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Multi-Domain Modeling of Lithium-Ion Batteries EncompassingMulti-Physics in Varied Length Scales

Gi-Heon Kim,*,z Kandler Smith,* Kyu-Jin Lee,* Shriram Santhanagopalan,*

and Ahmad Pesaran

National Renewable Energy Laboratory, Golden, Colorado 80401, USA

This paper presents a general multi-scale multi-physics lithium-ion battery model framework, the Multi-Scale Multi-Dimensionalmodel. The model introduces multiple coupled computational domains to resolve the interplay of lithium-ion battery physics in var-ied length scales. Model geometry decoupling and domain separation for the physicochemical process interplay are valid where thecharacteristic time or length scale is segregated. Assuming statistical homogeneity for repeated architectures typical of lithium-ionbattery devices is often adequate and effective for modeling submodel geometries and physics in each domain. The modularizedhierarchical architecture of the model provides a flexible and expandable framework facilitating modeling of the multiphysics behav-ior of lithium-ion battery systems. In this paper, the Multi-Scale Multi-Dimensional model is introduced and applied to a model anal-ysis that resolves electrochemical-, electrical-, and thermal-coupled physics in large-format stacked prismatic cell designs.VC 2011 The Electrochemical Society. [DOI: 10.1149/1.3597614] All rights reserved.

Manuscript submitted March 17, 2011; revised manuscript received May 10, 2011. Published June 17, 2011.

Lithium-ion batteries (LIBs) have been widely accepted andused in consumer applications such as laptop computers and perso-

nal communication and entertainment devices because of their highpower and energy density. LIBs are also getting increasing attentionfor electrical energy storage in electric drive vehicles (EDVs). LIBsfor EDVs are much larger in capacity and physical size than thoseused in consumer applications. However, scale-up of the batteriesraises the complexity of physical phenomena that do not play a sig-nificant role in small battery systems. Since the physical phenomenaoccur over a wide range of time and length scales from atomic var-iations to heat transfer over an entire device, it is important to under-stand how these different mechanisms are related. For LIB cellmanufacturers, the lack of understanding of how macroscopicdesign features impact local microscopic electrochemical processeshas been one of the major obstacles in scaling up their consumerelectronics cell technologies to build large automotive batteries. Forautomotive companies and system integrators, the lack of a battery

model to use in their established computer-aided engineering designprocesses in order to predict thermal, electrical, electrochemical,and mechanical response of the battery under various system opera-tion strategies and management system designs is recognized as anurgent barrier to overcome for expediting EDV development andproduction.

Model-based investigations promote theoretical understanding ofbattery physics beyond what is possible from experiments only.Modeling of a system where the response is critically affected byinteraction between the physics at varied scales, however, is chal-lenging in terms of computational cost. In the early 1990s, Newmanand his colleagues suggested a LIB model utilizing porous electrodetheory.1 The model solves lithium diffusion dynamics and chargetransfer kinetics to predict the electrical response of a cell in apaired intercalation electrode system. This model has been widely

used in academia and industry to describe the performance of a LIBbased on material properties and electrode design.

Newmans approach toward paired intercalation composite elec-trodes is fairly adequate for prediction of small battery behavior. Inlarge-format cells, however, non-uniformity of the electric potentialalong the current collectors in cell composites and the temperaturethroughout the cell volume become severe enough to impact batteryresponses. The same is true for large battery packs consisting ofmultiple cells. Potential and temperature imbalances cause certainlocations in a cell to be cycled more than the rest of the cell. Thelocal excess use generates more heat and stress, causing severe deg-radation in the performance and life of the cell. Therefore, for a bet-ter understanding of the behavior of a large LIB system, the trans-

port of electrical current and heat must be evaluated not only in thecomposite electrode matrices at the length scale of electrode pair,

but also across the highly anisotropic cell composite medium at thecell-dimension length scale.

Through the multi-year effort supported by U.S. Department ofEnergy (DOE), the National Renewable Energy Laboratory (NREL)has developed a modeling framework for predictive computer simu-lation of LIBs known as the Multi-Scale Multi-Dimensional(MSMD) model that addresses the interplay among the physics invaried scales.27 In this paper, we introduce NRELs MSMD modeland present an example model analysis that evaluates largeformat,stacked, prismatic cell designs.

Multi-Domain Model Framework

It is computationally expensive to perform a predictive numeri-cal simulation of battery performance, degeneration, and safetyresponse while capturing the interactive coupling among the differ-ent physicochemical processes in varied characteristic length andtime scales in complex geometries using a single computational do-main. The MSMD model achieves computational efficiency forresolving multi-physics interactions occurring over a wide range oflength scales by introducing separate solution domains, at the parti-cle, electrode, and cell level. Figure 1 shows the conceptual diagramof the multi-scale multi-domain approach used in the MSMD model.Each domain uses its own independent coordinate system for spatialdiscretization of the variables solved in that domain. Separation ofthe model domain and adoption of the statistical homogeneityassumption are enabled based on the intrinsic nature of typical LIBsystems where physics with significant time-scale differences

interplay.The MSMD model framework has a hierarchical structure. Solu-tion variables defined in a lower hierarchy domain have finer spatialresolution than those solved in a higher hierarchy domain. Conse-quently, physical and chemical quantities of smaller length-scalephysics are evaluated with a finer spatial resolution to resolve theimpact of corresponding small-scale geometry. Larger-scale quanti-ties are calculated with coarser spatial resolution, eliminating thecomplications of the smaller-scale geometric features. In addition tocomputational efficiency, the MSMD approach provides a modular-ized framework, enabling model flexibility by allowing multiplesubmodel options with arbitrary physical and computational com-plexities. This model flexibility comes from the fact that each levelsubmodel is independent from the choice of models and solverschemes used in other domains as long as the model input fromother domains is properly transferred through the specified inter-do-main information exchange. The modularized MSMD framework is

*Electrochemical Society Active Member.z E-mail: [email protected]

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A955

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well suited for multi-physics investigations as one can easily addnew physics of interest or drop physics of insignificant phenomena.

Multiscale model for a lithium-ion battery.To model the per-formance of LIBs, the MSMD model adapts the electrochemical andthermal physics introduced by other researchers.1,8,9 Charge transferkinetic reactions are solved at the electrode-electrolyte interface.

Lithium transport in active particles is modeled with a diffusionmechanism. Lithium-ion migration and diffusion through a liquidelectrolyte are evaluated. Charge balances are resolved in the posi-tive and negative solid matrices and in the liquid electrolyte, respec-tively. Temperature is solved for in the overall cell and pack geome-try. The model predicts the electrical and thermal behavior of a LIBcell in operation for given electrical loads and thermal boundaryconditions. As mentioned earlier, the battery geometry is resolvedinto three coupled domains: a) the particle domain, b) the electrodedomain, and c) the cell domain. In the following sections, the detailsabout the physics represented in each level domain and the inter-do-main coupling of the solution variables for general modeling of aLIB will be described.

Particle domain model. In conventional LIB designs, lithium-

hosting electrode materials are prepared as particulates with con-trolled shape and size. Solid-phase lithium transport is one of therate-determining steps for overall LIB performance and is affectedby the particle geometry as well as thermodynamic and kinetic prop-erties of the materials. Using a separate coordinate system for solid-phase diffusion from that for other variables has been well practicedwith the so-called pseudo-2d model by Newman and others.10,11 Inthe MSMD model, charge transfer kinetics and solid-phase lithiumdiffusion are solved for in the particle domain. The schematic inFig. 2a shows the solution variables and the input and the output forthe particle domain submodel. Field variables fed in from thehigher-level electrode submodel are treated as averaged lumped val-ues in the particle domain. Those inputs from the electrode domainsubmodel are liquid-phase concentration, ce, electrical potential atthe negative or positive electrode, s,a or s,c, and in the liquid

phase, e, and temperature, T. This approach is reasonable wheretransport in the liquid phase is much faster than in the solid phase:e.g., De >> Ds. Transfer current density at the particle surface, i

00

n,and the lithium concentration within a solid electrode particle, cs,are the main solution variables in the particle domain. Surface heatflux at the electrode-electrolyte interface, q

00

n, including heat fromthe charge transfer reaction, is a major heat source in this domain,while other volumetric heat sources, q

000

n , such as the heat of mix-ing12,13 within a particle, may also contribute to the total heat gener-ation. Additional complexities of particle physics, such as mechani-cal failure and surface kinetics, can be modeled in this domain asnecessary.

