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Computers and Chemical Engineering 35 (2011) 1937– 1948

Contents lists available at ScienceDirect

Computers and Chemical Engineering

jo u rn al hom epa ge : www.elsev ier .com/ locate /compchemeng

odeling and simulation of lithium-ion batteries

rnesto Martínez-Rosasa, Ruben Vasquez-Medranob, Antonio Flores-Tlacuahuacb,∗

Facultad de Química, Universidad Nacional Autónoma de México, Ciudad Universitaria, México D.F. 04510, MexicoDepartamento de Ingeniería y Ciencias Químicas, Universidad Iberoamericana, Prolongación Paseo de la Reforma 880, México D.F. 01210, Mexico

r t i c l e i n f o

rticle history:eceived 26 October 2010eceived in revised form 9 May 2011ccepted 10 May 2011vailable online 27 May 2011

a b s t r a c t

In this work the dynamic one-dimensional modeling and simulation of Li ion batteries with chemistryLixC6−− LiyMn2O4 is presented. The model used is robust in terms of electrochemical variables predictionrather than only the electrical ones. This enables us to analyze the internal behavior of the battery underdifferent discharge rates. The method of lines (MOL) was used for predicting the behavior from the model

ithium-ion batteriesodeling

imulationnergy

without any loss of exactitude for regular geometries. The boundary conditions were modified to achievea better convergence of the solver. The simulation results were compared to experimental data fromthe research literature. Some examples of application are also presented that include the simulation forthe optimization of design parameters, the evaluation of the behavior of the battery under dynamicdischarge rates simulating real simplified conditions of operation and the simulation of the paralleldischarge of different capacity pairs of batteries.

. Introduction

The race for alternative energy sources and green technologiess speeding worldwide, mainly due to global warming and the fore-ast run out of fossil fuels. However, these are not the only reasons.rom an economic profit point of view, there is a growing sector ofnvironmental aware buyers, willing to spend something extra toeduce their impact to the planet. The real concern is that many ofhese green technologies simply propose to move the CO2 gener-tion from our vehicles to a power plant. An appropriate solutiono prevent CO2 generation would be to replace internal combus-ion vehicles with alternative energy sources. Among these sourceshere are: biofuels, compressed air, electric fuel cells and recharge-ble batteries.

There are many options for electric vehicle batteries, each sys-em offers unique features with advantages and disadvantages. The

ost promising batteries are lithium ion (Li-ion) which provideigh energy density (Scrosati & Grache, 2010). However, Li-ion bat-eries have problems with the sensitivity to the overload that mayeduce its life cycle. Other options under study include fuel cellsnd mechanically rechargeable batteries. In any case, it should beoted that these options do not offer the same amount of energy

s fossil fuels do (∼40 MJ/kg for fossil fuels versus 1.5–0.25 MJ/kgor fuel cells and advanced batteries respectively (Linden & Reddy,002)).

∗ Corresponding author. Tel.: +52 55 59504074; fax: +52 55 59504074.E-mail address: [email protected] (A. Flores-Tlacuahuac).

098-1354/$ – see front matter © 2011 Elsevier Ltd. All rights reserved.oi:10.1016/j.compchemeng.2011.05.007

© 2011 Elsevier Ltd. All rights reserved.

Although EVs were designed and built in the past century, inter-nal combustion (IC) vehicles were the best choice because of low oilprices. Currently there is no power source that equals the capacityof acceleration (power) and range (energy) of the IC engine. Never-theless, researches are conducted in the hope that a robust system,capable of meeting reasonable acceleration and range capabilities,can displace the IC engine (Cairns & Albertus, 2010). Fuel cells andLi-ion batteries are good alternatives for use in electric vehicles.

Lithium ion batteries where introduced commercially in the1990s for portable applications such as camcorders and cameras(Ozawa, 1994). They offered a greater capacity than nickel (Ni) bat-teries for portable devices but their cost was prohibitive for largerapplication (20,000 kWh) (Cairns & Albertus, 2010). The lithiumion batteries use graphite as the anode while the cathode is acobalt oxide, the two electrodes are porous. The electrolyte is amixture of organic solvents and a lithium salt. Different organiccompounds are used as electrolytes because lithium is highly reac-tive in water (Linden & Reddy, 2002). Most common electrolytesinclude: Dimethyl carbonate (DMC), ethyl methyl carbonate (EMC)and diethyl carbonate (DEC) (Huggins, 2009). In terms of elec-trodes, several compounds have been proposed since the originalLiC6–LiCoO2 (Broussely & Archdale, 2004; Whittingham, 2004).The use of graphite as the anode material is universally accepted(Chung, Jun, Lee, & Kim, 1999; Michio, 1996; Tran, Spellman, Pekala,Goldberger, & Kinoshita, 1995), while the following compounds:LiNiO2, LiMn2O4, LiMn1/3Ni1/3Co1/3O2 and LiFePO4 have been used

as cathodes (Delmas & Saadoune, 1992; Seung-Taek et al., 2008;Shina, Basak, Kerr, & Cairns, 2008; Wang, Bradhurst, Dou, & Liu,1998). Li-ion batteries employ very thin (10–30 �m), microporousfilms to electrically isolate the negative and positive electrodes.

Page 2

1938 E. Martínez-Rosas et al. / Computers and Chem

Nomenclature

List of symbolsa specific interfacial area, m2/m3

c concentration of Li ions in the electrolyte, mol/m3

Cs concentration of Li in the electrode, mol/m3

D diffusion coefficient of the salt in the electrolyte,m2/s

Ds diffusion coefficient of Li in the electrode, m2/sF Faraday’s constant, 96,487 C/molf± activity of the salt in the electrolyte, mol/m3

i1 current density in the electrode, A/m2

i2 current density in the electrolyte phase, A/m2

I total current density, A/m2

jLi pore wall flux of Li ions, mol/cm2 sN mass transport flux, mol/m2

R reaction term of the mass balance equation,mol/m3 s

R gas constant, 8.314 J/mol KRf film resistance, � m2

Rs radius of electrode spherical particle, mT temperature of the systemt time of operation, sto+ transport number of the positive ion

u0 open circuit voltage, V

Greekı region widthε porosity of the composite electrodes� over-potential, V� ionic conductivity of electrolyte, S/m� electronic conductivity of solid matrix, S/m

Indexa anodes separatorc cathodeT maximum concentration in intercalation material

Apcpl2el

rtd1latpicN

b1s

0 initial condition

ll commercially available liquid electrolyte cells use microporousolyolefin materials as they offer excellent mechanical properties,hemical stability and satisfactory cost. Currently the microporousolyolefin materials are made of polyethylene, polypropylene or

aminate of polyethylene and polypropylene (Linden & Reddy,002). The separator operates as the solvent for a lithium salt. Prop-rties for different salts and polymer combinations are cited in theiterature (Doyle, Fuller, & Newman, 1993).

