A physical solution for solving the zero-flow singularity in static thermal-hydraulics

Transcription

1 A ysicl solution for solving t zro-flow singulrity in sttic trml-ydrulics miing modls Dnil Bouskl Blig El Hfni EDF R&D 6, qui Wtir F-784 Ctou Cd, Frnc dnil.ouskl@df.fr lig.l-fni@df.fr Astrct For t D-D modlling of trml-ydrulics systms, it is common rctic to us sttic miing modls to comut t miing scific ntly in fluid junctions suc s mrgrs or slittrs. Howvr, tis simlifiction lds to wll known singulrity wn t mss flow rt insid t junction gos to zro. T origin of t singulrity is lind, nd rigorous ysicl solution is roosd to limint t singulrity. A rototy imlmnttion s n dvlod in t TrmoSysPro lirry for owr lnt modlling tt illustrts t intrst of t roosd solution, sows t imct on t structur of t lirry nd nls to vlut t comuting ovrd wit rsct to svrl ossil vrints. Kywords: trml-ydrulics; miing modls; convction; diffusion; TrmoSysPro Introduction Wn modlling trml-ydrulics t t systm lvl, suc s owr lnts, it is common rctic to us sttic qutions to comut fluid quntitis in miing quimnts suc s mrgrs nd slittrs. Tis simlifiction stms from t fct tt t volum of miing is oftn nglctd in junctions, trfor liminting t diffrntil trm in t lnc qutions. It lso occurs wn comuting isoltd orting oints tt only rquir sttic modls. Nglcting diffusion is vry common wn on dls wit lrg mss flow rts, s diffusion is only significnt wn mss flow rts roc zro. Wn diffusion is nglctd, t only trml nomnon rmining in t modl is convction. Howvr wn mss flows go to zro, convction disrs. So if diffusion is nglctd, wn mss flow rts go to zro, s convction lso disr, tr is no trml nomn lft in t modl, lding to ossil indtrmintion of t ntly. Tis indtrmintion rsults in singulrity wn sttic modls r usd, cus in suc cs tr is no diffrntil vril to ct s mmory for t ntly wn mss flow rts r qul to zro. In susqunt ctrs, t mtmticl origin of t singulrity is lind. Tn rigorous mtmticl formultion is roosd sd on ysicl insigt to rmov t singulrity. T id is to rintroduc diffusion in sttic miing modls. Finlly, rformnc ncmrk is givn, sd on rototy imlmnttion in TrmoSysPro. TrmoSysPro is Modlic lirry dvlod y EDF for t modlling of owr lnts of ll tys []. Comuting t stt of trmlydrulics systm As t ojctiv is to find t origin of t ysicl singulrity for giving solution for rmoving t singulrity, it is usful to undrstnd ow t ysicl stt of trml-ydrulic systm suc s volum is dfind. A volum is n strct ysicl comonnt wr incoming flows mi. Figur fturs four incoming flows. Flows r ositiv wn ty ntr t volum nd ngtiv otrwis. 3 3 Figur : volum In gnrl, t stt of ysicl systm is givn y t st of indndnt ysicl quntitis tt comltly dfin t stt. Tr r mny wys to coos t stt vrils for givn ysicl sys- 4 4 Procdings of t t Intrntionl ModlicConfrnc Mrc -, 4, Lund, Swdn 847

