268747

Hindawi Publishing CorporationInternational Journal of ElectrochemistryVolume 2013 Article ID 268747 10 pageshttpdxdoiorg1011552013268747

Research ArticleFaster-Than-Real-Time Simulation of Lithium Ion Batterieswith Full Spatial and Temporal Resolution

Sandip Mazumder and Jiheng Lu

Department of Mechanical and Aerospace Engineering The Ohio State University Columbus OH 43210 USA

Correspondence should be addressed to Sandip Mazumder mazumder2osuedu

Received 26 June 2013 Accepted 4 November 2013

Academic Editor Sheng S Zhang

Copyright copy 2013 S Mazumder and J Lu This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A one-dimensional coupled electrochemical-thermal model of a lithium ion battery with full temporal and normal-to-electrodespatial resolution is presented Only a single pair of electrodes is considered in the model It is shown that simulation of a lithiumion battery with the inclusion of detailed transport phenomena and electrochemistry is possible with faster-than-real-time computetimesThe governing conservation equations of mass charge and energy are discretized using the finite volumemethod and solvedusing an iterative procedure The model is first successfully validated against experimental data for both charge and dischargeprocesses in a Li

119909C6-Li119910Mn2O4 battery Finally it is demonstrated for an arbitrary rapidly changing transient load typical of a

hybrid electric vehicle drive cycle The model is able to predict the cell voltage of a 15-minute drive cycle in less than 12 seconds ofcompute time on a laptop with a 233GHz Intel Pentium 4 processor

1 Introduction

Lithium ion batteries by virtue of having high powerdensity and high open circuit voltage have evolved as thefrontrunner among energy storage systems Today they areused in virtually all electronic devices as well as in hybridelectric vehicles among other applications Modeling of suchbatteries provides a means to better understand the coupledphysical and chemical phenomena that occur within thebattery Simulation results can shed light on phenomena suchas thermal runaway transient response of the battery andeffect of external thermal conditions on the performanceSimulations can also lead to design of better thermal man-agement strategies

The literature is full of several well-established validatedmodels for the prediction of the performance lithium ionbatteries with a single electrode pair Some of these modelsaccount for the detailed electrochemistry and transportphenomena within the battery [1ndash12] while others do not[13 14] Similarly some models account for spatial non-uniformity of temperature within the battery [3 7 11] whileothers do not [1 2 4ndash6 8ndash10 12] To the best of the authorsrsquoknowledge the earliest model was developed by Newman

and coworkers [1] and later extensively used and improvedby White and coworkers [4ndash8] Although the governingequations proposed by Newman and coworkers are generalconservation equations of mass charge and energy andare applicable to any geometry until the late 1990s mostof the modeling effort focused on simple one-dimensional(1D) geometries in which variations along the electrodesare neglected and only variations across the electrodes andseparator are considered Although somewhat simplisticsuch models have found widespread use because they arestill able to capture the coupling of the electrochemistryand transport phenomena within the electrodes In the pastdecade or so with the advent of faster computers researchershave embarked upon multidimensional modeling of lithiumion batteries [3 15] In particular it has been brought tolight by Wang and coworkers [3] that to fully understand thethermal response of a battery it is necessary to account formultidimensional effects

The vast majority of the transport-electrochemistrymod-els presented in the literature have been validated and demon-strated for constant load charge and discharge processes withrelatively low charge or discharge rates (up to about 4C)For relatively low chargedischarge rates diffusion of Li ions

2 International Journal of Electrochemistry

X

Negative electrode Positive electrodeSeparator

Loademinus

eminus

eminus

eminus

Iapp

Li+

Li+ Li+

Ln Ls Lp

LixC6 activeparticle

LiyMn2O4

active particle

Figure 1 Schematic one-dimensional representation of a lithiumion battery with a single pair of electrodes under discharge condi-tions

within the active particles can keep up with the reactionsoccurring at the surface of the particle and therefore thediffusion resistance to Li ions within the active particles doesnot play a significant role in the performance of the battery Insuch scenarios the active particles can be treated as lumpedmasses and a simplistic subgrid scale model can be usedfor the solid phase concentration field Smith and Wang [2]pointed out that while such an approximation is adequatefor batteries used in electronic devices it is inadequate forbatteries used in hybrid electric vehicles because they oftenencounter much higher chargedischarge rates than 4C

With growing interest in using lithium ion batteries forhybrid electric vehicles control of these batteries has becomean important issue When the load (current drawn) on abattery changes suddenly as in an automotive drive cyclethe voltage does not respond instantaneously The delayin response is caused by the combination of a variety oftransport and electrochemical processes within the batteryIdeally a physics-based mathematical model of the typesdiscussed earlier is able to predict such delays in responseUnfortunately it is widely believed that such detailed physics-based models that resolve the battery both temporally andspatially are not efficient enough to be used for controlapplications Consequently so-called equivalent circuit mod-els of batteries that are used either in time or frequencydomain have found widespread development and use [16ndash19] particularly within the hybrid electric vehicle community[20ndash22] While these models are very efficient they providelittle or no insight into the fundamental processes that occurwithin the battery In recent years efforts have been made todevelop reduced-order models that include some of the keyphysicalchemical aspects while being more amenable to usefor control purposes [14 23]

In this paper it is demonstrated that it is possible tosimulate in real time (as warranted by control applications)the performance of a lithium ion battery for realistic hybridvehicle drive cycles with the inclusion of detailed transportphenomena and electrochemistry and full temporal andnormal-to-electrode spatial resolution

2 Mathematical Model

A typical lithium ion battery with a single pair of electrodesis comprised of three regions a negative electrode a positiveelectrode and a separator A one-dimensional schematicrepresentation is shown in Figure 1 The negative electrodeis a porous structure comprised of lithium carbide particlestightly packed together while the positive electrode is aporous structure comprised of lithium metal oxide particlestightly packed together The most commonly used metaloxides are LiMn

2O4 LiCoO

2 or LiFePO

4The two electrodes

are separated by a separator (or ionic conductor) that allowlithium ions to migrate across but prevents any electrontransport across it The entire cell is filled with an electrolytethat occupies any space not occupied by the active particles orthe binder and filler (not shown in Figure 1) The electrolyteis comprised of a lithium salt dissolved in an organic solvent

Under normal operating conditions the following reac-tions occur at the two electrodes (assuming a manganeseoxide positive electrode)

Negative electrode Li119909C6

discharge999445999468

charge6C + 119909Li+ + 119909eminus (R1)

Positive electrode

Li1minus119909

Mn2O4

+ 119909Li+ + 119909eminusdischarge

999445999468

chargeLiMn

2O4

(R2)

During the discharge process lithium ions that are generatedat the negative electrode migrate across the separator to thepositive electrode where they recombine with the electronsthat flow through the external circuit to form themetal oxideThe reverse occurs during the charging process

21 Macroscopic Model The macroscopic mathematicalmodel employed in this work is themodel proposed by Doyleet al [1] and subsequently used by numerous researchers[2ndash12] While material properties kinetic constants andother associated properties change depending on the exactchemical composition of the electrodes (aswill be pointed outlater) the model proposed by Doyle et al is general enoughto be applicable to any lithium ion battery since it is basedon basic conservation laws The governing equations areequations of conservation of mass (of lithium ions) chargeand energy Under the assumption of electroneutrality thecharge conservation equation reduces to a current conser-vation equation In general vector notations the governingequations are written as

120597

120597119905

(120576119890119862119890) = nabla sdot (120576

119890119863

eff119890

nabla119862119890) +

1 minus 119905+

119865

119895Li (1)

nabla sdot (120581eff

nabla120601119890) + nabla sdot (120581

eff119863

nabla ln119862119890) = minus119895Li (2)


nabla120601119904) = 119895Li (3)

