A Single Particle Thermal Model for Lithium Ion Batterie R. Painter* 1, B. Berryhill 1, L. Sharpe 2 and S. Keith Hargrove 2 1 Civil Engineering, Tenneee State Univerity, Nahville, TN, USA 2 Mechanical Engineering, Tenneee State Univerity, Nahville, TN, USA *Correponding author: 3500 John A. Merritt Blvd., Nahville TN, 37209, rpainter@tntate.edu Abtract: COMSOL Single Particle Model for Lithium-Ion Batterie (Model ID: 14527) i generalized to include an energy balance. Thi i accomplihed by approximating the olution phae polarization a a function of current and temperature. The theoretical approach for thi work i imilar to Guo et al. [Journal of the Electrochemical Society, 158, (2) A122-A132 (2011)] for modelling lithium ion cell. The COMSOL iothermal model run at 0.1 A provided the baeline for the open circuit potential (OCP). Keyword: Lithium Ion Batterie, Single Particle Model. In thi work, the exiting ingle-particle model: COMSOL Single Particle Model for Lithium- Ion Batterie (Model ID: 14527) i extended to include thermal effect by adding the energy balance equation to the SP model. 2. Mathematical Model Generally, a lithium ion battery conit of the current collector, the poitive electrode, the eparator and the negative electrode. A lithiated organic olution fill the porou component and erve a the electrolyte. A chematic of a lithium ion battery i hown in Figure 1[Ref. 7]. 1. Introduction A diadvantage common to fully dimenional lithium ion battery model i the long imulation time due to the large number of nonlinear equation, o thee model become computationally inefficient for imulating condition in real time [Ref. 1 and 2]. To improve computational run time without compromiing accuracy, the ingle particle model (SP model) ha been propoed [Ref. 3 and 4]. The ingle particle model for a lithium-ion battery i a implification of the one dimenional (1d) model, ubject to everal implifying aumption. In the ingle particle model formulation, the local potential and concentration gradient in the electrolyte phae are ignored and accounted for uing a lumped olution reitance term [Ref. 5 and 6]. Similarly, the potential gradient in the olid phae of the electrode are alo neglected and the porou electrode i treated a a large number of ingle particle all being ubjected to the ame condition. The ingle particle formulation account for olid diffuion in the electrode particle and the intercalation reaction kinetic. Figure 1: Schematic of lithium ion battery The material balance for lithium ion in the batterie olid electrode i governed by Fick econd law in pherical coordinate: c (,) rt,j 1 2,j(,) c rt = D, j r 2 t r r r Eq. 1 The flux i zero at the center of the particle and i equal to the flux from the pherical particle at the particle urface. Thi give boundary condition c c D, j = 0@ r = 0 and D, j = j( t)@ r = R r r Eq. 2 1
and initial condition c j( r,0) = c.,, j,0 Where j = p, n for the poitive and negative electrode repectively, D i the olid phae lithium ion diffuion coefficient, R i the olid particle radiu and j (t) i the pore wall molar flux of lithium ion. The molar flux i related to the applied cell current a: where k i the reaction rate contant, c l i the olution phae concentration which i taken to be equal to a contant value in the ingle particle model, and c l, ref i the reference olution phae concentration (taken to be equal to 1 mol/m3). The Butler-Volmer kinetic expreion i rewritten in term of the invere hyperbolic function in order to improve computational efficiency. An expreion for the olid phae potential for each electrode can be obtained a: ɸ = E eq + ɸ l + RT 0.5F ainh (i loc 2i 0 ) Eq. 8 j(t) = i loc F = i applied (F(3/R )ԑ L) Eq. 3 The potential drop in the olution phae between the poitive and negative electrode i: Where i applied i the applied current denity, ԑ i the volume fraction of the olid phae active material in the electrode, and L i the thickne of the electrode. In thi model formulation, dicharge current denitie are taken to be poitive and charge current denitie are taken to be negative. A tate of charge (SOC) variable for the olid electrode particle i defined a follow: SOC = c urf cmax Eq. 4 where c urface max and c are the urface and maximum concentration, repectively, of lithium in the electrode particle. The Butler-Volmer expreion decribe the intercalation reaction kinetic, i loc = i 0 (exp ( 0.5Fɳ 0.5Fɳ exp )) Eq. 5 RT RT and the overpotential ɳ i defined a: ɳ = ɸ ɸ l E eq Eq. 6 where ɸ and ɸ l are the potential of the olid and olution phae, repectively. The exchange current i 0 i defined a: i 0 = Fk (c max c )c ( c l c l,ref ) Eq. 7 ɸ l,poitive ɸ l,negative = i applied R olution Eq. 9 where R olution i the olution phae reitance which i actually determined from coupled ma and charge tranfer procee. In the ingle particle model R olution i an adjutable parameter which could depend on cell temperature and applied current. The cell potential i determined a follow: E cell= ɸ,poitive ɸ,negative Eq. 10 In the energy balance ued for thi model formulation, the patial temperature ditribution in cell i neglected, o the cell temperature T i a function of time only. The general energy balance equation i given a Eq. 24 in Ref. 8, and i written a: ρυc p dt dt = IT U p T SOC p,urface U n T SOC n,urface +I(ɳ p ɳ n + IR olution ) Eq. 11 The initial cell temperature i aumed to be ame with the ambient temperature where T i the cell temperature, v i the volume of cell, p i the denity of cell, C p i the pecific heat capacity of cell, and q i the rate of heat tranfer between cell and urrounding. In thi work, heat flux at cell urface are aumed to follow Newton law of cooling, o the term q i expreed a q = ha(t T amb ) Eq. 12 2