2 A ysicl solution for solving t zro-flow singulrity in sttic trml-ydrulics miing modls tm. For trml-ydrulic volum, common coic is to us t vrg rssur P nd t vrg scific ntly insid t volum. Tn t stt of t volum will dfind if P nd cn comutd. In t squl, w r only intrstd in comuting wic is clld t miing ntly in volum. To comut, on must considr t nigoring volums of wic r collctivly dnotd (s Figur ). Volum P, ( ) Volum P, Volum Volum 4 Volum 3 P, P, 3 3 P 4, 4 ( ) ( 4 ) Figur : grid scm ( 3 ) Ec volum is ssumd to in trmodynmic quilirium, so tt tir trmodynmic stt is ysiclly dfind. Howvr nigoring volums my v diffrnt ysicl stts, so tt rssur nd tmrtur grdints my ist tt cus mss nd nrgy flows twn nigoring volums troug tir common limiting oundry. Mss flowing from volum to volum is dnotd ( ). Trfor ( ) is ositiv if t flow ctully occurs from to, nd ngtiv otrwis. So ( ) ( ). Notic tt t rltion ( ) ( ) is not mss lnc qution twn volums nd, ut mrly stts t fct tt ( ) nd ( ) dnot t sm ysicl quntity wit oosit sign convntions. T scific ntly of flow ( ) is dnotd :. T mning of nottions is rclld in ctr 7. T dynmic mss nd nrgy lnc qutions r givn y d( V ) ( ) dt d( V u ) : ( ) J ( ) W dt J ( ) is t nrgy flow troug diffusion. J ( ) A : k : T ( ) () T sttic mss nd nrgy lnc qutions r otind y liminting t dynmic trms on t lft nd sids. ( ) () ( ) J ( W : ) (3) As t quntity dos not r licitly in t sttic nrgy lnc qution, it must comutd toug t quntitis. So t rltion twn : nd : must stlisd. To tt nd, t fluid vin twn volums nd is considrd (s Figur 3). Volum P d ( ) Figur 3: fluid vin L Volum T sttic mss nd nrgy lnc qutions in t volum limitd y d is (4) T A k W (5) Eq. (4) stts tt is constnt. Trfor: ( ) ( ) In ordr to find n nlyticl solution to Eq. (5), t following rltion twn d nd dt is considrd: P L 848 Procdings of t t Intrntionl ModlicConfrnc Mrc -, 4, Lund, Swdn

3 Sssion 5D: Trmofluid Systms, Modls nd Lirris d cp dt (6) In t squl, it is ssumd tt t rltion givn y Eq. (6) is vlid (i.. outsid of t sturtion lin, for isoric trnsformtions, or for idl gss, or wn t contriution of t rssur vrition to t vrition of t scific ntly is ngligil s comrd to t contriution of t vrition of tmrtur). Undr t dditionl ssumtion tt W, t nrgy lnc qution writs: A k ( ) (7) c Eq. (7) cn solvd nlyticlly []: P P L P ( ) (8) wit ( ) (9) P A k () cp L L : is t vlu of for : : L P ) P ) () wit ) () Figur 4 givs lot of ŝ. Figur 4: lot of ŝ if s ( ) if (4) if Tis is t wll known uwind scm roimtion for flow rvrsl. Tis rltion is widly usd, vn if t ssumtions usd in tis drivtion r not fulfilld. Not tt s is discontinuous t, wrs ŝ is continuous nd diffrntil vrywr. 3 Origin of t singulrity in sttic miing modls T ojctiv of tis ctr is to sow tt t singulrity in sttic miing modls riss wn diffusion is nglctd. So in t squl diffusion is nglctd, wic mns tt J ( ). Also ( ) is dnotd to simlify t nottion. T mss nd nrgy lnc qutions com (5) (6) : T vlu of t ntly : is givn y t uwind scm (s Eq. (3)): : ) ) (7) In t squl, t following rltions r usd: sgn( ) (8) sgn( ) ) ) (9) ) ) () wr sgn is t sign function. Tn using Eq. (5), (6), (7) nd () ) ) : ) ) ) ) If diffusion is nglctd, tn, P nd : coms: : ( )) ( )) (3) wr s is t st function: Trfor s ( ) ) () Procdings of t t Intrntionl ModlicConfrnc Mrc -, 4, Lund, Swdn 849