120597

120597119905

(120588119888119901119879) = nabla sdot (119896

effnabla119879) + 119902 (4)

International Journal of Electrochemistry 3

In the above equations 119862119890is the molar concentration of

lithium ions in the electrolyte while 120576119890is the volume fraction

of the electrolyte containing part of the porous electrodesandor separator 119863

eff119890

is the effective diffusion coefficientof lithium ions within the porous electrodesseparator It isrelated to the free-stream diffusion coefficient of the lithiumions through the so-called Bruggeman relationship

119863eff119890

= 11986311989012057615

119890 (5)

where 119863119890is the free-stream diffusion coefficient of lithium

ions in the electrolyte 119905+is the so-called transference number

and physically represents the charge carried by the lithiumions relative to the solvent due to drift 119865 is the Faradayconstant The electrolyte (ionic) and solid (electronic) phasepotentials are denoted by 120601

119890and 120601

119904 respectively while the

effective ionic and diffusional conductivities are denoted by120581eff and 120581

eff119863 respectively They are also related to their free-

stream values through the Bruggeman relationship (5) Thediffusional conductivity (electrical conductivity in the elec-trolyte phase due to diffusion of lithium ions) is related to theionic conductivity (electrical conductivity in the electrolytephase due to drift or electromigration of lithium ions) by therelationship

120581eff119863

=

2119877119879120581eff

119865

(119905+

minus 1) (1 +

120597 ln119891

120597 ln119862119890

) (6)

where 119877 is the universal gas constant and 119879 the absolutetemperature The mean molar activity coefficient of themixture is denoted by 119891 which is assumed to be constant inthis study due to lack of better information

Due to electrochemical reactions at the surface of theactive particles electrons are transferred to the solid (orelectronic) phase while positively charged lithium ions aretransferred to the electrolyte phase This process is repre-sented by a current known as the transfer current and isdenoted here by 119895Li The transfer current depends on anumber of factors such as the lithium ion concentrationsboth in the solid and electrolyte phase the temperature andthe surface overpotential It is customary to express the rateof generation of this current using the Butler-Volmer kinetics

119895Li = 1198861199041198940

[exp

120572119886119865

119877119879

(120578 minus

119877SEI119886119904

119895Li)

minus exp

minus120572119888119865

119877119879

(120578 minus

119877SEI119886119904

119895Li)]

(7)

where 119886119904is the active surface area to volume ratio of the

electrode and 1198940is the so-called exchange current density and

is given by

1198940

= 1198960119862120572119886

119890(119862119904max minus 119862

119904surf)120572119886

119862120572119888

119904surf (8)

where 119862119904max is the maximum concentration of lithium ions

in the solid phase and 119862119904surf is the concentration of lithium

ions at the surface of the active particle that is at thesolid-electrolyte interface 120572

119886and 120572

119888are kinetic constants

that depend on the chemical composition of the electrodes

and the electrochemical reactions that occur on the activeparticles The exchange current also depends on the kineticconstant 119896

0 In (7) the quantity 119877SEI represents resistance

due to irreversible film formation at the solid-electrolyteinterface It is generally believed [4ndash6 9 10] that these filmsform in all batteries during the initial assembly process butgrow only if the battery is subjected either to extremely highrates of chargedischarge andor deep discharge Aging of thebattery is often attributed to growth of this film [10] at theanode which causes increase in the internal resistance ofthe battery In the present model although there is scope toinclude the effect of this film (as indicated by (7)) due to lackof understanding of the exact mechanism of film formationand growth this resistance is neglected in this preliminarywork

The overpotential 120578 drives the electrochemical reactionsand is given by

120578 = 120601119904minus 120601119890

minus 119880 (9)

where 119880 is the open circuit (or equilibrium) potentialThe open circuit potential is dependent on the chemicalcomposition of the electrode its state of chargedischargeand temperature Details on how this quantity is calculatedare presented later

Equation (4) represents the conservation of energy andconduction is assumed to be the only mode of heat transferThe average (including both phases) density specific heatcapacity and thermal conductivity are denoted by 120588 119888

119901 and

119896 respectively The volumetric heat generation rate withinthe battery is a net result of three effects Joule heatingirreversible heat generation and reversible heat generationand is written as [3]

119902 = 119886119904119895Li120578 + 119886

119904119895Li119879

120597119880

120597119879

+ 120590eff

nabla120601119904sdot nabla120601119904

+ 120581eff

nabla120601119890

sdot nabla120601119890

+ 120581eff119863

nabla120601119890

sdot nabla ln119862119890

(10)

The first term on the right hand side of (10) represents irre-versible heating and the second term represents reversibleheating The third term represents Joule heating due to thetransport of electrons within the two electrodes while the lasttwo terms represent Joule heating due to the transport of ionswithin the entire battery The boundary conditions for thegoverning conservation equations [(1)ndash(4)] are well known[1ndash3] and are summarized in Table 1

22 Subgrid Scale Active Particle Model Equations (1)ndash(10) along with the boundary conditions shown in Table 1represent the macroscopic model that can predict the perfor-mance (voltage-current) characteristics of a typical lithiumion battery for both charge and discharge cycles Howeverthe model is not complete in terms of closureThis is becausethe solid-phase concentration at the surface of the activeparticles 119862

119904surf (as appearing in (7)) is an unknown andrequires additional equations to be determined While thedependent variables in the macroscopic model namely 119862

119890

120601119890 120601119904 and 119879 are defined within the entire battery the

solid phase concentration is defined only within the active


Table 1 Summary of boundary conditions required for the solution of the governing conservation equations

Dependent variable Negative currentcollector

Negativeseparatorinterface

Positiveseparatorinterface

Positive currentcollector

119862119890(1) Zero flux Not needed Not needed Zero flux


120601119904(3) Applied current Zero flux Zero flux Applied current

119879 (4) Isothermal adiabaticor Newton cooling Not needed Not needed Isothermal adiabatic

or Newton cooling

particles Since the active particles are much smaller than thecharacteristic dimensions of the electrodes (active particleshave a diameter of less than 1 120583m while the electrodes areseveral tens of 120583ms thick) they cannot be resolved by thesame computational mesh that is employed to solve themacroscopic model equations (1)ndash(10) The active particle isessentially at a length scalemuch smaller than the grid scale ofthe macroscopic model Therefore any equation (or model)used to describe the lithium ion concentration field withinthe active particles is at the so-called subgrid scale and ishenceforth referred to as the subgrid scale model

In reality the active particles are bound together withinthe electrodes using filler and binder materials As a resulteven though they start as spherical nanoparticles once boundtogether they form a complex network better represented bysets of overlapping spheres of various diameters Doyle et al[1] proposed treating the active particles as nointeractingperfect spheres of constant radius In this work we adoptthe same assumption although with significantly largercomputational effort it is possible to treat more complexshapes as demonstrated by Kamarajugadda and Mazumderfor fuel cell electrodes [24] and by Wang and Sastry [15] forbattery electrodes Under the isolated sphere assumption andassuming that there is no variation of any quantity along thesurface of the sphere (polar and azimuthal uniformity) theconcentration of lithium ions within a single active particlecan be described by an unsteady diffusion equation writtenin spherical coordinates with only radial dependence

120597119862119904

120597119905

=

119863119904

1199032

120597

120597119903

(1199032120597119862119904

120597119903

) (11)

where119863119904is the diffusion coefficient of lithium ions within the

active particle (or in the solid phase) Equation (11) requirestwo boundary conditions which are written as

