An experimentally validated equivalent circuit model for ... - UBC Math

An experimentally validated equivalent circuit model for lithium-ion batteries systematically derived from porous electrode theory I. R. Moylesa,1,∗, M. G. Hennessyb,c,1 , T. G. Myersc,d , B. R. Wettone a Mathematics

Applications Consortium for Science and Industry, University of Limerick, Limerick, Ireland Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom c Centre de Recerca Matem` atica, Campus de Bellaterra, Edifici C, 08193 Bellaterra (Barcelona), Spain d Departament de Matem` atiques, Universitat Polit` ecnica de Catalunya, 08028 Barcelona, Spain e Department of Mathematics, University of British Columbia, Vancouver, Canada

b Mathematical

Abstract We present an electrochemical model for lithium-ion batteries based on porous electrode theory that is derived by taking volume averages of microscale equations. Using a rigorous scaling analysis, we then show how the model can be systematically reduced to a pair of equivalent circuit models and, importantly, provide three simple conditions that are required for the reduction to be valid. Since the reduced model is derived from a detailed electrochemical model, it is based on physical parameters rather than empirical equivalent circuit parameters. The reduced model consists of two first-order ordinary differential equations for the solid-phase electric potentials and can be straightforwardly simulated using traditional fast methods. This makes the reduced model ideal for use in battery management systems and for parameter estimation and optimisation. Finally, we validate the reduced model against three sets of experimental data and large-scale numerical simulations and find excellent agreement in all cases. Keywords: lithium-ion batteries, equivalent circuit model, model reduction, electrochemical modelling, porous electrode theory

1. Introduction Continual improvements in the design of lithium-ion batteries (LIBs) have made them one of the most popular technologies for energy storage [1]. The wide-spread use of LIBs can be attributed to their long lifetime, low self-discharge rates, high energy density, and lack of memory effect [2]. While the use of LIBs in portable electronic devices and electric cars has been the driving force behind much of their recent development [3], these two application areas place very different demands on LIB technology [4]. On the one ∗ Corresponding

author Email addresses: [email protected] (I. R. Moyles), [email protected] (M. G. Hennessy), [email protected] (T. G. Myers), [email protected] (B. R. Wetton) 1 These authors share senior author contributions Preprint submitted to Journal of Power Sources

October 20, 2018

hand, the portable electronics industry requires a battery design that is compact, lightweight, inexpensive, and able to supply a small current to a low-power device for as long as possible. The automotive industry, on the other hand, pushes LIBs to their limits, where they are subjected to high-power charging and discharging rates over much shorter periods of time. Both of these applications areas are, however, unified in their need for efficient and accurate battery management systems that are able to monitor, in real time, the state of charge (SOC), state of health (SOH), and temperature of a battery [5]. The vastly different demands that are placed on LIBs means there is no single design that operates effectively in all circumstances. Therefore, substantial research efforts are focused on optimising battery designs for a given application. This requires detailed understanding of the thermodynamic and electrochemical processes taking place under a range of possible operating conditions, information that is both expensive and time consuming to obtain by experimental means. Mathematical modelling of LIBs can alleviate these issues by providing rapid and cost-effective insights into physical processes that might be difficult or even impossible to probe experimentally [6]. Theoretical models of LIB operation also play a fundamental role in the design of onboard management systems that are responsible for performance optimisation, temperature regulation, and maintaining the overall health of the battery. The need to produce accurate and fast predictions of LIB performance and health characteristics has prompted the development of a broad range of mathematical models of varying complexity. These models can be categorised as equivalent circuit models or electrochemical models [7, 8]. Equivalent circuit models, which describe the battery as a series of resistors, capacitors, and current sources, are the simplest of the LIB models and little computational effort is required to simulate them. Equivalent circuit models are therefore ideal for real-time SOC and SOH monitoring as well as parameter estimation and optimisation [9, 10]. However, these models are often based on empirical parameters that must be experimentally calibrated for each battery design. As a result, equivalent circuit models can become increasingly inaccurate with time due to the battery ageing process. Electrochemical models are based on comprehensive theoretical descriptions of the reaction kinetics, species transport, and electrochemistry that occurs within the cell of an LIB. The aim of these models is to provide detailed insights into battery operation and accurate predictions of quantities that are inaccessible to experimental measurement. Due to the complexity of electrochemical models, significant computational resources are required to numerically solve them, making them unsuitable for use in battery management systems. However, electrochemical models offer greater accuracy and robustness than equivalent circuit models and are based on well-defined physical parameters rather than empirical parameters. With numerical simulations of battery models forming the foundation of real-time management systems that are crucial in LIBs, there is now a major focus on bridging the gap between fast equivalent circuit models and accurate electrochemical models using order-reduction methods. Typically, the order-reduction process begins with the formulation of an electrochemical model based on concentrated solution theory and 2