4 A ysicl solution for solving t zro-flow singulrity in sttic trml-ydrulics miing modls wn s ( ). To find out wn tis condition is stisfid, using Eq. (5), (8), (9) nd (): sgn( ) ) ) * Hnc: ) ) ) wn. ) * ) ) ) () So wn ll mss flow rts r qul to zro, t miing ntly is indtrmint ( / ). Altoug t indtrmintion occurs only t n isoltd oint (ll mss flow rts qul to zro), it is not ovious to tnd in ordr to rmov t singulrity t zro (contrry to otr functions wit isoltd singulritis suc s sin( ) / ). In rticulr, it is not sufficint to rlc s y ŝ (or in otr words gt rid of t uwind scm y introducing diffusion in t flow rvrsing formul givn y Eq. (7)) cus tn wit s ˆ ˆ( ˆ ) sˆ( ˆ ) (3) (4) : T singulrity still rmins sinc s ˆ( ˆ ) wn ll mss flow rts r qul to zro. Howvr, noticing tt ) m(,) Eq. () my writtn s Trfor, if on is not intrstd in t corrct vlu of nr zro flows, wic is in gnrl t cs wn diffusion is nglctd, tn s suggstd in [3] on cn rlc y m(, ) wr is smll ositiv mss flow rt. Tn wn ll mss flow rts r low ( ): (5) N wr N is t numr of nigoring volums of volum, so t singulrity is rmovd for zro flows. Noticing tt m(, y) y) ( y) y, on cn vn v C wy of rmoving t singulrity y () considring t function s ( ) (6) y () rlcing s y s in t m function ov smootm(, y, ) s ( y) ( y) y nd y (3) rorly djusting t vlu of wrt.. Tis solution will not dvlod r ny furtr, s full ysicl solution is sougt. T rson is tt rlcing y wn for comuting t miing ntly violts t nrgy lnc fundmntl lw, so rquirs ror coic of wrt. t rolm t nd. As suggstd in [3] tr r lso otr wys to writ smootm. Notic lso tt sˆ s. 5 nd s s (s Eq. (), (4) nd (6). 4 Rmoving t singulrity t zro flows Diffusion is rinstlld in t nrgy lnc qution. Tn ( ) (7) : ( ) J ( ) (8) ssuming witout loss of gnrlity tt W. Using Eq. () nd (6): 85 Procdings of t t Intrntionl ModlicConfrnc Mrc -, 4, Lund, Swdn

5 Sssion 5D: Trmofluid Systms, Modls nd Lirris J ( ) A k c : A k : T L / L / Tking t drivtiv of Eq. (8) wrt. t yilds: wit L J ( ) r( P ) J ( ) (9) J ( : ) ( ) (3) A k (3) : cp L : P ( ) (3) : if r( ) (33) if J ( ) is t nrgy flu wn ( ). For sir comuttion r () my roimtd y t Gussin.33 r ( ) (34) T lot low comrs r in rd wit rˆ in lu. Figur 5: lot of rˆ nd r Wn ll flows r qul to zro, t nrgy lnc qution writs J ( ) ( ) : As t cofficints : r lwys strictly ositiv ( : ), wn ll flows r qul to zro t miing ntly is dfind nd tks t vlu: : : (35) If ll cofficints : r qul, tn is t ritmtic mn (s lso Eq. (5)) (36) N wr N is t numr of nigoring volums of volum. As conclusion to tis ctr, wn diffusion is tkn into ccount, t nrgy lnc qution is : ( ) ( ) ( ) (37) wit ( ) r ( ) : (38) : T trms ( ) r in gnrl smll ut r lwys strictly ositiv nd v t sm ysicl unit s mss flow rt (kg/s). So ty nvr go to zro, vn wn ll mss flow rts go to zro. Ty ct trfor s smll ositiv mss flow rts tt rmov nturlly in C wy t singulrity of t miing ntly t zro flows. 5 Bncmrk of t roosd solution To vlut t comuting ovrd of introducing diffusion to solv t singulrity rolm, t ncmrk consists in comring two ltrntivs for t sttic nrgy lnc qution (s Eq. (8)): Altrntiv : witout diffusion, wit or witout uwind scm. Altrntiv : wit diffusion, wit or witout uwind scm. Witout diffusion mns tt J ( ) Wit diffusion mns tt J ( ) rˆ( ˆ ) : ( ) Wit uwind scm mns tt : ) ) Witout uwind scm mns tt sˆ( ˆ ) sˆ( ˆ ) : Procdings of t t Intrntionl ModlicConfrnc Mrc -, 4, Lund, Swdn 85