120597119862119904

120597119903

10038161003816100381610038161003816100381610038161003816119903=0

= 0 (12a)

minus119863119904

120597119862119904

120597119903

10038161003816100381610038161003816100381610038161003816119903=119903119904

=

119895Li119886119904119865

(12b)

Equation (12a) represents symmetry at the center of theactive particle while (12b) represents a diffusion-reaction fluxbalance at the surface of the active particle (119903 = 119903

119904)The active

surface area to volume ratio is dependent on the radius of theactive particles and their overall volume fraction

119886119904

=

3120576119904

119903119904

(13)

where 120576119904is the volume fraction of the active particles in the

electrodeSince the transfer current 119895Li is a nonlinear function of

the solid phase concentration 119862119904 as indicated by (6) and (7)

(11) represents a partial differential equation whose boundarycondition (12b) is non-linear Thus closed-form analyticalsolution of (11) is not possible Under the assumption thatthe mass transport by diffusion within the active particle ismuch faster than the reaction occurring at its surface (iesmall mass transport Biot number) the active particle can betreated as a lumped mass and the solid phase concentrationof lithium ions at the surface of the active particle is given by[2]

119862119904surf minus 119862

119904avg =

minus119895Li5119886119904119865119863119904

[1 minus exp(minus

20

3

radic119863119904119905

119903119904

)] (14)

where 119862119904avg is the average solid-phase concentration of

lithium ions within the active particle Unfortunately thelumpedmass approximation breaks down in situations wherethe load is large as demonstrated by Smith and Wang [2]Therefore for hybrid vehicle applications it is desirable tosolve (11) in its original form which is done in the presentwork

Solution of (11) in its original form can be computa-tionally expensive This is because of two reasons Firstas discussed earlier its boundary condition is non-linearSecondly the equation has to be solved at each computationalnode (or cell) of the macroscopic model Thus if 100 nodesare used in the macroscopic battery model (11) has to besolved 100 times because for each cell the transfer currentwhich goes in as an input to the model via the surfaceboundary condition (12b) may be different Smith andWang[2] solve this equation using a finite-element method In thiswork (11) is solved using the finite-difference method withnominally 11 grid points

3 Numerical Procedure

Thebasic procedure used in the present work to discretize thegoverning partial differential equations is the finite-volume


Start of time-marching

TDMA solver

Calculate all transport and thermodynamic properties

Converged No

Out

er it

erat

ion

Yes

New

tim

e ste

p

TDMA solver

TDMA solver

(subgrid scale model) TDMA solver

TDMA solver

Solve linearized equation for 120601s

Solve linearized equation for 120601e

Update 120578 in all cells


Solve linearized equation for T

Initial conditions for Ce Cs and T

Guess Ce Cs 120601e 120601s and T of whole computational domain

Solve linearized equation for Ce

Solve linearized equation for Cs

Figure 2 Algorithm employed for solution of the governing equations

method This method is chosen because it guarantees bothlocal and global conservation irrespective of mesh size whileother methods are inherently nonconservative Furthermoreit is amenable to handling discontinuities in material proper-ties because the method is basically an integral method Fortemporal discretization the backward Euler method is usedbecause it is unconditionally stable Since all of the governingPDEs have nonlinear source terms appropriate source termlinearization techniques were employed to improve diagonaldominance of the resulting discrete equations sets Eachgoverning PDE when discretized and linearized resultedin a set of tridiagonal equations that were solved using theThomas algorithm (TDMA solver) The PDEs were solvedsequentially Coupling between the PDEs was addressedby using an outer iteration loop over the five governingPDEs This iteration loop was also instrumental in address-ing nonlinearities in the system Within each time stepconvergence was deemed to have been achieved when theresiduals for each conservation equation decreased by at least4 orders of magnitude The residual of each PDE was definedas the l2norm (inner product) of the discretized equationswithout source term linearization The overall algorithm isdepicted in Figure 2 At each time step once the solutionreaches convergence the results are postprocessed to extractquantities of engineering interest One of these is the cellvoltage which is given by

119881cell = 120601119904

1003816100381610038161003816119909=119871119899+119871119904+119871119901

minus 120601119904

1003816100381610038161003816119909=0

minus

119877contact119860collector

119868app (15)

where 119877contact is the contact resistance between the electrodesand the current collectors and 119860collector is the collector plate

surface area The applied (drawn) current or load is denotedby 119868app

4 Results and Discussion

41 Model Validation A validation study was first under-taken In this study the entire charge and discharge cycleof a Li

119909C6-Li119910Mn2O4battery was simulated The battery

considered here has a nominal rating of 6Ah and achargedischarge rate of 1C was used Experimental data isavailable for the same set of conditions Other necessarygeometric parameters material properties and operatingconditions are summarized in Table 2 The data presentedin Table 2 are extracted from Smith and Wang [2] whoalso provide the experimental data used for the validationstudy The only data that were not directly extracted out ofSmith andWang [2] are the two kinetic constants The valuesreported in Table 2 for these constants were estimated toobtain an exchange current density of 36Am2 in the negativeelectrode and 26Am2 in the positive electrode as reportedby Smith and Wang [2]

In addition to the data summarized in Table 2 Smithand Wang [2] also provided curve-fits to experimental mea-surements for the ionic conductivity and the open circuitpotential The free stream ionic conductivity (electrical con-ductivity of the electrolyte phase due to drift of ions) in Smis given by [2]

120581 = 158 times 10minus4

119862119890exp[085(

119862119890

1000

)

14

] (16)


Table 2 Values of various parameters for the Li119909C6-Li119910Mn2O4battery simulated in the present work

Parameter Negative electrode Separator Positive electrodeGeometry and mesh

Thickness 119871119899 119871119904 119871119901

50120583m 254 120583m 364 120583mActive particle radius 119903

1199041 120583m NA 1 120583m

Electrolyte volume fraction 120576119890

0332 05 033Active particle volume fraction 120576

119904058 NA 05

Number of control volumes (nominal) 50 25 36Initial state of battery

Maximum solid phase concentration 119862119904max 161 times 10

3molm3239 times 10

3molm3

Stoichiometry at 0 state of charge 119909 119910 0126 NA 0936Stoichiometry at 100 state of charge 119909 119910 0676 NA 0442Electrolyte concentration 12 times 10

3molm312 times 10

3molm312 times 10

3molm3

Kinetic and transport properties

Kinetic constant 1198960

138 times 10minus4

(Am2)sdot(m3mol)32 NA 064 times 10minus4

(Am2)sdot(m3mol)32

Charge transfer coefficients 120572119886 120572119888

05 05 NA 05 05SEI film resistance 119877SEI 0 NA 0Lithium ion diffusion coefficient in solid phase 119863

1199042 times 10

minus16m2s NA 37 times 10minus16m2s

Solid phase electrical conductivity 120590 100 Sm NA 10 SmLithium ion diffusion coefficient in electrolyte 119863

1198902 times 10

minus10m2s 2 times 10minus10m2s 2 times 10

minus10m2sElectrolyte phase ionic conductivity 120581 (15) (15) (15)Transference number 119905

+0363 0363 0363

Open circuit potential (16) NA (16)Density 2500 kgm3 1200 kgm3 1500 kgm3

Thermal conductivity 5WmK 1WmK 5WmKSpecific heat capacity 700 JkgK 700 JkgK 700 JkgK

where the electrolyte concentration has units of molm3 Theopen circuit potential of the two electrodes was also fitted toexperimental data [2] to yield

119880119899

= 800229 + 50647119909 minus 1257811990912

minus 86322 times 10minus4

119909minus1

+ 21765 times 10minus5

11990932

minus 046016 exp [15 (006 minus 119909)]

minus 055364 exp [minus24326 (119909 minus 092)]

