Modeling of the Charge-discharge Behavior of a 12-V Automotive Lead-acid Battery

Korean Chem. Eng. Res., Vol. 45, No. 3, June, 2007, pp. 242-248 y k 12-V }nsm } n m Çn ÇnksÇ Ço * Ç * kt l vedš 443-749 oe m o} 5 * q 445-706 e q 772-1 (2007 2o 9p r, 2007 3o 1p) Modeling of the Charge-discharge Behavior of a 12-V Automotive Lead-acid Battery Ui Seong Kim, Sehoon Jeon, Wonjin Jeon and Chee Burm Shin, Seung Myun Chung* and Sung Tae Kim* Ajou University, Division of Energy Systems Research, San 5, Wonchun-dong, Yeongtong-gu, Suwon 443-748, Korea *Hyundai Motor Company, 772-1, Jangduk-dong, Hwaseong 445-706, Korea (Received 9 February 2007; accepted 1 March 2007) k q r q edšp r o n rvp r r p m p p n. l l o n p pn l n 12-V rvp r r p m p 2 o p m. p l l n r l r p, r vp o, l p p mp r, r p e l p p l. p e p v o l r re p m. re p 25 o Cl C/5, C/10 C/20p rpl l m, re p 25 o Cl rr -rrk p (r r 30A, r rk 14.24 V) m. l l m r p r e m q p m. 2 o p l rp v k er rp r v p r, r kp r ˆ(state of charge; SOC)p m pl. h Abstract For an optimal design of automotive electric system, it is important to have a reliable modeling tool to predict the charge-discharge behaviors of the automotive battery. In this work, a two-dimensional modeling was carried out to predict the charge-discharge behaviors of a 12-V automotive lead-acid battery. The model accounted for electrochemical kinetics and ionic mass transfer in a battery cell. In order to validate the modeling, modeling results were compared with the experimental data of the charge-discharge behaviors of a lead-acid battery. The discharge behaviors were measured with three different discharge rates of C/5, C/10, and C/20 at operating temperature of 25 o C. The batteries were charged with constant current of 30A until the charging voltage reached to a predetermined value of 14.24 V and then the charging voltage was kept constant. The discharge and charge curves from the measurements and modeling were in good agreement. Based on the modeling, the distributions of the electrical potentials of the solid and solution phases, the porosity of the electrodes, and the current density within the electrodes as well as the acid concentration can be predicted as a function of charge and discharge time. Key words: Lead-Acid Battery, Charge, Discharge, Model, Finite Element Method 1. 12-V rv 2 rvp s, PbO 2 p k Pbp p r v l p. rv 2 rvl p r kr p l qr l, q v p e (starting), r (lighting), r (ignition) p l p r q p n p n l To whom correspondence should be addressed. E-mail: cbshin@ajou.ac.kr p l v l p p. l s (steering), r (braking), s(air-conditioning) p r n kr, r, k p po p p r r n ~ p. p rvl n r v p p. pm p s v r l rv krrp q l l rop plk. p rp o l er l n p rvp m p e m p lr m. p l e p 242

