An Improved Mathematical Model of Lead-Acid Batteries for Simulation of VRLA Batteries

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1 Journal of Powr Sourcs Syposu, England, 7 An Iprovd Mathatcal Modl of Lad-Acd Battrs for Sulaton of VRLA Battrs Vahd Esfahanan Assocatd Profssor vahd@ut.ac.r Farschad Torab Ph. D. Canddat ftorab@ut.ac.r Vhcl, Ful and Envronnt Rsarch Insttut Faculty of Mchancal Engnrng Collg of Engnrng Unvrsty of Thran Al Mosahb M.Sc Studnt aosahb@ut.ac.r Abstract Prdctng transnt bhavor of lad-acd battrs durng charg and dscharg procss s an portant factor n any applcatons ncludng ybrd Elctrc Vhcls (EV) whr sd ractons lk oxygn rcycl and hydrogn voluton ar prsnt n battry. In ths study, an provd athatcal odl for VRLA battrs basd on Coputatonal Flud Dynacs (CFD) and Equvalnt Crcut Modl (ECM) has bn ntroducd. Ths odl nhrts accuracy of CFD odl and physcal undrstandng of ECM. Morovr, t s vry fast whch aks t qut sutabl for ral-t sulatons. Th prsnt approach s vrfd by prvous CFD odls and xprntal data. 1. Introducton Lad-acd battrs ar wdly prcvd as th bst nar-tr powr sourc for lctrc vhcls (EV) du to thr low cost, vrsatlty, and xcllnt rchargablty and rcyclablty. Iportant charactrstcs of EV battrs nclud hgh spcfc nrgy and nrgy dnsty, hgh spcfc powr and powr dnsty, and long cycl lf. A long cycl lf s a ky rqurnt f advancd EV battrs ar to hav rasonabl lf cycl costs. In ordr to prdct and furthr prov ths charactrstcs, t s portant to undrstand th dynac bhavor of EV lad-acd battrs durng charg, rst and dscharg cycls. To coplnt th convntonal tral-and-rror thod of provng th prforanc and cycl lf of th lad-acd battry, athatcal odls hav bn dvlopd to prdct dscharg and charg bhavor, as wll as th ct of cyclng. Nwan and Tdann frst dvlopd a coprhnsv porous lctrod thory and appld t to sulat th dscharg bhavor of a lad-acd cll [1]. Brnard t al. proposd a twodnsonal odl to sulat th cts of th currnt collctor tabs []. Most rcntly C.Y. Wang and W.B. Gu dvlopd th athatcal govrnng quatons of battrs whch ar obtand fro th consrvaton laws [3]. In thr odl, th ct of fr convcton has bn consdrd and t was shown that flud dynacs hav a vry portant rol n th battrs dynac bhavor. Ths quatons can b usd n all knds of battrs (VRLA or Vntd lad-acd battrs) wth lttl odfcaton. Coputatonal flud dynacs (CFD) can b usd to solv ths govrnng quatons of battry dynacs to obtan an nsghtful undrstandng of battry prforanc and dtald charactrstcs of ts bhavor. W. B. Gu t al. usd CFD tchnqus and nurcally solvd th govrnng quatons of VRLA battrs [4]. Esfahanan and Torab [5] solvd ths govrnng quatons usng

2 Fgur 1: Schatc llustraton of a lad-acd cll. Kllr-Box thod and nvstgatd th ct of acd concntraton dpndncy of opn crcut voltag on battry bhavor. Snc th govrnng quatons of battry dynacs ar athatcally nonlnar and stff, th soluton of th syst rqurs any tratons whch aks t t consung. Thrfor, CFD odls ar norally usd for dsgn purposs and ar not sutabl for ral t applcatons. thr sulaton thods such as quvalnt crcut odl (ECM) [6] and ngnrng thods [7] ar fast nough for ral-t applcatons but th forr suffrs fro lack of accuracy and th ngnrng thod dos not provd any nsghtful nforaton about th battry. To ovrco th abov ntond probls of prvous odls, a nw athatcal odl s dvlopd whch s a cobnaton of CFD and ECM. Th nw dvlopd odlng nhrts accuracy of CFD odls and physcal undrstandng of ECM. Morovr, t s fast nough for ralt sulatons. Th prsnt approach s vrfd by prvous CFD odls and xprntal data.. Mathatcal Modl A typcal lad-acd cll s shown schatcally n Fgur 1 whch conssts of th followng rgons: a lad-grd collctor at x whch s at th cntr of th postv lctrod; a postv Pb lctrod; a porous sparator; a ngatv Pb lctrod; and fnally, a lad-grd collctor at xl whch s at th cntr of th ngatv lctrod. Th postv and ngatv lctrods consst of porous sold atrcs whos pors ar floodd by a bnary sulfurc acd, S 4, and th gasous phas that s consstd of, and N. Durng charg and dscharg, th followng lctrochcal ractons occur n th postv lctrod (Pb /PbS 4 ): Pb S 3 + PbS Dscharg (s) Charg 4(s) and th ngatv lctrod (Pb/PbS 4 ): Dscharg Pb + S PbS (s) 4 Charg 4(s) (1) ()