119880119901

= 866811199106

minus 35771199105

+ 613891199104

minus 555651199103

+ 281061199102

minus 76648119910

minus 030987 exp (5657119910115

) + 131983

(17)

where 119909 and 119910 are the stoichiometries in the negative andpositive electrodes respectively The open circuit potentialscomputed using (17) are in Volts For the external contactresistance 119877contact a value of 20Ω cm2 was used Althoughthe present model has the ability to predict the temperaturefield for the validation study a constant temperature of 288Kwas used

In order to compute the voltage characteristics of theentire chargedischarge cycle the calculations were per-formed for a net duration of 3800 seconds with a time stepof 1 second Within each time step it took approximately 50iterations to achieve convergence by 4 orders of magnitudeA typical convergence plot is shown in Figure 3

Figure 4 shows the comparison of the computed cellvoltage withmeasured voltage for the entire charge-dischargecycle with a chargedischarge rate of 1CThe computed resultsagree quite well with the experimental data except towardthe extreme end of the discharge cycle where the predictedvoltage drop occurs slightly later than the measured voltagedrop and at a much sharper rate This discrepancy could bedue to a number of factors including uncertainties inmaterialproperties and kinetic constants neglecting the formationand growth of the SEI film and thermal effects

Since the present model resolves the battery both tempo-rally and spatially it is instructive to carefully examine theevolution of some of the dependent variables Figure 5 showsspatiotemporal evolution of lithium ion concentration in theelectrolyte phase the electrolyte phase electric potential andthe overpotential It is seen that with time the lithium ionconcentration distribution becomes more lopsided while theelectric potential in the electrolyte phase is fairly uniformThis is because the ionic conductivity of the entire cell is quite


0 10 20 30 40 50 60

10minus4

10minus3

10minus2

10minus1

100

Nor

mal

ized

resid

ual

Iterations

Ce

Cs

120601s120601e

Figure 3 Convergence behavior at the tenth time step

high resulting in a small potential variation The overpoten-tial though quite small at 1C discharge rate shows significantspatial variation At the negative electrode towards the endof the discharge cycle it rises rapidly This is consistent withthe sharp drop in the electrolyte phase concentration nearthe negative-separator interface The simulation of an entirecharge cycle or an entire discharge cycle (3800 time steps witha time step of 1 s) required about 42 seconds on a laptopwith a23 GHz Intel Pentium4processorOf these 42 seconds about15 seconds after used to write data to files for postprocessingimplying that a 3800-second cycle was simulated in less than30 seconds of real time

42 Demonstration for HEV Driving Cycle In order todemonstrate the model for real-time control applicationsa driving cycle typical of that of a hybrid electric vehicle(HEV) was simulated Current (load) pulses as large as 10Cand lasting over only 10 seconds was used The load cyclewas commenced with the battery at 100 state of chargeFigure 6 shows the computed voltage response of the batterysubjected to the prescribed pulse load The predicted voltageshows that the numerical algorithm is able to cope up withthe rapidly changing loadThe jumps in cell voltage when thecurrent is reversed (ie when the operation mode is changedfrom charge to discharge or vice versa) are caused by thedifferences in the output voltage for the same current duringcharge versus discharge as observed in Figure 4 The 900-second drive cycle simulation required about 12 seconds ofCPU time (which is almost the same as wall-clock time) on alaptop with a 23 GHz Intel Pentium processor out of whichabout 3 secondswere used for writing files for postprocessingThis implies that the execution time of the current modelis about 100 times faster than real time It is to be notedthat the aforestated performancewas recordedwhen the codewas written of Fortran90 When the same code was rewritten

0 02 04 06 08 1Normalized capacity

3

35

4

Volta

ge (V

)

ExperimentModel

(a)


3

35

4

Volta

ge (V

)

ExperimentModel

(b)

Figure 4 Computed cell voltage versus measured cell voltage forthe entire charge-discharge cycle of a Li

119909C6-Li119910Mn2O4battery with

1C chargedischarge rate (a) charge (b) discharge

using MATLAB and executed the performance deterioratedsignificantly and compute times of only about 5ndash8 timesfaster (as opposed to 100 times faster) than real time wereachieved

43 Thermal Effects In order to demonstrate the modelrsquosability to simulate thermal effects the same load cycle con-sidered in the preceding study was now simulated but with


0 20 40 60 80 100X (120583m)

1180

1200

1220

Elec

troly

te p

hase

conc

entr

atio

n (m

olm

3)

Separator

10 s1800 s3600 s

(a)

0 20 40 60 80 100X (120583m)

10 s1800 s3600 s

Separator

212

214

216

218

Elec

troly

te p

hase

pot

entia

l (V

)

(b)

0 20 40 60 80 100X (120583m)

10 s1800 s3600 s

Separator

minus00002

minus00001

0

00001

00002

Ove

rpot

entia

l (V

)

(c)

Figure 5 Spatiotemporal evolution of key quantities during the beginning middle and end of the discharge process with 1C rate of discharge(a) electrolyte phase concentration (b) electrolyte phase potential and (c) overpotential

the inclusion of the energy equation Prior to performing thesimulation however the thermal properties of the materialshad to be estimated Table 2 already shows the thermalproperties used for the simulations These were extractedfrom Srinivasan and Wang [3] These authors also providedcurve-fits for the rate of change of the open circuit potentialwith temperature (as appearing in (10)) for both Li

119909C6and

Li119910Mn2O4electrodes During a discharge cycle reaction

(R1) is endothermic while reaction (R2) is exothermic Thenet effect of the two reactions however is to generate heatirrespective of charge or discharge This is clearly evident ifthe battery is operated under adiabatic conditions Figure 7shows the temperature evolution of the battery under suchconditions A 45 K rise in temperature is observed during the15-minute hybrid drive cycle considered earlier Although notshown here even with Newton cooling at one of the ends


200 400 600 800Time (s)

0

10

20

30

40

50

60

Curr

ent (

A)

Current

34

35

36

37

38

39

Volta

ge (V

)Voltage

Figure 6 Computed voltage response of a 6 Ah battery subjected toan arbitrary load (current) over a 15-minute period The prescribedload (current) is shown by the dotted red line while the predictedvoltage is shown by the solid green line Positive current denotesdischarge

the temperature distribution was found to be very uniformbecause the battery is only 111 120583m thick and the resultingconduction resistance across it is very small Therefore eventhough the heat generation within the battery as shownby (10) is not uniform a very small spatial variation oftemperature is observed within the battery because the heatspreads very quickly over a distance of 111120583m To trulyunderstand thermal effects multidimensional modeling iswarranted

5 Summary and Conclusions

Amodel for predicting the voltage characteristics of a lithiumion battery subjected to a prescribed load was developedThe model completely resolves both time and space acrossthe battery and is therefore capable of capturing any spatialnonuniformity of relevant quantities such as concentra-tions electric potentials and temperature within the batteryAlthough the model is validated and demonstrated herefor a Li

119909C6-Li119910Mn2O4battery in principle it is general

enough to be applicable to any lithium ion battery providedthat the material properties and kinetics are appropriatelycharacterized and known The model was validated againstexperimental measurements for the entire charge and dis-charge cycle of a Li

119909C6-Li119910Mn2O4battery and was able to

accurately predict the voltage characteristics It was thensuccessfully demonstrated for a load input typical of that ofan HEV automotive drive cycle

The computational efficiency of the model was notewor-thy A 900-second drive cycle was computed in less than 12

0 20 40 60 80 100X (120583m)