porous electrode theory [11]. Then, the complexity of the model is reduced through the use of simplifying assumptions or mathematical techniques, as described below. Many reduced-order models are derived from the pseudo-two-dimensional (P2D) electrochemical model proposed by Doyle et al. [12]. In the P2D model, each electrode is envisioned as a collection of solid spherical particles that interact through the electrolyte. Comparisons with experimental data have shown that the P2D model is able to achieve high accuracy over a range of operating conditions [13, 14]. In an effort to reduce the computational complexity of the P2D model, Zhang et al. [15] proposed a simplified model called the single particle model (SPM). Each electrode in the SPM is considered to be a single spherical particle and variations in the electrolyte concentration and electric potential are neglected. Wang and Gu [16] took an alternative approach to reducing the P2D model by assuming a polynomial form for the concentration of intercalated lithium. Several researchers have followed in this direction and used polynomial approximations [17, 18] or the Galerkin projection method [19–21] to convert partial differential equations into large systems of ordinary differential algebraic equations (DAEs). While this approach is effective at reducing model complexity and the resulting systems of DAEs can be simulated with relative ease, there is a need for a-priori knowledge of the solution behaviour and some pre-processing is required. In the works of Di Domenico et al. [22, 23], the P2D model is simplified by neglecting spatial variations in the concentration of intercalated lithium and the Butler–Volmer current. However, these simplification are presented in a rather ad-hoc manner and the authors do not make an effort to determine their range of validity. Other reduction approaches include proper orthogonal decomposition [24], Kalman filtering [25, 26], Pad´e approximations [27, 28], and residue grouping [29]. Jokar et al. [30] provide a comprehensive overview of strategies for simplifying electrochemical models of LIBs and describe the advantages and disadvantages of each. Despite the extensive literature on LIB model reduction, only a few of the reduced models have been validated against experimental data [21, 23, 31, 32]. The majority of the comparisons are made between reduced and P2D models or P2D models and experimental data [33, 34]. Moreover, much of the reduction strategy is based on the immediate application of mathematical techniques without giving consideration to whether the detailed electrochemical model can first be simplified based on physical grounds (e.g. due to processes having vastly different time scales). Finally, nearly all of the reduced models are derived from the P2D model and thus inherit some of the same shortcomings in terms of complexity. The goal of this paper is to present a new form of electrochemical model that is based on porous electrode theory yet is distinct from the P2D model. This model is derived by taking volume averages of microscale equations and, as a result, it depends only on physical parameters associated with the LIB cells. Using a rigorous scaling analysis, we then show how the model can be systematically reduced to a pair of equivalent circuit models and, importantly, provide three simple conditions that are required for the reduction to be valid. Since the model is derived from a detailed electrochemical model, there is no need to conduct experiments to measure equivalent circuit parameters. No pre-processing is required to use the model and 3

it can be simulated straightforwardly using traditional methods for nonlinear ordinary differential equations (ODEs). Finally, we validate the reduced model against three sets of experimental data, finding excellent agreement in all cases, and discuss the implications of the results.

2. Model We model the electrochemical processes inside a one-dimensional cell of an LIB with distance L between the two current collectors. Our choice of a one-dimensional geometry is consistent with previous literature [33, 35, 36] and justified by the large aspect ratio of real batteries [37]. The cell is composed of a positive (P ) electrode, a separator (S), and a negative (N ) electrode. The domain of the cell, 0 < x < L, can be decomposed so that the positive electrode exists on 0 ≤ x ≤ Xp , the separator on Xp ≤ x ≤ Xn , and the negative electrode on Xn ≤ x ≤ L. Each electrode is a porous material consisting of an electrically conductive solid matrix that is filled with an electrolyte. The electrolyte is able to carry ions but not lone electrons. The active component of the solid matrix carries electrons and contains intercalated lithium, the latter of which is released as ions into the electrolyte. Upon discharge, the positive electrode undergoes lithiation while the negative electrode delithiates. The charge/discharge process for the positive electrode is summarised as charge

+ − −−− −− −− −− −− * LiI ) − I + Li + e , discharge

(1)

where I is an intercalation material that hosts lithium storage in the electrode. Examples of this include CoO2 , Mn2 O4 , FePO4 , and NiO2 for the positive electrode and graphite for the negative electrode [38]. The reaction arrows are reversed for the negative electrode. The electrolyte is composed of a solvent with a lithium salt that dissociates following + − − * LiA − ) − − Li + A ·

(2)