Lithium ion cells operation is based on a process known asocking-chair, owing its name to the extraction and intercala-ion of lithium ions from the electrodes that correspond to theonation and acceptance of electrons respectively (Doyle et al.,993). The overall charge/discharge cycle is then seen as a swing of

ithium ions between electrodes. Intercalation/extraction presentsdvantages and disadvantages compared to the traditional oxida-ion/reduction scheme. Intercalation and extraction are topotacticrocesses of moderate to high reversibility, that present a change

n the volume of the host, depending of its nature, which overycling life leads to degradation of the matrix (Liaw, Jungst, &agasubramanian, 2011).

The use of LiMn2O4 has the advantages of low cost, good sta-ility and adequate specific capacity (120 mAh/g, compared to40 mAh/g that provides the LiCoO2). In this work, we have cho-en to simulate the LiC6–LiMn2O4 battery for two main reasons.

ical Engineering 35 (2011) 1937– 1948

First, because of its low cost and their appropriate characteristicsmaking it a promising alternative for using in electric vehicles. Sec-ondly, because there are published data for different parametersneeded to develop the mathematical model.

A simplified model for this type of battery has been solvedanalytically by partial differential and algebraic equations, butthis model is only valid for a limited set of conditions (Baker, &Verbrugge, 2011). There are also some proposed equivalent cir-cuits for the description of the electrical variables of charge anddischarge of the battery but the accuracy of these variables arenot robust to describe the electrochemical phenomena (Liaw et al.,2005). On the other hand, different numerical simulation methodshave been used to solve models of lithium ion batteries. Approachesto this problem include the linearization of the system using theBAND program (Doyle & Newman, 1996), finite element (Tang,Albertus, & Newmana, 2009) and Newton–Krylov algorithms (Wu,Srinvasan, Xu, & Wang, 2002). With the use of numerical methodsit has been possible to solve different models related to batteries.Among these we mention the optimization of the thickness of theelectrode (Fuller, Doyle, & Newman, 1994), the performance of bat-teries with electrodes of complex geometry (Garca & Chiang, 2007)and determining the capacity and discharge rate of batteries (Ning& Popov, 2004.

The contribution of this work lies in formulating and experimen-tally validating a mathematical model suitable for approaching thedesign, optimization and control of Li-ion batteries. We used a mod-ified version of the model presented by Doyle and Newman (1996)for the lithium ion battery with chemistry LiC6–LiMn2O4. Modifi-cations were made in the boundary conditions to determine thepotential of both electrodes and the electrolyte. The original modelused flux boundary conditions whereas we transformed some ofthese conditions to specified values, improving the convergence ofthe model. The transformation was made based on the fact thatthe current supplied by each portion of the anode should sum upto the total current. This also applies to the accepted current inthe cathode side. We also included a consistent initialization strat-egy, similar to the one proposed in Boovaragavan and Subramanian(2007), to promote the quick start of the solver used for getting thedynamic and spatial behavior of the battery. The model was solvedby the numerical method of lines (MOL) (Schiesser, 1991), which iseasier to deploy than the finite element or finite volume especiallyfor systems with regular geometries as the one addressed in thepresent work.

In the first section we present the mathematical model in detailand modifications from the original model (Doyle & Newman,1996). Secondly we deal with the reduced version of the modelassuming constant and null terms for the LiC6–LiMn2O4 chem-istry. The next section is the validation of the model where wepresent the comparison between simulated and experimental datafrom previous works (Doyle & Newman, 1996). Finally we presentsome examples of application which include: (1) variation of designparameters for the optimization of battery performance, (2) thedynamic response to a varying rate of discharge and (3) the dis-charge simulation of different capacity batteries connected inparallel.

2. Mathematical model

The electrochemical system under study has the chemistryLixC6–LiyMn2O4. In this type of batteries, electricity is produced bythe simultaneous insertion/deinsertion of lithium ion in the porous

cathode and anode respectively. Deinsertion process releases elec-trons that go into the electric circuit, on the other side, theseelectrons are accepted to cause insertion of lithium ions. Insidethe battery, positively charged lithium ions move from the anode

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E. Martínez-Rosas et al. / Computers and Chem

wrcaFaovt

eimi

tfl

∇

io

∇

i

ts

∇

∇

fltctto

ions in the electrolyte:

Fig. 1. LixC6−− LiyMn2O4 battery.

here they where released towards the cathode where electronseturn closing the electric circuit, this process is known as “rocking-hair” because of the overall effect of lithium ions, which move fromnode to cathode during discharge and backwards upon charging.ig. 1 shows a scheme of this system. The model assumes that thective material of both electrodes is spherical and it is supportedn an inert material. The electrolyte is LiPF6 in a EC/DMC 2:1 sol-ent. There is an insertion/deinsertion phenomenon occurring athe electrodes as the charge and discharge processes occur.

The Faradic process deals with everything concerned to the het-rogeneous oxidation-reduction reactions and the electric chargesnvolved in this. The formulas governing this process are the nor-

al and modified Ohm’s law and the insertion/deinsertion of Li ionsnto the active materials.

Faraday’s laws express the relationship between the inser-ion/deinsertion of Li-ions into electrodes with the electrical chargeow (all the variables are described in Nomenclature section).

