Modeling and simulation of the thermal behavior of

Modeling and simulation of the thermal behavior of Li-ion cells Andreas Melcher, Carlos Ziebert, Boxia Lei, Magnus Rohde, Hans Jürgen Seifert Institute of Applied Materials - Applied Materials Physics (IAM-AWP) Karlsruhe Institute of Technology Abstract: The thermal behavior of Li-ion cells is an important issue with regard to safety considerations. Therefore, in order to construct a safe Lithium based battery system for use in electrical cars or large stationary electrochemical storage the behavior of the Li-ion cell has to be known under varying conditions. At present the control of the charge and discharge process, respectively, as well as the temperature monitoring and control is achieved by so called battery management systems (BMS) which mostly rely on relative simple charging and discharging algorithms due to the lack of a complete data base. One of the objectives of this work is to gain a better understanding of the temperature increase within the cell considering different heat sources under normal working conditions but also under abuse condition with regard to the so called “thermal runaway”. Therefore a mathematical multi-scale multi-domain model (MSMD) has been developed based on a three-dimensional thermal model coupled with a one dimensional electrochemical model. This model was realized using the commercial finite element (FEM) package COMSOL Multiphysics. First simulations were performed to calculate the generated heat and temperature increase under various charge and discharge rates. In order to study the so called “thermal runaway” additional heat sources were implemented into the model which describes the exponential temperature increase using a phenomenological approach in which an adapted Arrhenius equation was used as a heat source term in the heat transport equation.1 .

1 Introduction Lithium-ion batteries (LIB) have been found a wide application in many consumer products in the last 25 years. The range is spread from cell phones over Tablet and laptop computers until the usage in electric (EV) or hybrid electric (HEV) driven vehicles. The size and the capacity of the batteries differs with the application. A battery used in an EV or HEV is much bigger than one used for a cell phone. But the scale up of the batteries leads to a complexity of the physical phenomenons that do not play any significant role in single or small battery systems. Physical phenomena can occur over a wide range of time and length scales, starting with atomic variations up to the heat transfer of the whole battery. It is of imminent importance how these different phenomenons are related to each other especially in the context of the safety of the battery systems. The usage of model-based investigations promises a theoretical understanding of the battery physics even beyond the point what is possible in experiment. Due to their general structure the behavior of the batteries is strongly affected by the interaction of physics on varying length and time scales. Many attempts have been made in the last 25 years beginning with electrochemical considerations in the work of Newman [DO93, FU94, DO96]. This model is based on the theory of porous electrode theory [NE75] describes the diffusion dynamics and charge transfer kinetics of a battery and is able to predict the electric response of a cell in an intercalation electrode system. This model is adequate in small battery systems. In large formatted battery cells however the uniformity of the electrical potential along the current collectors in the cell composites are not longer given as well as the temperature distribution must also be taken into account due to its inhomogeneity along the cell geometry. Therefore the electrochemical model must be extended with thermal and multi-scale contributions. First thermal models for Lithium batteries are given by Newman and Pals [NE95, PA95-1, PA95-2] using a heat equation with the heat generation as inhomogeneity, where the coupling of the thermal and electrochemical model is realized with the heat generation terms. This idea is used for single battery cells as well as for whole battery packs [HA99, CH05, IN07, KU08, ZH13, HU14]. Parallel the multi-scale and multi-domain character of the electrochemical model is developed [WA98-1, WA98-2] by using the mathematical theory of homogenization resp. the spatial averaging theorem [HO85]. Similar newer attempts can be found for example in [LA11, RI12] and [LA13]. First coupling between the multi-scale, multi-domain electrochemical-thermal model is introduced by [GU00, WA02] and are under development until now. A general short overview can be found in [CA11] and actual some results in 1 Typesetting

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2 The multi-scale multi-domain electrochemical-thermal model

[KI11, GU13, LE13]. The purpose of this modeling ansatzes are to gain a better understanding of the behavior of large LIB systems because the transport of electrical current and heat must be considered on the length scales of the electrode particles up to the length scale of the cell composite which is in general an anisotropic medium. One central point in the modeling of LIBs are the aspects of cell safety which correspond to the thermal stability of a LIB. Several exothermic reactions can occur inside a cell during operation as its inner temperature is increasing if the heat creation is bigger than the dissipated heat in the surrounding space. This may cause heat to accumulate inside the cell and the chemical reactions will be accelerated which yields in a further temperature increase until a thermal runaway is reached. This phenomenon is not captured with the common mathematical models describing the dynamics of a LIB. A first attempt to overcome this disadvantage is given in [HA01]. The heat equation is coupled with ordinary differential equations (ODEs) describing the temporal evolution of the concentration of the exothermic reactions based on an Arrhenius-type law. In [KI07, PE14] this model is extended with additional contributions. But until now no framework is given describing the abuse of LIB in a general way. In the next section 2 we give an short overview of the multi-scale, multi-domain electrochemical-thermal modeling of LIBs [KI11, CA11]. In section 3 the corresponding partial differential equations (PDEs) with respect to the cell geometry are discussed. These equations are extended with additional equations coming from mathematical combustion theory [BE89, VO14] in section 4 to describe abuse behavior and thermal runaway. In the section 5 some simulation result from Comsol Multiphysics are shown. In the final section an outlook of possible future work is given.

