simulation of lithium battery discharge

Edisi Khusus Desember 2008, hal : 292 - 298 ISSN : 1411-1098

Jurnal Sains Materi Indonesia Indonesian Journal of Materials Science

Akreditasi LIPI Nomor : 536/D/2007 Tanggal 26 Juni 2007

SIMULATION OF LITHIUM BATTERY DISCHARGE Pratondo Busono1 and Evvy Kartini2 1

Center of Pharmaceutical and Medical Technology (PTFM) - BPPT Jl. MH Thamrin No. 8, Jakarta 10340 2 Center for Technology of Nuclear Industry Material (PTBIN) - BATAN Kawasan Puspiptek, Serpong 15314, Tangerang

ABSTRACT SIMULATION OF LITHIUM BATTERY DISCHARGE. Aset of mathematical models was developed to predict the battery performance of lithium battery under discharge condition. The model not only account for coupled processes of electrochemical kinetics and mass transport occurring in a battery cell, but also consider heat generation within the battery. Numerical solution based on finite volume method was used for the simulation. The potential and current distribution at positive and negative electrode were calculated for various discharge times. Key words : Lithium battery, Solid electrolyte, Battery modeling

ABSTRAK SIMULASI PENGOSONGAN BATERAI LITHIUM. Serangkaian model matematik dikembangkan untuk memprediksi unjuk kerja baterai Lithium pada kondisi pengosongan. Model tidak hanya memperhitungkan proses kinetika elektrokimia dan transport massa yang terjadi dalam sel baterai, tapi juga mempertimbangkan pembangkitan panas dalam baterai. Solusi numerik berbasis metode volume terbatas digunakan untuk simulasi. Distribusi tegangan dan arus pada elektroda positif dan negatif dihitung untuk berbagai waktu pengosongan baterai. Kata kunci : Baterai Lithium, Elektrolit padat, Pemodelan baterai

INTRODUCTION Commercial rechargeable batteries are widely used in medical and consumer application today [1]. Most of these cells employ non aqueous liquid electrolyte and gelled polymer which consist of a non aqueous liquid suspended in the pores of polymer. The use of traditional polymer electrolyte is still an important research area due to advantages of design flexibility and safety [2]. The disadvantages of polymer electrolyte include lower conductivity and lower diffusion coefficient and difficult in fabrication process. In addition, future application of battery place an increasing demand for developing miniaturized power sources for mounting an energy source on the electronic board, micro machine and implantable medical devices which can not be met by gelled polymer electrolyte based batteries. Superionic conducting glass is one of promising material for electrolyte. There are several aspects that must be considered to model the battery behavior. The electrodes are generally porous, which means that the distribution of the reaction in the depth of the electrodes must 292

be considered. The basic modelling frame work consist of porous electrode theory, concentrated solution theory, Ohm’s law, kinetic relationship, and charge and material balance. Porous electrode theory treats the porous electrode as a superposition of active material, electrolyte and filler. The material balances are averaged about a volume with respect to the overall dimensions of the electrode. Ohms’s law describes the potential drop across the electrode and also in the electrolyte. In the electrolyte, Ohm’s law is modified to include the diffusion potential. Finally, the Buttler Volver equation is used to relate the rate of electrochemical reaction to the difference in potential between the electrode and electrolyte. The dependent variables of concentration, potential, reaction rate and current density each appear in more than one governing equations, and therefore the coupled governing equations must be solved simultaneously. The material properties may vary with concentration.

Simulation of Lithium Battery Discharge (Pratondo Busono)

In battery development process, it is essential to proceed with detailed mathematical modelling of battery system in order to produce the optimum cell design and configuration. Several mathematical models have been developed to predict the performance of battery cell [3-6]. In this work, a set of mathematical models was developed to predict the battery performance under charge and discharge condition. The model not only account for coupled processes of electrochemical kinetics and mass transport occurring in a battery cell, but also consider heat generation within the battery.

MATHEMATICAL MODEL The lithium cell as shown in Figure 1 consists of the positive electrode current collector (Al), negative electrode current collector (Cu), negative electrode (LixC6), a positive electrode (Liy Mn2 O4) and a layer of separator made from lithium phosphate. This separator is acting as an electronic insulator.