6 A ysicl solution for solving t zro-flow singulrity in sttic trml-ydrulics miing modls r ˆ( ˆ ) nd s ˆ( ˆ ) r dfind y Eq. (), (4) nd (34). T qutions r imlmntd s rototy in t TrmoSysPro lirry using t scm sown in Figur 6. P, P, ( ) ( ) P, ( 3 ) ( 4 ) P, 4 4 P, 3 3 Figur 6: grid scm in TrmoSysPro T quntitis ( P, P ) nd (, ) r comutd in t multi-ort lmnts wit t mss nd nrgy lnc qutions. Multi-ort lmnts rrsnt t control volums nd. For tis rson, ty r lso clld volums. T quntitis J ( ) nd : r lso comutd in t volums. T quntitis ( ) r comutd in t two-ort lmnts wit t momntum lnc qutions. Two-ort lmnts rrsnt t intrfcs : twn nd. T intrfc : is orintd ositivly from t lu ort to t rd ort of t two-ort lmnt tt rrsnts :. So t mss flow rt is ositiv wn t fluid flows long t ositiv dirction of t intrfc orinttion, i.. from t lu ort to t rd ort. To rflct tis sign convntion for mss flow rts, t lu ort is clld inut ort, nd t rd ort is clld outut ort. T comonnts r connctd togtr vi t inut nd outut orts tt corrsond to Modlic connctors. Inut nd outut connctors v t sm structur. In ordr to ndl diffusion, ty r somwt diffrnt from t usul fluid connctors usd in Modlic fluid lirris, nd in rticulr in t currnt distriution of t TrmoSysPro lirry. T mning of t vrils in t connctor dnds on wtr t connctor is ttcd to volum or to two-ort lmnt. If t connctor is ttcd to two-ort lmnt rrsnting t intrfc : : P Prssur P in volum, or rssur P in volum, dnding on wtr t connctor is on t sid of or on t sid of. Mss flow rt ( ) troug intrfc :. of t fluid going Scific ntly : of t fluid going troug intrfc :. _vol_ Scific ntly or of t fluid in volum or loctd on t sid of t inut ort of t two-ort lmnt tt rrsnts :. _vol_ Scific ntly or of t fluid in volum or loctd on t sid of t outut ort of t two-ort lmnt tt rrsnts :. If t connctor is ttcd to multi-ort lmnt tt rrsnts volum, nd t connctor is connctd to t two-ort lmnt tt rrsnts intrfc : : P Prssur P of t fluid in volum. Mss flow rt ) troug intrfc :. ( of t fluid going Scific ntly : of t fluid going troug intrfc :. _vol vol_ If t connctor is n inut ort: scific ntly of t fluid in t nigoring volum loctd in t dirction of ngtiv flow rts. If t connctor is n outut ort: scific ntly of t fluid in volum. If t connctor is n inut ort: scific ntly of t fluid in volum. If t connctor is n outut ort: scific ntly of t fluid in t nigoring volum loctd in t dirction of ositiv flow rts. 85 Procdings of t t Intrntionl ModlicConfrnc Mrc -, 4, Lund, Swdn

