Mathematical Modeling of a Lithium Ion Battery

Transcription

1 Ecert from the Proceedngs of the COMSOL Conference 9 Boston Mathematcal Modelng of a Lthum Ion Battery Long Ca and Ralh E. Whte * Deartment of Chemcal Engneerng Unversty of South Carolna *Corresondng author: 1 Man Street Columba SC 98 whte@cec.sc.edu Abstract: The estng lthum on battery model n COMSOL s Multhyscs (MP) software s etended to nclude the thermal effects. The thermal behavor of a lthum on battery s studed durng the galvanostatc dscharge rocess wth and wthout a ulse. Keywords: Lthum on battery Thermal model Pulse dscharge Temerature 1. Introducton The estng lthum on battery model n COSMOL.5a s etended here by addng an energy balance and the temerature deendence of roertes of the battery. Ths thermal model s develoed based on the seudo two-dmensonal (PD) model whch was descrbed n [1 ] and a thermal electrochemstry couled model. The dffuson coeffcent of L ons n the sold hase and electrolyte the reacton rate constants of the electrochemcal reactons the oen crcut otentals and the thermal conductvty of the bnary electrolyte deend on the temerature n the model resented here.. Mathematcal Model A schematc of a lthum on battery s shown n Fgure 1. Current Collector L Postve Electrode L s Searator Fgure 1. Schematc of a Lthum on battery L n Negatve Electrode Current Collector Generally a lthum on battery conssts of the current collector the ostve electrode the searator and the negatve electrode. A lthated organc soluton flls the orous comonents and serves as the electrolyte. The materal balance for the L ons n an actve sold materal artcle s governed by Fck s second law n shercal coordnates: cs 1 cs = Ds r (1) t r where = for the ostve electrode and =n for the negatve electrode. At the center of the artcle there s no flu that s: cs at r = : D = On the surface of the artcle the flu s equal to the consumng/roducng rate of L ons due to the electrochemcal reacton occurrng at the sold/lqud nterface: cs at r = Rs : Ds = J where J s the flu of lthum ons away from the surface of the shercal artcles. The materal balance for the bnary electrolyte n the lqud hase s gven by: c c = D + t a J () s ( ) ε eff 1 + t where = s and n and a s the electrode surface area er unt volume of the electrode. In the searator the ore wall flu J s s equal to zero. At the two ends of the cell n the -drecton there s no mass flu that s: c Deff = cn Deff n = L+ Ls+ At the nterfaces between the ostve electrode/searator and searator/negatve electrode the concentraton of the bnary electrolyte and ts flu are contnuous c = c c s + L L ( ) = cn s ( s) + s L + L L + L

2 c D = eff Deff s L c + L cs cn Deff s = Deff n ( L Ls) + = + ( L+ Ls) The effectve dffuson coeffcent of the bnary electrolyte n the lqud hase s corrected by orosty []: brugg D = D ε where D e = 1 s eff e c T c The charge balance n the sold hase s governed by Ohm s law: φ1 σ eff = afj () where = and n. The effectve conductvty s determned by σeff = σ( 1 ε εf ). The secfc area can be calculated by a = ( 1 ε εf ). At the nterface of the Rs current collector and the ostve electrode the charge flu s equal to the current densty aled to the cell: φ1 σ eff = Ia where I a s the aled current densty. At the nterfaces of the ostve electrode/ searator and the searator/negatve electrode there s no charge flu: φ1 φ1 n σ eff = σ eff n = L L+ Ls The otental of the sold hase at the rght end of the cell s set to zero φ 1 = n = L + L L s+ n and the otental of the sold hase at s equal to E cell (the cell voltage). φ 1 = The charge balance n the lqud hase s based on Ohm s law and s gven by: φ κeff + (4) RT ( 1 t+ ) ln c κeff = afj F where = s and n and the secfc conductvty of the bnary electrolyte s a functon of temerature and the concentraton of the bnary electrolyte n the lqud hase []: brugg κ = κ ε eff c c +.74T where κ = c 17.8cT ct T.8 1 ct + At the two ends of the cell there s no charge flu n the lqud hase: φ κeff = and φ κ = n eff n L+ Ls+ The otental n the soluton hase and ts flu are contnuous at the nterfaces of the electrodes and the searator. In the above equatons the ore wall flu J s determned by the Bulter-Volmer equaton: ( ) ma surf surf J = k cs cs cs c.5f.5f (5) e η e η RT RT and the over otental of the ntercalaton reacton s gven by: η = φ1 φ U The energy balance s gven by [4 5]: dt T ρc = λ + Q rn + Qrev + Qohm (6) dt wth the boundary condtons determned by Newton s coolng law: T λ = ht ( T) T λ = ht L+ Ls+ ( T) where h s the heat transfer coeffcent T s the envronmental temerature Q rn s the total reacton heat generaton rate Q rev s the total reversble heat generaton rate Q ohm s the total ohmc heat generaton rate. The heat flues are defned by: Qrn = FaJ( φ1 φ U) U Qrev = FaJT T