2 The multi-scale multi-domain electrochemical-thermal model Following [KI11, LE13] a survey of the multi-scale multi-domain (MSMD) electrochemical-thermal model depending on the geometry of a single cell is given in this section. The corresponding partial differential equations are depending on the geometry of the LIB. The most common used single cells consist of cylindrical geometry and of rectangular geometry for pouch cells. We consider the different models on the different length scales, derive coupling conditions between the models with respect to the electrochemical nature of the problem for a single cell. Finally the electrochemical model is coupled with the thermal model. The MSMD model possesses a hierarchical domain structure, consisting of several model parts on his own length scale for every subdomain. Variables defined on a lower hierarchical domain have a finer spatial resolution than variables on higher hierarchical domain levels. Each domain level is connected with the next finer and next coarser domain level due to coupling conditions. This approach provides a modular framework which is well suited for multiphysics investigations arising in the research of LIB. It can be applied for single cells as well as for whole battery stacks and packs with rising computational complexity. In the sequel the considerations are restricted to a single cell, but the extending to whole battery packs and stacks is straightforward.

2.1 The multi-scale multi-domain model for Lithium-ion batteries The modeling of the dynamic phenomena of LIBs with the MSMD approach is an extension of existing simpler models from several authors [DO93, FU94, NE95, HA99, WA02, CH05, IN07, KU08]. The general scheme of the approach can be seen in Figure 1. The three domains of interest and their inclusion in the LIB are identified as the particle domain (a), electrode domain (b) and (c) cell domain. The coordinate system in every domain are associated to the corresponding length scale in each domain. Charge transfer kinetics have to be solved on the electrode-electrolyte interface. The transport of the Li-ions is modeled with a diffusion mechanism and the migration and diffusion of the Li-ions through the liquid electrolyte can be evaluated. The charge balances in the solid cathode and anode as well as the liquid electrolyte are resolved in the corresponding matrices too. The temperature is solved over the whole cell geometry. The model is able to predict the electrical and thermal behavior of a LIB cell for given electrical load with respect to given thermal boundary conditions.

2.2 The particle domain model In LIB design the electrode material that include the Lithium ions are particulates of given size and shape. The Lithium transport in solid phase is the important physical process that governs the LIB performance. This process depends on the particle geometry i.e. shape, size and on the thermodynamic and kinetic properties of the solid phase materials. Using an own coordinate system for the solid state diffusion separate this process from the other variables. In the MSMD model the process of solid-phase Lithium diffusion and the charge transfer kinetics are solved in the particle domain submodel. In Figure 2(a) and Table 1(a) the state variables, input and output variables are shown. The input of the particle domain are the state variables of the electrode domain submodel, in detail the liquid phase concentration ce , the electrical potential at positive and negative electrode φs,{c,a} , Temperature T and potential in liquid phase φe . These variables are treated 2

2 The multi-scale multi-domain electrochemical-thermal model

(c) Cell Domain (b) Electrode Domain (a) Particle Domain

ξ2 x2 X2

ξ1 ξ3 x3

x1 X1 X3

Figure 1: Scheme of the multi-scale multi-domain approach. as averaged lumped values in the particle domain. Since the transport in the liquid phase is much faster than in the solid phase this approach in reasonable. The state space variables of the particle domain are the concentration of Lithium cs and the volumetric heat source in the solid electrode particle qξ000 . Further variables are the transfer current density at particle surface i00ξ and surface heat flux at electrode-electrolyte interface qξ00 . Output variables to the electrode domain model are spatial averaged volumetric heat source in the solid electrode particle q¯ξ000 , the surface heat flux at the electrode-electrolyte interface q¯ξ00 and the transfer current density ¯i00ξ .

2.3 The electrode domain model The structure of the electrode is a porous one, to form a structure possessing a good electrical network and mechanical properties due to the solid matrix. The active particles are mixed with a conductive agent and a polymer binder and coated on a thin metal current collector. In a conventional battery the cathode and anode are separated with a thin porous polymeric layer called separator to prevent an electronic short between the two electrodes. The transport of the Lithium ions through the liquid electrolyte is strongly influenced by the pore structure of the composite electrodes and separator and the wetting of the pore surfaces. The charge balance across the composite electrodes, their corresponding matrices, the liquid electrolyte as well as the Lithium-ion transport in the liquid electrolyte is resolved in the electrode subdomain model. In Figure 2(b) and Table 1(b) the state, input and output variables are shown. The static electric potential φ± along the current collectors and the temperature T are the input from the cell domain submodel and are again treated as lumped averaged values in this submodel. This will be justified by the argument that lumped values of the electrode potential at the interface with the current collectors are without spatial dependence in the electrode domain coordinate system. This is reasonable because the current collectors on the boundary with positive and negative electrode have an electrically higher conductance than the composite electrodes. The state variables in the electrode subdomain model are the concentration of Lithium ions in the liquid electrolyte ce , the electric potential in the positive, negative solid matrices φs,{c,a} and electric potential in the electrolyte φe . Additional variables are the volumetric reaction current at the electrode composite volume jx000 , current density at the current collector interface i00x , the volumetric heat generation in electrode domain qx000 including Joule heat due to the current flow in the solid composite electrodes and electrolyte and the surface heat generation qx00 due to the contact resistance between electrode and metal current collector surface.