where h i the advection heat tranfer coefficient and A i the cell urface area. 2.1 Model Parameterization All the model parameter required by the thermal ingle particle model are identical to the parameter ued in the COMSOL iothermal model example except for parameter impacted by the thermal balance. Thi wa done for the purpoe of comparion of the dicharge voltage profile between the two formulation. The temperature dependence of the diffuion coefficient, reaction rate contant and the open circuit potential (OCP) are given by D,j (T) = D,j exp [ Ea d,j R (1 T 1 T ref )] Eq. 12 k j (T) = k jexp [ Ea r,j R (1 T 1 T ref )] Eq. 13 Figure 3: The entropy change coefficient for the negative electrode. 3. Reult and Dicuion Figure 4 how the 1C dicharge curve at different cooling rate for the thermal ingle particle model a compared with the correponding dicharge profile for the iothermal model. Note that the dicharge data from the iothermal model i imported into COMSOL a text file for the purpoe of comparion. U j SOC j, T = U j SOC j, T ref + U j T (T T ref) Eq. 14 where j = p, n for the poitive and negative electrode. The entropy change coefficient profile from Ref. 9 are hown in Figure 2 and 3. Thee curve were incorporated into the COMSOL model a interpolation in SOC. Figure: 4: Cell Voltage for 1C dicharge at different cooling rate. Figure 5 how the temperature profile on the kin of the cell for a 1C dicharge rate and different cooling rate. Figure 6 how the reverible and irreverible heat generation and the heat lo to the cell urrounding at a cooling rate given by ha = 0.25 (W/m 2 ). Figure 2: The entropy change coefficient for the poitive electrode. 3
Figure 5: Temperature profile on the cell kin for a 1C dicharge and different cooling rate. Figure 6: Cell Heat Balance for different Cooling Rate. Figure 7: Comparion of 1C Dicharge Profile for Thermal Model and Iothermal Model. Figure 8: Comparion of 2C Dicharge Profile for Thermal Model and Iothermal Model. The COMSOL model wa ued to conduct parameter weep with repect to ambient temperature (T amb ) applied dicharge rate (i applied ) and the rate of cooling a reflected by the magnitude of the advection heat tranfer coefficient (h). The reulting temperature profile from the thermal balance were then fitted by conducting multiple nonlinear regreion in term of i applied and h uing Oakdale Engineering Datafit 9.0 oftware. The lowet order model with good approximation of the ingle particle thermal model i: T = 298 + (1 ha)1.0 6 t 2 i applied 1.0 5.0054 Eq. 15 Thi allowed the COMSOL model to repreent R olution (i applied, T) a a imulation in time ince there i no patial component aociated with R olution for the ingle particle thermal model and becaue the dependence of R olution on i applied and T i reflected by the regreion in T. The reulting tatitical model i capable of approximating the thermal behavior of lithium ion batterie a reflected in the preceding figure. The model i not capable of directly modelling the macro patial battery thermal characteritic but the heat balance depicted in Figure 6 can be ued in conjunction with geometric and thermal conductivity information of battery component to approximate a batterie patial thermal behavior. Thi modelling approach i tochatic in nature and the implifying aumption that form the bai of the ingle particle model repreent a gro departure from exiting robut model. However, even robut dimenional lithium ion 4
battery model rely heavily on empirical data becaue the complex ma tranfer and electrochemical kinetic involved are not completely undertood. The virtue of the ingle particle model and other reduced order model i that they have the potential to provide real time modelling and control of battery procee where the accuracy of more ophiticated model i not required. 4. Reference 1. J. Newman, T. William, AIChE J. 21 (1) (1975) 25 41. 2. M. Doyle, T. Fuller, J. Newman, J. Electrochem. Soc. 140 (1993) 1526 1533. 3. S. Santhanagopalan, Q. Guo, P. Ramada and R. E. White, J. Power Source, 156, 620, (2006) 4. G. Ning and B. N. Popov, J. Electrochem. Soc., 151, A1584 (2004) 5. M. Guo, G. Sikha, and R. E. White, Single Particle Model for a Lithium Ion Cell: Thermal Behavior, J. Electrochem. Soc., vol 158, p. A122, (2011) 6. Long White and Ralph White, Mathematical Modelling of a Lithium Ion Battery, Proceeding COMSOL Conference 2009 Boton, (2009) 7. L. Cai, R.E. White, J. Electrochem. Soc. 156 (3) A154 A161, (2009). 8. C. R. Pal, J. Newman, J. Electrochem. Soc., 142, 3282 (1995) 9. O. Egokina and A. Skundin, J. Solid State Electrochem, 2. 216, (1998) 5