Electrode domain model.The active particles are typicallymixed with a conductive agent and a polymer binder, and thencoated on thin metal current collector sheets to form porous com-posite electrodes with a good electrical network and mechanical in-

tegrity through the solid matrix. In a conventional electrode pair, athin polymeric porous separator is inserted in between the cathodeand anode electrode composites to prevent an electronic shortbetween the pair of electrodes. The pore structures of the compositeelectrodes and separator, as well as wetting of the pore surfaces,impact lithium ion transport through the liquid electrolyte. The elec-trode domain submodel resolves the charge balance across the com-posite electrode pair in the solid matrices of electrode composites

and in the liquid electrolyte and lithium-ion transport in the liquidelectrolyte. The schematic in Fig. 2b shows the solution variablesand the submodel input and output for the electrode domain submo-del. Electrostatic potential along the current collectors, U and U-,and temperature, T, are input from the cell domain submodel andtreated as averaged lumped values in the electrode domain. Usingthe lumped value of the electrode potentials at the interface withcurrent collectors without spatial dependence in the electrode do-main coordinate system is reasonable because the current collectorsheets on the boundaries of both the positive and negative electrodesare electrically much more conductive than the composite electrodelayers. For the same reason, an electrode domain submodel is oftenrepresented as a one-dimensional problem with the in-plane statisti-cal homogeneity assumption. The main solution variables in theelectrode domain are lithium concentration in the liquid electrolyte,c

e, and electric potentials in negative- and positive-biased solid mat-

rices and liquid electrolyte, s,a, s,c, and e. The volumetric

Figure 1. (Color online) Conceptual diagram for the multi-scale multi-do-main approach of the MSMD model.

Figure 2. (Color online) Summary of the model solution variables in eachcomputational domain and the coupling variables exchanged between the ad-jacent length scale domains.

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reaction current at the electrode composite volume, j000

x , is evaluatedwith averaged charge transfer current density over the active particlesurfaces, i

00

n , calculated in a lower- level hierarchy particle domainsubmodel. Current density at the electrode composite boundary atthe current collector interface, i

00

x, is defined either as flux of chargeout from the electrode into the positive current collector or as theinflux of charge from the negative current collector. Volumetric

heat generation in the electrode domain, q

000

x , includes joule heat dueto current flow in solid composite electrodes and liquid electrolyte,as well as heat sources averaged at the lower hierarchy particle do-main, q

00

n, and q000

n . Heat generation due to the contact resistancebetween electrode and metal current collector foil surface can beconsidered a surface heat source term, q

00

x, in this domain.

Cell domain model. Unit assemblies of paired electrode layersare stacked or wound to build prismatic or cylindrical cell devices.In this stage, major cell design issues are related to thermal and elec-trical optimization of internal cell configurations. As the cellcapacity and size increase, the spatial non-uniformity of the temper-ature and the electric potentials becomes sufficiently large to causethe performance and life of a battery to degenerate. Building a mod-ule or a pack with multiple individual cells is another extension of

designing a thermal and electrical configuration for a larger batterysystem, often involving external controls. The schematic in Fig. 2cshows the solution variables and the submodel input and output forthe cell domain model. The spatial distribution of temperature, T,and electric potential at the current collectors and other passivecomponents, U and U-, in a system must be resolved with the sys-tem geometry and the boundary conditions. Therefore, they aresolved in the cell domain to resolve macroscopic electrical and ther-mal transport with a given electrical load and thermal ambient con-ditions. On the other hand, the volumetric electric current source atthe positive current collector phase, j

000

H, in the cell composite vol-ume, which comprises composite electrodes, separator, electrolyte,and current collector foils, is evaluated using the average electrodeplate current density, i

00

x, passed from the electrode domain submo-del. The volumetric heat generation in a cell domain, q

000

H, includes

joule heat for carrying the electrical current in the current collectorphase in cell composites and in other passive components of a cell,in addition to the contribution from the heat sources passed from thelower-hierarchy electrode domain, q

00

x, and q000

x .

Inter-domain coupling. In the MSMD model framework, mul-tiple physics models defined in the corresponding domains aresolved simultaneously through two-way inter-domain couplings, asdescribed in Fig. 2. In general, information from a higher hierarchydomain to a lower hierarchy domain is delivered using field varia-bles. When upper hierarchy domain field variables are given as inputto a lower hierarchy domain submodel, they are considered as aver-aged lumped values in the lower hierarchy domain. In other words,inputs from a higher hierarchy domain do not have spatial depend-ence in the coordinate system of the lower hierarchy domain. On the

other hand, information delivery from a lower to a higher hierarchydomain is done through source terms. The volumetric or surfacesources evaluated in a lower hierarchy domain are averaged overthat domains geometry, eliminating the coordinate system depend-ence before they are passed to a higher hierarchy domain. The aver-aged sources delivered to a higher hierarchy domain are thenconverted to a volumetric source term in the higher hierarchy sub-model. For example, the transfer current density, i

00

n, is averagedover the electrode-electrolyte interfaces in a particle domain

i00

n x;H

An

i00

nn; x;HdAn

An[1]

where An is electrode-electrolyte interface area of the particle do-main. The average transfer current density, i

00

n, is delivered to an

electrode domain submodel and then converted to a volumetric

transfer reaction current at the electrode composite volume usingEq. 2

j000

x x;H i00

n x;Has;x [2]

where the specific area of active interface between the electrode par-ticles and the electrolyte in the electrode composite volume, as,x, is

an electrode domain parameter. Consequently, the current density atthe electrode composite boundary, i00

x, is averaged over the area ofinterface with the current collector in the electrode domain

i00

x H

Ax

i00

xx;HdAxAx

[3]

where Ax is the electrode composite current collector interface areaof the electrode domain geometry, and Ax is the electrode plate area.They are identical in many model approaches. The average elec-trode plate current density, i

00

x , is delivered to a cell domain, andthen converted to a volumetric current source at the current collectorphase of the cell composite jelly volume using Eq. 4

j000

HH i00

x Has;H [4]

where the specific electrode plate area in the cell composite volume,as,H, is a cell domain parameter. The expressions shown in Eqs. 2and 4 are useful to keep consistency and simplicity of formulationfor coupling quantities of the current source in each domain. Thecoupling of the heat sources is also done in a similar manner. Whilethe actual reaction current source exists only at the electrodeelectrolyte interfaces, heat release and absorption occur due to vari-ous mechanisms12,14,15 in all length-scales. Therefore, heat sourcesshould be properly modeled in each hierarchical submodel. Surfaceheat sources at electrode-electrolyte interfaces, q

00

n, and volumetricheat sources within a particle volume, q

000

n , are averaged over the par-ticle domain geometries and delivered to the electrode domainsubmodel

q00

nx;H An q

00

nn; x;HdAnAn

[5]

q000

n x;H

Vn

q000

n n; x;HdVn

Vn[6]

where An and Vn are the electrodeelectrolyte interface area and thevolume of the particle domain geometry, respectively. In the elec-trode domain, a volumetric heat source that originates from the par-ticle domain is evaluated using Eq. 7

q000

x;nx;H q00

nx;Has;x q000

n x;Hes [7]

where as,x is the specific area of active interface between electrodeparticles and electrolyte, and es is the volume fraction of active par-ticles in the composite electrode volume. The volumetric heat forcontribution from the particle domain heat sources evaluated withEq. 7 is added into the net electrode domain volumetric heat sourceterm with other heat sources, such as joule heat due to the electroniccurrent flow in the composite electrode matrix and the ionic currentflow in the electrolyte

q000

x x;H q000

x;nx;H X

k

q000

x;kx;H [8]

Surface and volume heat sources at the electrode domain, q00

x and,q

000

x , are averaged over the geometries of the electrode domain anddelivered to the cell domain

q

00

xH Ax q00xx;HdAx

Ax [9]

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q000

x H

Vx

q000

x x;HdVx

Vx[10]

where Vx is volume of the electrode domain geometry. A cell do-main submodel uses these terms to evaluate the heat source from thelower hierarchical domains using Eq. 11

q000

H;xH q00

xHas;H q000

x Heasc [11]

where as,H is the specific electrode plate area, and easc is the volumefraction occupied by the electrode composites and the separator in

the cell composite volume. The net volumetric heat source at thecell domain volume is the sum of q

000

H;x and other volumetric heatsources such as electrical heating at current collector foils and otherpassive conductors

q000

HH q000

H;xH X

k

q000

H;kH [12]

The schematics shown in Fig. 2 summarize the model solution vari-ables in each computational domain and the coupling variablesexchanged between the adjacent length-scale domains.