· i2 = FajLi (1)

Based on the law of conservation of charge, current densitys preserved between the electrode and electrolyte, therefore webtain the following expression:

· (i1 + i2) = 0 (2)

1 + i2 = I (3)

his provides a relation between Li-ions flux with both current den-ities:

· i1 = −FajLi (4)

· i2 = FajLi (5)

The model assumes that both electrodes are porous and non-at surfaces such as in most common batteries. As mentioned,he model assumes that the active materials are spherical parti-les supported in the filling material. Moreover, we assume that

he solid phase diffusion coefficient is independent of concentra-ion. However, there are some cases wherein they are functionf concentration. Therefore, the Li-ion concentration in the active

ical Engineering 35 (2011) 1937– 1948 1939

material is simulated via molecular diffusion in spherical coordi-nates according to the following equation:

∂Cs

∂t= Ds

[∂2Cs

∂r2+ 2

r

∂Cs

∂r

](6)

Charge and discharge of the battery is only possible in a closedcircuit. In the electrodes and the external electrical circuit, electri-cal current is generated by the flow of electrons, whereas in theelectrolyte, the electric current is due to ions flow. In both cases,the electric current is the same. The electrode potential (�1) iscalculated with Ohm’s law:

i1 = −�∇�1 (7)

In the electrolyte, under the effect of concentration gradients,Ohm’s law does not describe accurately the ionic transport ofcharges. A modification of Ohm’s law can be used instead (Doyle& Newman, 1996):

�∇�2 = −i2 − 2�RT

F

(1 + ∂ lnf±

∂ ln c

)(1 − to

+)∇ln c (8)

This equation relates the potential with the local current density.Nevertheless, it is more convenient to relate the potential to the Liion flux. This can be done by differentiating (in x) the equation ofcurrent density:

∇ · i2 = ∇ ·(

−�∇�2 − 2�RT

F

(1 + ∂ lnf±

∂ ln c

)(1 − to

+ )∇ln c

)(9)

Finally we substitute the current density gradient for its equiv-alent in terms of Li ion flux according to Faraday’s laws, leading tothe following equations:

−�∇2�1 = −FajLi (10)

−�∇2�2 = FajLi + 2�RT

F

(1 + ∂ lnf±

∂ ln c

)(1 − to

+ )∇2ln c (11)

On the other hand, when the heterogeneous reaction (elec-trode/electrolyte interface) is controlled by the activation, thekinetics can be described by the Butler–Vomer equation:

jn = io

{exp

(F

2RT(� − uo)

)− exp

( −F

2RT(� − uo)

)}(12)

2.1. Mass transport process

Electrolyte is present in the three battery regions (anode,separator and cathode). The mass balance is deduced by the con-centrated solution theory (Newman & Thomas-Alyea, 2004) andincludes the migration and diffusion effects. The total flux is thusrepresented by:

NTotal+ = −D

(1 − d ln co

d ln c

)· ∇c + i2to

+F

(13)

On the other hand, the reaction term that occurs at the elec-trodes is expressed as:

R = a

+(1 − to

+ )jLi (14)

The overall mass balance is:

∂c

∂t= −∇ · N + R (15)

If we substitute the previous definitions in the overall mass bal-ance we have the general formula for the concentration of lithium

∂c

∂t= −∇ ·

(−D

(1 − d ln co

d ln c

)∇c + i2to

+F

)+ a

+(1 − to

+ )jLi (16)

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1940 E. Martínez-Rosas et al. / Computers and Chem

Table 1Constant and null terms (Doyle & Newman, 1996).

Term Status Explanation

to+ Constant It has been proved that the transport number is a

function of the Li-ions concentrationin the electrolyte, however there is not reliabledata for these dependence

n Constant These specific battery and in general all the basedin lithium, are restricted to thetransference of one electron per atom of reacting Li

+ Constant The electrolyte used in this battery is dissociated ina 1:1 relation

s+ Constant In both anodic and cathodic reactions the Licoefficient is 1

d(ln c0)d(ln c) Null This term is neglected due to the lack of

experimental data

otta

2

a

−

−

2

2

ms&Tg

•

∂ lnf±∂ ln c

Null This term is neglected due to the lack ofexperimental data

∂c

∂t= D

(1 − d ln co

d ln c

)∇2c − i2∇ · to

+F

+ a

+(1 − to

+ )jLi (17)

The previous formula is only valid for the continuous regionf the battery i.e. the separator. In order to make it applicable tohe electrode regions we have to take into account the porosity ofhe medium. In this way we obtain a more general formula that ispplicable to all three regions (Doyle & Newman, 1996):

∂c

∂t= D

(1 − d ln co

d ln c

)∇2c − i2∇ · to

+F

+ a

+(1 − to

+ )jLi (18)

.2. Governing equations

Li-ion diffusion in the electrolyte

∂c

∂t= D

(1 − d ln co

d ln c

)∇2c − i2∇ · to

+F

+ a

+(1 − to

+ )jLi (19)

Faraday’s laws

jLi = 1F

∇ · i2 (20)

Ohm’s law (electrode)

�∇2�1 = −FajLi (21)

Modified Ohm’s law (electrolyte)

�∇2�2 = FajLi + 2�RT

F

(1 + ∂ lnf±

∂ ln c

)(1 − to

+)∇2ln c (22)

Li ion diffusion in the active material

∂Cs

∂t= Ds

[∂2Cs

∂r2+ 2

r

∂Cs

∂r

](23)

.3. Simplification

.3.1. Constant termsThe presented model is general and thorough. However, the

odel can be simplified by means of the consideration of con-tant or null terms related to the LixC6−− LiyMn2O4 system (Doyle

Newman, 1996). An explanation of these terms is presented inable 1. Considering the simplification of terms, we can rewrite theoverning equations as follows:

Li ion diffusion in the electrolyte

∂c

∂t= D∇2c + a(1 − to

+ )jLi (24)

ical Engineering 35 (2011) 1937– 1948

• Modified Ohm’s law (electrolyte)

−�∇2�2 = FajLi + 2�RT

F

(1 − to

+)∇2ln c (25)

Eqs. (19)–(23) can be rewritten as follows assuming a singledimension (1D). (For the active material, spherical diffusion occursonly in the r dimension)

• Li-ion diffusion in the electrolyte

∂c

∂t= D

∂2c

∂x2+ a(1 − to

+ )jLi (26)

• Faraday’s laws

1F

di2

dx= ajLi (27)

• Ohm’s law (electrode)

−�∂2�1

∂x2= −FajLi (28)