Typical anions A include PF6 , AsF6 , ClO4 , and BF4 [38]. The separator is also a porous material that is filled with the same electrolyte as the electrodes. This allows ions, and hence electrical current, to be transferred between the two electrodes. However, the solid component of the separator is electrically insulated to prevent short circuiting. We will model the conservation of charge and mass in each of the electrodes and separator following the seminal work of Newman and Tobias [39]. We thus consider a continuum model that is valid on the macroscale of the cell. The derivation of the model is based on taking volume averages [40–43] of a microscopic model, details of which can be found in Moyles et al. [37]. We note that the volume average approach used here is a form of homogenisation which incorporates microscopic structure such as porosity into a macroscopic model, grouping the individual solid particles into a single solid continuum. This is in 4

contrast to the volume average method used, for example, by Subramanian et al. [18] who apply it to each individual solid particle as a means of removing the spatial dependence of variables. We further note that the volume average method does not produce the P2D model by Doyle et al. [12], and is instead equivalent to averaging over several solid particles. When we compare to data, we will see this is a sufficient formulation to consider. For our model, roman subscripts n, p, s on variables are used to denote quantities in the negative and positive electrodes and the separator, respectively. Thus, the variable ψj,i represents the quantity ψj in component i of the cell. We define t as time; ca,i and cL,i are the concentration (mol m−3 ) of lithium in the solid and the liquid phases; ia,i and ie,i are solid and liquid current densities (A m−2 ), and Φa,i and Φe,i (V) the electric potential of the solid and electrolyte phases. The corresponding bulk equations are   ∂ ∂ ∂ ca,i 1 ∂ (φa,i ca,i ) = φa,i Da,i + (φa,i ia,i ) , ∂t ∂x ∂x F ∂x ∂ Φa,i ia,i = −σa,i ,   ∂x ∂ ∂ (φa,i ia,i ) = −ai gi + CΓ,i (Φa,i − Φe,i ) , ∂x ∂t for the (active) solid phase of each electrode;   ∂ ∂ ∂ cL,i ∂ Φe,i 1 ∂ (φe,i cL,i ) = φe,i DL + φe,i µL F cL,i + (φe,i ie,i ) , ∂t ∂x ∂x ∂x F ∂x ie,i = F (NL,i − NA,i ),   ∂ ∂ (φe,i ie,i ) = ai gi + CΓ,i (Φs − Φe ) , ∂x ∂t

(3a) (3b) (3c)

(4a) (4b) (4c)

for the fluid phase of each electrode; and ∂ ∂ (φe,s cL,s ) = ∂t ∂x

  ∂ cL,s ∂ Φe,s φe,s DL + φe,s µL F cL,s , ∂x ∂x

ie,s = F (NL,s − NA,s ), ∂ (φe,s ie,s ) = 0, ∂x

(5a) (5b) (5c)

for the separator, where the ionic fluxes Ni,j are given by   ∂ Φe,j ∂ ci,j + zi µi F ci,j . Ni,j = − Dj ∂x ∂x

(6)

Furthermore, Da,i is the diffusivity (m2 s−1 ) of lithium in the solid matrix; DL and DA are the diffusivity of lithium and anion in the liquid electrolyte, respectively; F is Faraday’s constant (96487 C mol−1 ), and φa,i and φe,i are the volume fraction of active solid material and electrolyte (assumed constant). The electrical conductivity of the active solid area is σa,i (S m−1 ), CΓ,i are the electrode capacitances (F m−2 ), and µi the ionic mobilities (m2 mol J−1 s−1 ). The specific area ai (m−1 ) is defined as the area of active electrode 5

material per unit volume, ai = Aae,i /V,

(7)

with Ai,ae the interfacial surface area formed between active solid material and electrolyte and V the volume of the electrode. The surface-averaged electrochemical kinetics are denoted gi and following, for example, Refs. [11, 12, 44, 45], we will take Butler–Volmer-type kinetics,      (1 − βi )F −βi F gi = j0,i exp ηi − exp ηi , RTa RTa

(8)

where η is the surface overpotential, ηi = Φa,i − Φe,i − Ui ,

(9)

with Ui being the open-circuit potential (OCV), R the universal gas constant (8.314 J mol−1 K−1 ), and Ta the ambient temperature. Each term in the exponential of (8) considers anodic and cathodic reaction currents with β acting as a symmetry factor. The exchange current density is denoted j0,i which has the form similar to that in Refs. [33, 37], j0,i (ca,i , cL,i ) =

βi 1−βi βi F Ka,i KL,i ca,i

cmax a,i − ca,i cmax a,i

!1−βi i c1−β L,i ,

(10)

where Ki (m s−1 ) are heterogeneous reaction constants. The maximal density, cmax a,i , is included to capture the limited capacity of the solid matrix to store lithium. We do not include a similar maximum for the electrolyte concentration under the assumptions of infinitely dilute solution theory and in agreement with previous models [33, 36, 37]. The Butler–Volmer kinetics (10) induce a theoretical OCV [11], something that was considered in Moyles et al. [37]. However, quite often the OCV of real batteries does not fit this theoretical curve and curve fitting is used to generate the OCV in terms of the ratio of solid lithium concentration to its maximal value (denoted the state of charge) [33, 35, 36]. This deviation from the theoretical OCV is indicative of real batteries not following exact Butler–Volmer kinetics, making (10) an approximate expression at best. Nevertheless, consistent with other literature, we will utilise the Butler– Volmer theory but with a generic OCV, Ui = Ui (ca,i ).