3 Th sd ractons consttut th ntrnal oxygn and hydrogn cycls n th cll. In th prsnc of sd ractons, a VRLA cll s a thr-phas syst consstng of th sold atrx, th lqud lctrolyt and a gas phas. Durng chargng and ovr chargng, th oxygn s gnratd at th Pb lctrolyt ntrfac and ay volv nto th gas phas aftr xcdng ts solublty lt n th lctrolyt. Th oxygn can thn b transportd, va th lqud and gas phas, fro th postv to th ngatv lctrod whr th oxygn gas ay dssolv back n th lctrolyt and b rducd at th Pb lctrod ntrfac. Ths procss fors an ntrnal oxygn cycl n VRLA clls. Kntc rat Equatons η φ φ U j s Consrvaton of Charg Consrvaton of Spcs Consrvaton of Mass j Tabl 1: Suary of govrnng quatons Prary ractons (j1 and 4): xygn gnraton rcobnaton (j and 5): ydrogn voluton (j6): γ c αaj F αcj F nj j,rf xp η xp j η j c rf RT RT c γ cj F nj j,rf xp α η j c rf RT.( k φ) +.[ kd (ln c )] + S c.( σ φ ) S c γ δ c α F aj c F αcj nj j,rf xp η xp j η RT j RT rf,rf c c In lctrolyt: In sold lctrod: s In lctrolyt: ( εc ).( D c ) + S In gas phas: In lctrolyt: In sold lctrod: ( εc ).( D, c ) + S ( ε c ).( Dg, cg ) + J ε V S εs V S s g g g Tabl : Sourc trs n consrvaton quatons Trs Postv Elctrod Sparator Ngatv Elctrod a + a a + a + a S c 1 n1 n 4 n4 5 n5 6 n6 S S J g V S V S s F + t + a 1 n1+ a n 3 t 1 1 a 4F n F g J D g lg k ( c c ), k ag ( ) an V t V o + F ( ) an V t+ V o F MPb M PbS a 4 1 n 1 ρpb ρ PbS F 4 1 t+ 1 t+ a 4 n4 + ( a 5 n5+ a 6 n6) F F 1 J g a 5 n5 Jg 4F an ( ) ( ) an V t V t V o F + F a ( ) n + V t F MPbS M Pb a 4 n 4 ρpbs ρ Pb F

4 ydrogn rcobnaton occurrng at th postv lctrod s nglgbl bcaus of ts poor kntcs. Accuulaton of oxygn and hydrogn n th gas phas contrbuts to th cll prssur buld-up, thrby causng vntng. Gnral govrnng quatons of battry dynacs hav bn dvlopd by Wang and Gu [3] whch ncluds all chcal and lctrochcal ractons as wll as th flow oton. Gu and Wang usd ths quatons to sulat VRLA battrs [4]. In th prsnt study, ths quatons ar splfd for lad-acd battrs and suarzd n Tabl 1, along wth sourc trs lstd n Tabl. Fgur : Currnt dstrbuton n lctrod and lctrolyt phass. Fgur 3: Schatc llustraton of (a) coputatonal volu and (b) volu crcut odl. In gnral, th appld currnt should b consrvd throughout th cll. Ths currnt ntrs th cll fro xtrnal crcut and ntrs th sold phas. As t can b sn schatcally n Fgur, th currnt ntrs th lctrolyt through surfac ractons whos rat s dtrnd by kntcs of th ractons. At th nd of sold phas, all th currnt ntrs th lctrolyt phas through whch s carrd by ons to th othr lctrod. n th othr lctrod surfac, ths currnt ntrs th sold phas agan n a rvrs annr. In ths odl, th currnt n th gas phas s nglctd. To obtan th dynac bhavor of battry syst, th govrnng syst of coupld lctrochcal quatons should b solvd nurcally. In th prsnt study, a two stp sch s ntroducd. In th frst stp, th consrvaton of charg quatons ar consdrd. Mathatcally, ths two coupl quatons hav llptc natur whos soluton rqurs a lot of coputatonal t. To rduc th coputatonal t, ths quatons ar cobnd by Krchhoff's currnt and voltag laws (KCL and KVL) and ar solvd wth an ECM basd fnt volu thod. Usng ths nw for of dscrtzaton, th dstrbuton of φ s and φ l can b obtand uch fastr than th prvous works whch wr basd on fnt dffrnc or fnt volu thods [4-5]. nc th dstrbuton of φ s and φ l s obtand (at ach t lvl), n th scond stp, th quatons of consrvaton of spcs ar solvd usng Kllr-Box thod [5]. Snc ths quatons ar parabolc and as t can b sn n th govrnng quatons, both functons and thr drvatvs ar prsnt.