• Modified Ohm’s law (electrolyte)

−�∂2�2

∂x2= FajLi + 2�RT

F

(1 − to

+) ∂2ln c

∂x2(29)

• Li-ion diffusion in the active material

∂Cs

∂t= Ds

[∂2Cs

∂r2+ 2

r

∂Cs

∂r

](30)

2.3.2. Dimensionless equationsThis model was made dimensionless in the x variable and also

in the r coordinate of the spherical particles. The variable changeobeys to the following formulas:

x∗ = x

ı(31)

r∗ = r

Rs(32)

As an example we take the Ohm’s law:

−�∂2�1

∂x2= −FajLi (33)

Applying the dimensionless formula we get:

−�∂2�1

ı2 ∂x∗ 2= −FajLi (34)

we can adopt the convention of defining x∗ as x to avoid visuallyovercharging the equations. In this way we have:

−�∂2�1

ı2 ∂x2= −FajLi (35)

When we proceed in the same way for all the equations we have:

• Li-ion diffusion in the electrolyte

∂c

∂t= D

ı2

∂2c

∂x2+ a(1 − to

+ )jLi (36)

• Faraday’s laws

1ıF

di2

dx= ajLi (37)

• Ohm’s law (electrode)

−�

ı2

∂2�1

∂x2= −FajLi (38)

• Modified Ohm’s law (electrolyte)

−�

ı2

∂2�2

∂x2= FajLi + 2�RT

ı2F

(1 − to

+) ∂2ln c

∂x2(39)

Page 5

Chem

•

2

rt

•

•

•

•

E. Martínez-Rosas et al. / Computers and

Li-ion diffusion in the active material

∂Cs

∂t= Ds

R2s

[∂2Cs

∂r2+ 2

r

∂Cs

∂r

](40)

.4. Boundary conditions

This model presents three regions in the x dimension and twoegions in the r sub-dimension. In the next section we shall discusshe boundary conditions of all equations between regions.

Li-ion diffusion in the electrolyte Diffusion takes place in the threeregions of the cell (anode, separator and cathode as seen in Fig. 1).Current collectors present an impermeable wall to the electrolyte,therefore the Li-ion flux is null in these boundaries. The interfacesof the three regions show a continuity condition that is expressedas an equality of the mass fluxes on both sides of the interface.Mathematically we have:

In the current collectors

∂c

∂x

∣∣∣∣x=0

= 0 (41)

∂c

∂x

∣∣∣∣x=L

= 0 (42)

At the interfaces:

∂c

∂x

∣∣∣∣x+ =ıa

= ∂c

∂x

∣∣∣∣x− =ıa

(43)

∂c

∂x

∣∣∣∣x+ =ıa+ıs

= ∂c

∂x

∣∣∣∣x− =ıa+ıs

(44)

Li-ion diffusion in the active material

∂Cs

∂r

∣∣∣∣r=0

= 0 (45)

∂Cs

∂r

∣∣∣∣r=Rs

= − jLi

Ds(46)

However there is a problem in applying the boundary conditionat the sphere center, this is due to the term 1/r that becomes unde-termined when r = 0. To avoid this problem we apply L’Hopitalrule which leads us to:

∂Cs

∂t=

⎧⎪⎨⎪⎩

3Ds

R2s

∂2Cs

∂r2if r = 0

Ds

R2s

[∂2Cs

∂r2+ 2

r

∂Cs

∂r

]if r > 0

∂cs

∂r= 0

∂cs

∂r= − jLi

Ds

Faradic processFaraday’s laws relate current and mass. The majority of bib-

liography indicates that the conditions at the boundary ofthe electrodes are of null potential flux (Neumann problem,(Schiesser, 1991). This imposes a difficulty because the problemdoes not possess a unique solution, rather it presents a family ofsolutions. To avoid this problem we can transform the Neumanncondition to a Dirichlet one. For this purpose we can use Faraday’slaw in its integrated form:

1F

di2

dx= ajLi (47)

i2|b − i2|a =∫ b

a

FajLidx (48)

Ohm’s law

ical Engineering 35 (2011) 1937– 1948 1941

Null potential flux conditions (Doyle & Newman, 1996) arenormally used. However to promote better convergence in thesolution of the model, we decided to transform the Neumannconditions to Dirichlet ones.

• ElectrolyteThe potential in the electrolyte is solved for the three regions;

the potential at the negative electrode can be fixed to grant theaccomplishment of Faraday’s laws, this is expressed as:

�2 = �2, 0 (49)

provided that:

i2|b − i2|a =∫ b

a

FajLi(�2, 0)dx (50)

Current density is not calculated in this model. However, it iseasily demonstrated that in the separator region it is constant.Taking into account that in the electrolyte region, jLi equals zero,Eq. (47) reduces to:

di2

dx= 0 (51)

i2 = constant (52)

As Ohm’s law is conservative we have:

I = i1 + i2 (53)

Combining the current density conservation (Eq. (3)) and thefact that in the separator there is only electrolyte (which impliesthat i1 = 0) we have the following:

i2 = I (54)

Following the same logic, we have that in the current collectorsall the current flows trough the solid phase, this is expressed asi1 = I or in terms of i2; i2 = 0.

Considering the negative electrode the condition reads as fol-lows:∫ ıs

0

FajLidx = i2|x=ıs − i2|x=0 (55)

under the latter deductions it becomes:∫ ıs

0

FajLidx = I (56)

For the positive electrode, we keep Neumann null flux condi-tion.

• ElectrodesThe null flux can also be used as a boundary condition in the

separator and the Ohm’s law at the current collectors (Doyle &Newman, 1996). However it is more convenient to adopt theconvention of the electrode potential (�1) equal to zero at theinterface between the current collector and the negative elec-trode, by doing this the potential of the positive electrode equalsthe cell potential; we kept the other conditions unchanged. Themathematical expressions for this boundary conditions are:

�1|x=0 = 0 (57)

d�1

dx|x=ıs = 0 (58)

For the positive electrode we assume a similar condition as the

ones used for the electrolyte potential and kept Ohm’s law at thecurrent collector potential (Doyle & Newman, 1996):

�1|x=ıa+ıs = �1, 0 (59)

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1942 E. Martínez-Rosas et al. / Computers and Chemical Engineering 35 (2011) 1937– 1948

Table 2Summary of boundary conditions.