(11)

Adding (3c) and (4c) maintains global conservation of charge, ∂ (φa,i ia,i + φe,i ie,i ) = 0, ∂x

(12)

as could be expected. The cell voltage, ∆V , is the difference in the electric potentials in the active solid phase of the positive electrode at x = 0 and negative electrode at x = L, ∆V (t) = Φa,p (0, t) − Φa,n (L, t). 6

(13)

2.1. Boundary and initial conditions The liquid electrolyte provides a means for ions to be transported between the electrodes and the separator. Therefore, at the electrode-separator interfaces, we impose continuity of lithium concentration, molar flux of lithium, electrolytic current, and electrolyte potential: cL,i − cL,s = 0,

x = Xp and x = Xn ;

(14a)

φe,i NL,i − φe,s NL,s = 0,

x = Xp and x = Xn ;

(14b)

φe,i ie,i − φe,s ie,s = 0,

x = Xp and x = Xn ;

(14c)

Φe,i − Φe,s = 0,

x = Xp and x = Xn .

(14d)

The volume fractions that appear in (14) are due to the volume averaging of the underlying microscopic model. Since the material of the separator is electrically inactive, there can be no transport of ions or current between the solid phases of the separator and the electrodes. This results in the conditions Na,i = 0,

x = Xp and x = Xn ,

(15a)

ia,i = 0,

x = Xp and x = Xn .

(15b)

The boundaries at x = 0 and x = L are assumed to correspond to the interfaces between the electrodes and current collectors. The current collectors are impermeable and therefore the molar fluxes and the corresponding currents must vanish at electrode-collector interfaces, yielding ie,p = 0,

x = 0,

(16a)

ie,n = 0,

x = L,

(16b)

x = 0 and L.

(16c)

NL,i = NL,s = 0,

Without loss of generality, we impose a grounding condition on the electrolyte potential at the interface formed by the negative electrode and current collector: Φe,n = 0,

x = L.

(17)

Finally, we assume that a current is being drawn from the positive electrode and thus impose the condition φa,p ia,p = −iapp ,

x = 0,

(18)

where iapp = Iapp /Acell is the discharge current density, Iapp is the actual discharge current, and Acell is the electrode area. We will assume that each electrode and the electrolyte has a prescribed concentration, ca,i (x, 0) = ca0,i and cL,i (x, 0) = cL0 while also imposing that the solid electric potential is the open circuit potential (11), Φa,i (x, 0) = Ui , while the liquid electric potential is Φe,i (x, 0) = 0. 7

3. Model reduction The complexity of the model can be considerably reduced by first identifying the dominant terms in the governing equations through a scaling analysis and then neglecting sub-dominant terms. To proceed with the scaling analysis, the order of magnitude of each variable in the model is estimated using a characteristic scale that is chosen based on the physics of battery operation. For instance, we take the distance between the current collectors, L, to be the relevant length scale in the model, which is used to estimate the size of gradients. Moreover, it is physically reasonable to assume that the applied current density iapp can be used to estimate the order of magnitude of the local current densities in the cell. Finally, we assume the thermal voltage RTa /F to be the characteristic scale for the electric potentials. Using this choice of scales, the change in concentration of lithium ions in the electrolyte, ∆cL , can be estimated by considering a quasisteady scenario whereby diffusion balances the generation and consumption of lithium from electrochemical reactions. Thus, by making the approximations DL ∂ 2 cL /∂x2 ' Dl ∆cL /L2 and F −1 ∂ie,i /∂x ' iapp /(F L) and then equating the results, we find that ∆cL = (iapp L)/(F DL ). The change in concentration of lithium ions relative to the initial value is given by γ=

∆cL iapp L = . cL0 F DL cL0

(19)

The time scales of lithium diffusion in the electrolyte and solid electrode materials are given by L2 /DL and L2 /Da,i , respectively. Thus, the relative rate at which lithium diffuses across the solid and liquid phases of the electrodes can be estimated from the ratio Di =

Da,i . DL

(20)

The magnitude of the gradient in electrical potential that is generated due to Ohmic resistance in the solid component of the electrodes can be estimated from (3b) as iapp /σa,i . However, our choice of scalings for the potential and space for the imparts an intrinsic estimate for the gradient of the electrical potentials given by (RTa )/(F L). The ratio of these two estimates leads to the definition of a dimensionless resistivity given by νa,i =

iapp LF . RTa σa,i

(21a)

Similarly, by introducing the ionic conductivity of the electrolyte, σe = F 2 cL0 (µA + µL ), the relative magnitude of gradients in the electrolyte potential can be estimated through the dimensionless ionic resistivity νe =

iapp LF . RTa σe

(21b)

The following simplifications can be applied to the model when one of the dimensionless parameters in (19)–(21) are small: 8

Simplification 1. If the dimensionless resistivities given by (21) are small, then, to a first approximation, the electric potentials are uniform in space. In particular, Φa,i (x, t) = Φa,i (t),

Φe,i (x, t) = 0.