3

2 The multi-scale multi-domain electrochemical-thermal model

(a) Particle Domain i00ξ (ξ) qξ00 (ξ)

cs (ξ) qξ000 (ξ)

(b) Electrode Domain ¯i00 , ξ q¯ξ000

q¯ξ00 ,

ξ2 ξ3

φs,a , φs,c , φe , c e , T

ξ1

(c) Cell Domain ¯i00x , q¯x000

i00x (x) qx00 (x) x2 x1 x3

φs,c (x) φs,a (x) ce (x) φe (x) jx000 (x) qx000 (x)

q¯x00 ,

φ+ (X), φ− (X) T (X) 000 jX (X) 000 qX (X) X2

φ+ , φ − , T

X3

X1

Figure 2: Scheme of model variables in each domain and corresponding coupling. (a) State Variables of particle domain Lithium concentration within solid electrode particle Transfer current density at particle surface Surface heat flux at electrode-electrolyte interface Volumetric heat source within solid electrode particle

Symbol cs i00ξ qξ00 qξ000

Input to

Symbol

Electrode domain Electrode domain Electrode domain

¯i00 ξ q¯ξ00 q¯ξ000

(b) State Variables of electrode domain Lithium concentration in electrolyte Potential in liquid phase Potential at positive/negative electrode Current density at electrode boundary Surface heat source in electrode domain Volumetric current source at electrode domain Volumetric heat source in electrode domain

Symbol ce φe φs,{c,a} i00x qx00 jx000 qx000

Input to Particle domain Particle domain Particle domain Cell domain Cell domain Cell domain

Symbol ce φe φs,{c,a} ¯i00x q¯x00 q¯x000

(c) State Variables of cell domain Electric potential at positive/negative current collector Temperature Volumetric current source in cell domain Volumetric heat source in cell domain

Symbol Φ± T 000 jX 000 qX

Input to Electrode domain Particle/Electrode domain -

Symbol φ± T -

Table 1: Variables used in MSMD model.

2.4 The cell domain model To build prismatic or cylindrical LIBs, unit assemblies of the electrode layers are stacked or wound. The cell domain model describes the battery cell as a complete device. This model is able to resolve the spatial inhomogeneities in temperature and potential with respect to a given current load and ambient temperatures conditions, and has a big influence on the overall performance of LIBs. In Figure 2(c) and Table 1(c) the corresponding state, input and output variables are shown. The spatial distribution of the temperature T and the electric potential φ± at the current collectors must be resolved with respect to the battery geometry and suitable initial and boundary conditions. The cell domain model is a macroscopic approach to describe the electrical and thermal transport properties of a LIB. Furthermore the volumetric current source 000 000 jX and the volumetric heat generation qX is described in this model depending on corresponding variable from the lower domain models.

2.5 Domain coupling In the MSMD approach the underlying multiphysics model for temperature T , electric potential φ and concentration c of Lithium ions is solved in the corresponding subdomain simultaneously. This needs a communication between the different subdomain levels as in Figure 2 described. It is a two-way inter-domain coupling. The information flow from a higher domain level to a lower domain level is realized in terms of average lumped variables as input for the lower level model. 4

2 The multi-scale multi-domain electrochemical-thermal model

The flow of information from a lower domain level to higher one is realized due to source terms which will be averaged in a suitable way using the spatial average theorem [HO85]. This approach eliminates the dependence on coordinate system of lower domain level before passed to the higher domain level. In the sequel ξ, x, X are the coordinate systems of the particle, electrode and cell domain respectively and ξ = (ξ1 , ξ2 , ξ3 )T , x = (x1 , x2 , x3 )T , X = (X1 , X2 , X3 )T denotes the corresponding cartesian coordinates. With X = (r, θ, z)T cylindrical coordinates are denoted for cylindrical cell. Starting from the bottom up on particle domain level the transfer current density is averaged over the electrode-electrolyte interface Aξ Z i00ξ (ξ, x, X)dAξ A ξ ¯i00ξ (x, X) = , (2.1) Aξ and delivered to the electrode domain level. On this level this variable will be transferred into a volumetric transfer reaction current at the electrode composite volume using jx000 (x, X) = ¯i00ξ (x, X)as,x ,

(2.2)

with the specific area of active interface between electrode particles and the electrolyte in the electrode volume as,x . On the electrode domain level the current density of electrode boundary is averaged in the same way as in equation 2.1 over the area of interface between current collector and electrode domain Ax Z i00x (x, X)dAx ¯i00x (X) = Ax , (2.3) A¯x where A¯x is the electrode plate area. This averaged value is delivered to the cell domain level and transferred in a volumetric current source at the current collector 000 jX (X) = ¯i00x (X)as,X ,

(2.4)

where as,X is the specific electrode plate area in the cell volume. In a cylindrical geometry the volumetric current sources on cell domain level are given as 000 j± (X) = ± ((¯i00x (X)as,X )iep + (¯i00x (X)as,X )oep ) ,

(2.5)

where as,X is again the specific electrode plate area in the cell volume and iep, oep denotes the inner and outer electrode pair in the spiral wound inner geometry of the cylindrical cell. For more details we refer to [LE13]. The coupling of the heat sources is done in a similar way and occur on all length scales. The surface heat sources at the electrolyte-electrode interface and the heat source within a volumetric particles are averaged with respect to the particle domain boundary and delivered to the electrode domain submodel Z q¯ξ00 (x, X)