p n. r p m p p, pl l rvp pn pp, n n sl n p rvp kp p p n l [1-3]. Tiedemann [4]p rvp r p m o l r p r p, s v r l p vr p 1 o p rk m. Vaaler [5]p p p q l, r m k Žp ro m m. Tiedemann [6]p k Ž k p Žl p r m ro m p r p rk m. Morimoto [7] p r p v r[4-6]p r 2 o p r m ro m m. Bernadi [8]p r m ro k r kp, p, m m. op l p r pl l. Gu [1]p rr r p(25 o C, -18 o C)l p rk m r k m p m. Maja [9]p vp p k, rk p r l l m p v m. Gu [3]p r pl 2 o p re m. l l l l ˆ o l r p, r vp o l p pmp r, r p p l p rp r p r m. o n p pn l 2 o p p r ol l n ol pn l m p l. q~ rq rv k n 12-V rv( l p 90 Ah rv) pn l re p m, p e m l p e p v m. 2-1. }ns 2. n Fig. 1p rv p p. rv k p PbO 2 m p p Pb pm p l PbSO 4 m p r n 12-V rvp r 243 pp l v r l v. PbSO 4 m p l PbO 2 m Pb o r pp p r ˆ. pp rv l pl pp e pep. ( rv k l p p) discharge PbO 2 3H + + + HSO 4 + 2e PbSO 4 + 2H 2 O (1.685 V) ( rv p l p p) discharge charge Pb+ HSO 4 PbSO 4 + H + + 2e (1.685 V) charge ( rv r~p p) discharge PbO 2 + Pb+ 2H 2 SO 4 2PbSO 4 + 2H 2 O (2.041 V) charge 2-2. s o 12-V rvp rr (rrk) r p o l, rv l l v r [3] l l m. rvl n ~ ˆp r l p p sq. p kl r kp n nkp l p. l l r r l r p ~ r l r v nkp r s, o l p pmp v s, Butler-Volmerp r p, r ˆ ˆ SOC(state of charge)p, r p ˆ re p l p. r r p ~ r l r v nkp r sp ˆ rel l. r r p p pl r (i) ~ ˆp r pl r (i s )m r v nkl r p p p r (i l )p p. ~ r l m r e p k p rp, r p divergence 0p. p r p. i = i s + i l (1) i s + i l = 0 r v nkl r d(flux) r v r p r k p p(a)m Butler-Volmerp r p ep rp r r (transfer current density, j)p p. (2) i l = Aj (3) Fig. 1. Schematic illustration of a lead-acid cell. j i c 0 -------- exp α a F --------η α c F = --------η RT exp RT c ref l ro η PbO 2 l l η = φ s - φ l - U PbO2, Pbl l η=φ s - φ l rp. PbO 2 p n, ro( U PbO2 ) r r kp s m p p. r l p o l p ro (2.05) r m. α a (α c ) k (p )p r (anodic(cathodic) apparent (4) Korean Chem. Eng. Res., Vol. 45, No. 3, June, 2007

244 p Ër ËrovËe Ërd Ë ˆ transfer coefficient)p (α a =1.5, α c =0.6). ~r p r (i s ) mp l ~ p ro n l. i s = σ eff Φ s l ~r p r (σ) (ε =0.6) r o r p n m. σ eff = σ 1 ε ( ) 1.5 r p r kp r (i l ) ro n m r kp n l. i l = κ eff Φ l κ eff D ( lnc) eff l κ D pmp l p p rpq p r r p. p Š, ~ r l r v nkp r sp ˆ rep ˆ p. ( ~r p r s re) (5) (6) (7) Aj p. l ----- 2F pm d(flux) p, Faraday p r pl r r (A j)l p. a 1 p r vp, PbO 2 l l a 1 = M PbSO4 M PbSO2 --------------- ---------------, Pbl l a 1 = M Pb M PbSO2 --------- --------------- p. ρ PbSO4 ρ PbSO2 ρ Pb ρ PbSO2,, p (migration)l p vr l pmp v sp ˆ ep εc ( ) ------------ + v c D eff Aj = ( c) + a 2 ----- 2F (14) p. l a 2 PbO 2 l l a 2 =3 2t +0, Pbl l a 2 =1 2t + 0 p. t + r (transference number)p. v p r kp o p Boussinesq m l rep Navier-Stokes ep p. v ----- + v v 1 ν = -- + p + ( ν v) + g1 [ β( c c ρ 0 )] + --- ( εv) K (15) v = 0 (16) ν l κ -- ( εv) o~p r m Œ l p (drag) l p p. (εv) p Œ v r l Darcyp p. ( σ eff Φ s ) Aj = 0 (8) K εv = --- { p ρg[ 1 β( c c µ 0 )]} (17) (r v nkp r s re) l Œ (K) Kozeny-Carman ep p. ( k eff F 1 ) + [ k eff D ( lnc) ] + Aj = 0 r rp r(a)p rvp p ˆ ˆpq tl p. po r pp v r r v p pl p vp p pl l, r pl r rp rp r pl p r p v k p. r rp rp k m p SOC p pn ep. (9) ε 3 d 2 K = ------------------------- 180( 1 ε) 2 l d r p pq p v p. (18) 2-3. { o o ~m k~ p ro r ˆp Poisson ep v p s p n. r v rn rv(cell)p l ~m k~ p rol rr s p ( r) A = A max SOC 0.6 (10) Φ -------- l Φ s = -------- = 0 (19) ( r) A = A max ( 1 SOC) 0.6 (11) l A max rp p. e l SOCp ˆ ep ( SOC) ------------------ i l = ± ---------- ( + for PbO 2, for Pb) Q max o45 o3 2007 6k (12) p. l Q max r ˆl p r p pn p r p ˆ. e (5)~(9)l n o (o ~q eff) p rvp p ˆ ˆpqp r l p v p. e l p ˆ ep ε ---- Aj a 1 ----- = 0 2F (13) p. l np l vp o p. r v l p s p ( r rkp p lp n) Φ s = 0 or V ( r r r lp n) σ eff Φ s -------- = I (20) (21) p. l Ip p (+)p n r p, ( )p n r p p np. rv p l, p s p c ----- = 0 (22)