(22)

In physical terms, the applied current is too small, or the electrical and ionic conductivities are too large, for significant gradients in the electrical potential to develop across the electrodes. In this case, Eqns. (3b) and (4b) can be eliminated from the model. Simplification 2. If the relative rate of lithium diffusion in the solid given by (20) is small, Di  1, then the electrode model simplifies to 1 ∂ ia,i ∂ ca,i = , ∂t F ∂x   ai ∂ ∂ ia,i =− gi + CΓ,i (Φa,i − Φe,i ) . ∂x φa,i ∂t

(23a) (23b)

Simplification 3. If the parameter γ, defined by (19), is small, then the concentration of lithium ions in the electrolyte does not undergo a substantial change from its initial value; thus, the approximation cL,i (x, t) ' cL0 is valid and Eqns. (4a) and (5a) can be eliminated from the model. The consequence of this approximation is that the exchange current densities j0,i , defined in (10), become functions of the concentration of lithium in the solid component of the electrode (unknown) and the initial concentration of lithium ions in the electrolyte (known): j0,i (ca,i , cL,i ) = j0,i (ca,i , cL0 ).

(24)

The full mathematical justification of these simplifications can be found in Sec. 3 of Moyles et al. [37]; however, we highlight two important points. Firstly, Simplification 3 does not imply that the liquid-phase lithium concentration is constant. Rather, it implies that deviations from the initial concentration are negligible. Moyles et al. compute the concentration of lithium in the electrolyte and show that it has a piecewise quadratic profile throughout space, similar to what was identified numerically by Li et al. [33]. However, the Butler–Volmer kinetics are unaffected by the profile of the electrolyte concentration allowing Simplification 3 to hold. Secondly, Simplifications 1–3 arise from a leading-order asymptotic analysis. A first-order correction will capture the spatial dependence of the solid-phase lithium concentration induced by the spatial profile of the concentration of lithium in the electrolyte. When all three simplifications are simultaneously satisfied, case studies of which will be shown below, the governing equations can be systematically reduced to a system of two ordinary differential equations (ODEs). The reduction begins by noticing that if the initial concentration of lithium in the solid component of the electrode, ca0,i , is spatially uniform, then it will remain uniform for all time: ca,i (x, t) = ca,i (t). Thus, integrating (23a) across the electrode domains and imposing the relevant boundary conditions for the 9

current density yields a differential equation for ca,i (t), which can be solved to find iapp t , F φa,p Xp iapp t ca,n (t) = ca0,n − . F φa,n (L − Xn )

ca,p (t) = ca0,p +

(25a) (25b)

Thus, the concentration of intercalated lithium is expected to change linearly with time. By substituting the solutions for the lithium concentration (25) into (23a), the current densities in the solid phase of the electrodes are found to be iapp (Xp − x) , φa,p Xp iapp =− (x − Xn ) . φa,n (L − Xn )

ia,p = −

(26a)

ia,n

(26b)

The current density in the electrolyte can then be obtained by integrating (12) to find that φa,i ia,i + φe,i ie,i = −iapp .

(27)

Finally, by substituting the solid current densities given by (26) into (23b), we find that the solid potential in the electrode satisfies the system of decoupled, nonlinear ODEs iapp dΦa,p + gp (Φa,p , ca,p ) = − , dt ap Xp iapp dΦa,n + gn (Φa,n , ca,n ) = , CΓ,n dt an (L − Xn ) CΓ,p

(28a) (28b)

where the gi are defined in (8)–(11) with Φe,i = 0, cL,i = cL0 and ca,i = ca,i (t) given by (25). The initial conditions for (28) are given by Φa,i (0) = Ui (ca0,i ). An examination of the equations in (28) reveals that the continuum model based on porous electrode theory has been systematically reduced to an equivalent circuit model. The circuit model shows that the electrodes can be compartmentalised into three elements consisting of a capacitor and two current sources representing the applied current and the current generated by electrochemical reactions. The circuit equations (28) decouple because Simplifications 1 and 3 imply that both the electric potential of the electrolyte and the concentration of lithium ions in the electrolyte are, to a first approximation, uniform across the cell. These features make our model similar to the SPM; however, rather than assuming the liquid-phase lithium concentration and electric potential have uniform spatial profiles, we are able to rigorously derive the physical conditions that lead to this behaviour. Our model is, in fact, even simpler than the SPM because the solid-phase lithium concentration and electric potential are uniform on the macroscale as well, which was naively assumed to be the case in the electrode-averaged model (EAM) proposed by Di Domenico et al. [22]. Thus, our model may be considered as a hybrid SPM-EAM that accounts for capacitance effects at the electrode-electrolyte interfaces. 10

4. Results and discussion We first compare our reduced model to data and simulations presented by Li et al. [33]. The battery used in their study has graphite for the negative electrode and lithium iron phosphate (LFP) for the positive electrode. They prescribe a fitted OCV for each electrode given by   ξn − 0.1958 Un = 0.6379 + 0.5416 exp(−305.5309ξn ) + 0.044 tanh − 0.1088       ξn − 1.0571 ξn + 0.0117 ξn − 0.5692 − 0.1978 tanh − 0.6875 tanh − 0.0175 tanh , 0.0854 0.0529 0.0875