10 s450 s900 s

280

285

290

295

300

Tem

pera

ture

(K)

Figure 7 Spatiotemporal evolution of temperature within thebattery under adiabatic conditions during a 15-minute hybrid drivecycle

seconds of CPU time on a laptop indicating that the modelcan be integrated with real-time control algorithms

Future work will involve incorporation of appropriatesubmodels to describe aging of the battery so that thelife cycle of a battery can be predicted Extension of themodel to complex three-dimensional geometries is currentlyunderway

Nomenclature

119886119904 Active surface area to volume ratio (mminus1)

119862119890 Electrolyte phase lithium ion concentration

(molmminus3)119862119904 Solid phase lithium ion concentration

(molmminus3)119862119904max Maximum concentration of lithium ions in

the solid phase (molmminus3)119862119904surf Concentration of lithium ions at the surface

of the active particle (molmminus3)119888119901 Specific heat capacity at constant pressure

(J kgminus1 Kminus1)119863

eff119890 Effective diffusion coefficient of lithium ions

in electrolyte (m2 sminus1)119865 Faraday constant (=9648533 Cmol)1198940 Exchange current density (Amminus2)

119895Li Transfer current (Amminus3)119896eff Effective thermal conductivity (Wmminus1 Kminus1)119902 Heat generation per unit volume (Wmminus3)

119903119904 Radius of active particle (m)


119877 Universal gas constant (=8314 Jmolminus1 Kminus1)119905 Time (s)119905+ Transference number (dimensionless)

119879 Absolute temperature (K)119880 Open circuit potential (V)

Greek

120572119886 120572119888 Kinetic constants (dimensionless)

120576119890 Volume fraction of electrolyte

(dimensionless)120576119904 Volume fraction of the active particles in

the electrode (dimensionless)120578 Overpotential (V)120581eff Effective ionic conductivity (Smminus1)

120581eff119863 Effective diffusional conductivity (Smminus1)

120601119890 Electrolyte (ionic) phase potential (V)

120601119904 Solid (electronic) phase potential (V)

120588 Density (kgmminus3)120590eff Effective electronic conductivity of solid

phase (Smminus1)

Acknowledgment

Financial support provided to Jiheng Lu by the Undergradu-ate Fellowship Program of the College of Engineering of theOhio State University is gratefully acknowledged

References

[1] M Doyle T Fuller and J Newman ldquoModeling of galvanostaticcharge and discharge of the lithium polymerinsertion cellrdquoJournal of the Electrochemical Society vol 140 no 6 pp 1526ndash1533 1993

[2] K Smith and C-Y Wang ldquoSolid-state diffusion limitations onpulse operation of a lithium ion cell for hybrid electric vehiclesrdquoJournal of Power Sources vol 161 no 1 pp 628ndash639 2006

[3] V Srinivasan and C Y Wang ldquoAnalysis of electrochemical andthermal behavior of Li-ion cellsrdquo Journal of the ElectrochemicalSociety vol 150 no 1 pp A98ndashA106 2003

[4] G Ning R E White and B N Popov ldquoA generalized cycle lifemodel of rechargeable Li-ion batteriesrdquoElectrochimicaActa vol51 no 10 pp 2012ndash2022 2006

[5] P Ramadass B Haran R White and B N Popov ldquoMathemat-ical modeling of the capacity fade of Li-ion cellsrdquo Journal ofPower Sources vol 123 no 2 pp 230ndash240 2003

[6] Q Zhang and R E White ldquoCapacity fade analysis of a lithiumion cellrdquo Journal of Power Sources vol 179 no 2 pp 793ndash7982008

[7] K Kumaresan G Sikha and R E White ldquoThermal model fora Li-ion cellrdquo Journal of the Electrochemical Society vol 155 no2 pp A164ndashA171 2008

[8] K Kumaresan G Sikha and R E White ldquoComparison ofthermal-electrochemical model predictions with experimentaldischarge data for Li-ion batteriesrdquo ECS Transactions vol 3 no27 pp 173ndash190 2007

[9] J Christensen and J Newman ldquoCyclable lithium and capacityloss in li-ion cellsrdquo Journal of the Electrochemical Society vol152 no 4 pp A818ndashA829 2005

[10] J Christensen and J Newman ldquoEffect of anode film resistanceon the chargedischarge capacity of a lithium-ion batteryrdquoJournal of the Electrochemical Society vol 150 no 11 pp A1416ndashA1420 2003

[11] W B Gu and C Y Wang ldquoThermal-electrochemical modelingof battery systemsrdquo Journal of the Electrochemical Society vol147 no 8 pp 2910ndash2922 2000

[12] B V Ratnakumar M C Smart and S Surampudi ldquoEffects ofSEI on the kinetics of lithium intercalationrdquo Journal of PowerSources vol 97-98 pp 137ndash139 2001

[13] J Marcicki G Rizzoni A T Conlisk and M CanovaldquoA reduced-order electrochemical model of lithium-ion cellsfor system identification of battery agingrdquo in Proceedingsof the ASME Dynamic Systems and Control Conference andBathASME Symposium on Fluid Power and Motion Control(DSCC rsquo11) Arlington Va USA October-November 2011

[14] J C Forman S Bashash J L Stein andH K Fathy ldquoReductionof an electrochemistry-based li-ion battery model via quasi-linearization and Pade approximationrdquo Journal of the Electro-chemical Society vol 158 no 2 pp A93ndashA101 2011

[15] C-W Wang and A M Sastry ldquoMesoscale modeling of a Li-ionpolymer cellrdquo Journal of the Electrochemical Society vol 154 no11 pp A1035ndashA1047 2007

[16] S Al Hallaj H Maleki J S Hong and J R Selman ldquoThermalmodeling and design considerations of lithium-ion batteriesrdquoJournal of Power Sources vol 83 no 1-2 pp 1ndash8 1999

[17] P Nelson D Dees K Amine and G Henriksen ldquoModelingthermal management of lithium-ion PNGV batteriesrdquo Journalof Power Sources vol 110 no 1 pp 349ndash356 2002

[18] P Nelson I Bloom K Amine and G Henriksen ldquoDesignmodeling of lithium-ion battery performancerdquo Journal of PowerSources vol 110 no 2 pp 437ndash444 2002

[19] E Karden S Buller and R W De Doncker ldquoA frequency-domain approach to dynamical modeling of electrochemicalpower sourcesrdquo Electrochimica Acta vol 47 no 13-14 pp 2347ndash2356 2002

[20] W Fang O J Kwan C Wang and Y Ishikawa ldquoModeling ofLi-ion battery performance in hybrid electric vehiclesrdquo SAETechnical Paper 2009-01-1388 2009

[21] Y Hu and S Yurkovich ldquoLinear parameter varying batterymodel identification using subspace methodsrdquo Journal of PowerSources vol 196 no 5 pp 2913ndash2923 2011

[22] Y Hu S Yurkovich Y Guezennec and B J YurkovichldquoElectro-thermal battery model identification for automotiveapplicationsrdquo Journal of Power Sources vol 196 no 1 pp 449ndash457 2011

[23] K A Smith C D Rahn and C-Y Wang ldquoControl oriented 1Delectrochemical model of lithium ion batteryrdquo Energy Conver-sion and Management vol 48 no 9 pp 2565ndash2578 2007

[24] S Kamarajugadda and S Mazumder ldquoGeneralized floodedagglomerate model for the cathode catalyst layer of a polymerelectrolyte membrane fuel cellrdquo Journal of Power Sources vol208 pp 328ndash339 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy


Carbohydrate Chemistry

International Journal of


Journal of

Chemistry


Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of


Chromatography Research International


Applied ChemistryJournal of



Theoretical ChemistryJournal of


Journal of

Spectroscopy

Analytical ChemistryInternational Journal of


Journal of


Quantum Chemistry


Organic Chemistry International

ElectrochemistryInternational Journal of



CatalystsJournal of


X

Negative electrode Positive electrodeSeparator

Loademinus

eminus

eminus

eminus

Iapp

Li+

Li+ Li+

Ln Ls Lp

LixC6 activeparticle

LiyMn2O4

active particle

Figure 1 Schematic one-dimensional representation of a lithiumion battery with a single pair of electrodes under discharge condi-tions

within the active particles can keep up with the reactionsoccurring at the surface of the particle and therefore thediffusion resistance to Li ions within the active particles doesnot play a significant role in the performance of the battery Insuch scenarios the active particles can be treated as lumpedmasses and a simplistic subgrid scale model can be usedfor the solid phase concentration field Smith and Wang [2]pointed out that while such an approximation is adequatefor batteries used in electronic devices it is inadequate forbatteries used in hybrid electric vehicles because they oftenencounter much higher chargedischarge rates than 4C

With growing interest in using lithium ion batteries forhybrid electric vehicles control of these batteries has becomean important issue When the load (current drawn) on abattery changes suddenly as in an automotive drive cyclethe voltage does not respond instantaneously The delayin response is caused by the combination of a variety oftransport and electrochemical processes within the batteryIdeally a physics-based mathematical model of the typesdiscussed earlier is able to predict such delays in responseUnfortunately it is widely believed that such detailed physics-based models that resolve the battery both temporally andspatially are not efficient enough to be used for controlapplications Consequently so-called equivalent circuit mod-els of batteries that are used either in time or frequencydomain have found widespread development and use [16ndash19] particularly within the hybrid electric vehicle community[20ndash22] While these models are very efficient they providelittle or no insight into the fundamental processes that occurwithin the battery In recent years efforts have been made todevelop reduced-order models that include some of the keyphysicalchemical aspects while being more amenable to usefor control purposes [14 23]

In this paper it is demonstrated that it is possible tosimulate in real time (as warranted by control applications)the performance of a lithium ion battery for realistic hybridvehicle drive cycles with the inclusion of detailed transportphenomena and electrochemistry and full temporal andnormal-to-electrode spatial resolution

2 Mathematical Model

A typical lithium ion battery with a single pair of electrodesis comprised of three regions a negative electrode a positiveelectrode and a separator A one-dimensional schematicrepresentation is shown in Figure 1 The negative electrodeis a porous structure comprised of lithium carbide particlestightly packed together while the positive electrode is aporous structure comprised of lithium metal oxide particlestightly packed together The most commonly used metaloxides are LiMn

2O4 LiCoO

2 or LiFePO

4The two electrodes

are separated by a separator (or ionic conductor) that allowlithium ions to migrate across but prevents any electrontransport across it The entire cell is filled with an electrolytethat occupies any space not occupied by the active particles orthe binder and filler (not shown in Figure 1) The electrolyteis comprised of a lithium salt dissolved in an organic solvent

Under normal operating conditions the following reac-tions occur at the two electrodes (assuming a manganeseoxide positive electrode)

Negative electrode Li119909C6

discharge999445999468

charge6C + 119909Li+ + 119909eminus (R1)

Positive electrode

Li1minus119909

Mn2O4

+ 119909Li+ + 119909eminusdischarge

999445999468

chargeLiMn

2O4

(R2)

During the discharge process lithium ions that are generatedat the negative electrode migrate across the separator to thepositive electrode where they recombine with the electronsthat flow through the external circuit to form themetal oxideThe reverse occurs during the charging process

21 Macroscopic Model The macroscopic mathematicalmodel employed in this work is themodel proposed by Doyleet al [1] and subsequently used by numerous researchers[2ndash12] While material properties kinetic constants andother associated properties change depending on the exactchemical composition of the electrodes (aswill be pointed outlater) the model proposed by Doyle et al is general enoughto be applicable to any lithium ion battery since it is basedon basic conservation laws The governing equations areequations of conservation of mass (of lithium ions) chargeand energy Under the assumption of electroneutrality thecharge conservation equation reduces to a current conser-vation equation In general vector notations the governingequations are written as

120597

120597119905

(120576119890119862119890) = nabla sdot (120576

119890119863

eff119890

nabla119862119890) +

1 minus 119905+

119865

119895Li (1)


nabla120601119890) + nabla sdot (120581

eff119863

nabla ln119862119890) = minus119895Li (2)


nabla120601119904) = 119895Li (3)

120597

120597119905

(120588119888119901119879) = nabla sdot (119896

effnabla119879) + 119902 (4)





eff119890


119863eff119890

= 11986311989012057615

119890 (5)




119890and 120601





120581eff119863

=

2119877119879120581eff

119865

(119905+

minus 1) (1 +

120597 ln119891

120597 ln119862119890

) (6)



119895Li = 1198861199041198940

[exp

120572119886119865

119877119879

(120578 minus

119877SEI119886119904

119895Li)

minus exp

minus120572119888119865

119877119879

(120578 minus

119877SEI119886119904

119895Li)]

(7)



is given by

1198940

= 1198960119862120572119886

119890(119862119904max minus 119862

119904surf)120572119886

119862120572119888

119904surf (8)




119886and 120572







120578 = 120601119904minus 120601119890

minus 119880 (9)



119901 and


119902 = 119886119904119895Li120578 + 119886

119904119895Li119879

120597119880

120597119879

+ 120590eff

nabla120601119904sdot nabla120601119904

+ 120581eff

nabla120601119890


+ 120581eff119863

nabla120601119890


(10)




119890













or Newton cooling



120597119862119904

120597119905

=

119863119904

1199032

120597

120597119903

(1199032120597119862119904

120597119903

) (11)



120597119862119904

120597119903

10038161003816100381610038161003816100381610038161003816119903=0

= 0 (12a)

minus119863119904

120597119862119904

120597119903

10038161003816100381610038161003816100381610038161003816119903=119903119904

=

119895Li119886119904119865

(12b)


119904)The active


119886119904

=

3120576119904

119903119904

(13)





119862119904surf minus 119862

119904avg =

minus119895Li5119886119904119865119863119904

[1 minus exp(minus

20

3

radic119863119904119905

119903119904

)] (14)








TDMA solver


Converged No

Out

er it

erat

ion

Yes

New

tim

e ste

p

TDMA solver

TDMA solver


TDMA solver












119881cell = 120601119904

1003816100381610038161003816119909=119871119899+119871119904+119871119901

minus 120601119904

1003816100381610038161003816119909=0

minus


119868app (15)








120581 = 158 times 10minus4

119862119890exp[085(

119862119890

1000

)

14

] (16)




Thickness 119871119899 119871119904 119871119901


1199041 120583m NA 1 120583m



119904058 NA 05



3molm3239 times 10

3molm3


3molm312 times 10

3molm312 times 10

3molm3



138 times 10minus4


(Am2)sdot(m3mol)32



1199042 times 10



1198902 times 10



+0363 0363 0363




119880119899

= 800229 + 50647119909 minus 1257811990912


119909minus1


11990932

minus 046016 exp [15 (006 minus 119909)]


119880119901

= 866811199106

minus 35771199105

+ 613891199104

minus 555651199103

+ 281061199102

minus 76648119910

minus 030987 exp (5657119910115

) + 131983

(17)






0 10 20 30 40 50 60

10minus4

10minus3

10minus2

10minus1

100

Nor

mal

ized

resid

ual

Iterations

Ce

Cs

120601s120601e





3

35

4

Volta

ge (V

)

