criicl/subcriicl ouflow of ir. hermodymic equio of subcriicl mss flow ws simlified o ge more friedly she of relev equios, which ebles esier soluio of he roblem. Absrk Čláek je změře umerickou simulci escioárího rouděí v orubí. Zvláší ozoros je věová umerickému modelu zvzdušňovcího veilu, kerý musí obsáhou všechy možé režimy: kriické/odkriické sáí kriický/odkriický výfuk vzduchu. ermodymická rovice odkriického rouděí lyu byl zjedoduše, kže její vr je mohem jedodušší ro dlší řešeí. Keywords Air vlve, umericl model, oe-dimesiol flow, wer hmmer, Lx-Wedroff. INODUCION Flow i ielie sysems is secil cse of fluid moio. Axil comoe of he velociy domies over he rdil d geil oe. As vrious kids of liquid re, usully, rsored by ielie, i is ecessry o y eio o his heomeo. Jeordy of sudde velociy chge hs o be reed i ech sysem s i c cuse ruure or collse of ie wll or c dmge oher comoes of he sysem [7]. Air vlve is oe wy how o suress sudde dro of he ressure, which is coeced wih he rsie flow. I is lced o locio, where his eve is mos likely o occur. I is, usully, o os of he ielie rofile, dowsrem of emergecy vlve d so o. Whe he ressure goes uder he mosheric vlue, he ir vlve oes d les ir ge io he ielie. Whe he ressure is greer h he mosheric oe, he ir vlve llows ir o leve he ielie, bu kees liquid iside. Whe ir vlve is well desiged, i mkes ressure ulsios lower d roecs he hydrulic sysem gis imcs of he wer hmmer (e. g. [8]). o judge effec of he ir vlve o he sysem behvior, desiger hs o simule he rsie flow wih oe-dimesiol versio of coiuiy d momeum equios () d (). * Ig., Ph. D., Derme of Hydromechics d Hydrulic Equime, Fculy of Mechicl Egieerig, VŠB echicl Uiversiy of Osrv, 7. Lisodu 5, Osrv, el. (+40) , e-mil diel.himr@vsb.cz 9

2 D dimeer m, K Q 0, x Q Q Q x D g, () () m g rojecio of grviy ccelerio o ie xis, s K bulk modulus P, ressure P, m Q flow re 3, s ie cross-secio m, ime s, x sce coordie m, fricio coefficie -, kg desiy 3. m hese equios hve o be solved ogeher wih boudry codiios. Mhemicl descriio of he ir vlve is oe of hem. NUMEICAL OLUION Equios () d () re subjec o umericl soluio. here re my ossibiliies. Probbly, mehod of chrcerisic (or geerl mehod of chrcerisic) is he mos oulr [], [5], [0], bu Lx-Wedroff mehod is used i his er. he umericl scheme is drw i he figure. Fig. Lx-Wedroff umericl scheme [3] he mehod is bsed o he ylor s exsio i he ime direcio d shows umericl viscosiy, which mkes resuls more similr o rel eves h resuls by he mehod of chrcerisic. Oe c fid derivio of he mehod i [6]. he resul is he give by iiil codiios, which re imor he begiig of comuio d he heir ifluece disers d by boudry codiio, which corol ll comuio. 30

3 Models of vrious hydrulic elemes serve s boudry codiios d c be very simle (e. g. rescribed ressure, flow re, resisce...) or more comlex (surge k, um, urbie...), see [9]. Adio of boudry codiios for Lx-Wedroff mehod is described i [].. Air vlve Model of he ir vlve c be udersood s ir ocke wih vrible mss. Whe he ressure is lower h he mosheric ressure he mss icreses, becuse ir flows io he ie. Whe he ressure exceeds he mosheric vlue, ir is beig exelled d he mss decreses ill o ir remis i he ie. he se equio of gs c be wrie i followig form: m mss of ir i he ie kg, V m, (3) J gs cos, kgk emerure i ie K, 3 V volume of gs i he ie m. he, derivive of volume wih resec o ime is: V m m. (4) his volumeric chge corresods o differece bewee wer flowig ou of he comuiol ode rereseig he ir vlve d wer flowig i. he ime derivive of he mss obeys hermodymic lw for flowig gses d hs four ossible shes:. io of he ressure i he ie d mosheric ressure is lower h criicl vlue: r k criicl rio -, r k, (5) olyrohic exoe (vlue i limis o.4 for ir) -, he he ir flow is criicl d obeys followig equio: C i m Ci i 3, iflow coefficie of he vlve (vlue i limis 0 o ) -, mosheric ressure P, i ile cross-secio of he ir vlve m, emerure ouside of ie K.. ubsoic iflow srs whe rio of ressures is lower h, bu greer h r k : (6)