(29a)

Up = 3.4323 − 0.8428 exp(−80.2493(1 − ξp )1.3198 ) − 3.2474 × 10−6 exp(20.2645(1 − ξp )3.8003 ) + 3.2482 × 10−6 exp(20.2646(1 − ξp )3.7995 ),

(29b)

where ξi is the state of charge of the battery defined as ξi =

ca,i . cmax a,i

(30)

The battery used by Li et al. has a capacity of 2.3Ah and data in their manuscript is available for discharge at rates of 0.1C, 0.5C, 1C, and 2C. The electrode cell area is 0.6194 m2 and the battery cell has a distance of 129 µm between the current collectors. The conductivity for each electrode is σa,p = 2.15 Sm−1 and σa,n = 100 Sm−1 while the conductivity of the electrolyte is σe = 3.23 Sm−1 . We take an ambient temperature of 298.15 K. From (21) we have that the key dimensionless parameters are 3.2 × 10−3 ≤ νa,p ≤ 6.3 × 10−2 , 6.8 × 10−5 ≤ νa,n ≤ 1.4 × 10−3 , and 2.1 × 10−3 ≤ νe ≤ 4.2 × 10−2 depending on the particular C-rate. These values are all small thus satisfying Simplification 1. Furthermore, using the other parameters in Li et al., we find from (20) and (19) that Dp = 1.9 × 10−4 , Dn = 1.5 × 10−4 , and γ = 5.8 × 10−2 thus satisfying the conditions for Simplifications 2 and 3 as well. Therefore, the reduced model (28) is appropriate. The parameters required to solve this model are in Table 1. However, Li et al. do not directly provide the rate constants Ki,j and instead provide values for the expression βi 1−βi Ka,i KL,i 1−βi cmax a,i

≡ Cˆi ,

(31)

from which we can extract the rate constant product. This product is necessary for computing j0,i in (10), but the individual rate constant values are not needed. We first validate the concentration approximation (25) with simulation data from Li et al. [33], who simulate a 1C discharge pulse with a duration of 200 s and plot the solid-phase concentration in each of the electrodes. Their modelling approach uses the P2D formulation which means they have different concentration curves at each radial point r in the spherical electrode particles. However, the curves are very tightly bound aside from the small deviations that occur near r = 0. To compare our results with Li 11

Table 1: Parameters for the reduced model (28) from Li et al. [33].

Parameter (unit)

Value Positive Electrode

Negative Electrode

ai (m−1 )

3.53 ×107

4.71 ×105

φe,i (–)

0.33

0.33

φa,i (–)

0.43

0.55

−3 cmax ) a,i (mol m

22806

31370

cs0,i (mol m−3 )

0.022 cmax a,p

0.86 cmax a,n

cL0,i (mol m−3 )

1200

1200

βi (–)

0.5

0.5

βi 1−βi Ka,i KL,i (m s−1 )

2.1×10−10

CΓ,i (F m−2 )

0.2

a

a

5.3×10−9

a

0.2

see discussion surrounding (31)

et al., we arbitrarily select their simulation results for r = R/2, where R is the particle radius, to be as far away as possible from either of the radius boundaries. In Figure 1, we present the solid-phase lithium concentrations computed from (25) at a 1C discharge rate and compares them to the r = R/2 simulation data. We emphasise that the comparisons in Figure 1 are between large-scale simulation results and a significantly reduced model of the same type. Li et al. do not provide any experimental data on the lithium concentration. In fact, data of any kind on lithium concentrations is only recently being collected through magnetic resonance imaging and nuclear magnetic resonance spectroscopy techniques [46]. We now solve for the solid-phase electrical potential in each electrode using the reduced model (28) with the ode15s function in MATLAB for the various C-rates. We then compute the cell voltage via (13). The differential equations we solve are functions of time whereas discharge curves are generally presented as a function of capacity. As the battery discharges and one of the electrodes either becomes completely full or completely empty, the cell voltage will tend to negative infinity. In theory this leads to model solutions blowing up after a finite amount of time has passed, which has been predicted by Moyles et al. [37]. As the asymptote is approached in simulations of (28), either the time step required to continue computing decreases below some selected threshold value or the asymptote is overstepped and the battery artificially charges. In either case, the simulation is terminated and the discharge time, tdis , is taken to be the time at which the minimum cell potential is achieved. We then define a normalised time as T = t/tdis so that T = 1 corresponds to a fully discharged battery. Discharge curves computed from the reduced model are compared to those obtained from simulations and experiments by Li et al. in Figure 2. The average time needed to solve the reduced model using a 12

(a) Positive electrode.

(b) Negative electrode.

Figure 1: Comparison of the simplified solid-phase lithium concentration (25) and the r = R/2 simulation data presented in Figure 6 of Li et al. [33]. The rationale for only displaying data up to 200 s is that Li et al. consider only a discharge pulse duration of 200 s after which the cell relaxes.