=

Aξ Z

q¯ξ000 (x, X)

=

qξ00 (ξ, x, X)dAξ

Aξ

,

(2.6)

,

(2.7)

qξ000 (ξ, x, X)dVξ

Vξ

Vξ

where Aξ , Vξ are the electrode-electrolyte interface area and the volume of particle domain geometry respectively. In the electrode domain a volumetric heat source originate from the averaged heat sources of particle domain 000 qx,ξ (x, X) = q¯ξ00 (x, X)as,x + q¯ξ000 (x, X)s ,

(2.8)

where as,x is the specific area of active interface between electrode particle and electrolyte and s the volume fraction of active particles in electrode volume. The volumetric heat source from equation 2.8 is incooperated with other heat contributions on this domain level like Joule heat due to electronic current flow in the electrode matrix and ionic current flow in the electrolyte. X 000 000 qx000 (x, X) = qx,ξ (x, X) + qx,k (x, X). (2.9) k

5

3 The electrochemical and thermal partial differential equations for rectangular and cylindrical geometry

(a) Particle Domain

(b) Electrode Domain

(c) Cell Domain

x

r

X2 Rs

ls

la

1D spherical particle model

X3 X1

lc

3D continuum model

1D porous electrode model

Figure 3: The submodel in each domain for a prismatic cell. In electrode domain level surface and volumetric heat sources are averaged over the corresponding geometries and delivered to cell domain Z q¯x00 (X)

=

Ax

A¯x Z

q¯x000 (X)

=

qx00 (x, X)dAx ,

(2.10)

,

(2.11)

qx000 (x, X)dVx

Vx

Vx

where Vx the volume of electrode domain geometry. In the cell domain level equations 2.10, 2.11 are used to form the heat source 000 q¯X,x (X) = q¯x00 (X)as,X + q¯x000 (X)asc ,

(2.12)

where as,X is the specific electrode plate area, asc is the volume fraction occupied by electrode and separator in the cell volume. Finally the volumetric heat source in the cell domain is given by X 000 000 000 qX (X) = qX,x (X) + qX,k (X), (2.13) k

where several other heat sources like electric heating at current collector and other passive conductors are also taking into account.

3 The electrochemical and thermal partial differential equations for rectangular and cylindrical geometry The macroscopic design of LIB features the thermal and electric behavior concerning the overall cell performance with respect to the geometry, the underlying electrochemistry, thickness of current collector foils, particle size in the electrode and many other parameters. To gain a better knowledge and understanding of temperature and potential distribution analytical models like the MSMD model and their numerical implementation are of vital importance. In this section we consider the governing partial differential equations, algebraic equations and the corresponding initial, transition and boundary conditions with respect to cylindrical and rectangular geometry. In general on every domain level a coupled system of elliptic generalized Poisson equations for the electrical potentials and parabolic diffusion equations of corresponding concentrations of Lithium ions must be solved [DO93, FU94]. On macroscopic scale these equations are coupled with a heat equation, which represent also an parabolic differential equation [NE95, HA99, CH05, IN07, KU08]. For some newer publications we refer to [KI11, LE13]. The modeling in the sequel is implemented in a similar way in Comsol Multiphysics in the Batteries and Fuel Cells Module. The governing equation and boundary conditions for prismatic and spiral wound cylindrical geometry can be found in Table 2.

6

3 The electrochemical and thermal partial differential equations for rectangular and cylindrical geometry

3.1 The particle domain submodel In the particle domain the geometrical shape of the particles is assumed to be perfect spheres with a radius r = Rs (Figure 3(a)). This leads to a 1D spherical particle model. The solution variables of this model are the concentration cs of the Lithium ions inside the particle and the exchange current density i00ξ on the surface of the particle. Following Fick’s law of diffusion one has to solve a radial-symmetric diffusion equation for the concentration cs in the spherical particle. The transfer current density is governed by a Butler-Volmer kinetic expression, where the electrochemical reaction rate is driven by the overpotential η. The overpotential is defined as the deviation of the potential difference between solid and liquid phase φs , φe and the thermodynamic equilibrium potential Ueq . This two state variables are coupled via the flux condition for the concentration on the surface of the particle. The diffusion coefficient Ds is assumed to be constant with respect to the radial coordinate r. The temperature T is a given lumped value in particle domain as well as the electrolyte concentration ce , solid and liquid phase potential φs , φe . The heat source at the surface of a particle is modeled by considering heat from the charge transfer reaction and ohmic losses.

3.2 The electrode domain submodel On the electrode domain level a simplified porous electrode model in cartesian coordinates is used similar to the 1D spherical particle on particle domain level by assuming lumped potentials and liquid concentrations. It resolves the charge balance in the liquid and solid phase and Lithium ions will be conserved in the electrolyte [SA06, KI09]. One has to take into account three volumes for positive, negative electrode and separator as shown in Figure 3(b). In these volumes the solid and liquid phase are assumed as superimposed continua. The Lithium concentration ce and the potential φe are evaluated in the liquid phase in these three volumes. The potential for the anode φs,a and cathode φs,c are solved in the solid phase of cathode and anode matrix. φ− , φ+ , T are provided by the cell domain model and are treated as spatial invariants. The electronic current density at electrode boundary is equal to the average electrode plate current density i.e. ¯i00x = i00x while jx000 is coming from equation 2.2. The averaged heat sources from the particle domain are converted in a 000 following equation 2.8. Furthermore the Joule heat qx,Ω for the electric current flow in the volumetric heat source qr,x presence of potential gradients in solid and liquid has also being taking into account. These last two quantities are needed for the volumetric heat source q¯x000 for passing to the cell domain model. The assumption of statistical homogeneity is also valid in electrode domain.