4 3. C m i i (7) 3. Whe he rio of ressures is greer h bu lower h iverse vlue of r k, subsoic ouflow of ir srs:, C m ou ou (8) C ou ouflow coefficie of he vlve (vlue i limis 0 o ) -, ou oule cross-secio of he ir vlve m. 4. Criicl ouflow srs whe he rio of ressures is greer h iverse vlue of r k :. C m ou ou (9) Mss flow of ir hrough he ir vlve c be, he, loed s fucio of ressure or ressure rio, see fig.. Of course, whe here is o ir i he ie, he flow equls zero. Fig. Mss flow hrough he ir vlve Bu here is oe difficuly: subsoic mss flow is quie comliced fucio d mkes roblems i umericl model, becuse ukow ressure hs he exoe, which is o whole umber. o, here is effor o simlify i. Lee d Leow [4] sli subsoic re io iervls d, i ech iervl, relced he fucio wih rbol. he more iervls he beer ccurcy, bu icresed requiremes o comuiol ime, becuse i is ecessry o solve ll rbols d look for soluio lyig righ iervl.

5 33 Whe oe looks he subsoic iflow (ressure rio iervl 0.58 o.0), he fucio is similr o ellise, so i could be ossible o use i he eire iervl. hus, equio (7) is relced by:. k k i i r r C m (0) Equio for subsoic ouflow (8) c be relced i similr wy by:. k k ou ou r r C m () Comriso of he origil fucios (7) d (8) wih subsiuig fucios (0) d () is show i he fig. 3. Equios (0) d () re friedlier for furher soluio, becuse cois oly firs d secod ower of he ressure ulike equios (7) d (8). Fig. 3 Comriso of subsiuio wih he origil fucio he error of subsiuio deeds o he olyrohic exoe d is lower h 3.5% for vlues from o.4. Whe he exoe hs vlue.449, he subsiuio is he mos ccure. ee figure 4. Now, he umericl model of he vlve c be wrie i she () usig eqs. (4), (5), (6), (0) d (). Fucio m(+)=m((+))., V m Q Q i ou () Q i wer flow io he ir vlve ode s m, Q i wer flow ou of he ir vlve ode s m,

6 ime se of he comuio s. x Fig. 4 Error of subsiuio for vrious vlues of he olyrohic exoe Iflow d ouflow come from equio () i form: Q ou sce se of he comuio m. Q i x (3) K Q, x (4) K Q, Vribles Q d Q - re flow res give by Lx-Wedroff mehod (oe sce se dowsrem d oe sce se usrem he ir vlve resecively), Q d Q re he sme s Q ou d Q i resecively, see fig. 5. he oly ukow is he ressure i he followig ime se (+), whe equios () o (4) re beig solved.. imulio Fig. 5 Numericl scheme of he ir vlve ode imle sk of ielie wih ir vlve ws used o es he roosed umericl model. Figure 6 shows ielie rofile. Air vlve is lced five meers from he usrem ed, where he ielie becomes horizol. his lce is he mos dgerous, becuse he colum serio is mos likely o er here. 34