Flexible submodel and solver choices. The MSMD model is ageneric model framework allowing the submodel and the solver ineach domain to be independently chosen. As long as the predefined

coupling variables are properly transferred between the submodelsat the designated physical time, the choice of the model equations,geometries, spatial discretization and the solution schemes, and thetime step size in each domain is independent of those in the otherdomains. During the exchange of model inputs between submodels,information regarding the geometry or the computational grid is nottransferred. In this regard, the MSMD model approach decouples

the geometries of each domain, while the physics solved in thedomains are still coupled. Therefore, the rigorousness of the submo-del solved in each domain can be independently adjusted for thepurpose of modeling. As shown in Fig. 2, information is exchangedonly between the adjacent scale domain submodels. For example, anMSMD cell domain submodel does not directly communicate with aparticle domain submodel. This enhances the flexibility in the total

number of domains simultaneously solved in the MSMD model.There have been efforts to solve charge transfer kinetics, particlelithium diffusion, and transports across electrode composites in asingle computational domain to explicitly resolve the influence ofcomposite electrode morphology and composition.16,17 An MSMDcell domain submodel could be linked to this type of single-domainmesoscale geometry electrode submodelat the expense of highcomputational costor alternately coupled with a simple circuitmodel to mimic electrode voltage-current response, maximizingcomputational efficiency.

Submodel Choice for Large-Capacity Stacked Prismatic CellPerformance Prediction

Macroscopic cell design features regarding the thermal and elec-trical configuration, such as the number of unit stacks of the elec-trode pair, area of the unit electrode stack layer, thickness of the cur-rent collector foils, size and location of current tabs, electric busgeometries, and external heat transfer conditions, are known togreatly impact the microscopic electrochemical processes and deg-radation mechanisms, and, in consequence, the overall cell perform-ance and life, especially in large battery systems. Therefore, forwide acceptance of LIBs in large-capacity applications such ashybrid electric and full electric vehicles, the need to enhance knowl-edge of heat and electric current transport in a LIB system and theirimpacts on the performance, ageing, and safety behavior is critical.In this paper, the MSMD model is employed to perform thermal andelectrical design evaluations for a large-format stacked prismaticcell. Microscopic cell design parameters, including material compo-sitions, electrode loading thicknesses, and porosities, are held con-stant. Rather, the impact of large-format cell design features such as

the location and size of electrical tabs and the electrode area of theunit stack layer are varied.

The schematic in Fig. 3 together with the data in Table I summa-rize the four different cell designs investigated in this study.Because the focus of the study is to evaluate the impact of designfeatures in the cell domain, we use computationally efficient submo-dels and solution schemes in the particle and the electrode domains.The cell domain submodel resolves the complexity of the three-dimensional geometry for the cell designs investigated. The submo-del chosen for the electrode domain is a one-dimensional porouselectrode transport model. The submodel chosen for the particle do-main is a spherical particle model. The temperature dependence ofsome of the physiochemical properties used in this study is consid-ered using a general Arrhenius form

W Wref expEWactR

1Tref

1T

![13]

where Wref is the property value defined at the reference tempera-ture, Tref 298 K. The activation energy, E

Wact, determines the tem-

perature sensitivity of a general physiochemical property, W.

Figure 3. (Color online) Schematic description of the 20-Ah stacked pris-matic cell designs investigated: (a) ND cell, (b) CT cell, (c) ST cell, (d) WS

cell.

Table I. Description of the investigated cell form factor designs for 20-Ah stacked prismatic cell.

Case Description Lx (mm) Ly (mm) Lz (mm) Tab width (mm) Tab configuration

ND Nominal design 200 140 7.5 44 Adjacent tabsCT Counter tab design 200 140 7.5 44 Counter tabsST Small tab design 200 140 7.5 20 Adjacent tabs

WS Wide stack-area design 300 140 5.0 44 Adjacent tabs



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Further details of the governing equations and solution schemesused in each domain are described in the following sections. Toimprove computation speed, a model-reduction scheme18 is adopted tosolve the particle and electrode domain submodels. This reduced-ordermodel describes the electrochemical/thermal governing equations as aquasi-linear system in state-variable model (SVM) form. Appendix Adescribes the SVM model-reduction procedure and the numerical setupused in this study. Appendix B compares numerical simulation resultsfor the present parameters set to a higher-order solution using a com-mercial finite-element method solver. Note that this SVM method isnot an exclusive submodel for the present MSMD model. Other possi-ble methods for fast computation of the particle and electrode domainsubmodels include proper orthogonal decomposition,19 circuit-analogrepresentation,20 or empirical polarization function.21

Submodel choice in particle domain.The particle domain sub-model selected in this study solves lithium diffusion in the solid par-ticle volume and charge-transfer kinetics at the particle surface. TheButler-Volmer kinetic expression provides transfer current densityat the particle surface, where the rate of electrochemical reaction isdriven by the overpotential, g, which is defined as the deviation ofthe potential difference between the solid phase and the liquid phaseat the reaction site from the thermodynamic equilibrium potential, U

i00n i00

o exp aaF

RTg

! exp acF

RTg

!& '[14]

g s e i00

nRfilm U [15]

In Eq. 16, the exchange current density, i00

o, is given as a function ofboth the solid phase and the electrolyte concentrations according to

i00

o ki ce aa cs;max cs;e aa

cs;e ac

i00 ref

o

ce

crefe

aa cs;max cs;ecs;max c

refs;e

aacs;e

crefs;e

ac[16]

where ki is a kinetic rate constant and aa and ac are the anodic and

cathodic transfer coefficients, respectively. In this study, the refer-ence exchange current density at 100% state of charge (SOC), i00 ref

o ,is evaluated as a function of temperature using Eq. 13. The lithiumconcentration in the active particles is obtained using Ficks law ofdiffusion

ocs

ot rn Dsrncs

[17]

with a surface flux boundary condition for the rate of charge transferreaction at the surface

rncsAn

nn i

00

n

DsF[18]

where nn

is an outward surface normal unit vector at the particlesurface. The dependence of lithium diffusivity on the lithium con-

tent or on the mechanical phase of the material is not considered inthis study. Introducing the spherical coordinate system, shown inFig. 4a, in the particle n-domain with symmetry assumption in theinclination and azimuth angles simplifies Eq. 17. The simplifiedone-dimensional form of the equation is shown in Table II, wheren1 r. The solid phase diffusion coefficient, Ds, is treated invariantwith the r-coordinate, even though it varies with temperature; tem-

perature, T, is given a lumped value in the particle domain submodelas are the electrolyte phase lithium concentration, ce, the solid phasepotential, s, and the electrolyte phase potential, e. The surfaceheat source at the active particle surfaces is modeled as shown inEq. 19 considering heat from the charge transfer reaction and theohmic loss at particle surface layer

q00

n i00

n s e U ToU

oT

[19]

Particle volumetric heat sources such as heat from a phase changeor the heat of mixing are ignored in this study. The average currentand heat sources for inter-domain coupling parameters passed to theelectrode domain are simplified from Eq.1, Eqs. 5 and 6 into Eq. 20,Eqs. 21 and 22 in the submodel chosen

i00

n i00

n [20]

q00

n q00

n [21]

q000

n 0 [22]

Submodel choice in electrode domain. A computationallyefficient electrode domain submodel is a preferred choice for thepurposes of this study. A simplified electrode domain transportmodel such as a single-particle model22,23 assuming lumped poten-tials and liquid concentration has been used and performs reason-ably well for thin electrode cells in moderate charge and dischargeconditions. However, in the present work, high pulse response andend of discharge behavior of different cell designs are of interest.

Therefore, the porous electrode model is adopted in the electrodedomain to resolve the charge balance in the solid and liquid phasesand conservation of lithium in the electrolyte phase. Instead, theSVM method is applied, compensating for the computational costfor solving the relatively complex model. The computational do-main for this model is composed of three contiguous volumes wherethe solid phase and the liquid phase are treated as superimposedcontinua as shown in Fig. 4b. Lithium concentration and electricpotential in the liquid electrolyte, ce and e, are evaluated in thecontinuous liquid phase in these three volumes, while negative andpositive electrode potentials in the composite electrodes, s,a ands,c, are solved in the solid phase of the anode and cathode electrodematrix, respectively.

o eece

ot r

xDeff

er

xc

e 1 to

Fj

000

x

ie rxto

F[23]

rxce nx 0 at all boundaries

rx jeffrx/e

rx j

effD rx ln ce

j

000

x 0 [24]

rxe nx 0 at all boundaries

rx reffrxs

j

000

x 0 [25]

s U orU at the interface with a current collector,rxs nx 0 at other boundaries

Incorporating the in-plane statistical homogeneity assumption inthe electrode domain, the one-dimensional forms (x1 x) of thegoverning equations and boundary conditions used in the submodelchoice of this study are summarized in Table II.

Figure 4. Choices of submodel in each domain for large-format stackedprismatic cell design evaluation study presented.