ExperimentModel

(a)


3

35

4

Volta

ge (V

)

ExperimentModel

(b)







0 20 40 60 80 100X (120583m)

1180

1200

1220

Elec

troly

te p

hase

conc

entr

atio

n (m

olm

3)

Separator

10 s1800 s3600 s

(a)

0 20 40 60 80 100X (120583m)

10 s1800 s3600 s

Separator

212

214

216

218

Elec

troly

te p

hase

pot

entia

l (V

)

(b)

0 20 40 60 80 100X (120583m)

10 s1800 s3600 s

Separator

minus00002

minus00001

0

00001

00002

Ove

rpot

entia

l (V

)

(c)



119909C6and




200 400 600 800Time (s)

0

10

20

30

40

50

60

Curr

ent (

A)

Current

34

35

36

37

38

39

Volta

ge (V

)Voltage










0 20 40 60 80 100X (120583m)

10 s450 s900 s

280

285

290

295

300

Tem

pera

ture

(K)




Nomenclature















Greek









phase (Smminus1)

Acknowledgment


References


































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of





eff119890


119863eff119890

= 11986311989012057615

119890 (5)




119890and 120601





120581eff119863

=

2119877119879120581eff

119865

(119905+

minus 1) (1 +

120597 ln119891

120597 ln119862119890

) (6)



119895Li = 1198861199041198940

[exp

120572119886119865

119877119879

(120578 minus

119877SEI119886119904

119895Li)

minus exp

minus120572119888119865

119877119879

(120578 minus

119877SEI119886119904

119895Li)]

(7)



is given by

1198940

= 1198960119862120572119886

119890(119862119904max minus 119862

119904surf)120572119886

119862120572119888

119904surf (8)




119886and 120572







120578 = 120601119904minus 120601119890

minus 119880 (9)



119901 and


119902 = 119886119904119895Li120578 + 119886

119904119895Li119879

120597119880

120597119879

+ 120590eff

nabla120601119904sdot nabla120601119904

+ 120581eff

nabla120601119890


+ 120581eff119863

nabla120601119890


(10)




119890













or Newton cooling



120597119862119904

120597119905

=

119863119904

1199032

120597

120597119903

(1199032120597119862119904

120597119903

) (11)



120597119862119904

120597119903

10038161003816100381610038161003816100381610038161003816119903=0

= 0 (12a)

minus119863119904

120597119862119904

120597119903

10038161003816100381610038161003816100381610038161003816119903=119903119904

=

119895Li119886119904119865

(12b)


119904)The active


119886119904

=

3120576119904

119903119904

(13)





119862119904surf minus 119862

119904avg =

minus119895Li5119886119904119865119863119904

[1 minus exp(minus

20

3

radic119863119904119905

119903119904

)] (14)








TDMA solver


Converged No

Out

er it

erat

ion

Yes

New

tim

e ste

p

TDMA solver

TDMA solver


TDMA solver












119881cell = 120601119904

1003816100381610038161003816119909=119871119899+119871119904+119871119901

minus 120601119904

1003816100381610038161003816119909=0

minus


119868app (15)








120581 = 158 times 10minus4

119862119890exp[085(

119862119890

1000

)

14

] (16)




Thickness 119871119899 119871119904 119871119901


1199041 120583m NA 1 120583m



119904058 NA 05



3molm3239 times 10

3molm3


3molm312 times 10

3molm312 times 10

3molm3



138 times 10minus4


(Am2)sdot(m3mol)32



1199042 times 10



1198902 times 10



+0363 0363 0363




119880119899

= 800229 + 50647119909 minus 1257811990912


119909minus1


11990932

minus 046016 exp [15 (006 minus 119909)]


119880119901

= 866811199106

minus 35771199105

+ 613891199104

minus 555651199103

+ 281061199102

minus 76648119910

minus 030987 exp (5657119910115

) + 131983

(17)






0 10 20 30 40 50 60

10minus4

10minus3

10minus2

10minus1

100

Nor

mal

ized

resid

ual

Iterations

Ce

Cs

120601s120601e





3

35

4

Volta

ge (V

)

ExperimentModel

(a)


3

35

4

Volta

ge (V

)

ExperimentModel

(b)







0 20 40 60 80 100X (120583m)

1180

1200

1220

Elec

troly

te p

hase

conc

entr

atio

n (m

olm

3)

Separator

10 s1800 s3600 s

(a)

0 20 40 60 80 100X (120583m)

10 s1800 s3600 s

Separator

212

214

216

218

Elec

troly

te p

hase

pot

entia

l (V

)

(b)

0 20 40 60 80 100X (120583m)

10 s1800 s3600 s

Separator

minus00002

minus00001

0

00001

00002

Ove

rpot

entia

l (V

)

(c)



119909C6and




200 400 600 800Time (s)

0

10

20

30

40

50

60

Curr

ent (

A)

Current

34

35

36

37

38

39

Volta

ge (V

)Voltage










0 20 40 60 80 100X (120583m)

10 s450 s900 s

280

285

290

295

300

Tem

pera

ture

(K)




Nomenclature















Greek









phase (Smminus1)

Acknowledgment


References


































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of











or Newton cooling



120597119862119904

120597119905

=

119863119904

1199032

120597

120597119903

(1199032120597119862119904

120597119903

) (11)



120597119862119904

120597119903

10038161003816100381610038161003816100381610038161003816119903=0

= 0 (12a)

minus119863119904

120597119862119904

120597119903

10038161003816100381610038161003816100381610038161003816119903=119903119904

=

119895Li119886119904119865

(12b)


119904)The active


119886119904

=

3120576119904

119903119904

(13)





119862119904surf minus 119862

119904avg =

minus119895Li5119886119904119865119863119904

[1 minus exp(minus

20

3

radic119863119904119905

119903119904

)] (14)








TDMA solver


Converged No

Out

er it

erat

ion

Yes

New

tim

e ste

p

TDMA solver

TDMA solver


TDMA solver












119881cell = 120601119904

1003816100381610038161003816119909=119871119899+119871119904+119871119901

minus 120601119904

1003816100381610038161003816119909=0

minus


119868app (15)








120581 = 158 times 10minus4

119862119890exp[085(

119862119890

1000

)

14

] (16)




Thickness 119871119899 119871119904 119871119901


1199041 120583m NA 1 120583m



119904058 NA 05



3molm3239 times 10

3molm3


3molm312 times 10

3molm312 times 10

3molm3



138 times 10minus4


(Am2)sdot(m3mol)32



1199042 times 10



1198902 times 10



+0363 0363 0363




119880119899

= 800229 + 50647119909 minus 1257811990912


119909minus1


11990932

minus 046016 exp [15 (006 minus 119909)]


119880119901

= 866811199106

minus 35771199105

+ 613891199104

minus 555651199103

+ 281061199102

minus 76648119910

minus 030987 exp (5657119910115

) + 131983

(17)






0 10 20 30 40 50 60

10minus4

10minus3

10minus2

10minus1

100

Nor

mal

ized

resid

ual

Iterations

Ce

Cs

120601s120601e





3

35

4

Volta

ge (V

)

ExperimentModel

(a)


3

35

4

Volta

ge (V

)

ExperimentModel

(b)







0 20 40 60 80 100X (120583m)

1180

1200

1220

Elec

troly

te p

hase

conc

entr

atio

n (m

olm

3)

Separator

10 s1800 s3600 s

(a)

0 20 40 60 80 100X (120583m)

10 s1800 s3600 s

Separator

212

214

216

218

Elec

troly

te p

hase

pot

entia

l (V

)