F = Faraday constant R = Universal gas constant T = Absolute temperature The exchange current density, ioj ,is a function of silver concentration in both electrolyte and electrode [3-4] , i oj

oj

k (ce )

( c s , max

c se )

oj

oj

( c se )

.... (2)

where : c = Volume averaged silver concentration, with subscript e and s referring for electrolyte and electrode cse = Area averaged concentration at the interface between electrolyte and electrode cs,max = Maximum concentration at the solid phase k = The Constant and determined by the initial exchange current density and species concentrations The local surface over potential of reaction j is determined as follow,

-

e

Re

s

Anode

+

Ion+

Catode

Figure 1. Schematic diagram of lithium cell

The assumption used in the model development can be written as follow : during the chemical reaction, there is no gas phase occurred, the charge transfer kinetic follows the Butler Volmer equation, the anode and cathode are porous and consist of spherical particles with uniform size, ionic movement process is due to diffusion and migration and volume change due to chemical reaction process (insertion- de-insertion process) is neglected. Using such assumptions, the basic governing equations for species and charge balance are given below.

Electrode Kinetics Based on the previous assumptions, the reaction process occurred at the electrode can be expressed by Butler Volmer equation [3], a ioj

exp

F RT a

j

j

i nj R

..................

f

(3)

The film resistance was used as adjustable parameter to get better agreement between simulation and experiments . Open circuit potential is a function of local state of charge and temperature T. It is usually approximated as a linear function of temperature, namely,

j

exp

F RT c

j

U

j , ref

(T

T ref )

U

j

T

............

(4)

where : Uj,ref = Open circuit potential at reference temperature Uj,ref = Obtained from the experiments

Mass Balance Using the mass balance averaging method to represent the concentration of species, the mass balance in the electrolyte and the electrode can be written respectively as [3], (

J

U

where : Uj = Open circuit of electrode reaction-j Rf = Finite film resistance on the interface

Separator Ion

e

ce )

e

t

.. (1)

( D eff

ce )

( D eff

ce )

t0

1 F

J ...(5)

and

where : aj cj

= Transfer coefficient at anode for j-th reaction = Transfer coefficient at cathode for j-th reaction

(

s

cs ) t

J 2F

.... (6) 293



where

and

= Porosity of electrode s = Porosity of electrolyte e Deff = The effective diffusion in the electrode t+ = Transference number of cation F = Faraday constant c = Concentration J = Reaction current

1.5 e

De

.........................................

(7)

The current resulted from the ionic reaction process at the interface [4], a s1in1 0

at anode .......................

at separator

a s 2 in 2

(8)

eff

j as F

cs )

.............................

(9)

................................................

s

..................................................

(10)

Energy Conservation

Cp T )

(

t

................................................

Q

a sj i nj (

To determine the electrical potential in the electrode and electrolyte can be written respectively as, j

0

(15)

s

Uj

e

T

Uj

i nj R f )

T

.....

eff s

ln c e

e

(16)

e

e

Initial Conditions At t=0, uniform initial conditions are assumed in both electrode (s) and electrolyte (e) ce , 0 , c s

c s ,0 , T

T0

.........

(17)

Boundary Conditions There is no chemical reaction in the current collector, 0,

................................................

(18)

The electrolyte is confined within the cell, e

)

...........

First term of Eq. 16 represents the heat effect due to electrode reaction, the second and third term results from the joule heating in the active material and electrolyte phase.

cs n

s

Q

(11)

Charge Balance

eff

T)

where heat generation Q can be expressed as

ce

where : = Volume fraction of electrode s

294

(14)

where : s = The material used in the anode and cathode

eff D

and the specific interfacial area of the electrode can be written as [4],

(

, can

The conductivity of the electrolyte is dependent on the is the effective eff conductivity of electrode. This parameter is strongly dependent on the electrode.

eff

Assuming that an insertion electrode is composed of spherical particles with a radius of r s, therefore the parameter lse can be written as [4],

as

eff D

where : f = the mean molar activity coefficient of the electrolyte

(

where : l se = Length of microscopic diffusion.

3 s rs

0 .....(12)

j

d ln f ) ... (13) d ln ce

t0 ) ( 1

(1

s

rs 5

ln c e )

The energy conservation equation can be written as

Interfacial balance in either anode or cathode can be expressed as,

l se

eff D

(

at cathode

where : a s = Area of the interface

Ds (c se l se

)

2RT F

eff D

where : De = Mass diffusion coefficient of anion in the electrolyte

j

e

where effective diffusion conductivity of ion, be written as [7],

Effective diffusion coefficient can be expressed using the formula developed by Bruggeman [3],

Deeff

eff

(

n

0

at all boundaries

Simulation of Lithium Battery Discharge (Pratondo Busono) s

Table 1. Geometry of Cell

0

y

at y =0 and y=H

0

s

Symbol Value Unit

x=0

0.130 cm

Thickness of anode

0.75

Thickness of separator

cm

At x=L

V

s

eff

s

eff

I , A

s

x

Thickness of the negative current collector

0.015 cm

Thickness of the negative current collector

0.190 cm

Thickness cathode

a

0.95

Porosity of anode

s

0.95

Porosity of separator

c

0.835

Porosity of cathode

(constant current)

s

(constant load)

RL A

The temperature boundary condition, T n

0.010 cm

(constant potential )

x

h (T

Ta )

..........................