3.3 The cell domain submodel On the cell domain level a model was developed to solve for the temperature T and the pair of electrical potentials φ± in the three dimensional cell geometry. Such a model is called Single-Potential-Pair Continuum (SPPC) Model. Using the statistical homogeneity assumption the cell volume is assumed to be a continuum having orthotropic thermal and electrical conductivities. The layered structure inside the cell is not necessarily resolved for numerical computations. In typical battery designs the conduction transport in the cell composite is anisotropic and one has to distinguish between in-plane and transverse diffusivity. Therefore one has to introduce a conductivity tensor σ ef f from which the effective material properties can be computed. Depending on the cell geometry more information can be found in [KI11, LE13]. The effective material properties depends also on the volume fractions ± of negative and positive current collector and the volume fraction of anode, separator and cathode asc in the cell volume. The boundary conditions for the electric potential 000 is computed from equation 2.4. are governed via the electric load of the cell. The volumetric electric charge source jX 000 The energy conservation yield a inhomogeneous heat equation, where the volumetric heat source qX on cell domain level includes the heat from electrode domain submodel and Joule heat at all passive components carrying electric current in 000 the cell domain. The cell output current Io is then the volumetric average of qX . (a) Particle domain Variables cs (r, x, X) i00ξ (r = Rs , x, X)

7

Governing equations   ∂cs Ds ∂ 2 ∂cs = 2 r ∂t r ∂r ∂r α

qξ00 (r = Rs , x, X)

i00ξ = ki (ce )αa (cs,max − cs,e ) a (ce,s )αc      αa F αc F · exp η − exp η RT RT  ∂U eq qξ00 = i00ξ φs − φe − Ueq − T ∂T

η((r, x, X)

η = φs − φe − i00ξ R − Ueq

Boundary conditions/Notations ∂cs =0 ∂r r=0 ∂cs 1 00 =− i ∂r r=Rs Ds F ξ

3 The electrochemical and thermal partial differential equations for rectangular and cylindrical geometry

ce (x, X) φe (x, X) φs,{a,c} (x, X)

¯i00x (x, X)

000 qx,Ω (x, X)

(b) Electrode domain   1 − t0+ 000 i00e ∂t0+ ∂e ce ∂ ∂ce = Deef f + jx − ∂t ∂x  ∂x F  F ∂x ∂ ∂ ef f ∂ ef f ∂φe κ + κD ln ce + jx000 = 0 ∂x  ∂x ∂x ∂x  ∂ ef f ∂φs,{a,c} σ{a,c} = jx000 ∂x ∂x

 Z l  ef f ∂φs,c  =− jx000 dx −σc ∂x x=l l +l a s ¯i00x = i00x = Z la  ∂φ  −σaef f s,a jx000 dx = ∂x x=0 0 X ef f 000 qx,Ω = σi k∇x φs,i k2 + κef f k∇x φe k22

∂ce ∂ce = 0, =0 ∂x x=0 ∂x x=l ∂φe ∂φe = 0, =0 ∂x x=0 ∂x x=l ∂φs,a = Φ− , =0 φs,a ∂x x=l x=0 a ∂φs,c = Φ+ , = 0, φs,a ∂x x=la +ls

l = la + ls + lc  T ∂ ∂ ∂ ∇x = , , ∂x1 ∂x2 ∂x3 k • k2 =

q¯x000 =

3 X

•2i

i=1

i∈{a,c}

q¯x000 (x, X)

x=l

f +κef ln ce ∇x φe D ∇ x R l  000 000 qξ,x + qx,Ω dx 0

l

(c) Cell domain (Cartesian coordinates) (

φ± (X)

T (X)

000 qX (X)

∇X˜ · (± σ± ∇X˜ φ± ) =

1 if i = j 0 else 3 X et (X) = etiˆii

000 jX

δij =

i=1

∂ρcp T 000 = ∇X · (k∇X T ) + qX ∂t  T ∂ ∂ ∇X˜ = , ∂X1 ∂X2  T ∂ ∂ ∂ ∇X = , , ∂X1 ∂X2 ∂X3 000 000 000 000 qX = qx,X + qΩ,X = q¯X asc +

kij = (kt − kp )eti etj + kp δij ˆi: Basis in X space kt,p transversal/planar componenent ef f X σi,± k∇X Φi k2 i σi

ef f σ±,ij = δij − eti etj ± σ±

i∈{±}

Z I0

I0 =

jx000 dVX

VX

(d) Cell domain(Cylindrical coordinates) φ± (r, θ, z) T (r, θ, z) 000 qX (r, θ, z)