(b)

0 20 40 60 80 100X (120583m)

10 s1800 s3600 s

Separator

minus00002

minus00001

0

00001

00002

Ove

rpot

entia

l (V

)

(c)



119909C6and




200 400 600 800Time (s)

0

10

20

30

40

50

60

Curr

ent (

A)

Current

34

35

36

37

38

39

Volta

ge (V

)Voltage










0 20 40 60 80 100X (120583m)

10 s450 s900 s

280

285

290

295

300

Tem

pera

ture

(K)




Nomenclature















Greek









phase (Smminus1)

Acknowledgment


References


































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of



TDMA solver


Converged No

Out

er it

erat

ion

Yes

New

tim

e ste

p

TDMA solver

TDMA solver


TDMA solver












119881cell = 120601119904

1003816100381610038161003816119909=119871119899+119871119904+119871119901

minus 120601119904

1003816100381610038161003816119909=0

minus


119868app (15)








120581 = 158 times 10minus4

119862119890exp[085(

119862119890

1000

)

14

] (16)




Thickness 119871119899 119871119904 119871119901


1199041 120583m NA 1 120583m



119904058 NA 05



3molm3239 times 10

3molm3


3molm312 times 10

3molm312 times 10

3molm3



138 times 10minus4


(Am2)sdot(m3mol)32



1199042 times 10



1198902 times 10



+0363 0363 0363




119880119899

= 800229 + 50647119909 minus 1257811990912


119909minus1


11990932

minus 046016 exp [15 (006 minus 119909)]


119880119901

= 866811199106

minus 35771199105

+ 613891199104

minus 555651199103

+ 281061199102

minus 76648119910

minus 030987 exp (5657119910115

) + 131983

(17)






0 10 20 30 40 50 60

10minus4

10minus3

10minus2

10minus1

100

Nor

mal

ized

resid

ual

Iterations

Ce

Cs

120601s120601e





3

35

4

Volta

ge (V

)

ExperimentModel

(a)


3

35

4

Volta

ge (V

)

ExperimentModel

(b)







0 20 40 60 80 100X (120583m)

1180

1200

1220

Elec

troly

te p

hase

conc

entr

atio

n (m

olm

3)

Separator

10 s1800 s3600 s

(a)

0 20 40 60 80 100X (120583m)

10 s1800 s3600 s

Separator

212

214

216

218

Elec

troly

te p

hase

pot

entia

l (V

)

(b)

0 20 40 60 80 100X (120583m)

10 s1800 s3600 s

Separator

minus00002

minus00001

0

00001

00002

Ove

rpot

entia

l (V

)

(c)



119909C6and




200 400 600 800Time (s)

0

10

20

30

40

50

60

Curr

ent (

A)

Current

34

35

36

37

38

39

Volta

ge (V

)Voltage










0 20 40 60 80 100X (120583m)

10 s450 s900 s

280

285

290

295

300

Tem

pera

ture

(K)




Nomenclature















Greek









phase (Smminus1)

Acknowledgment


References


































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of




Thickness 119871119899 119871119904 119871119901


1199041 120583m NA 1 120583m



119904058 NA 05



3molm3239 times 10

3molm3


3molm312 times 10

3molm312 times 10

3molm3



138 times 10minus4


(Am2)sdot(m3mol)32



1199042 times 10



1198902 times 10



+0363 0363 0363




119880119899

= 800229 + 50647119909 minus 1257811990912


119909minus1


11990932

minus 046016 exp [15 (006 minus 119909)]


119880119901

= 866811199106

minus 35771199105

+ 613891199104

minus 555651199103

+ 281061199102

minus 76648119910

minus 030987 exp (5657119910115

) + 131983

(17)






0 10 20 30 40 50 60

10minus4

10minus3

10minus2

10minus1

100

Nor

mal

ized

resid

ual

Iterations

Ce

Cs

120601s120601e





3

35

4

Volta

ge (V

)

ExperimentModel

(a)


3

35

4

Volta

ge (V

)

ExperimentModel

(b)







0 20 40 60 80 100X (120583m)

1180

1200

1220

Elec

troly

te p

hase

conc

entr

atio

n (m

olm

3)

Separator

10 s1800 s3600 s

(a)

0 20 40 60 80 100X (120583m)

10 s1800 s3600 s

Separator

212

214

216

218

Elec

troly

te p

hase

pot

entia

l (V

)

(b)

0 20 40 60 80 100X (120583m)

10 s1800 s3600 s

Separator

minus00002

minus00001

0

00001

00002

Ove

rpot

entia

l (V

)

(c)



119909C6and




200 400 600 800Time (s)

0

10

20

30

40

50

60

Curr

ent (

A)

Current

34

35

36

37

38

39

Volta

ge (V

)Voltage










0 20 40 60 80 100X (120583m)

10 s450 s900 s

280

285

290

295

300

Tem

pera

ture

(K)




Nomenclature















Greek









phase (Smminus1)

Acknowledgment


References


































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


0 10 20 30 40 50 60

10minus4

10minus3

10minus2

10minus1

100

Nor

mal

ized

resid

ual

Iterations

Ce

Cs

120601s120601e





3

35

4

Volta

ge (V

)

ExperimentModel

(a)


3

35

4

Volta

ge (V

)

ExperimentModel

(b)







0 20 40 60 80 100X (120583m)

1180

1200

1220

Elec

troly

te p

hase

conc

entr

atio

n (m

olm

3)

Separator

10 s1800 s3600 s

(a)

0 20 40 60 80 100X (120583m)

10 s1800 s3600 s

Separator

212

214

216

218

Elec

troly

te p

hase

pot

entia

l (V

)

(b)

0 20 40 60 80 100X (120583m)

10 s1800 s3600 s

Separator

minus00002

minus00001

0

00001

00002

Ove

rpot

entia

l (V

)

(c)



119909C6and




200 400 600 800Time (s)

0

10

20

30

40

50

60

Curr

ent (

A)

Current

34

35

36

37

38

39

Volta

ge (V

)Voltage










0 20 40 60 80 100X (120583m)

10 s450 s900 s

280

285

290

295

300

Tem

pera

ture

(K)




Nomenclature















Greek









phase (Smminus1)

Acknowledgment


References


































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


0 20 40 60 80 100X (120583m)

1180

1200

1220

Elec

troly

te p

hase

conc

entr

atio

n (m

olm

3)

Separator

10 s1800 s3600 s

(a)

0 20 40 60 80 100X (120583m)

10 s1800 s3600 s

Separator

212

214

216

218

Elec

troly

te p

hase

pot

entia

l (V

)

(b)

0 20 40 60 80 100X (120583m)

10 s1800 s3600 s

Separator

minus00002

minus00001

0

00001

00002

Ove

rpot

entia

l (V

)

(c)



119909C6and




200 400 600 800Time (s)

0

10

20

30

40

50

60

Curr

ent (

A)

Current

34

35

36

37

38

39

Volta

ge (V

)Voltage










0 20 40 60 80 100X (120583m)

10 s450 s900 s

280

285

290

295

300

Tem

pera

ture

(K)




Nomenclature















Greek









phase (Smminus1)

Acknowledgment


References


































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


200 400 600 800Time (s)

0

10

20

30

40

50

60

Curr

ent (

A)

Current

34

35

36

37

38

39

Volta

ge (V

)Voltage










0 20 40 60 80 100X (120583m)

10 s450 s900 s

280

285

290

295

300

Tem

pera

ture

(K)




Nomenclature















Greek









phase (Smminus1)

Acknowledgment


References


































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of




Greek









phase (Smminus1)

Acknowledgment


References


































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of










Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of

268747

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