Equation x = 0 x = ıa x = ıa + ıs x = L

Li+ diffusion a ∂c∂x

= 0 Continuity Continuity ∂c∂x

= 0

Ohm’s law a �2 = �2, 0 Continuity Continuity ∂�2∂x

= 0

Ohm’s law b �1 = 0 d�1dx

= 0 �1 = �1, 0d�1dx

= − I�

Equation r = 0 r = Rs

Li+ diffusion b ∂cs = 0 ∂cs = − jLi

3

aodc

4

tapssifcds

acattfsabutaa(

dssnanA

0 0.5 1 1.52

2.5

3

3.5

4

4.5

Capacity (mAh)

Cel

l vol

tage

(V

)

0.1 C0.5 C1.0 C2.0 C3.0 C4.0 C0.1 C0.5 C1.0 C2.0 C3.0 C4.0 C

∂r ∂r Ds

a Electrolyte.b Electrodes.

provided that:∫ L

ıa+ıs

FajLi(�1, 0)dx = −I (60)

d�1

dx

∣∣∣x=L

= − I

�(61)

Table 2 presents a summary of all boundary conditions withinthe anode, separator and cathode regions.

. Model summary

The simplified and dimensionless model and respective bound-ry conditions are presented in Table 3. It constitutes a systemf Partial Differential and Algebraic Equations. The boundary con-itions are also presented. Notice that only c and �2 feature aontinuity condition, but �1 does not.

. Discretization

There are different approaches to solve partial differential equa-ion systems. Probably the best well known are finite differencesnd finite volume. The basic idea behind these methods consists inroviding an algebraic approximation to the differential terms inome or all the variables. The method of lines (MOL), lies under theame basic principle of substituting the spatial variables and keep-ng the first derivative in time, and rely on sophisticated solversor the ordinary differential equations system obtained after dis-retization. In more specific cases, a system of partial and ordinaryifferential equations becomes a differential and algebraic equationystem (DAEs).

Discretization is made by means of Taylor approximation and series of algebraic manipulation. There are many types of dis-retization depending mainly on the number of discretized pointsnd the degree of the derivative term; there are also some specialypes of discretization, for example the ones used to avoid fluctua-ions in the convective term. Because partial differential equationseaturing convective and diffusive terms tend to be hard to beolved by numerical integration techniques, different manners ofpproximating the embedded first and second derivatives shoulde used. Therefore, in this work two discretization routines weresed: dss010 (a 11 point, first order derivative discretization rou-ine) and dss044 (a three point, second order derivative which canccept Dirichlet or Neumann type boundary conditions for the leftnd right boundaries). More information is available in Schiesser1991).

The diffusion into the porous electrodes presents a pseudo-2-imension problem, accounting for the spatial coordinate x and thepherical coordinate r of the active material sphere. It was neces-ary a gathering of variables leading to two matrices of dimension

pa× npp and npc× npp for each electrode, where npa and npc

re the number of points of discretization in each electrode andpp is the number of discretized points in the spherical particles.fter some trials we came to the conclusion that 11 points were

Fig. 2. Comparison between simulated (continuous line) and experimental (discon-tinuous points) data for different values of the discharge rate.

enough for representing the system behavior for the three regions.The resulting system of DAEs was solved using the ode15i solverembedded in Matlab, which is an implicit solver. Other solverswhere tested but this specific problem presented a considerabledependence between variables and could only be initialized withimplicit solvers. The previous mentioned dependence betweenvariables makes it hard for some solvers to achieve a consistent ini-tialization, the approach we adopted was modified from previousworks (Boovaragavan & Subramanian, 2011). Although each vari-able is solidly interconnected to the others, the ones that presentthe highest dependence are jLi, �1 and �2. We first solved thesethree variables in their algebraic form for t = 0 assuming all theconcentrations equal to their initial values provided by the user,and then feed this vector corresponding to y0 along with a vectorof zeros with equal size that corresponds to y′

0. The equations cor-responding to the three more dependent variables will be solvednumerous times as time moves forward in the solver, but now withvarying concentrations in the electrodes and electrolyte. The com-plete Matlab code can be consulted elsewhere (Martinez-Rosas,2010).

5. Experimental validation

The results obtained from Matlab simulations were comparedversus experimental data (Doyle & Newman, 1996). The agree-ment between simulated and experimental data is presented inFig. 2. A film resistance coefficient was proposed in the same workto promote a better concordance and somehow explain the for-mation of undesirable compounds in the electrodes. The samecoefficient value (Rf = 900�/cm2) was used in the anode side. Theover-potential equation is modified as follows:

�a = �1 − �2 − uo − FjLiRf

Fig. 2 presents the comparative values of experimental and sim-ulated data for different discharge rates in C notation (1.0 C is thecurrent that exhaust the battery to its cut-off voltage in 1 h). Per-centage errors are presented in Fig. 3. It can be noticed that themajority of these errors lies between ±5% of the real value, whichis considered an acceptable error.

6. Base simulation

In this part we compare the results obtained using our modelagainst other published results (Doyle et al., 1993). We claim that

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E. Martínez-Rosas et al. / Computers and Chemical Engineering 35 (2011) 1937– 1948 1943

Table 3Summary of the mathematical model.

Governing equation Boundary condition

Anode x = 0 x = ıa

∂c∂t

= Dı2

∂2c∂x2 + a(1 − to

+ )jLi∂c∂x

= 0 Continuity

− �ı2

∂2�1∂x2 = −FajLi �1 = 0 d�1

dx= 0

− �ı2

∂2�2∂x2 = FajLi + 2�RT

ı2F

(1 − to

+

)∂2 ln c∂x2 �2 = �2, 0 Continuity

Separator x = ıa x = ıa + ıs∂c∂t

= Dı2

∂2c∂x2 Continuity Continuity

− �ı2

∂2�2∂x2 = FajLi + 2�RT

ı2F

(1 − to

+

)∂2 ln c∂x2 Continuity Continuity

Cathode x = ıa + ıs x = L

∂c∂t

= Dı2

∂2c∂x2 + a(1 − to

+ )jLi Continuity ∂c∂x

= 0

− �ı2

∂2�1∂x2 = −FajLi �1 = �1, 0

d�1dx

= − I�

− �ı2

∂2�2∂x2 = FajLi + 2�RT

ı2F

(1 − to

+

)∂2 ln c∂x2 Continuity �2

∂x= 0

Active material (anode and cathode) r = 0 r = Rs

∂Cs

∂t=

⎧⎨⎩

3Ds

R2s

∂2Cs

∂r2if r = 0

Ds

R2s

[∂2Cs

∂r2+ 2

r

∂Cs

∂r

]if r > 0

∂cs∂r

= 0 ∂cs∂r

= − jLiDs

Auxiliary equations { (F

2RT(�

) ( )}1, 0)dx

ofwkp

•••••

tiDfp

a coincidence, since we previously fixed the anodic potential at itsinterface with the current collector to zero.