n 12-V rvp r 245 n (finite element method)p n m [10]. e l r p implicit predictor-multicorrector method n m [11]. rvp l n o n q(finite element mesh) Fig. 3 l ˆ l. l m p, p ˆ ml(, )p s, v k p mll l. o n q 4,032 p nodem 3,875 p bilinear quadrilateral element l. 3. n }n Fig. 2. Input and output parameters for the modeling of the lead-acid battery. p. (c)p s p r kp l. c=c 0 (23) l c o 5Ë10-3 (mol/cm 3 )p. 2-4. m} mm rvp l n p pq vpq, ˆpq, q pq, p pq p. vpq r kl p pmp m r kp r, r, p r v r vp r kp l p. ˆpq r vp, Žp, Ž p, Žp p. q pq q r m m p, p p q s p p. pq rv rk(cell voltage) r, SOC, r kp p. Fig. 2l p pqm p q r m. 2-5. r l n rep l o l o l l l l rs n 12-V(90 Ah ) rv n m. e p ml lp, er l p k r ˆ p C/5, C/10, C/20(5e, 10e, 20e )p rpl l s rkp r 10.5 V v re p m. r p 130% v r rr -rrk p (r r 30A, r rk 14.24 V) re p m. rp r p l lrp p p l rvp p r r eˆ l rs rkp 10.5 V r m. 4. y 4-1. n Fig. 4 rpp C/5, C/10, C/20p re m p, e e p, p rp e m. re m r rp p m. e l p rrkp p n o, r e p p l p r vp rp r p q p. 9A r C/10 re 4.5A r C/20 re mll p rrk p p ˆ. po l [1, 3, 12] p p, l rk 12.5 V r rkp r rkp m p. e m 18 A r C/5 re p mll, rkp lr wp e Fig. 3. Finite element mesh used for the lead-acid battery. Fig. 4. Comparison between experimental and modeling discharge curves at discharge rates of C/5, C/10, and C/20. Korean Chem. Eng. Res., Vol. 45, No. 3, June, 2007

246 p Ër ËrovËe Ërd Ë ˆ Fig. 5. The distribution of the current density within the cell after 1, 4, and 8 hours of discharge with C/10 rate. Fig. 7. The distribution of the state of charge within the cell after 1, 4, and 8 hours of discharge with C/10 rate. kl e coup de fouet p rp rk (voltage-dip) p ˆ. p rk p r tol p ƒv l p, rr ƒ v. p p r rkp m e, r p l m p p p [12]. l p rk p v kk. Fig. 5 rv p r ˆ (contour plot)p ( rp = C/10). rv p r rp v l m m. r (a)l Pb r p p p p r k. p p r l p v p l p. r (c)l r p lr p pl. rp v l l p vp kp tl, r p v v p. Fig. 6p rv p r k ˆ p ( rp = C/10). r l rv p r kp pl l, rp v rv p r kp p l l p m l. rv p r k v p, p tp p p rp n p oyp p, rp n p k yp p l p. PbO 2 r yp r k Pb r yp r k p m l. p p Pb r p PbO 2 r r r p, v pn ƒr m ep qkv p. Žl r p r p p l l, p l. Fig. 7p rv p SOC ˆ p ( rp = C/10). rp v l SOC p, PbO 2 r p SOC p Pb r p SOC, rv p SOC p p SOC p p p m l. r l rv p v p. Žl r p r p ll SOC p 0p. 4-2. }n Fig. 8p r r 30 A, r rk 14.24 V p re m p, e }ov (rk; Ë, r ; Ë)p, p (rk; Ë, r ; Ë)p e m. re m r rp p m. rr r l rkp n r rkl p Fig. 6. The distribution of the concentration of sulfuric acid within the cell after 1, 4, and 8 hours of discharge with C/10 rate. o45 o3 2007 6k Fig. 8. Comparison between experimental and modeling charge curves. The battery was charged with the constant current of 30 A for 1 hour and then with the constant voltage of 14.24 V.