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The volumetric source of the electric charge for the charge trans-fer lithium ion into the liquid phase, j

000

x , is evaluated from i00

n usingEq. 2. U-, U, and T, given from the cell domain, are treated as spa-tial invariants in the electrode domain. In the one-dimensionalmodel, the electronic current density at the boundary of electrodecomposite is evaluated with Eq. 26.

i00

x reffc

os;cox

xlalslc

lalslclals

j000

x dx

reffaos;aox

x0

la0

j000

x dx [26]

The average electrode plate current density, i00

x , the predefined inter-domain coupling term passed to the cell domain submodel, is equiv-alent to i

00

x from Eq. 3 for the one-dimensional electrode model

i00

x i00

x [27]

The averaged heat sources, q00

n and q000

n , delivered from the particledomain submodel are converted into a volumetric heat source in theelectrode domain, q

000

x;n, using Eq. 7. The joule heat for the electriccurrent flow in the presence of the potential gradient in solid matri-ces and in the liquid electrolyte is calculated as

q000

x;X reffrs rs j

effre re jeff

D r ln ce re [28]

The averaged volumetric heat source, q000

x, a quantity for passing to

the cell domain submodel, is evaluated using Eq. 29

q000

x

lalslc0

q000

x;n q000

x;X

dx

la ls lc[29]

The surface heat source at the electrode domain is not considered inthis study

q00

x 0 [30]

Submodel choice in cell domain. For the cell domain sub-model in the present study, the Single Potential-Pair Continuum(SPPC) model was developed to solve for the temperature, T, and apair of electric potentials, U, and U, in a three-dimensional cellgeometry. Through the statistical homogeneity assumption, theSPPC model treats the cell composite jelly volume as a continuumhaving orthotropic thermal and electrical conductivities. Therefore,the layered geometry of the cell jelly is not necessarily resolved in acomputational mesh of the SPPC model. In typical designs, conduc-tion transport in the cell composite volume occurs with distin-guished in-plane and transverse diffusivities. The SPPC modelresolves the anisotropic conduction in the cell composite volumeusing the conductivity tensor, ~c

cij ct cp

etiet

j cpdij [31]

where cp and ct are planar and transversal conductivities, respec-tively, and dij is the Kronecker delta. The unit transverse directionvector is determined as a function of location in the cell compositevolume

Table II. Summary of solution variables and governing equations of the submodel choices.

Domain Solution variable Governing equation

Particle Submodel Choice: 1D spherical particle model

i00

nr Rs;x;X; Y;Z i00

n ki ce aa cs;max cs;e

aa

cs;e

ac

expaaF

RTg

! exp

acF

RTg

!& '

csr;x;X; Y;Zocs

ot

Ds

r2o

orr2ocs

or

;ocs

or

r0

0,ocs

or

rRs

i00n

DsF

Electrode Submodel Choice: 1D porous electrode model

cex;X; Y;Zo eece

ot

o

oxDeffe

o

oxce

1 toF

j000

x i

00

e

F

otoox;

oce

ox

x0

0,oce

ox

xlals lc

0

ex;X; Y;Zo

oxjeff

o

oxe

o

oxj

effD

o

oxln ce

j

000

x 0

oeox

x0

0,oeox

xlals lc

0

s;ax;X; Y;Zo

oxreff o

oxs

j

000

x 0; s;ax0

U,os;aox

xla

0

s;cx;X; Y;Z

o

ox reff o

ox s

j000

x 0;

os;cox

xla ls

0, s;cxla lslc U

Cell Submodel Choice: 3D Single Potential-Pair Continuum (SPPC) model

UX; Y;Zo

oXer

oU

oX

o

oYer

oU

oY

j

000

H 0

UX; Y;Zo

oXer

oU

oX

o

oYer

oU

oY

j

000

H 0

TX; Y;Zo qcpT ot

o

oXkp

oT

oX

o

oYkpoT

oY

o

oZktoT

oZ

q

000

H



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etH et1 i1 et2 i2 e

t3 i3 [32]

where ii is the basis ofH space, and eti is the corresponding compo-

nent of the transverse direction vector. In order to apply the SPPCmodel, the following conditions should be met in the system ofinterest:

i) In a finite volume of cell composite jelly in H space, et can beuniquely determined.

ii) In a finite volume of cell composite jelly in H space, U andU can be uniquely determined.

Aligned stack cells, which are investigated in this study,and spirally wound cells with extended foil type continuous cur-rent tabs are the examples of systems where the SPPC model isapplicable. The computational domain of the SPPC model can bedivided into several zones. U is computed from charge conserva-tion in the negative current collector phase of the cell compositeand the negative-bias passive components outside the cellcomposite

rH ~reff rHU

j000

H 0 [33]

while U is solved in the positive current collector phase of the cellcomposite and the positive-bias passive components outside the cellcomposite

rH ~reff rHU

j

000

H 0 [34]

where j000

H is the volumetric electric charge source in the positivecurrent collector phase and is evaluated with Eq. 4. Effective elec-trical conductivity tensors in Eqs. 33 and 34 are reduced fromEq. 31 as

reff ij dij etie

tj

er [35]

reff ij dij e

tie

tj er [36]

where e and e are the volume fractions of negative and positivecurrent collectors in the cell composite volume, and r and r arethe electrical conductivity values of the conductor components car-rying electronic current. If the transverse direction vector is definedas a zero vector (et 0) outside the cell composite jelly, Eqs. 35and 36 are effective both in the cell composite volume and in otherpassive conductor volumes. Boundary conditions for Eqs. 33 and 34are evaluated from the electrical load by which the cell is operated,such as output current, voltage, or power. Energy conservationyields

o qcpT ot

rH ~krHT

q

000

H [37]

where the thermal conductivity tensor ~k is given with Eq. 31. Thevolumetric heat source in the cell domain submodel, q

000

H, includesthe heat evaluated from the electrode domain submodel and jouleheat at the passive components carrying electronic current in thecell domain

q000

H q000

H;x q000

H;X

q000

x easc ~reff rHU ~r

eff rHU

er

~reff rHU ~r

eff rHU

er

[38]

The cell output current Io is given as

Io

VH j

000

HdVH [39]

In this study, the SPPC model computational domain only includesthe cell composite jelly volume (i.e., complex geometries of othercell parts are neglected), so that the structured solver can be used.

By introducing the Cartesian (X, Y, Z) coordinate system in H-space and aligning the cell-layer normal direction to the Z axis, thetransverse direction vector becomes invariant over the entire compu-tational domain, i.e., et 0; 0; 1. The reduced governing equations

are summarized in Table II. The differential forms of governingequations are discretized using the finite volume method (FVM).Structured orthogonal hexahedral meshes are constructed and usedfor simulations with 22, 17, and 7 for the numbers of discretizationin the X, Y, and Z directions, respectively. The intrinsic volumetricconservation in the FVM has the advantage in the solution of theMSMD model.

Model Analysis and Results

Four different stacked cell designs are numerically investigatedusing the MSMD model setup described in Section 3. The theoreti-cal capacity of the proposed cell designs is 20.46 Ah, based on refer-ence stoichiometries at 0 and 100% SOC- and the active materialcontent in the cells presented in Table III. However, considering ki-

netic and transport limitations, the nominal capacity of the cells isregarded as 20 Ah in this study. The dimensions of the nominaldesign (ND) cell are 200 mm in length, 140 mm in width, and 7.5mm in thickness. It has 44-mm-wide terminal tabs located on thesame side of the cell, as shown in Fig. 3a. The counter-tab (CT) cell,shown in Fig. 3b, is identical to the ND cell except for the configura-tion of the electrical tab, which are located on opposite ends. Thesmall-tab (ST) cell (Fig. 3c) has smaller, 20-mm-wide tabs, whilethe wide-stack (WS) cell (Fig. 3d) has a stack area 1.5 times larger,and therefore has fewer stack layers, than the ND cell. With the pro-posed cell designs, the dependence of the performance response ofstacked cells on the location and size of the tabs and the aspect ratioof the cell stack is explored against the reference ND cell. Constantcurrent discharge and hybrid pulse power characterization (HPPC)tests are simulated to compare the capacity and rate capability of the

proposed designs. A US06 driving profile test is simulated toexplore application of the proposed cell designs to a mid-size sedan10-mile electric range plug-in hybrid electric vehicle (PHEV10)energy storage system. A convective heat transfer boundary condi-tion was applied on the top face of each design using 25 W/m2K forthe heat transfer coefficient and 25C as the ambient temperature inall cases, reflecting single-side cooling thermal management. In thepresent study, all three domain submodels were run with identicaltime steps. The HPPC test simulation uses 100-ms fixed time steps,and the other cases use 1-s time steps. Solution variables over thethree domains are tightly coupled through the first-order implicittemporal discretization and Picard iteration addressing the nonli-nearity of the equations. The MSMD model input parameters for thesubmodel choices in this study are summarized in Table III.