∂ 2 φ± ± σ ± ∂ 2 φ ± 000 +  σ = j± ± ± 2 r± ∂θ2 ∂z 2     1 ∂ ∂T kp ∂ 2 T ∂ρcp T ∂2T 000 = kt + 2 + k + qX p ∂t r ∂r ∂r r ∂θ2 ∂z 2 000 000 qX = qx000 + qΩ

l± r± = r + ± (ls + la ) 2   ± σ ± ∂ 2 φ ∂ 2 φ + ± σ ± 2 2 r± ∂θ2 ∂z Θ1 =0,π   ± σ± ∂ 2 φ ∂ 2 φ = + ± σ ± 2 2 r± ∂θ2 ∂z

Θ2 =π,0

qx000 = (¯ qx000 iep )ep + (¯ q 000 ep )oep  2  2 ! X X ∂φi 1 ∂φi 000 000 + qΩ = qi = i σi ri ∂θ ∂z i∈{±}

i∈{±}

Table 2: Governing equations for 3D model in cartesian and cylindrical coordinates.

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4 Modeling thermal abuse and thermal runaway

Internal energizing events and exothermric reactions

External abuse conditions

External heating Over-charging Over-discharging High current charging Nail penetration Crush External short

Electrode-electrolyte reactions Decompostions Electrochemical Reactions

Thermal runaway

Heating rate > Dissipation rate

Leak Smoke Gas venting Flames Rapid disassembly

Figure 4: Thermal abuse for LIBs.

4 Modeling thermal abuse and thermal runaway The model introduced in the last sections is able to resolve the normal use of Lithium-ion batteries. It is not able to resolve phenomena resulting in an abusive usage of LIBs. The worst case scenario in abusive usage of LIBs is the thermal runaway which leads to a complete destruction of the battery. In this chapter abuse mechanisms are investigated and a first mathematical model will be derived for the contributions coming from abuse mechanisms. These mechanisms will be described with an system of reaction-diffusion equations coming from mathematical combustion theory [BE89, VO14] and first abuse simulations are performed in Comsol Multiphysics.

4.1 Thermal abuse mechanisms The thermal stability of LIBs is a very important issue for cell safety. Several exothermic chemical reactions can occur inside a battery as the temperature rise. This may generate heat that accumulates inside the cell and accelerates the chemical reaction between the cell components, if the heat transfer to the surroundings is not sufficient [GU10], i.e. if the heat generation rate exceeds the dissipation rate. In Figure 4 some abuse mechanisms are shown and the corresponding chemical answer. Several authors have given models to describe abuse behavior and thermal runaway. First attempts to describe the exothermic reaction kinetics are given by [HA01]. A simplified reaction-diffusion model is given in the works of [KI07, PE14, PE14a]. The case of nail penetration is given in [CH14]. A catastrophe theoretic approach is given in [WA10] which based on a simplified ODE model and is extended in [WA12]. To describe the abuse behavior resp. the thermal runaway one has to identify the main exothermic chemical reactions. Following [LI11] the general mechanism that lead to a thermal runaway can be described with respect to rising temperature in four steps which are given in Table 3. Temperature

Reaction

T > 90◦ C:

Start of anodic reaction

Rate limiting step

T > 120◦ C:

Decomposition of solid electrolyte interface (SEI)

Reduction of electrolyte at negative electrode

T > 140◦ C:

Exothermic reaction starts at positive electrode

Oxygen rapidly evolves

Positive electrode decompose

High rate exothermic process ◦ C Temperature rise > 100 min

◦

T > 180 C:

Electrolyte decomposition due to oxidation

Table 3: Abuse mechanisms depending on cell temperature.

4.2 The reaction-diffusion model Since the abuse of LIBs could lead to thermal runaway, this can result in fire and explosion of the battery. The mathematical theory of choice for a proper modeling of this phenomena is the combustion theory[BE89] resp. a solid fuel model which represents a description in form of reaction diffusion equations for the temperature T and the n concentrations 9

4 Modeling thermal abuse and thermal runaway

c = (c1 , . . . , cn )T of the chemical species on the corresponding domains in the battery[VO14]. Under the assumption that the density ρ and the heat capacity cp are constant one yields

ρcp

n X

∂T ∂t

= ∇ · (κ∇T ) +

∂c ∂t

= ∇ · (D∇c) + ΓR,

Qi ,

(4.1)

i=1

(4.2)

where κ is the thermal conductivity, Qi = qi Ri , qi, Ri , i = 1, . . . , n are the heat sources, the heat release and reaction rate respectively. D is the diffusion matrix, which represents a diagonal matrix since the chemical reactions considered here are independent from each other. Γ is the matrix of the stoichiometric coefficients which is also a diagonal matrix. Since we consider exothermic chemical reactions in the abuse of LIBs the reaction rates are governed by an Arrhenius law, which have the general form   Ea,i mi , i = 1, . . . , n, (4.3) Ri = Ai ci exp − RT where Ai is a frequency factor, ci the concentration, mi the reaction order, Ea,i the activation energy and R the universal gas constant. Following [WA10] the different contribution can be found in Table 4. Temperature

Reaction

T > 90 − 120◦ C:

SEI decomposition reaction: SEI layer is metastable Decompose exothermic

T > 120◦ C:

Negative solvent reaction: Exothermic reaction between intercalated Lithium and electrolyte

T > 140◦ C:

Positive solvent reaction Exothermic reaction between positive cathode material and electrolyte