Potential values of the electrolyte are shown in Fig. 5. Electrolytepotential values grow in absolute value as discharge occurs. This

Table 4Parameters of the base simulation (all from Doyle and Newman (1996)).

Parameters Description Value

Operation variablesI Discharge current (1.0 C) 17.5 A/m2

T Temperature 298 KDesign variablesCe,0 Initial concentration of LiPF6 in the

electrolyte2000 mol/m3

Csa,0 Initial concentration of Li ions in the anode 14,870 mol/m3

Csc,0 Initial concentration of Li ions in the 3900 mol/m3

� = �1 − �2 − uo jn = io exp∫ ıs

0FajLi(�2, 0)dx = I

∫ L

ıa+ısFajLi(�

ur results are in good agreement with expected results and outper-orm previous reported solutions. The base case of the simulationas the one simulated by Doyle et al. (1993). The thermodynamic,

inetic and transport parameters as well as the design variables areresented in Table 4. The variables computed are:

Anodic and cathodic Potentials (�a,c1 ).

Electrolyte potential (three regions) (�a,s,c2 ).

Li-ion flux in the electrodes (ja,cLi

).Li-ion concentration in the electrolyte (Ca,s,c

e ).Li-ion concentration in the both active materials (Ca,c

s ).

The electrode potentials are presented in Fig. 4. It can be noticedhat the electrodes are practically equipotential surfaces. This is ofnterest given the fact that they are not metallic plates but porous 3-

materials. Also the anodic potential remains almost constant. Thisact allows us to calculate the cell potential (Vcell) as the cathodicotential in the interface with the current collector. This fact is not

0 20 40 60 80 100−15

−10

−5

0

5

10

15

Capacity (%)

Err

or (

%)

0.1 C0.5 C1.0 C2.0 C3.0 C4.0 C

Fig. 3. Percentage errors of the simulation

−Uo) − exp −F2RT(�−Uo)

= −I

cathodeıa Anode width 100 �mıc Cathode width 174 �mıs Separator width 52 �mεa Anode porosity 0.357εc Cathode porosity 0.444Kinetic, thermodynamic and transport parametersC a

TMax. conc. of Li ions in the anode 26,390 mol/m3

C cT

Max. conc. of Li ions in the cathode 22,860 mol/m3

Das Particle diffusivity coefficient in the anode 3.9 × 10−14 m2/s

Dcs Particle diffusivity coefficient in the

cathode1.0 × 10−13 m2/s

D Electrolyte diffusivity coefficient 7.5× 10−11 m2/ska

r Reaction rate constant in the anode 2 × 10−11 m5.5/mol0.5 skc

r Reaction rate constant in the cathode 2 × 10−11 m5.5/mol0.5 sRa

p Radius of the active material sphere in theanode

12.5 �m

Rcp Radius of the active material sphere in the

cathode8.5 �m

to+ Transport number 0.363

�a Conductivity in the anode 100 S/m2

�c Conductivity in the cathode 3.8 S/m2

Description FormulaCalculated parametersaa,c Specific area of the electrodes (m2/m3) 3 εs

Rp

εas Volume fraction of the solid phase in the

anode1 − εa − 0.172

εcs Volume fraction of the solid phase in the

cathode1 − εc − 0.259

Page 8

1944 E. Martínez-Rosas et al. / Computers and Chemical Engineering 35 (2011) 1937– 1948

0 0.5 1 1.5 2 2.5 3−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Ele

ctro

de p

oten

tial (

V)

t o 1/4 t f 1/2 t f 3/4 t f

t f

etto

iatT(de(

Fftttcmav

F

0 0.5 1 1.5 2 2.5 3−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

−5

Dimensionless width

Por

e w

all f

lux

of L

i+ io

ns (

mol

/m2 s)

t o 1/4 t f 1/2 t f 3/4 t f

t f

Fig. 6. Li-ion flux in the electrodes (jLi).

0 0.5 1 1.5 2 2.5 3500

1000

1500

2000

2500

3000

Li+ io

ns c

once

ntra

tion

in th

e el

ectr

olyt

e(m

ol/m

3 ) t o 1/4 t f 1/2 t f 3/4 t f

t f

Dimensionless width

Fig. 4. Electrode potentials (�1), anodic (left) and cathodic (right).

ffect can be attributed mainly to the emergence of the concentra-ion gradient. If we had chosen to use the dilute theory instead ofhe concentrated theory, this phenomenon would not have beenbserved adequately (Doyle & Newman, 1996).

The pore-wall flux of lithium ions in the electrodes is shownn Fig. 6. In the anode side this variable does not change consider-bly. For the cathode we can see some quasi-stability followed by aransition period after which quasi-stability is achieved once more.his variable is considered to be the more important of the modelBoovaragavan & Subramanian, 2011), due to the level of inter-ependence with respect to the other variables both implicit andxplicit (jLi = (c, cs, �)) and is also present in virtually all equations9 out of 10).

Lithium-ions concentration in the electrolyte is presented inig. 7. The results obtained in this work are considerably differentrom those reported by Doyle and Newman (1996). Nevertheless,he ones presented in this work are more realistic somehow thathe latter, which present zones of constant concentration along theime, while our solution presents a continuous evolution withoutonstant points. This phenomenon can be explained by the treat-ent given to the concentration of lithium ions in the electrolyte

s a continuum across the three regions. For this effect we considerectors of the specific parameters (a, Deff, εandıi) of each region.