Constant current discharge test simulation. Figure 5 comparesthe voltage curves of the investigated cell designs for 100A(5C-rate), 60A (3C-rate), and 20A (1C-rate) constant current dis-charge cases. During discharge, the output voltages of the four cellsshow several millivolts of variation. The discharge capacities at the2.5 V cut-off voltage are very similar: 18.9 Ah in 5C discharge, 19.6Ah in 3C discharge, and 20.2 Ah in 1C discharge. Even though sev-eral millivolts difference in cell voltage can cause meaningfulchanges in battery kinetics, uncertainties from typical measurementerrors and manufacturing variability make it difficult to distinguisha few millivolts discharge voltage variability among the differentdesigns from experimental variability. The apparent cell output volt-age does not seem to be affected significantly by the design changesbetween the comparable stacked cells. However, the model predictsthat the internal cell kinetics, which is not easily measured, is quitedifferent among the cells and closely follows the thermal and elec-trical configurations of the cell designs.



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Table III. Summary of model parameters in the submodel choices.

Domain Parameter Value/Model

Particle LixC6 Liy(NCA)O2Ref # Maximum Li capacity, cs,max (mol m

3) 2.87 104 4.90 104

Characteristic diffusion length, Rs (m) 3es/as 3es/as

Stoichiometry at 0% SOC, x0%, y0% 0.0712 0.98Stoichiometry at 100% SOC, x100%, y100% 0.63 0.41[27],[25] Reference exchange current density at 100%

SOC, ioref[(A m2)

36.0 4.0

- activation energy, Eioact (J/mol) 3.0104 3.0 104

Charge-transfer coefficients, aa, ac 0.5, 0.5 0.5, 0.5[26],[10] Film resistance, Rfilm (X m

2) 0.023 0.015[26],[10] Solid diffusion coefficient, Ds (m

2 s1) 9.010-14 3 1015

- activation energy, EDsact(J/mol) 4.0103 2.0 104

[26] Negative electrode, U- (V) U x 0:124 1:5exp 70x 0:0351 tanhx 0:286

0:083

0:0045 tanhx 0:9

0:119

0:035 tanh

x 0:99

0:05

0:0147 tanhx 0:5

0:034

0:102 tanh

x 0:194

0:142

0:022tanh x 0:980:0164

0:011 tanh x 0:124

0:0226

0:0155 tanhx 0:105

0:029

Fit to [24] Positive electrode, U (V) U x 1:638x10 2:222x9 15:056x8 23:488x7 81:246x6

344:566x5 621:3475x4 554:774x3 264:427x2

66:3691x 11:8058 0:61386exp 5:8201x136:4

Electrode Negative electrode composite Separator Positive electrode compositeRef # Thickness, la,ls, lc (m) 70.0 10

6 25 106 50.0 106

Thickness unit stack, lasc (m) la ls lc[25],[24] Volume fraction inert, ef 0.09 0.6 0.19

Volume fraction electrolyte, ee 0.4 0.4 0.4Volume fraction active material, es 0.51 0.41

Specific active surface area, as,x (m2 m3) 3.010 106 0.753 106

[26],[24] Solid electronic conductivity, ra,, rc (S m1) 100.0 10

Bruggeman tortuosity exponent, p 2.0 2.0 2.0

Electrolyte concentration, ce (mol m3) 1.2 103

[25] Electrolyte Li diffusion coefficient,De (m

2 s1)De 5:84 10

7 exp2870=Tce=10002 33:9 107 exp2920=Tce=1000

129 107 exp3200=T[25] Electrolyte ionic conductivity, j (S m1)

j 3:45 exp798=Tc=10003 48:5exp1080=Tc=10002

244exp1440=Tc=1000

Effective solid phase conductivity,reff (S m1)

resp

Effective liquid phase diffusion coefficient,De

eff (m2 s1)Deee

p

Effective liquid phase ionic conductivity,jeff (S m1)

jeep

Effective liquid phase diffusional ionicconductivity, j

effD (S m

1)j

effD

2RTjeff

F t0 1 1

dlnf6dln ce

[25] Li transference number, to t

0 0:000267exp883=Tce=1000

2 0:00309exp653=Tce=1000

0:517exp49:6=TThermodynamic factor, o lnf6=o ln ce 0

Cell Negative current collector Positive current collector

Thickness, d, d (m) 10.0106 15.0 106

Specific electrode plate area, as,H (m2 m3) (lasc0.5d0.5d)

1

Volume fraction, e, e 0.5d as,H 0.5d as,HConductivity, r, r (S m

1) 59.6 106 37.8 106

Electrode plate area, AH,p (m2) 1.33

Volume, cell composite, VH,cc (m3) AH,p/as,H

Volumetric heat capacity, cell composite,qcp, (J K

1 m3)2.04 106

Planar thermal conductivity, cell composite,

kp (W m1 K1)

27

Transversal thermal conductivity, cellcomposite, kt (W m

1 K1)0.8



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Figure 6 shows the contours of electrode plate current density, i00

x,at the cell composite volume near the bottom plane of the cells after

6 min of 5C discharge. An electric charge capacity of 10 Ah wasdischarged from each cell, and the average electrode plate currentdensity is constant at 75 A/m2. Electrical current in the current col-lector foils converges to or diverges from the tabs of the cells.Therefore, a fast electrostatic potential change happens in the cellcomposites near the tabs, causing locally larger deviations from thethermodynamic equilibrium. In consequence, the cell compositesnearest the tabs are preferentially discharged initially. In addition,higher temperatures for higher currents can further energize thelocal transfer reaction. This positive feedback promoting non-uni-form discharge reaction over a cell is limited by thermodynamicequilibrium shifts at the reaction sites. As inferred from Eq. 15,steep changes in the open circuit potential, U, with the lithium con-tent help to mitigate the non-uniformity caused by the thermal-elec-trical imbalance. In the ND cell shown in Fig. 6a, the electrode plate

current density is larger near the tabs. The CT cell achieves a moreuniform electric potential difference across the positive and negativecurrent collector foils, and subsequently the most uniform distribu-tion of electrode plate current density among the designs can becompared, as seen in Fig. 6b. The electrode plate current densityfield in the ST cell, shown in Fig. 6c, is very similar to that of the

ND cell except that even higher electrode plate current densitiesappear near the tabs. The most uneven kinetics is observed in theWS cell design, shown in Fig. 6d. The WS cell is 1.5 times wider inthe stack area than the ND cell; therefore, each current collectorsheet carries a proportionally larger electric current for a longer dis-tance. This results in an uneven electric potential field along the cur-rent collectors and uneven transfer reaction over a cell compositevolume.

During discharge of the cells, a cell-internal SOC imbalanceoccurs as a result of non-uniform discharge in the investigated cells.The contours of the electrode plate SOC at the cell composite vol-ume near the bottom plane of the cells after 6 min of 5C discharge

are presented in Fig. 7. Because cell degradation involved with lith-ium loss or active site loss is not considered in this model study, themean SOC of the electrode plate is directly determined from thestoichiometry numbers averaged over particles and electrode com-posites. The evolution of the imbalance in the electrode plate SOCwithin the cell during 5C discharge in each cell design is plotted inFig. 8a. The internal deviation of the SOC of the simulated cellsincreases until about half of the cell capacity is discharged, then itstarts to decrease and quickly converges at the end of discharge withsharp drops. This behavior can be explained with the voltage charac-teristics of the cell. Figure 8b shows dQ/dV, which is an inverse ofthe voltage slope calculated from the 5C discharge curves shown inFig. 5. The curves shown in Fig. 8b mainly represent the combined

Figure 5. (Color online) Discharge voltage comparison among the investi-gated cell designs for 100A (5C), 60A (3C), and 20A (1C) constant currentdischarge events.

Figure 6. (Color online) Contour of electrode plate current density at the

cell composite volume near bottom plane of the cells after 6 min of 5C dis-charge: (a) ND cell, (b) CT cell, (c) ST cell, (d) WS cell.

Figure 7. (Color online) Contour of electrode plate SOC at the cell compos-

ite volume near bottom plane of the cells after 6 min of 5C discharge: (a) NDcell, (b) CT cell, (c) ST cell, (d) WS cell.

Figure 8. (Color online) (a) Instantaneous maximum difference of electrodeplate SOC in the cells simulated during 5C constant current discharge, (b)

Inverse of discharge voltage slope of the cells simulated during 5C constantcurrent discharge.



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effect from the open circuit potentials of the cathode and the anodematerials of the cell. The maxima of the SOC imbalance in themiddle of the discharge of the simulated cells correspond to themaximum of dQ/dV at around 10-Ah discharge. Near the end of dis-

charge, the SOC distribution in the cells quickly converges becauseof the large slope of the voltage curve. The resemblance of thesetwo plots implies that cycling a battery along a slowly varying volt-age plateau can enhance non-uniformity of material use in a cell.Increasing a voltage slope by modifying the thermodynamic charac-teristics of the electrode materials for a better balanced use of a cellmay require considerable effort. Alternately, improving the electri-cal configuration of a cell can greatly lessen the cell use imbalance,as shown in Fig. 8a. In the WS cell, the difference in SOC reaches2.89% during a constant current discharge at 5C, while the SOC dif-ference of the CT cell only reaches 0.56%.