7 Sssion 5D: Trmofluid Systms, Modls nd Lirris T uros of _vol_ nd _vol_ is to rovid ot nd to volums nd vn if ty r srtd y lin of connctd two-ort lmnts. Wn conncting togtr two connctors, t vrils insid t connctors r md qul cus ty rrsnt t sm ysicl quntitis. So connctors r usd to ssml t modl from t diffrnt comonnts, nd not to gnrt tr ysicl qutions (suc s lnc qutions for instnc). Tis scm for distriuting t qutions twn multi-ort nd two-ort lmnts nd conncting tm togtr nls to connct togtr svrl two-ort lmnts witout ving to srt tm y volums. T connctd lin of two-ort lmnts is tn quivlnt to singl two-ort lmnt. Also, tr r no infinitsimlly smll volum lmnts imlid twn two connctd two-ort lmnts, so t connctions do not gnrt t kind of singulrity dlt wit in tis r. T tst modl is sown in Figur 7. Figur 7: tst modl A miing volum (VolumA) is connctd to two fluid sourcs (Sourc, SourcP) nd fluid sink (SinkP) vi i (Tu) nd two control vlvs (Vlv nd Vlv). T tst scnrio consists in rforming flow rvrsl, nd tn stting ll mss flow rts to zro. SourcP T scific ntly is constnt qul to.5 J/kg. T rssur is constnt qul to 3 rs. SinkP T tmrtur is constnt qul to 3 K T rssur is constnt qul to r. Vlv T osition vris from % to % in sconds strting from t = s. Vlv T osition is constnt qul to % Four simultion runs r rformd: Run.: witout diffusion, wit uwind scm Run.: witout diffusion, witout uwind scm Run.: wit diffusion, wit uwind scm Run.: wit diffusion, witout uwind scm For c run r lottd: T mss flow rts t c connctd ort of t miing volum (3 curvs) T scific ntly insid t miing volum ( curv) T scific ntlis insid c sourc nd sink (3 curvs). Figur 8 givs t mss flow rts for ll runs (no diffrnc in rsults for ll runs). Sourc T scific ntly is constnt qul to.5 J/kg. T mss flow rt follows t following curv (kg/s vs. s). Figur 8: mss flow rts for runs nd Procdings of t t Intrntionl ModlicConfrnc Mrc -, 4, Lund, Swdn 853

8 A ysicl solution for solving t zro-flow singulrity in sttic trml-ydrulics miing modls Runs. nd. Figur 9 givs t scific ntlis for run., nd Figur givs t scific ntlis for run.. Figur : scific ntlis for runs. nd. Figur 9: scific ntlis for run. Figur : scific ntlis for run. For run., wn ll mss flow rts r st to zro (t t = 8 s), t scific ntly in t miing volum ks its lst vlu rior to t zro mss flow rts condition, just s toug tr wr som kind of mmory olding tis vlu wn ll mss flow rts com zro. Tis is roly n rtifct du to t numricl mtods usd to solv t lgric qutions. T rsult is ysiclly corrct, ut tis looks s sr luck s t tory rdicts tt t rsult is in fct mtmticlly undfind wn diffusion is nglctd. To t contrry, for run., wn ll mss flow rts r st to zro (t t = 8 s), t scific ntly in t miing volum continus to vry until it tks smingly finl constnt vlu. Tis is fls trnsint wic is of cours unysicl cus, sinc t modl is sttic, ll vlus sould sty constnt wn t oundry conditions r constnt (ftr t = 8 s). In ot css, t tory rdicts tt t miing ntly cn tk ny vlu wn ll mss flow rts r zro nd diffusion is nglctd, so t rsult is consistnt wit t tory. Runs. nd. Figur givs t scific ntlis for runs. nd. (no diffrnc in rsults for ot runs). Wn ll mss flow rts r st to zro, t scific ntly in t miing volum tks t vlu tt corrsonds to t trml quilirium twn t miing nd t sourcs nd sink it is connctd wit, wic is corrct ysicl rsult. T trnsition to trml quilirium is sr ut continuous. T following tl givs t comuting tims in sconds nd t sizs of t non-linr systms ftr mniultion for c run wit Dymol. Run CPU tim Sizs of