3 Q φ1 φ ohm = σeff + κeff κ eff RT 1 c φ + ( 1 t+ ) F c The temerature deendent oen crcut otental of electrode s aromated by Taylor s frst order eanson around a reference temerature: du U = Uref + ( T Tref ) dt where U ref s the oen crcut otental under the reference temerature T ref.. COMSOL Model The mathematcal model descrbed n secton s a mult-scale model. We develoed several geometres n COMSOL: a 1D geometry whch conssts of three sequentally connected lnes to reresent the ostve electrode the searator and the negatve electrode resectvely a D geometry whch conssts of two rectangles to denote the sold hase n the electrodes. The geometres are shown n Fgure. The vertcal coordnate n the D geometry ndcates the radal drecton of the sold artcles. Snce we gnore the dffuson of L ons n the -drecton n the artcle the corresondng dffuson coeffcent s set to zero n ths drecton. The concentraton of the bnary electrolyte the otental n the electrolyte the otental n the sold hase and the ore wall flu are solved n the 1D geometry. The concentraton of L ons n the sold hase s solved n the D geometry. The ore wall flu s etruded from the 1D doman and rojected to the to boundary of the D geometry by usng subdoman etruson coulng varables n COMSOL. The concentraton of L ons on the to boundary n the D geometry s rojected to the 1D doman by usng boundary etruson coulng varables. The thermal behavor of the L on battery durng ulse dscharge s also smulated n COSMOL. The battery s dscharged for s at C/ rate frst and then dscharged at C rate untl the cell voltage dros to.5v. The change of the aled current densty s mlemented by usng the smoothed Heavsde functon flsmhs and s shown n Fgure. J c ssurf J n c snsurf Postve Electrode Fgure. Geometres and varables coulng between the geometres n COSMOL (The to 1D geometry conssts of three segments whch denote the ostve electrode the searator and the negatve electrode. The bottum two rectangles reresent the sold hases n the ostve electrode and the negatve electrode resectvely. The ore wall flu s etracted from the 1D geometry and rojected to the to boundary of the D geometry. The concentraton of L ons on the to boundary of the D geometry s rojected to the 1D doman as the surface concentraton of the sold artcles.) Fgure. Current densty rofle n the dscharge rocess ncludng a C ulse 4. Smulaton Results Negatve Electrode Fgure 4 shows the temerature on the cell surface at 1C dscharge rocess under three dfferent coolng condtons where the heat transfer coeffcent s and.1 W/m /K resectvely and two lmtng condtons: the sothermal condton and the adabatc condton.

4 Fgure 4. Temerature on the cell surface durng 1C dscharge rocess under dfferent coolng condtons The thermal effect on the cell voltage s shown n Fugre 5. The cell rovdes more dscharge caacty when t s laced n a better heat solaton envronment (.e. adabatc condton). In a better solated envronment the cell temerautre ncreases faster durng the 1C dscharge rocess whch results n the hgher dffuson coeffcent for the bnary electrolyte and reduces the dffuson lmtatons. Fgure 6. Comarson of the concentraton rofles of the bnary electrolyte at the end of the 1C dscharge rocess under the sothermal condton and the adabatc condton Fgure 7 shows the cell temerature durng the 1C dscharge rocess at dfferent current rates as the heat transfer coeffcent s 1. W/m /K. As eected the cell gets hotter as the dshcarge current rate ncreases. It s also notced that the wave art whch aears n begnnng of the temerature curve at low current rate (less than C) does not est n the hgh current rate cases. The wave art on the temerature curve s charactersed by the reversble heat generaton durng dschargng. Under low current rate dschargng the reversble heat s roughly equvelant to the ohmc heat but becomes unmortant as the dscharge current rate ncreases. Fgure 5. Cell Voltage for 1C dscharge rocess under dfferent coolng condtons The reducton of the dffuson lmt n the bnary electrolyte can be verfed by comarng the concentraton rofle of the electrolyte under dfferent coolng condtons. Fgure 6 shows the concentraton rofles of the electrolyte at the end of 1C dscharge rocess under the two lmtng condtons. The concentraton rofle under the adabatc condton s flatter than that n the sothermal case whch ndcates a better dffuson roerty n the electrolyte under the adabatc condton than under the sothermal condton. Fgure 7. Temerature on the cell surface durng dscharge rocess under dfferent current rates whle the heat transfer coeffcent h=1. W/m /K where the DOD s defned as: DOD=tme * C rate / 6

5 The PD model mentoned n secton s also useful for smulatng the dscharge rocess wth ulse. Fgure 8 shows the cell voltage durng the C/ dscharge for s followed by a C ulse dscharge untl the cell voltage dros to.5v. The corresondng temerature on the surface of the cell s also lotted n Fgure 9. The surface temerature at the end of the C ulse s slghtly less than that n the ure C dscharge rocess. Fgure 1. Concentraton of the bnary electrolyte at the two ends of the cell (the bottom lne s at the to lne s at 1.65e-4) n the C/ dscharge rocess wth a C ulse 5. Conclusons Fgure 8. Cell Voltage at C/ dscharge for s followed by a C ulse dscharge The thermal behavor of a Lthum on battery durng galvanostatc dscharge w/o ulse can be redcted by usng COMSOL Multhyscs. Though better thermal solaton envronment mroves the dscharge caacty the hgher cell temerature rases the rsk of thermal runaway and more rad wear out of the cell. 6. References Fgure 9. Temerature on the cell surface n the dscharge rocess wth C ulse Fgure 1 shows the concentraton of the bnary electrolyte at the two ends of the cell durng the ulse dscharge rocess. At the begnnng of the ulse the concentraton of the electrolyte changes etremely after that t relaes and tend to a stable value. 1. M. Doyle T.F. Fuller and J. Newman Modelng of galvanostatc charge and dscharge of the Lthum/Polymer/Inserton cell Journal of Electrochemcal Socety (199). T.F. Fuller M. Doyle and J. Newman Smulaton and Otmzaton of the Dual Lthum Ion Inserton Cell Journal of Electrochemcal Socety (1994). Lars Ole Valoen and N. Remers Transort Proertes of LPF 6 -Based L-Ion Battery Electrolytes JES 15 A88-A891 (5) 4. W.B. Gu and C.Y. Wang Thermal- Electrochemcal Modelng of Battery System Journal of Electrochemcal Socety () 5. K. Kumaresan G. Skha and R.E. Whte Thermal Model for a L-Ion Cell. Journal of the Electrochemcal Socety 155 A164-A171 (8)