Heat generation rate during 5C discharge for each cell is plottedin Fig. 9a. The average temperature evolutions of the CT, ST andWS designs are presented with minimum and maximum cell internaltemperature range bounds, compared to the temperature response of

the ND cell in Figs. 9b, 9c, and 9d. The average temperature evolu-tions of the CT cell and the ST cell are very similar to that of theND cell, since these three designs have similar heat generationamounts and heat transfer conditions. However, the cell internaltemperature range is narrower in the CT cell and wider in the STcell as compared to the ND cell. The WS cell shows a distinct ther-mal behavior. It generates higher heat during discharge, but ends upwith a lower average cell temperature at the end of discharge. In theinitial stage of discharge, when heat generation dominates cell heat-ing behavior, the maximum temperature increases faster in the WScell than in the ND cell, and average temperatures evolve in a simi-lar fashion in both designs. But as the cells heat up, the impact ofheat transfer becomes significant. Since the WS cell has a 1.5 timeslarger cooling surface and a shorter thermal diffusion distance in thenormal direction to the stack layer, the enhanced heat rejection lim-its the temperature increase of the WS cell during the later stage ofdischarge. In Fig. 10, temperature contours at nine cross-sectioned

surfaces of each cell are presented to show details of the spatial tem-perature imbalance at the end of discharge. Due to preferentialkinetics and electric current convergence, a higher temperature isobserved near the tabs in all cell designs. Unlike the other designs,the tabs of the CT cell are not co-located on the same end of thecell. So, the CT cell, as shown in Fig. 10b, has the most uniformtemperature distribution among the compared designs. Its main tem-

perature gradient exists in the normal direction to the stack platewith the lowest temperature at the top surface, which is cooled byambient coolant. The spatial temperature profile of the ST cell,shown in Fig. 10c, appears similar to that of the ND cell because ofa similar distribution of kinetics over the electrode plates as impliedby the electrode plate current density distribution presented inFig. 6. The peak temperature, however, is higher near the smaller-sized tabs. Thanks to the larger surface area for cooling, the averagetemperature at the end of discharge of the WS cell, shown in Fig. 10d,is lower by a few degrees Celsius than those for the other designs;however, the difference in the internal temperature in the WS cell isstill high among the cells compared. The discharge efficiency of eachcell design is evaluated using heat generated and electric energy out-put during discharge and is summarized in Table IV.

Figure 11 shows the spatial variation of the electrode-domain av-

erage volumetric heat generation rate,q

000

x , which is evaluated in thisstudy using Eq. 29 from the electrode domain submodel and appearsin the first term on the right-hand side of Eq. 38 evaluating thesource term for the energy conservation equation solved in the celldomain submodel. The results are shown at the cell composite vol-ume near the cell bottom plane after 6 min of 5C discharge. This av-erage quantity over the electrode domain volume includes heat fromthe transfer reaction and ohmic loss for charge transport underelectric fields in the cathode and anode electrode compositematrices and in the liquid electrolyte. Therefore, q

000

x has a close corre-lation with the electrode plate current density, i

00

x, shown in Fig. 6. Thecell composite volume heat source, q

000

H, calculated with Eq. 38,includes joule heat in the current collector phase of the cell compositevolume on top of the contribution from q

000

x . Joule heat from electricalheating in the current collector foils contributes a relatively small

amount, just 510% of total heat generation in the simulated celldesigns. However, depending on the internal electrical path design of acell, this heat is highly localized, causing spatial non-uniformity in q

000

H,as seen in Fig. 12.

Hybrid pulse power characterization test simulation.Thepulse responses are compared through the HPPC test simulations forthe cell designs examined. Figure 13 presents the HPPC voltageresponses of the cells at 20% SOC. The HPPC profile consists of a10-s duration 5C discharge pulse, a 40-s rest period, and a 10-s3.75C charge pulse. The area-specific resistances for the electrodeplate are evaluated from the voltage curves and presented in the fig-ure. The lumped cell, reasonably applicable to small cells, plottedin Fig. 13 assumes that the current collectors carry electric currentwithout resistance and that heat is transferred faster inside the cell

compared to the external heat exchange rate, so that the entire elec-trode sandwich of a cell is considered as being operated with identi-cal potential difference at the same temperature. Therefore, the dif-ferences between the HPPC pulse resistance of the lumped cell andthose of the other three-dimensional cells (i.e., ND, CT, ST, andWS) originate from non-ideal electrical and thermal transport. Forthe stacked cells simulated in this study, the major portion of thepulse resistance increase from the lumped cell comes from ohmicresistance for current transport through the current collectors. Theinsets in Fig. 13 are magnified views of the cell output voltagebehavior during relaxation right after the discharge pulse. Thelumped cell shows several millivolts higher initial relaxation voltagethan the other three-dimensional cells. The voltage curves merge inthe later stages of relaxation. In the lumped cell model, relaxationresolves lithium diffusion within the particle and redistribution oflithium through the depth of each electrode composite plate. In thethree-dimensional cells, the 10-s discharge establishes an uneven

Figure 9. (Color online) (a) Net heat generation rate of the cells simulatedduring 5C constant current discharge, (b) Comparison of the evolution of av-erage temperature (solid) with maximum and minimum temperatures (dotted)of the CT cell against those of the ND cell during 5C constant current dis-charge, (c) Same for the ST cell, (d) Same for the WS cell.



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SOC distribution across the electrode plates. Therefore, three-dimensional cell models resolve an additional relaxation process notcaptured in the lumped cell model; i.e., lithium redistributionbetween the counter electrode composites accompanying local ioncurrent through the separator. Figure 14 presents the contours ofelectrode plate current density 1 s after the end of discharge pulse,showing local charging and discharging current driven by in-planeSOC imbalance. The model predicts sub-millivolt output voltagedifferences among the initial relaxation voltages of the cell designscompared. The more uneven in-plane SOC distribution results inlower initial voltage relaxation following interruption of the dis-charge pulse current.

Vehicle driving test simulation. Standard cell characterization

tests such as the constant current discharge test and the HPPC testprovide useful information about the cells electrical-thermal per-formance characteristics. The design of a cell, however, must beevaluated with application-specific use scenarios as well becausethe response of a battery system is largely affected by the attributesof the application and the operation strategies. The cell designsinvestigated here are examined for use in a battery system for amid-size PHEV10 sedan. Vehicle simulation was conducted over a

repetition of an aggressive vehicle speed profile, known as the US06cycle, to determine the power demand for the vehicles battery. Thesimulated PHEV10 vehicle consumes battery energy during charge-depleting (CD) mode for the initial 16 km (10 mile) of driving, and

Table IV. Cell efficiency during constant current discharge.

Discharge Rate ND (%) CT (%) ST (%) WS (%)

5C 93.74 93.78 93.64 93.433C 96.23 96.26 96.16 96.041C 98.63 98.64 98.62 98.58

Figure 10. (Color online) Contours of temperature at nine cross-sectioned surfaces in cell composite volume at the end of 5C constant current discharge: (a)ND cell, (b) CT cell, (c) ST cell, (d) WS cell. (Dimensions in Z direction of the contour surfaces are exaggerated for clear view of quantity variation in Zdirection.)

Figure 11. (Color online) Contour of electrode plate (electrode-domain-av-erage) volumetric heat generation rate at the cell composite near bottom

plane of the cells after 6 min of 5C discharge: (a) ND cell, (b) CT cell, (c)ST cell, (d) WS cell.



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subsequent cycling occurs in the charge-sustaining (CS) hybriddrive mode maintaining a steady battery charge level. Figure 15presents the battery power profile, which is scaled for an individual20-Ah cell in the pack and used as a cell model input for electricload profile. During CD mode driving, which displaces petroleumuse in the vehicle, about 60% of the charge capacity of the battery isconsumed, and then the battery is cycled at around 20% SOC in CSmode driving. As shown in Fig. 16a, the evolutions of the averageSOC of the ND, CT, ST, and WS cells over a 15-min vehicle driveare quite similar. However, the imbalance of the electrode SOCwithin each cell, shown in Fig. 16b, is greatly affected by the celldesign. In all the designs investigated, the maximum discrepancy in

SOC occurs at around 5 min, where the average cell SOCs corre-spond to the small voltage slope in the middle section of the voltagecurve, and the cells run on continual high-rate discharge pulses. Themodel results indicate that the cell response, such as SOC imbalanceimplying non-uniform cycling of cell materials, is determined

through the close interaction among the various length-scale physicsand designs from material properties to system control.