non-linr systms..5 { 3, }.. { 3, }..9 { 4 }..33 { 7 } T conclusion from tis rimnt is tt t st solution is to tk into ccount diffusion in t nrgy lnc qution, ut still us t uwind scm, i.. nglct diffusion in t flow rvrsl qution. T ovrd ovr t stndrd roimtion of nglcting diffusion vrywr is 75%. Mor divrs rimnts sould md in ordr to dcid wtr it is ttr to tk into ccount diffusion in t flow rvrsl formul or not, cus voiding t uwind scm nls to rmov t discontinuity du to t us of t st function. 6 Conclusion Nglcting diffusion in trml-ydrulics systms is common roimtion wn dling wit lrg mss flow rts, s diffusion is only significnt wn mss flow rts r nr zro. Howvr, tis roimtion lds to undfind vlus for t miing ntlis wn ll miing mss flow rts r qul to zro. Tis is du to t fct tt convction, wic is t only trml nomn tkn into ccount wn diffusion is nglctd, vniss wn mss flow rts go to zro, so tr is no ysicl nomnon lft to dscri t trml ysicl stt insid t miing volum. 854 Procdings of t t Intrntionl ModlicConfrnc Mrc -, 4, Lund, Swdn

9 Sssion 5D: Trmofluid Systms, Modls nd Lirris A rigorous mtmticl nd ysicl solution to tis rolm is to rinstll diffusion in t nrgy lnc qution. Tis solution indd rmovs t singulrity for zro flows in continuously diffrntil wy, s torticlly dmonstrtd in tis r. A rototy imlmnttion s n md in t TrmoSysPro lirry for owr lnt modlling, dvlod y EDF. T introduction of diffusion into t lirry s n imct on t structur of connctors. T rototy s n tstd on smll sttic modl tt fturs miing volum connctd to fluid sourcs. T tst scnrio consists in rforming flow rvrsl, tn ringing ll flows to zro. T rsults r consistnt wit t tory dvlod in tis r. Ty lso sow tt t uwind scm, wic is t qution for comuting flow rvrsl tt nglcts diffusion, cn kt, s rinstlling diffusion in t flow rvrsl qution s wll dos not mk ny diffrnc in t comuting rsults, ut rovoks significnt ovrd in comuting tim. Howvr, mor numricl rimnts sould md to confirm tis lst oint. 7 Nottions G : : mor gnrlly, vlu of quntity G t t intrfc twn volums nd 8 Acknowldgmnts Tis work ws rtilly suortd y DGCIS witin t ITEA EUROSYSLIB rojct. Rfrncs [] El Hfni B., Bouskl D., Lrton G., Dynmic modlling of comind cycl owr lnt wit TrmoSysPro, Modlic confrnc rocdings. [] Ptnkr S.V., Numricl Ht Trnsfr nd Fluid Flow, Hmisr Pulising Corortion, Tylor & Frncis, 98. [3] Frnk R., Csll F., Ottr M., Silmnn M., Elmqvist H., Mttson S.E., Olsson H., Strm Connctors An tnsion of Modlic for Dvic-Orintd Modling of Convctiv Trnsort Pnomn, Modlic 9 confrnc rocdings. P : fluid rssur in volum : fluid scific ntly in volum u : fluid scific intrnl nrgy in volum : fluid dnsity in volum V : volum of volum W : trnl nrgy rougt to t volum ( ) : mss flow rt of t fluid flowing from volum to volum T ( ) : tmrtur grdint from volum to volum : : scific ntly of t fluid flowing from volum to volum ( c ) : : scific t ccity of t fluid flowing from volum to volum k : : diffusion cofficint t t intrfc twn volums nd A : : r of t intrfc twn volums nd L : : distnc twn t cntrs of volums nd Procdings of t t Intrntionl ModlicConfrnc Mrc -, 4, Lund, Swdn 855