Microsoft Surface Pro 3 with a 2.50GHz processor and 8GB of RAM is less than 0.4 s. Figure 2 (a) shows that the reduced model is able to capture data from large-scale simulations and battery experiments at a 2C discharge rate with a remarkably high degree of accuracy. However, the agreement between the reduced model and the simulation and experimental data seems to worsen as the C-rate decreases. This is suspicious since the discharge curve should approach the OCV as the C-rate is decreased. Further inspection of the model used by Li et al. reveals that their description of the OCV is taken from Safari and Delacourt [35] who derive (29a) and (29b) by fitting to a discharge curve at a very low C-rate of 1/100 for a different battery. We therefore compare our reduced model (28) to the discharge data of Safari and Delacourt to see if an improvement at low C-rate is noted. The battery studied by Safari and Delacourt [35] is still based on graphite and LFP electrodes, and the parameters relevant to the reduced model (28) are the same as those in Li et al. [33] listed in Table 1 with the exception of Cˆi defined in (31). Using the values of Cˆi from Safari and Delacourt replaces the rate β

1−βp

1−βn p βn constant values in Table 1 with Ka,n KL,n = 1.45 × 10−9 m s−1 and Ka,p KL,p

= 8.28 × 10−12 m s−1 for

the negative and positive electrode respectively. Figure 3 compares the cell voltage results of the reduced model (28) using the new parameters along with the data for a 2.3Ah 26650 graphite/LFP cell studied by Safari and Delacourt [35] for C-rates of 1/10, 1, and 3. We note that even for the higher C-rate of 3, the largest value of νi from (21) is 0.09 which is still small and Simplifications 1-3 hold. The agreement at low C, which is approximately the OCV, has indeed improved. However, the excellent agreement persists even at the relatively high discharge rate of 3C. 13

(a) 2C

(b) 1C

(c) 0.5C

(d) 0.1C

Figure 2: Comparison of the cell voltage (13) between the reduced model (28) as well as data and large-scale simulations from an ANR26650m1-a battery [47], both presented in Figure 4 of Li et al. [33].

14

(a) 3C

(b) 1C

(c) 0.1C Figure 3: Comparison of the cell voltage (13) with the data from a 2.3Ah 26650 graphite/LFP battery from Figure 1 of Safari and Delacourt [35].

15

We now demonstrate that the model reduction is versatile to batteries besides the graphite/LFP cells by comparing to results of Zhao et al. [34] who consider several LiCOO2 batteries. Arbitrarily, we compare to battery 2 in their manuscript which has a capacity of 1.2 Ah and is discharged at a rate of 1C. The OCV for this battery is Un = 8.0023 + 5.0647ξn − 12.578ξn0.5 − 8.6322 × 10−4 /ξn + 2.1765 × 10−5 ξn1.5 − 0.4602 exp(15(0.06 − ξn )) − 0.5536 exp(−2.4326(ξn − 0.92)),

(32a)

Up = 85.681ξp6 − 357.7ξp5 + 613.89ξp4 − 555.65ξp3 + 281.06ξp2 − 76.648ξp − 0.30987 exp(5.657ξp115 ) + 13.1983.

(32b)

Using the other parameters reported by Zhao et al., we find that νa,p = 9.3 × 10−3 , νa,n = 5.6 × 10−4 , and νe = 1 × 10−2 satisfying Simplification 1. Furthermore Dp = 3.8 × 10−4 , Dn = 1.5 × 10−4 , and γ = 2.9 × 10−2 from (20) and (19); thus Simplifications 2 and 3 hold as well, validating the use of the reduced model (28). The parameters from Zhao et al. [34] that are required for solving the reduced model are listed in Table 2. Table 2: Parameters for the reduced model (28) from Zhao et al. [34].

Parameter (unit)

Value Positive Electrode

ai (m

−1

1.8 ×10

)

6

Negative Electrode 1.74 ×106

φe,i (–)

0.3

0.3

φa,i (–)

0.6

0.58

−3 cmax ) a,i (mol m

49943

31858

cs0,i (mol m−3 )

0.32 cmax a,p

0.6 cmax a,n

cL0,i (mol m−3 )

2000

2000

βi (–)

0.5

0.5

CΓ,i (F m a

−2

)

a

0.2

0.2a

Chosen

It should be noted in Table 2 that the capacitance is chosen to be the same as in Li et al. [33] since such a value is not reported in Zhao et al. [34]. More importantly, however, Zhao et al. do not report their values for the rate constants Ka,i or any other values from which the rate constants could be deduced. We instead will fit them based on the discharge data, something easily achievable because of the simplicity of the reduced model. β

1−βp

p We use lsqcurvefit in MATLAB to do the parameter fitting which yields Ka,p KL,p

= 3.11 × 10−10 m s−1

1−βn βn and Ka,n KL,n = 5.39 × 10−9 m s−1 , values similar to those reported by Li et al. [33]. The computational

time required to determine these parameters was approximately 10 s on a Microsoft Surface Pro 3 with a 2.50 GHz processor and 8 GB of RAM. Having obtained the missing parameter, we compare the results of 16

the reduced model (28) to the discharge data and simulation results of Zhao et al. in Figure 4 where, once again, excellent agreement is obtained.