Temperature control is especially important in LIB vehicle appli-cations to extend the battery life while maximizing the electric per-formance of a vehicle. To prevent rapid degradation, excursions ofthe battery temperature to a high temperature must be limited. Atthe same time, the battery temperature should be kept in a narrowrange across the system to avoid excessive non-uniform kinetics.Limiting cell-to-cell and cell-internal temperature variation is morecritical in high-voltage battery systems where a large number ofcells are connected in series. The thermal response of the celldesigns investigated in the PHEV10 US06 driving simulations are

compared for average temperature, shown in Fig. 17a, maximumtemperature, shown in Fig. 17b, and internal temperature difference,shown in Fig. 17c. Cumulative heat generation and heat transfer toambient from each cell are plotted in Fig. 17d. The WS cell gener-ates higher heat than the other cell designs, but it also rejects heat tothe ambient environment at a much faster rate. The larger coolingsurface area of the WS cell brings the advantage of a lower averagecell temperature over the other three designs. The average tempera-tures of the other three cells behave similarly. However, in terms ofmaximum temperature and internal temperature imbalance, each

Figure 12. (Color online) Contour of net volumetric heat generation rate atthe cell composite volume near bottom plane of the cells after 6 min of 5Cdischarge: (a) ND cell, (b) CT cell, (c) ST cell, (d) WS cell.

Figure 13. (Color online) Comparison of the voltage responses of the simu-lated cell designs at 20% SOC for the HPPC pulse profile consisting of a 10 s

duration 5C discharge pulse and a 10 s duration 3.75C charge pulse with 40 srest period between the pulses.

Figure 14. (Color online) Contour of electrode plate current density at the

cell composite volume near bottom plane of the cells at 1 s after the HPPCdischarge pulse: (a) ND cell, (b) CT cell, (c) ST cell, (d) WS cell.

Figure 15. (Color online) Battery power demand profile produced from ve-hicle simulation a US06 driving cycle for mid-size sedan PHEV10, scaledfor an individual 20-Ah cell in the vehicle battery pack. Aggressive accelera-tion and high driving speed characteristics of the US06 cycle result in highheat generation rates for the battery. The above power profile has an average

C-rate of about 2.5 and an RMS C-rate of about 5.0 for the cell designscompared.



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design behaves differently. Increasing the electrode stack-area with-out thickening the current collector foils, as is the case with the WScell, causes a larger electrical current to converge in the current col-

lectors toward the cells electrical tabs, and consequently energizeslocal charge transfer kinetics and generates more heat near the tabs.As a result, the maximum local temperature of the WS cell is high-est among the compared cells during the most of CD mode driving.While the average temperature increase from the initial cell temper-ature is less than 20C, the WS cell shows excessive internal tem-perature differences that reach up to about 10C, which would beunacceptable in the practice of vehicle battery system integrators.Therefore, in spite of its large stack plate area available for bettercooling, the WS cell is not the best thermal design for use in thePHEV10 application. However, it should be noted that the thermalimbalance can be alleviated in lower power applications or withmore precise thermal management. To achieve a better-performingthermal design and management strategy, the thermal behavior ofLIBs should be well understood, not only the characteristics of a

cell but also the characteristics of its application.

Figure 18 shows the contours of the electrode plate area-specificampere-hour throughput in the simulated cells. The absolute valueof charging and discharging electrode plate current density is inte-grated over time during the 15-min PHEV10 drive, revealing spatialvariation of the cumulative electrochemical cycling over the cellcomposite volume. In general, cell composites near the tabs arepreferentially cycled in all designs, but the unevenness of electrodecycling is also greatly affected by the cells electrical-thermal con-figurations. The average values of ampere-hour throughput per elec-trode plate area are similar across the four cell designs: 13.12,13.11, 13.15, and 13.20 Ah/m2 for the ND, CT, ST, and WS cells,

respectively. However, the relative magnitudes of their internal vari-ation compared to the average throughputs are significantly differ-ent. Ampere-hour throughput imbalances are 6.0, 2.5, 6.9, and12.7% for the ND, CT, ST, and WS cells, respectively. Non-uniformcycling of a cell is expected to bring subsequent effects in long-termperformance degradation of a LIB system. Therefore, the impact ofthe electrical and thermal design of a battery should be adequatelyconsidered in predicting the life of large battery systems. A study onlarge-format battery degradation using the MSMD modularizedmulti-physics LIB model framework is ongoing.5

Conclusions

To enhance the understanding of the interplay among the physicsoccurring in LIB systems at various length scales, a modularized

multiscale model framework, the MSMD model, was developed.The MSMD model introduces multiple computational domains forcorresponding length scale physics. While the model decouples sub-model geometries through the statistical homogeneity assumption, itstill couples the physics between the domains using the predefinedinter-domain information exchange. The MSMD model selectivelyenhances spatial resolution for the physics taking place at smallercharacteristic length scales, achieving high computational efficiencycompared to a single domain model approach. Thanks to its modu-larized hierarchical architecture, the MSMD model provides a flexi-ble and expandable framework facilitating multiphysics LIB model-ing with various levels of physical and computational complexities.In this study, the MSMD model is applied to evaluate the thermaland electrical design of large-format stacked prismatic LIB cells.Four different cell designs, ND, CT, ST, and WS, are investigated toevaluate the impacts of tab configuration and size and cell stack as-pect ratio for identical electrode-level designs. Apparent overall cell

Figure 16. (Color online) (a) Average SOC evolution of the cells simulatedduring 15-min PHEV10 drive with the US06 cycle, (b) Instantaneous maxi-mum difference of electrode plate SOC in the cells simulated during 15-minPHEV10 drive with the US06 cycle.

Figure 17. (Color online) Comparison of thermal response of the investi-gated cell designs from the PHEV10 US06 driving simulations: (a) for aver-

age temperature, (b) for maximum temperature, (c) for internal temperaturedifference, (d) for cumulative heat generation and heat transfer to ambient.

Figure 18. (Color online) Contour of electrode plate ampere-hour through-put at the cell composite volume near bottom plane of the cells during 15min PHEV10 drive with the US06 cycle: (a) ND cell, (b) CT cell, (c) STcell, (d) WS cell.



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responses such as cell output voltage, which are directly measurablewith typical experiments, did not show great variation among thecompared cell designs, because the cell designs simulated here arealready fairly optimized. However, the model results revealed thatthe cell internal battery kinetics is significantly influenced by themacroscopic cell design for electrical current and heat transport.The thermal, electrical, and electrochemical behaviors of LIB sys-

tems, for example, non-uniform electrochemical cycling over acell composite volume and subsequent internal SOC imbalances,are found to be affected by the complex interaction among thephysics at varied length scales from thermodynamics to systemcontrol.

Acknowledgments

The authors gratefully acknowledge David Howell, Brian Cun-ningham, and the U.S. DOE Office of Vehicle Technologies EnergyStorage Program for funding and support and Jeffrey Gonder ofNREL for generating the vehicle application power profile.

Appendix A: State Variable Representation of a1-Dimensional Porous Electrode Model

A reduced-order representation of the 1-dimensional electrochemical model ofDoyle and Newman1 improves the computational speed of the present MSMD model

by a factor of 100 to 1,000, depending on the chosen geometry. This appendix summa-

rizes the reduction method18 used here. The governing equations are treated as a quasi-

linear system. Nonlinearities are approximated by interpolating between reduced local-

linear models at various electrode-surface stoichiometries and temperatures. The

approximation generally works well in practice but should be verified for each new

electrochemical model parameter set as in Table III. The quasi-linear treatment loses

validity, for example, when severe spatial nonlinearities arise within the electrode sand-

wich, such as in the case of electrolyte depletion.18

A.1 Model order reduction.The reduction procedure follows Section 3.4 of

Ref. 28. High-order frequency model simulations are run to determine the output

response of electrochemical field variables j000

x , cs,e, e and ce to a small input perturba-

tion in the applied current i00

x . At discrete spatial locations across the electrode sand-

wich, each field variable is organized into an output vector, y. Applied current, a scalar

value, is model input u. Output/input transfer functions are constructed as ys=us,

where s is the Laplace variable. The complex frequency response is calculated by sub-stituting s jx where j is the square root of 1. For a full-order transfer function,

ys=us, its reduced-order counterpart is defined as

ys

us z

Xnk1

rks

s kk[A-1]

with steady-state vector z obtained from the full-order model as z lims!0 ys=us,

and eigenvalues kk and residue vectors rk numerically generated by minimizing the cost

function

J Xmk1

Xni1

Re yi jxk yijxk 2 Im yi jxk yijxk 2 [A-2]

across the frequency range x [ [0, 2pfc]. The cutoff frequency, fc, is chosen to beslightly faster than the smallest time step desired for the final time-domain model. In

this manner, fast dynamics such as double-layer capacitance and electrode film trans-

port, are automatically captured in the reduced model with steady-state representations.

A time-domain representation of the reduced frequency model in state variable

form29 is

_xt Axt But

yt Cxt Dut y0 [A-3]

where

A diag k1 kn ; B 1 1 T

C r1k1 rnkn ; D z Xnk1

rk

" #[A-4]

and static constant y0 gives output y* the proper value at the linearization point.