Concentration of lithium ions in the electrodes is presented inig. 8. The zig-zag shape or discontinuities represents two pseudo-

0 0.5 1 1.5 2 2.5 3−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

Dimensionless width

Ele

ctro

lyte

pot

entia

l (V

)

t o 1/4 t f 1/2 t f 3/4 t f

t f

Fig. 5. Electrolyte potential (�2).

Dimensionless width

Fig. 7. Li-ion concentration in the electrolyte (Ce).

dimensions; r dimension of spherical particles, and x dimension

corresponding to the cells width. In this way, each section ofthe curve represents the behavior inward the particle of a sin-gle x discretized point. Figure shows 121 discretized points (11

0 0.5 1 1.5 2 2.5 30

2000

4000

6000

8000

10000

12000

14000

16000

18000

Dimensionless width

Inse

tred

Li+

ions

con

cent

ratio

n (m

ol/m

3 )

t o 1/4 t f 1/2 t f 3/4 t f

t f

Fig. 8. Li ions concentration in the electrodes (Cs).

Page 9

E. Martínez-Rosas et al. / Computers and Chemical Engineering 35 (2011) 1937– 1948 1945

0 0.5 1 1.5 2 2.5 3−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Dimensionless width

Ele

ctro

des´

ove

rpot

entia

l (V

)

t o 1/4 t f 1/2 t f 3/4 t f

t f

pMtIlcbtsscAatvi1toFs

7

iebidtdce

7

iw

•••

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.02

0.04

0.06

0.08

0.1

0.12

0.14

max

(ηa* )

Ce,0

= 25%

Ce,0

= 50%

Ce,0

= 100% − Base case

Ce,0

= 125%

Ce,0

= 150%

Fig. 9. Over-potential (�).

oints in x-direction × 11 points in r-direction), for each electrode.uch of the computational effort is dedicated to the solution of

he electrodes concentration (242 variables of the 354 calculated).nformation from insertion and extraction is crucial to the calcu-ation of the performance of the cell, and provides the degree ofharge (DOC), that indicates the amount of available energy in theattery, that is difficult to estimate in any other way. Althoughhere are algebraic and differential approximations to the diffu-ion equation (Zhang & White, 2007) that would eliminate the rub-dimension, these approximations introduce a significant error,onsidering the amplification of the error for the whole model.pplying exact equations, instead of approximations requires a bal-nce between discretized points in r and in x directions. Previousests with increased number of r points, resulted in a non con-ergence problem of the calculations mainly because of stiffnessssues. Simplified attempts, increasing the number of points from1 to 20 did not alter the results significantly. This demonstrateshat MOL efficiency is achieved even with few points. Electrodever-potentials (calculated in the post-processing) is presented inig. 9. This variable was not calculated explicitly because is nottrictly necessary to calculate other variables.

. Case studies

After carrying out validation of our model against both exper-mental information and published results, we use the model toxplore some potential and practical applications of Li-ion dynamicattery models. First we present some sensibility results by chang-

ng some important design parameters. Then, we explore theynamic performance of the Li-ion battery by carrying out someests whose aim is to demonstrate the battery performance underifferent driving scenarios. Finally, because battery performancean be improved by using different battery arrangements, wexplore a parallel battery connection.

.1. Simulation of the variation in the design parameters

The performance of different Li-ion batteries is studied by vary-ng the design parameters. The parameters chosen for this analysis

here:

Ce,0: initial concentration of the salt in the electrolyte,ıa,c: electrodes width,εa,c: electrodes porosity

Capacity (mAh)

Fig. 10. Variation of the parameter Ce,0.

For the simulations the next scenario was considered. The nom-inal capacity of the battery was found. Each battery was dischargedat a 1.0 C-rate to a cutoff voltage of 2.6 V. The maximum dimen-sionless anodic over-potential was used as a measure of stress ofthe battery according to the following equation:

�∗a = �a

V

• Li-ion initial concentration in the electrolyteThe concentration of the Li-ions in the electrolyte affects its

conductivity (�) in a non-linear way. The parameter Ce,0 was var-ied in a range between 25 and 150% of its base value. Fig. 10shows the over-potential and capacity of batteries with differentinitial concentration of the Li ion in the electrolyte. It can be seenthat a battery with an initial condition under 2000 mol/m3 givesrise to a low capacity battery (∼0.6 mAh/m2). On the other hand,above this concentration the increase in the capacity of the cell isminor (1.65–1.8 mAh/m2). Then, we can conclude that this vari-able is susceptible of optimization, especially when consideringthat the electrolyte salt represents a considerable percentage ofthe battery manufacture cost.

• Electrodes widthThe electrodes width determines two main factors in the oper-

ation of the battery: the amount of active material and theresistance to the mass transport. The width of both electrodeswas varied uniformly in a range between 80 to 120 % of the basevalue. The results are shown in Fig. 11. In this case we founda slight increase in battery capacity when the electrode widthincreases.

• Electrodes porosityThe porosity of the electrodes affects the effective conductivity

of the electrolyte and is also a resistance to the mass transferenceprocess. The variation of these variables was made in the samerange as that of the widths. Results are presented in Fig. 12. As therelation is non-linear, there is an opportunity for optimization ofthis parameter.

7.2. Simulation of the dynamic operation of the battery

Dynamic operation refers to the changes in operating condi-tions, being these the discharge rate and temperature. The model

does not consider the dynamic changes in temperature, so that onlychanges in current are analyzed.

Electrical vehicles the same as internal combustion engines,are subject to different driving regimes. The broader classification

Page 10

1946 E. Martínez-Rosas et al. / Computers and Chemical Engineering 35 (2011) 1937– 1948

0 0.5 1 1.5 2 2.50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Capacity (mAh)

max

(ηa* )

δa,c

= 8 0%

δa,c

= 9 0%

δa,c

= 100% − Base case

δa,c

= 1 10%

δa,c

= 1 20%

Fig. 11. Variation of the parameter ıa,c .