T > 180◦ C:

Electrolyte decomposition reaction: Exothermic reaction

qsei = Hsei Wc   Ea,sei msei csei Rsei (T, csei ) = Asei exp − RT ∂csei = ∇ · (dsei ∇csei ) − Rsei ∂t qne = Hne Wc   Ea,ne mne Rne (T, cne ) = Ane exp − cne RT ∂cne = ∇ · (dne ∇cne ) − Rne ∂t qpe = Hpe Wp   Ea,pe mpe cpe Rpe (T, cpe ) = Ape exp − RT ∂cpe = ∇ · (dpe ∇cpe ) − Rpe ∂t q e = He W e   Ea,e me Re (T, ce ) = Ae exp − ce RT ∂ce = ∇ · (de ∇ce ) − Re ∂t

Table 4: Abuse mechanisms depending on cell temperature.

In the last Table qx , x ∈ {sei, ne, pe, e} represent the volumetric reaction heat, i.e. Hx , x ∈ {sei, ne, pe, e} the exothermic reaction heat and Wx , x ∈ {c, p, e} a specific volume content. In [HA01, KI07, PE14, PE14a, CH14] anode-electrolyte and cathode-electrolyte are modeled in a sightly different way. Additional anode and cathode decomposition reactions can also be taken into account [WA10]. Additional heat sources are heat due to entropy change Qs , Ohmic and Joule heat Qp resp. Qj [WA10, WA12]. This gives the additional heat source   dUeq Qadd = I Ueq + T , (4.4) dT where I is the current and Ueq the equilibrium voltage of the cell. The heat exchange between the battery cell and the environment is modeled due to heat convection and radiation.

10

5 Simulations

Figure 5: Applied current with an amplitude |Imax | = 8 A.

Qconv Qrad

= hA (Tsurf − Tamb ) ,
4 4 = σA Tsurf − Tamb ,

(4.5) (4.6)

where h is the convection heat transfer coefficient, A the area of the battery case,  the emissivity of cell surface and σ Stefan-Boltzmann constant.

5 Simulations In this section we present some first simulation results done in Comsol Multiphysics. We consider a cylindrical 18650 LMO cell. The internal spiral wounds is not resolved in this simulation model as well as radiation boundary conditions. The other physical parameters can be found in Table 5 and 6. The main parameters have been taken from a Comsol tutorial about Li-ion batteries and the parameter for the abuse contributions are taken from [KI07, PE14]. The simulations in Comsol where performed with two models, in the first model the conventional model from the sections 2 and 3 is considered, while for the second model we used additional exothermic terms for the electrolyte and the reaction at positive and negative electrode. The environmental temperature is chosen as Tenv = 343.15 K. In the simulation the heat transfer coefficient h is varied, i.e. h ∈ [−20, 4] W/(m2 · K) and the mean temperature is evaluated. The important boundary condition in the simulation is n · (κ ∇T ) = h (Tenv − T ) ,

(5.1)

which describes the inward flux condition. For vanishing h we have adiabatic boundary conditions, positive h correspond to isoperibolic boundary conditions where heat flow to the environment occur, while for negative h a heat flow from the enviroment in the battery cell occur. The simulations were performed with the current load profile as seen in Figure 5, between the charging and discharging pulses with duration of 250 s we have insert relaxation periods of 250 s. A complete current cycle lasts 1000 s and maximal 10 cycles, tsim = 104 s are simulated.

Parameter Neg. solid phase Li-diffusivity Pos. solid phase Li-diffusivity

11

Value 2 Ds,neg = 3.9 · 10−14 ms 2 Ds,pos = 1 · 10−13 ms

Parameter

Value

Neg. particle radius

rp,neg = 12.5 · 10−6 m

Pos. particle radius

rp,pos = 8 · 10−6 m

Pos. solid phase vol.-fraction

s,pos = 0.297

Neg. solid phase vol.-fraction

s,neg = 0.471

Pos. Electrolyte vol.-fraction

e,pos = 0.444

Neg. electrolyte vol.-fraction

e,neg = 0.357

Neg. max. solid phase concentration

cs,max,neg = 26390

Neg. reaction rate coefficient

kneg = 2 · 10−11

mol m3

m s

5 Simulations

Pos. reaction rate coefficient

kpos = 2 · 10−11

m s

Initial pos. State of Charge

cs,pos,0 = 16002

mol m3

Initial temperature

T0 = 343.15 K

rbatt = 9 mm

Battery height

hbatt = 65 mm

Mandrel radius

rmand = 2, mm

Thickness battery canister

dcan = 0.25 mm

Length pos./neg. electrode

Lpos/neg = 55 µm

Length separator

Lsep = 30 µm

Thickness neg. current collector

Lneg,cc = 7 µm

Thickness pos. current collector

Lpos,cc = 10 µm

Cell thickness

Lbatt = 157 µm

Pos. electrode thermal conductivity

κT,pos = 1.58

Neg. electrode thermal conductivity

κT,neg = 1.04

Neg. current collector thermal cond.

κT,cc,neg = 298.15

Separator collector thermal cond.