(19)

where : h = Heat transfer coefficient I = Current density RL = External load Ta = Absolute temperature

Table 2. Transport Parameters

Symbol Value Unit t+0

0.363

Description Trans. Number anion

10-5

Cm2

Permeability of separator

S/cm Ionic conductivity

-2

for anode

NUMERICAL PROCEDURES The algebraic equations involving the unknown values at chosen grid points, named as discretization equations, are derived from the differential equations. This algebraic equations expresses the same physical information as the differential equations. The value of variables at the grid points thereby influences the distribution of such variables only in its intermediate neighborhood. As the number of grid points become very large, the solution of the discretization equations is expected to approach the exact solution of the corresponding differential equations. Transient term of the governing equation is discretized by fully implicit scheme making used backward Euler method. The spatial term is discretized by finite volume formulation as described by Patankar [6]. In this calculation method domain is divided into a number of non overlapping control volumes such that there is one control volume surrounding a grid points. Each differential equation is integrated over control volume and the resulting first derivative expressed using two point difference, assuming a linear profile to yield algebraic equation, which depends on the dependent variables of the nodal points adjacent to the point under consideration. The material properties and the geometry used in the simulation are presented in Table 1 and 2, respectively.

Description

ity

Electronic conductivity for cathode De

7.5 10- Cm2/s Diffusion constant 7

Newton’s Method The resulting set of algebraic equations are solved using the Newton’s method for each time step. Let us represent the set of algebraic equations in vector notation as

F (x )

0 ................................................ (20)

where f: R4n R4n, where n is the number of vertices in the grids. Solving using the Newton’s method leads to a sequence of problems,

J ( xk ) ( x k

1

xk )

F ( x k ) ........... (21)

where J(xk) is the Jacobian evaluated in the current iteration xk and can be written as

J (x )

F x

..........................................

(22)

The solution of such system is the main bottle neck in many applications. Apart from improving solvers of individual systems there is a strong need for reduction 295



of costs for solving more subsequent linear systems by inexact solver and sharing some of the computational effort.

Generalized Minimum Residual Solver The generalized minimal residual iterative solver is robust because it exhibits monotonic error decreases, guarantee convergence even for poorly conditioned system, and is associated with lowest residual error among all solvers for a fixed number of matrix vector products. Also it allows modification for the pre conditioner during the iteration to speed up convergence. Generalized minimal residual method is a solver that belongs to the Krylov subspace methods and seeks an approximate solution [5]. The value of m refers to the subspace dimension or search vectors and is critical to the convergence and efficiency characteristic of the solvers. Although the number of search vector m is arbitrary, it is the main factor that controls convergence. As a rule of thumb, larger values of m lead to faster convergence. However, the memory requirement of GMRES solver is O(mn) and its complexity is O(m2n). Therefore, it is essential to have an approximate estimate for m before proceeding the GMRES iteration. If this estimate is to small, convergence will be slow and may not be achieved at all. On the other hand, if m is larger than necessary storage and CPU is wasted.

Preconditioners Preconditioners may be applied to speed up convergence of iterative solvers. The diagonal preconditioner has minimal CPU and memory costs but often delivers 0-69% convergence improvement [7]. For poorly conditioned systems, there is a certain need for stronger and more robust preconditioners. Use of the approximate inverse preconditioning scheme is one approached that has been found quite successful. It relies on finding a sparse matrix M which minimizes the Frobenius norm of the residual matrix R given by [7]

R

I

AM

.........................................

(23)

where I is the identity matrix. It can be determined by minimizing the objective function, n

F (M )

Ij j 1

AM j

Ij

AMj

....................................

(25)

where Rj represents the j-th column of the residual matrix R. An important aspect of this approach is that only few columns of M need to be constructed. We typically choose these columns to refer the rows of A that are the major cause of iteration error.