7 Fig. 6 Pielie rofile he flow re boudry codiio he begiig of ielie is loed i he fig. 7. Iiil flow re is m 3 /s, which is cos for oe secod. he, i srs decresig o zero followig rbolic fucio. his is similr o closig bll vlve. Cos ressure. kp is he oule boudry codiio. Prmeers of he ir vlve were chose s followig: mosheric ressure 0 5 P, gs cos 87 J/kg/K, emerure i ielie 88 K, emerure ouside of ie 98 K, ir ile cross-secio 0-3 m, ir oule cross-secio m, olyrohic exoe of ir.4, ouflow d iflow coefficies of he vlve re. Dimeer of ielie is 0.5 m, roughess mm, wve seed 000 m/s, desiy of wer 000 kg/m 3 d viscosiy 0-6 m/s. As he sce se is m he ime se of comuio is 0.00 s. Fig. 7 Ile boudry codiio 35

8 Figure 8 shows ressure surge he begiig of ie d o he locio of he ir vlve (which is o cosidered i his cse). Oe c see h he o of he highes ek exceeds vlue 3.5 kp d he lowes ressure is more h 0.5 kp below he bsolue zero, wh mes h cviio would er here. (his sk would deserve usig rorie cviio model, bu his is o he gol of his er). Fig. 8 Pressure ulsios wihou he ir vlve Whe he ir vlve is cosidered o is locio, he ressure ulsios re oicebly lower d miiml ressure is oly 0.5 kp below he mosheric vlue, hus here is o risk of cviio, see fig. 9. Figure 0 shows volume d mss of ir ocke, which origies whe ir is sucked io he ie. he differece bewee frequecies of ulsios i figures 8 d 9 is give by he ir ocke, which serves s ir vessel wih vrible cciy. Fig. 9 Pressure ulsios wih he ir vlve 36

9 Fig. 0 Volume d mss of he ir ocke durig he rsie eve 3 CONCLUION Desig of umericl model of ir vlve is described i his er. Model comes from hermodymic equio for flowig gs d se equio describig behvior of he ir i ie. ice he subsoic flow of he gs is described by quie comliced equio, which mkes furher comuio difficul, his fucio ws subsiued wih simler oe, which is similr o origil oe. he error of he subsiuio is low eough o jusify his se. he, he fil umericl model ws esed s boudry codiio i comuio of wer hmmer i simle ielie. 4 ACKNOWLEDGMEN his er hs bee elbored i he frmework of he rojec Ooruiy for youg reserchers, reg. o. CZ..07/.3.00/30.006, suored by Oeriol Progrmme Educio for Comeiiveess d co-ficed by he Euroe ocil Fud d he se budge of he Czech eublic. EFEENCE [] CALO, M. e l. Udersdig Air elese hrough Air Vlves. Jourl of Hydrulic Egieerig. 0, CXXVII, Nr. 4, IN [] HIM, D. oluio of No-lier Hydrulic Neworks. Bro Uiversiy of echology, Fculy of Mechicl Egieerig, 0. (I Czech). [3] HIM, D. & HABÁN, V. imulio of Low Pressure Wer Hmmer. IOP Coferece eries: Erh Eviromel ciece. 00, XII, Nr. 0087,. 8. IN [4] LEE,.. & LEOW, L. C. Numericl udy o he Effecs of Air Vlve Chrcerisics o Pressure urges durig Pum ri i Pumig ysems wih Air Erime. Ieriol Jourl for Numericl Mehods i Fluids. 999, XXIX, Nr. 6, IN

10 [5] LEE,.. & LOW, H.. & NGUYEN, D.. Effecs of Air Erime o Fluid rsies i Pumig ysems. Jourl of Alied Fluid Mechics. 008, I, Nr., IN [6] LEVEQUE,. J. Fiie Volume Mehods for Hyerbolic Problems. New York: Cmbridge Uiversiy Press, 00. IBN [7] PEJOVIC,. & BOLDY, A. P. Guidelies o Hydrulic rsie Alysis of Pumig ysems. Belgrde-Covery: P&B Press 97. IBN [8] EPHENON, D. Effecs of Air Vlves d Piework o Wer Hmmer Pressures. Jourl of rsorio Egieerig. 997, CXXIII, Nr., IN [9] ZÁUBA, J. Wer Hmmer i Pie-Lie ysems. Prh: Acdemi, 993. IBN [0] ZHANG, Y. & VAIAVAMOOHY, K. rsie Flow i idly Fillig Air-Ered Pielies wih Movig Boudries. sighu ciece d echology. 006, XI, Nr. 3, IN