A.2 Frequency submodel in the particle domain.Analytical Laplace transfer

functions are available to describe the impedance due to lithium intercalation in active

material particles, that is the perturbation of the lithium surface concentration for a per-

turbation in charge transfer current. For a spherical particle, that solid-state diffusion

impedance is Ref. 30cs;es

i00f s

1

F

Rs

Ds

tanhb

tanhb b

![A-5]

where b Rsffiffiffiffiffiffiffiffiffiffi

s=Dsp

. For linear charge transfer kinetics at the particle surface, the But-ler-Volmer equation reduces to

Rct og

oi00

n

i00

n0

RT

i00

oFaa ac[A-6]

A.3 Frequency submodel in the electrode domain.With the assumption of

uniform electrolyte concentration across each electrode, analytical transfer functions

are available to describe the perturbation of electrochemical field variables j000

x and cs,efor a perturbation in applied current i

00

x . Modify the dimensionless impedance variable v

from Ong and Newman31 to include solid-state diffusion impedance

ms d 1jeff

1reff

12 Rct

oU

ocs

cs;es

i00

ns

12

[A-7]

and define the dimensionless electrode position z x/l, where z 0 is the current collec-tor interface and z 1 is the separator interface. Following the derivation,18 the transfer

functions are

j000

x z; si

00

x s 1

d1

jeffreffms

sinh msjeff cosh vs z 1 reff cosh ms z

[A-8]

cs;e z; si

00

x s

1

as

j000

x z; si

00

x s

cs;es

i00

ns[A-9]

Equations A-8 and A-9 are written for the negative electrode. For the positive electrode,

multiply the right-hand sides of the equations by 1.

Analytical transfer functions for e and ce are unduly cumbersome. A numerical

approach is taken instead. Spatial discretization of Eq. 23 followed by Laplace transfor-

mation yields the numerical transfer matrix

cesi 00x s Kc

e sMce

1

Fcej

000

x si 00x s [A-10]

In Eq. A-10, Ki, Mi and Fi are the stiffness, mass, and forcing matrices defined by the

finite element method, and ce s and j000

x s are vectors representing field variables

cex; s and j000

x x; s at discrete node points xi. Similar treatment of Eq. 24 yields the

transfer matrix

/esi

00

x s Kje

1 KjDe

cesi

00

x s Fe

j000

x si

00

x s

[A-11]

A.4 Reduced-order submodel in the electrode domain.Applying the reduc-

tion procedure from Section A.1 to transfer functions Eqs. A-8, A-9, and A-11 yields

reduced-order state variable models for outputs cs;ex; t, j000

xx; t, and e x; tas a func-

tion of input i00

x t. Note that these reduced models are in the time domain in SVM

form. For a given applied current, the voltage drop across the 1-dimensional electrode

sandwich with thickness L la ls lc is thus approximated as

s;cL; t s;a0; t

e L; t

e 0; t U c

s;eL; t

U c

s;e0; t

g j

000

xL; t

g j

000

x0; t

[A-12]

Appendix B: SVM Performance Comparison

The SVM that serves as a reduced-order 1-dimensional porous electrode model is cre-

ated using the procedure in Appendix A using the electrode and particle domain submo-

del parameters in Table III. Those model parameters are chosen to represent a moderate

power cell for PHEV application. The model reduction procedure, minimization of Eq.

A-2, is used to create local linear models for discrete values of temperature and stoichi-

ometry, the latter non-evenly spaced to capture features of each electrodes open-circuit

potential curve. The model order, the value n in Eq. A-1, is chosen such that reduced-

order models closely match full-order transfer functions from 0 to 10 Hz. The present

SVM uses a third-order model to approximate negative electrode dynamics (Eqs. A-8and A-9), a third-order model for positive electrode dynamics (Eqs. A-8 and A-9), and



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a fourth-order model for electrolyte transport dynamics (Eqs. A-10 and A-11). Together

with one state to represent the SOC bulk dynamics, the complete SVM is an 11th order

set of ordinary differential equations.

Figure B 1 compares the reduced-order SVM with a high-order model, solved using

the commercial finite element software, COMSOL. Figure B 1 generally shows less

than 10-mV difference between the two models during 1C, 2C, and 4C constant current

discharges. An exception is during end-of-discharge where model differences increase

to as high as 30 mV. For the simulated cell, end-of-discharge occurs due to positive

electrode surface saturation causing a sudden drop-off in voltage. Slight differences in

the two models prediction of positive surface stoichiometry at the current collector are

responsible for the end-of-discharge model error as small differences in stoichiometry

cause large differences in voltage in this region of positive electrode operation.

In Ref. 18, a high-power HEV cell with graphite/nickel-cobalt-aluminum chemistry

was considered. Similar to that work, the present SVM results compare sufficiently well

with the high-order model to serve as a surrogate porous electrode model for numerical

studies of PHEV cell design. A deficiency of the SVM, however, is its linear treatment

of electrolyte transport. Figure B2 compares distributions of electrolyte-phase concen-

tration and potential between the two models at various times during the 4C discharge.Model agreement is good as long as perturbations in electrolyte salt concentration are

small, e.g., for times of 50 s and less in Fig. B2. Later during the 4C discharge, the elec-

trolyte salt concentration gradient grows large enough that non-linear electrolyte trans-

port becomes important. At end-of-discharge, the model mismatch in the electrolyte

phase potential is 4 mV. Future extensions to the SVM are desired to better accommo-

date nonlinear electrolyte transport and electrolyte depletion.

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Figure B1 (Color online) Comparison of constant current discharge voltagepredicted by reduced-order SVM (lines) and high-order COMSOL model(symbols).

Figure B2 (Color online) Comparisonof electrolyte distributions predictedby reduced-order SVM (lines) and high-order COMSOL model (symbols);(a) electrolyte salt concentration, (b) electrolyte phase potential.

http://dx.doi.org/10.1149/1.2221597http://dx.doi.org/10.1149/1.2817888http://dx.doi.org/10.1149/1.2817888http://dx.doi.org/10.1016/S0378-7753(02)00199-4http://dx.doi.org/10.1016/j.jpowsour.2008.09.045http://dx.doi.org/10.1149/1.1634273http://dx.doi.org/10.1149/1.2113792http://dx.doi.org/10.1149/1.2113792http://dx.doi.org/10.1016/S0378-7753(03)00283-0http://dx.doi.org/10.1149/1.1526512http://dx.doi.org/10.1149/1.2767839http://dx.doi.org/10.1149/1.2767839http://dx.doi.org/10.1149/1.1836132http://dx.doi.org/10.1149/1.1836132http://dx.doi.org/10.1016/j.enconman.2007.03.015http://dx.doi.org/10.1149/1.3049347http://dx.doi.org/10.1149/1.2128096http://dx.doi.org/10.1016/j.jpowsour.2008.10.019http://dx.doi.org/10.1016/j.jpowsour.2005.05.070http://dx.doi.org/10.1149/1.3043429http://dx.doi.org/10.1149/1.3129656http://dx.doi.org/10.1149/1.2939211http://dx.doi.org/10.1149/1.2939211http://dx.doi.org/10.1149/1.2939211http://dx.doi.org/10.1149/1.1785013http://dx.doi.org/10.1149/1.1569478http://dx.doi.org/10.1115/1.2807068http://dx.doi.org/10.1016/0013-4686(94)E0192-3http://dx.doi.org/10.1149/1.1392643http://dx.doi.org/10.1149/1.1392643http://dx.doi.org/10.1016/0013-4686(94)E0192-3http://dx.doi.org/10.1115/1.2807068http://dx.doi.org/10.1149/1.1569478http://dx.doi.org/10.1149/1.1785013http://dx.doi.org/10.1149/1.2939211http://dx.doi.org/10.1149/1.2939211http://dx.doi.org/10.1149/1.3129656http://dx.doi.org/10.1149/1.3043429http://dx.doi.org/10.1016/j.jpowsour.2005.05.070http://dx.doi.org/10.1016/j.jpowsour.2008.10.019http://dx.doi.org/10.1149/1.2128096http://dx.doi.org/10.1149/1.3049347http://dx.doi.org/10.1016/j.enconman.2007.03.015http://dx.doi.org/10.1149/1.1836132http://dx.doi.org/10.1149/1.1836132http://dx.doi.org/10.1149/1.2767839http://dx.doi.org/10.1149/1.2767839http://dx.doi.org/10.1149/1.1526512http://dx.doi.org/10.1016/S0378-7753(03)00283-0http://dx.doi.org/10.1149/1.2113792http://dx.doi.org/10.1149/1.1634273http://dx.doi.org/10.1016/j.jpowsour.2008.09.045http://dx.doi.org/10.1016/S0378-7753(02)00199-4http://dx.doi.org/10.1149/1.2817888http://dx.doi.org/10.1149/1.2221597

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