5 Thrfor, a sutabl thod to solv th probl nurcally would b th Kllr-Box thod. Ths two stps ar consdrd n or dtal n th nxt scton..1. Stp 1 To obtan th potntal n sold and lctrolyt and currnt transfr btwn ths two phass, ach lctrod s dvdd nto so control volus n whch all th paratrs ar assud to b unfor. Each control volu can b odld by an lctrc crcut whch s shown n Fgur 3(a). As t can b sn, on th juncton of sold surfac of th lctrod and lctrolyt, a voltag dffrnc xsts whch s known as ovrpotntal. Th surfac ovrpotntal s th drvn potntal whch causs th currnt ntrs th lctrolyt. Th aount of ths currnt can b dtrnd by kntc quaton whch s known as Butlr-Volr quaton. For an ractons (j1,4 n Tabl 1) ths quaton s gvn as follow: γ c αaf αcf j xp η xp η c rf RT RT (3) For sd ractons, ths quaton can b usd wth a lttl odfcaton as gvn n Tabl 1. Th lctro-nutralty ndcats that th su of th currnt that ntrs a control volu ust lav, bcaus no currnt s gnratd nsd th volu. Usng ths fact and th KCL n th lctrc crcut shown n Fgur 3(b), th consrvaton of currnt n nod A bcos s(-1) - s() -J (4) and consrvaton of currnt n nod C l(-1) - l() -J also fro KVL w hav ( φ - φ ) + ( φ - φ ) + ( φ - φ ) + ( φ - φ ) (6) n whch s(+ 1) s( ) l(+ 1) s(+ 1) l() l(+ 1) s() l() J s Δx, and th currnts n sold and lctrolyt can b obtand fro h's law: s s (7) σ φ x and φl l k k D (ln c) x x Equatons (6) and (7) can b dscrtzd as follows: s ΔVs Rs (5) (8) (9) n whch Rs Rs + Rs ( + 1), and Rs Δx. In th sa annr w hav σ

6 Fgur 4: Equvalnt Crcut Modl of th whol battry cll. l ΔVl Rl k D 1 c c x Rl + Rl ( + 1) Δx n whch Rl, and Rl. σ Fro Fgur 3, t s obvous that nstad of th potntal n nods, w can us th potntal dffrnc to obtan th currnts. Thus, w can dfn nw varabls as follows: Δ Vs φs( + ) φ 1 s Δ Vl φl( + 1) φl (11) Δ VJ φs φl Fnally, n ach volu, thr ar thr unknowns (Eq. (1)) and thr quatons (.. Eqs. (4)- (6)) whch can b solvd to obtan th soluton. To hav a good accuracy, th whol battry s dvdd nto a nubr of (naly n) volus n srs whch wll rsult a syst of 3n unknowns and 3n quatons (so of th nglctd n th sparator rgon). Ths odl s llustratd n Fgur 4. By solvng ths syst of quatons, th potntals n lctrods and lctrolyts can b obtand... Stp To obtan th othr battry paratrs, th spcs quatons n Tabl 1 ar solvd nurcally by Kllr-Box thod. To solv ths quatons, th potntals ar takn for stp 1 hnc ar not consdrd as unknowns. Th Kllr-Box thod and ts applcaton to battry odlng s xpland n [5] and s not ntond hr agan. Aftr obtanng th battry paratrs n ths t stp, th whol procdur s rpatd agan to advanc to th nxt t stp. 3. Rsults and dscussons To vrfy th abov ntond procdur, th dscharg probl of a floodd lad-acd battry has bn sulatd. Floodd lad-acd quatons can b obtand by lnatng th gas voluton quatons fro th abov syst. Ths sapl has bn studd by Gu t al. [1] and rproducd by Esfahanan and Torab. [5]. All th ncssary paratrs ar th sa as th ons usd by Gu t al. [1]. Fgur 5(a) shows th voltag varaton of th cll vs. t. As t can b sn, th rsults of ths thod atch th othr rsults vry wll. Fgur 5(b) shows th varaton of acd concntraton vs. t n dffrnt tstps. Th rsults show that ths procdur s also capabl to captur th varaton of spac dstrbutd paratrs as n CFD odls. Furthror, ths thod s about ts fastr than pur CFD odls. (1)