W m·K W κT,cc,pos = 170 m·K W κT,sep = 0.344 m·K

Pos. electrode density

ρpos = 2328.5

Neg. electrode density

ρneg = 1347.33

Neg. current collector density

ρneg,cc = 8933

Separator density

kg m3 kg ρpos,cc = 2770 m 3 kg ρsep = 1008.98 m3

Pos. electrode heat capacity

cp,pos = 1269.21

J kg·K J cp,cc,pos = 875 kg·K J cp,sep = 1978.16 kg·K

Pos. electrode heat capacity

cp,neg = 1437.4

Pos. current collector

cp,cc,neg = 385

Pos. max. solid phase concentration

c,smax,pos = 22860

Initial neg. State of Charge

cs,neg,0 = 7917

Initial electrolyte salt concentration

cl,0 = 2000

Battery radius

Pos. current collector thermal cond.

Pos. current collector density

Pos. current collector heat capacity Separator heat capacity

mol m3

mol m3

mol m3

W m·K W m·K

kg m3 kg m3

J kg·K J kg·K

Table 5: Simulation parameters.

The effective thermal conductivity (radial, angular direction), density and heat capacity are given as follows [KI07]:

κT,ang,batt

=

κT,rad,batt

=

ρbatt

=

Cp,batt

=

κT,pos Lpos + κT,neg Lneg + κT,cc,pos Lpos,cc + κT,cc,neg Lneg,cc + κT,sep Lsep Lbatt Lbatt Lpos Lneg Lpos,cc Lneg,cc Lsep + + + + κT,pos κT,neg κT,cc,pos κT,cc,neg κT,sep ρpos Lpos + ρneg Lneg + ρpos,cc Lpos,cc + ρneg,cc Lneg,cc + ρsep Lsep Lbatt cp,pos Lpos + cp,neg Lneg + cp,cc,pos Lpos,cc + cp,cc,neg Lneg,cc + cp,sep Lsep Lbatt

(5.2) (5.3)

(5.4) (5.5) (5.6)

The parameter for the exothermic reaction are taken from [PE14]:  x ∈ {pe, ne, e}, y ∈ {c, p, e}

Reaction heat Hx

Negative solvent reaction

1.714 · 106

J kg



3.14 ·

Electrolyte decomp. reaction

1.55 · 105

  1 s

2.5 · 1013

105

Positive solvent reaction

Reaction frequency factor Ax

6.667 ·

1013

5.14 · 1025

 Init value

Volume content Wy

0.75

1.39 · 103

0.04

1.3 · 103

1

5 · 102

kg m3



Table 6: Exothermic simulation parameters.

Heat transfer coefficient h

h

W m2 ·K

Thermal runaway time tT R [s]:

i

:

−20

−15

−10

−5

−4

−3

−2

−1

0

1

2

3

4

1207

1454

1755

2386

2621

2767

3089

3489

3802

4707

5826

8669

−

Table 7: Thermal runaway time tT R with respect to heat transfer coefficient h.

12

5 Simulations

In Table 7 and Figure 6 the time tT R where a thermal runaway occurs is given with respect to the heat transfer coefficient h. It shows that if the heat inside the battery can dissipate to the environment the time where thermal runaway occurs increases. The corresponding temperature profiles shows that for h > 0 in the relaxation intervals, i.e. Iload = 0 the temperature decreases while for h = 0 the temperature remains almost constant and for h < 0 the temperature increase in this intervals due heat dissipate from the exterior. But within the simulation time there exists for the most considered K , if the values of h a time where the temperature gradient with respect to the time becomes very high ≈ 90 − 100 min W W exothermic contributions are present in the model. In Figure 7 for h = 0 m2 ·K and h = 2 m2 ·K the differences in the model with and without exothermic terms are shown. The temperature in the model with exothermic contribution is always slightly higher than without these terms and of the order of 10−2 . In the area where the thermal runaway is starting this difference is growing very fast.

Figure 6: Thermal runaway time and corresponding thermal profiles.

Figure 7: Differences in the temperature profile with and without exothermic terms.

13

6 Conclusion and outlook

6 Conclusion and outlook In this work we have introduced the MSMD model for the simulation of LIBs with respect to the important state variables temperature T , the potential φ± and the corresponding concentration of the Lithium ions in the electrolyte ce and in the particles cs,{a,c} of positive and negative electrode for a given current profile. We have taken into account the length scales where these dynamics occurs, resulting on a coupled system of elliptic and parabolic partial differential equation on each length scale. These equations are coupled with the equations on the other length scales via homogenization procedures basing on the assumption of spatial homogeneity and the spatial averaging theorem. Furthermore we have done a first step to describe the thermal runaway of LIBs due to extending the MSMD model with additional contributions coming from various exothermic reaction inside the LIB during abuse and thermal runaway. In this simulation done in Comsol Multiphysics we have shown that these additional exothermic terms in the numerical FEM model can resolve this phenomena for a specific current load profile. For future work this first model for describing the thermal runaway of LIBs will be improved. Detailed consideration of the underlying battery chemistry and corresponding exothermic reaction kinetics are planned. Furthermore simplified models for the abuse mechanisms will be considered with respect to the application in equivalent circuits models (ECM) of LIBs.

Acknowledgement This R&D project is partially funded by the Federal Ministry for Education and Research (BMBF) within the framework “IKT 2020 Research for Innovations” under the grant 16N12515 and is supervised by the Project Management Agency VDI | VDE | IT. Special thanks to Anna Ossipova for creating the first Comsol model.

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