n 12-V rvp r 247 Fig. 9. The distribution of current density within the cell after 10, 20, and 40 minutes of charge with constant current of 30 A for 1 hour and then with the constant voltage of 14.24 V. Fig. 11. The distribution of the concentration of sulfuric acid within the cell after 10, 20, and 40 minutes of charge with constant current of 30 A for 1 hour and then with the constant voltage of 14.24 V. Fig. 10. The distribution of the concentration of sulfuric acid within the cell after 10, 20, and 40 minutes of charge with constant current of 30 A for 1 hour and then with the constant voltage of 14.24 V. v m. p p rvp r p ƒv l p. rrk r l p r n m p r n l p sq m. v p l p r p n r l q n p o. Fig. 9 re l rv p r ˆ p. Pb r p r rvp kv p m l. p p rvp PbSO 4 Pb o p p. Fig. 10p re l rv p r k ˆ p. r (a)p r k rv r ~rp pr. rp eq, r tab n rv o r. PbO 2 r p r k Pb r p r k v m. r r PbO 2 r tabp l r p r q p p. PbO 2 r -Reservoir p ml. p r ƒvl vr r p v p p [13]. Fig. 11 re l rv p SOC ˆ p. rp eq 20 p vl l 40 p l, SOC p Pb r l rv o, PbO 2 r l rv k d m. 5. l l rvp r r p m p r ol m. p l n r l rv l pl r p, r vp o l p pmp r, r p p l p rp l. p r r l rvp p l m pl. rp l p l n rv p r m r k, SOC p erp m p l. r ol lp m n 12-V rv pn r e l p e p v m. 25 o Cl k rp re m r rp p m. rrkp p n o, r e p p l p rvp rp r p q p. rv p r rp v l m p p pl. r l Pb r p p p p r v, r l r p lr p pl. r l rv p r kp pl l, rp v rv p r kp pl l p m l. rp v l SOC p, PbO 2 r p SOC p Pb r p SOC, rv p SOC p p SOC p p p ˆ. 25 o Cl r r 30 A, r rk 14.24 V re m r rp p m. rr r l rkp n r rkl p v m. Pb r p r rvp kv p m l. p p rvp PbSO 4 Korean Chem. Eng. Res., Vol. 45, No. 3, June, 2007

248 p Ër ËrovËe Ërd Ë ˆ Pb o p p. r p r k rv r ~rp pr. rp eq, r tab n r v o r. PbO 2 r p r k Pb r p r k v m. rp eq 20 p vl l 40 p l, SOC p Pb r l rv o, PbO 2 r l rv k d m. l q m (t)lv p vop ld. pl. o45 o3 2007 6k k A : active surface area of electrode [cm 2 /cm 3 ] a : coefficient A max : maximum specific active surface area of electrode [cm 2 /cm 3 ] c : concentration of binary electrolyte [mol/cm 3 ] c ref : reference concentration of the binary electrolyte [mol/cm 3 ] D : diffusion coefficient of the binary electrolyte [cm 2 /s] F : Faraday s constant [96,487 C/mol] g : gravitational vector [cm/s 2 ] i 0 : exchange current density [A/cm 2 ] j : transfer current density [A/cm 3 ] K : permeability [cm 2 ] M : molecular weight of species [g/ mol] p : pressure [Pa] Q : electrode capacity [Coulomb/cm 3 ] R : universal gas constant [8.3143 J/mol K] t : time [sec] + t 0 : transference number of H + with respect to solvent velocity T : absolute temperature [K] U PbO2 : equilibrium potential at c ref for positive electrode [V] v : velocity vector [cm/s] x : distance from centre of positive electrode [cm] k D eff : effective diffusion conductivity [A/cm] m α : transfer coefficient β : volume expansion coefficient [cm 3 /mol] γ : exponent for the concentration dependence of the exchange current density Γ : diffusion coefficient ε : porosity of electrode ξ : exponent for charge dependence of the specific active surface area η : electrode overpotential [V] κ : electrolyte conductivity [S/cm] µ : dynamic viscosity [kg/cm s] ν : kinematic viscosity [cm 2 /s] ρ : density [g/cm 3 ] σ : conductivity of the solid matrix [S/cm] Φ s : electric potential of solid phase [V] : electric potential of liquid phase [V] φ l lzm eff g zm D L max O Ref S : effective, corrected for tortuosity : pertinent to diffusion : liquid solution : maximum or theoretical : initial value : reference state : solid phase y 1. Gu, H., Nguyen, T. V. and White, R. E., A Mathematical Model of a Lead-Acid Cell, J. Electrochem. Soc., 134(12), 2953-2960(1987). 2. Nguyen, T. V. and White, R. E., The Effects of Separator Design on the Discharge Performance of a Starved Lead-Acid Cell, J. Electrochem. Soc., 137(10), 2998-3004(1990). 3. Gu, W. B., Wang, C. Y. and Liaw, B. Y., Numerical Modeling of Coupled Electrochemical and Transport Processes in Lead-Acid Batteries, J. Electrochem. Soc., 144(6), 2053-2061(1997). 4. Tiedemann, W. H., Newman, J. and DeSua, F., Power Sources 6, Collins, D. H.(Ed.), p.15, Academic Press, NY(1977). 5. Vaaler, L. E. and Brooman, E. W., Paper 780221, SAE Congress and Exposition, Detroit, MI(1978). 6. Tiedemann, W. H. and Newman, J., Battery Design and Optimization, Gross, S.(Ed.), p.23-39, The Electrochemical Society Softbound Proceedings Series, 79-1, Princeton, NJ(1979). 7. Morimoto, Y., Ohya, Y., Abe, K., Yoshida, T. and Morimoto, H., Computer Simulation of the Discharge Reaction in Lead-Acid Batteries, J. Electrochem. Soc., 135(2), 293-298(1988). 8. Bernadi, D. M., Gu, H. and Schoene, A. Y., Two-Dimensional Mathematical Model of a Lead-Acid Cell, J. Electrochem. Soc., 140(8), 2250-2258(1993). 9. Maja, M., Morello, G. and Spinelli, P., A Model for Simulating Fast Charging of Lead/acid Batteries, J. Power Sources, 40(1-2), 81-91(1992). 10. Hughes, T. J. R., The Finite Element Method, Prentice-Hall, NJ(1987). 11. Brooks, A. N. and Hughes, T. J. R., Streamline Upwind/Petrov- Galerkin Formulations for Convective Dominated Flows with Particular Emphasis on the Incompressible Navier-Stokes Equations, Comput. Math. Appl. Mech. Eng., 32(1/3), 199-259(1982). 12. Kim, S. C. and Hong, W. H., Analysis of the Discharge Performance of a Flooded Lead/acid Cell Using Mathematical Modeling, J. Power sources, 77(1), 74-82(1999). 13. Kim, S. C. and Hong, W. H., Effect of Electrode Parameters on Charge Performance of a Lead-acid Cell, J. Power sources, 89(1), 15-28(2000).