0 0.5 1 1.5 20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

max

(ηa* )

εa,c

= 80%

εa,c

= 90%

εa,c

= 100% − Base ca se

εa,c

= 110%

εa,c

= 120%

dctaawtau

TD

0 50 100 150 2003.4

3.6

3.8

4

4.2

Cel

l vol

tage

(V

)

0 50 100 150 200−1

0

1

2

3

Time (s)

Dis

char

ge c

urre

nt (

C)

Fig. 13. Simulation of a battery in city regime.

Table 6Dynamic operation in suburban regime.

I (C/m2) Time (s) Description

3.00 60 Start2.00 60 Disacceleration−1.50 10 Braking1.00 60 Acceleration3.80 25 Vigorous acceleration1.00 50 Disacceleration4.00 50 Vigorous acceleration3.20 120 Full stop

each battery is subject to and their states of charge are also different.One application of interest is to enhance the electric vehicle batteryperformance by the construction of an arrangement of different

Capacity (mAh)

Fig. 12. Variation of the parameter εa,c .

ivides them into city and suburban regimes. The city regime isharacterized by repetitive cycles of acceleration and braking. Onhe other hand, the suburban regime presents longer periods ofcceleration and deceleration with minimal full stops. The regimeffects the performance and duration of the battery in the same

ay as the fuel performance (i.e. km/gal) for the internal combus-

ion vehicles. The city regime includes short periods of accelerationnd braking. An idealized scheme is presented in Table 5. The sim-lation of this regime is presented in Fig. 13. The suburban regime

able 5ynamic operation in city regime.

I (C/m2) Time (s) Description

1.00 5 Start0.25 10 Disacceleration−0.25 7 Braking0.00 10 Full stop1.80 10 Acceleration0.60 30 Disacceleration−0.50 12 Braking0.00 15 Full stop2.20 25 Acceleration0.90 10 Disacceleration−0.60 10 Braking0.50 20 Acceleration0.00 10 Full stop

−2.00 20 Braking0.00 25 Full stop

is characterized by longer periods of acceleration and deceleration,some braking and virtually no full stops. A representation of thisscheme is shown in Table 6. The voltage profile under the suburbanregime is presented in Fig. 14

7.3. Parallel batteries simulation

Series and parallel arrangements of batteries are used toincrease the voltage and capacity of the system. However thesearrangements may present problems due to the lack of equalizationof the batteries. This refers to the different rates of discharge that

capacity batteries in parallel (Wu, Lin, Wang, Wan, & Yang, 2006).

0 50 100 150 200 250 300 350 400 450 5002.5

3

3.5

4

4.5

Cel

l vol

tage

(V

)

0 50 100 150 200 250 300 350 400 450 500−2

0

2

4

Time (s)

Dis

char

ge c

urre

nt (

C)

Fig. 14. Simulation of a battery in a suburban regime.

Page 11

E. Martínez-Rosas et al. / Computers and Chem

0 500 1000 150 0 2000 2500 300 0 3500

3

3.5

4

2.6

Vol

tage

(V

)

0 500 1000 150 0 2000 2500 300 0 35000

0.5

1

Par

titio

n co

eff.

(Cp)

0 500 1000 150 0 2000 2500 300 0 35000

0.05

0.1

max

(ηa* )

Tpqt

•

•

Afsrtb

otos

8

sTsusisnewrvewdAtms

Time (s)

Fig. 15. Simulation of batteries in parallel.

he basic idea is to connect a high energy density battery to a highower density one, expecting that the system would respond ade-uately to peak demands of current and also have a long dischargeime.

We present two criteria for simulation:

The total discharge current is equal to the sum of the individualbatteries.The voltage of both batteries remains equal at all times.

These conditions come from the basic theory of parallel circuits. simulation that fulfills both restrictions is proposed under the

ollowing considerations: both batteries are solved as a coupledystem; a term that relates the total current to the individual cur-ents is created and named partition coefficient, is calculated inhe simulation with the restriction of having equal voltages in bothatteries.

A simulation was carried out with the base case battery and thene of augmented width of 20%. The nominal capacities of both bat-eries are 17.5 and 20.4 Ah/m2 respectively. The expected capacityf the arrangement is 37.9 Ah/m2. Fig. 15 shows the results of theimulation at 1.0 C-rate (37.9 Ah) of the parallel arrangement.

. Conclusions

Fossil fuels have been for many years the main source of energyupply for many applications including the transportation sector.here are now some evidences that this strong dependence on fos-il fuels is one of the main reasons for climatic change problems andrban pollution. Moreover, it has been predicted that soon worldcale oil exploitation will reach its maximum peak. Therefore, theres a real need to explore the use of alternative sustainable energyources able to meet our energy demands. It is certainly true thatone of the new alternative energies being considered (i.e. biofu-ls, fuel cells, batteries, wind and solar power, etc) by themselvesill be capable of meeting world scale energy demands. What we

equire is the integration of all these alternative energies with con-entional fossil fuels (during the transition period from fossil fuelsnergy to alternative energies) (Weekman, 2010). Therefore, in thisork our aim was to explore the use of advanced energy storageevices (Li-ion batteries) to partially meet future energy demands.

lthough there is a great number of published works dealing with

he chemistry of Li-ion batteries, few works have addressed theodeling and simulation of Li-ion batteries to explore process sen-

itivity and their dynamic performance during different driving

ical Engineering 35 (2011) 1937– 1948 1947

scenarios. Hence, the modeling and simulation of the Li-ion bat-teries with chemistry LixC6−− LiyMn2O4 were addressed in thepresent work. The model used was modified from that of Doyleet al Doyle and Newman (1996) in the boundary conditions. TheMOL was used in the discretization of the system into a DAE sys-tem. A program in the form of a Matlab function was developedfor the simulation of the battery and the calculation of electro-chemical variables. The mathematical model was experimentallyvalidated and tested with different applications. These include thevariation of design parameters, the dynamic operation of batter-ies and the simulation of different capacity batteries connectedin parallel. Simulation with design purposes and the predictionof the degree of discharge are the following steps in the line ofinvestigation proposed by this work as well as the integration ofthe temperature dependence. Future work includes the determi-nation of optimal control policies to maximize the performanceof Li-ion batteries. With the advent of powerful CPU processorsand advances in non-linear predictive control techniques (Zavala& Biegler, 2009; Huang, Zavala, & Biegler, 2009) it may be feasibleto address the real time implementation of those optimal controlpolicies.

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