7 Fgur 5: (a) Cll potntal vs. t durng dscharg (b)acd concntraton dstrbuton n cll To nvstgat th ct of oxygn voluton n battry chargng, a sapl tst s solvd whch was frst ntroducd by Gu t al. [4]. Fgur 6 shows th varaton of cll voltag vs. t for chargng. As t can b sn, at th nd of charg, th voltag of th cll rans narly constant whl th oxygn cycl bgns to dvlop. Th dvaton of th two nurcal curvs s du to th dffrnt valus usd for sulaton. Fgur 6: Cll potntal vs. t durng charg and ovrcharg. In Fgur 7, th sold and dashd curv show calculatd cll voltag durng constant currnt dscharg, rst, charg and ovr charg of a battry wth and wthout gas cyclng, rspctvly. In th absnc of oxygn cyclng, th voltag rs vry sharply at th nd of charg (SC1). Ths happns bcaus all th avalabl PbS 4 has bn convrtd to thr Pb or PbS4 and snc thr

8 Fgur 7: Cll potntal vs. t durng dscharg,rst, charg and ovrcharg. Fgur 8: Dstrbuton of J, J durng (a) Dscharg, (b) Charg, (c) Transton and (d) vrcharg. s no sd racton, a sall aount of currnt causs a sharp jup n voltag. But n th prsnc of sd ractons, th xcss currnt gos nto th oxygn cycl ractons and th battry voltag rans narly constant.

9 Fgur 8 shows th dstrbuton of J and Jo n th cll for dffrnt stats of chargs and dschargs. Accordng to Fgur 8(a) and 8(b), th aount of Jo can b nglctd whn th battry s dschargng or chargng, bcaus th an ractons ar or donant than th othrs. Fgur 8(c) shows that whn th battry s nar fully chargd stat, th an ractons rats ar rducd whl th scondary ractons occur ntnsvly. Fnally whn th battry s fully chargd, only scondary ractons ar prsnt and thrfor, th currnt dnsty s only du to sd ractons. 4. Conclusons In ths papr an provd athatcal odl for VRLA battrs basd CFD and ECM has bn ntroducd. Ths odl nhrts accuracy of CFD odl and physcal undrstandng of ECM. Morovr, t s vry fast whch aks t qut sutabl for ral-t sulatons. Th prsnt odl not only prdcts battry dynacal charactrstcs but also unlk th classcal ECM odl, s capabl to solv dstrbutd paratrs such as acd concntraton dstrbuton vrsus t. Th prsnt approach s vrfd by prvous CFD odls and xprntal data. Th rsults shows that ths odl has a good accuracy and th xcuton t s qut fast nough for ral-t purposs. 5. Acknowldgnt Th authors wsh to gratfully acknowldg th fnancal support fro Nru Battry Manufacturng Co. and Vhcl, Ful and Envronnt Rsarch Insttut of Thran Unvrsty. 6. Rfrncs [1] J. Nwan and W. Tdann, AIChE J., 1, 5 (1975). [] D. M. Brnard,. Gu, and A. Y. Schon, bd., 14, 5(1993). [3] C. Y. Wang, W. B. Gu, "Mcro-Macroscopc Coupld Modlng of Battrs and Ful Clls: I Modl dvlopnt,'' J. Elctroch. Soc. 145 (1) (1998) [4] W.B Gu, GQ. Wang, C. Y. Wang, Modlng th ovrcharg procss of VRLA battrs" J. Powr Sourcs, 18 () [5] V. Esfahanan, F. Torab, "Nurcal Sulaton of Lad-Acd Battrs usng Kllr-Box Mthod,'' J. Powr Sourcs, 158 (6) [6]. J. Brgvld, W. S. Krujt, P.. L Nottn, Battry Managnt Systs Dsgn by Modlng, Publshd by Kluwr Acadc publshrs, [7] V. Esfahanan, F. Torab, R. Afzal, "Engnrng Sulaton of Lad-acd Cll Charactrstcs and Procsss n Battrs," 8 th ISAEST Confrnc, Inda, 6.