RESULTS AND DISCUSSION The solution of the 2 dimensional governing equation subject to the boundary conditions were obtained by finite volume method. Numerical simulations were conducted for electrode configuration of 150 mm length, 85mm width and thickness of 3 mm. The distance between the electrode is 5 mm. The electrodes are made from LiyC6 and Liy Mn2O4. The tabs for both positive and negative electrode were attached at the top of current collector. To demonstrate the capability of the model to predict the cell potential, galvanostatic discharge scenarios was used. For this discharge scenario, the current discharge was set to I=1 A/m2. In the beginning of the simulation, the species concentration is assumed to be homogenous in both separator and electrode. Volume fraction of the electrolyte in the electrode varies from 0.1 to 0.4. The surrounding temperature is assumed to be 293 0K. The same value is used for the temperature at the current collector. Time step used in the simulation varies from 0.3 second to 30 second. Cell potential was calculated using the following formula, Cell Potential

s (x

L) -

s (x

0)

........

(26)

Whereas electrode potential is the different between potential at the interface between current collector and electrode, and the interface between separator and the electrode, which can be written as Electrode Potential

e (x

Le )

e (x

0) ......(27)

The decrease of potential at the interface between electrolyte and electrode is calculated using the following formula,

2 2

...................

(24)

in which Ij is the j-th column of the identity matrix, Mj is the j-th column of the PC matrix and n is the system size. Here, ||.||2 denotes the Euclidian norm of the matrix. Since a complete solution of (18) is time consuming, we instead consider the construction of M in a column by column manner by minimizing 296

Rj

Electrolyte Potential =

e

e (x

Le ) .. (28)

Figure 2 shows the distribution of potential at negative electrode. It is shown that the gradient potential is small for all regions at the negative electrode. The potential close to the tab is zero and the remain regions is negative.

Simulation of Lithium Battery Discharge (Pratondo Busono)

Figure 4. The distribution of current density at the positive electrode for discharge time of 30 minutes. 4.20

Figure 2. The distribution of potential at negative electrode for discharge rate 1 C at the discharge time of 3 minute.

Volume Fraction of Electrolyte on Electrode Volume fraction of electrolyte = 0.2 Volume fraction of electrolyte = 0.3

4.10

Volume fraction of electrolyte = 0.4

Figure 3 shows the distribution of potential at positive electrode for discharge time of 30 minutes with constant discharge rate. The highest potential is observed at the tab. This is because all the current from the positive electrode plate flowing through this tab.

4.00

3.90

3.80 0.00

20.00

40.00

Discharge Time [minute]

60.00

Figure 5. Discharge characteristics of lithium battery.

the battery cell, the cell potential become decrease faster compared for lower volume fraction of electrolyte.

CONCLUSION The electrochemical model has been applied to study the discharge characteristic of lithium based battery. Numerical simulation based on finite volume method was performed on a plate type lithium battery. The software can also be used to calculate the potential and current distribution at negative and positive electrodes for various discharge time. Figure 3. The distribution of potential at positive electrode at the discharge time of 30 minutes

Figure 4 shows the distribution of current density at discharge rate of 1C for various discharge times of 30 minutes. The current is transferred from the positive electrode to the negative electrode. At the beginning of the discharge, the current density is higher at the regions near tabs and lower at the regions away from the tabs. However, after midway stage of discharge, the current density at the tab is lower compared to other regions at the electrode. It can be interpreted that the chemical reaction takes place at the electrode region away from the tabs. Figure 5shows the characteristics of lithium battery for volume fraction of electrolyte. It is shown that with the increasing volume fraction of electrolyte at

ACKNOWLEDGEMENT This work is supported by the Ministry of Research and Technology, Republic of Indonesia through Incentive Research Grant (Fiscal Year 2008).

REFERENCES [1]. [2]. [3]. [4].

T.F. FULLER, M. DOYLE, and J. NEWMAN, J. Electrochem. Soc., 141 (1) (1994) C.Y. WANG, W.B. GU, and B.Y. LIAW, J. Electrochem. Soc., 145 (10) (1998) C.Y. WANG, W.B. GU, and B.Y. LIAW, J. Electrochem. Soc., 145 (10) (1998) S.V. PATANKAR, Numerical Heat Transfer and Fluid Flow, Hemisphere Pub., (1980) 297


[5].

[6]. [7].

298

J. WU, VENKAT SRINIVASAN, J. XU, and C.Y. WANG, J. Electrochem. Soc., 149 (10) (2002) 1342-1348 C.Y. WANG, W.B. GU, and B.Y. LIAW, J. Electrocchem. Soc., 145 (1998) 3407 W.B. GU, C.Y. WANG, and B.Y. LIAW, J. Electrochem. Soc., 145 (1998) 3418