Figure 4: Comparison of the cell voltage (13) between the reduced model (28) as well as data and simulations for a 1.2Ah LiCOO2 battery in Zhao et al. [34].

5. Conclusion We have demonstrated through rigorous scaling analysis how a volume-averaged battery model derived from porous electrode theory can be reduced to an equivalent circuit model (28). The circuit model includes a capacitor and current sources from both the electrochemical reactions and the applied current. Solving the reduced model requires eight parameters (applied current density, area of active electrode per unit volume, electrode porosity, electrode active material fraction, maximal lithium concentration in each electrode, cathode-anode symmetry factor, Butler–Volmer reaction rate constant, and electrode capacitance), two initial conditions (initial lithium concentration in electrolyte and solid phase), and the OCV for each electrode. Our reduced model is robust. We have compared our reduced model simulations to three data sets, two for graphite-LFP batteries [33, 35] and one for LiCOO2 [34] showing excellent agreement for a variety of discharge rates. This is in distinct contrast to other reduced models which only compare to simulations of the full electrochemical model [18, 22, 37]. While this is essential for justifying the reduced model, it is important to state that the computational gains and other benefits made from the reduction are only as powerful as the reliability of the underlying model. By comparing to actual discharge curves, we are 17

validating both the large-scale and reduced models. Aside from data, we also compare to full simulation results accompanying the cited dataset when available. Primarily, this is to demonstrate good agreement between data, simulation, and our model. However, there is an interesting secondary effect. The models corresponding to the three datasets are all different. Li et al. [33] and Zhao et al. [34] take the most similar approach to us in that they follow porous electrode theory. However, they both use the P2D approach with spherical particles. Li et al. also use concentrated solution theory which introduces the electrolyte activity parameter. Furthermore, they consider current collectors in their model. The model from Safari and Delacourt [35] is quite different from the others in that they have only time-dependent differential equations and algebraic expressions arising from using threeparameter fourth-order polynomial approximations for the lithium concentration in the solid phase. This is a procedure taken from Subramanian et al. [17]. Despite the modelling differences, we are all able to generate results that compare favourably with data. It is natural to then wonder, what is the correct model? This is an important question to ask because many authors use the agreement with discharge curves as evidence that their model is correct: this is seemingly the justification for the universality of the P2D model [7, 13, 30]. At the very least, the P2D model is used as a benchmark when experimental data is limited [30]. It has been shown by Dalwadi et al. [48] that certain restrictions on diffusion can indeed require coupling microscopic and macroscopic transport models, justifying a P2D approach. However, it is not clear when such a separate treatment is required for LIB analysis and this is an avenue of future work. Indeed, we have shown three case studies where excellent results can be obtained without considering the P2D approach. At the very least, any model that describes the evolution of lithium concentration should be treated cautiously until more operational experimental data becomes available such as in Ref. [46]. When many models lead to the same results, the simplest should be employed. One unfortunate consequence of the P2D formulation is that the radial microscopic diffusion problem in the solid is coupled to the macroscopic transport of the electrolyte lithium. This means that either the radial problem must be solved at every x-position or simplifications such as polynomial approximations must be used [17, 35]. We avoid all of this complication by treating the solid as a volume-averaged continuum. An advantage of using a fast and simple model, such as the one derived here, is that very little computational resources and time are wasted if simulations do not match the data. When matching fails, this may be due to a problem with the parameters as we showcased in Figure 1 when comparing to data from Li et al. [33]. We saw an increasingly poor agreement as the C-rate was decreased. However, all models should converge to the OCV as the C-rate decreases. This is an important point to consider in model validation. We saw that the OCV chosen by Li et al. was based on the batteries studied by Safari and Delacourt [35], which likely have different electrochemical and physical characteristics. Comparing our modelling results to the experimental data obtained by Safari and Delacourt using a consistent OCV produced excellent agreement. 18

We cannot explain how Li et al. achieve near-perfect agreement between their simulations and data while using the OCV of Safari and Delacourt. If parameters are wrong or unknown, parameter fitting may be appropriate. The simple reduced model posed here is amenable to parameter fitting as we demonstrated when creating the modelling curves used in Figure 4. Parameter fitting can involve several calls to the forward model which is fast to run. This makes our reduced model ideal for use in fitting algorithms. Fitting on larger models can be cumbersome as these computations can take several minutes to run [18]. However, being able to parameter fit is important because parameters such as Butler–Volmer rate constants can be difficult to measure and their accuracy in simulations relies on a correct formulation of the underlying kinetics. If the reduced model fails to match data when all parameters are confidently measured or fitted as best they can be, then this could be an indication that neglected physics may play an important role. At this stage, running a full simulation of a standard P2D model may be warranted to see if the issues are resolved. Further failure indicates that new modelling should be considered. Overall we have demonstrated that our reduced model derived through scaling analysis is a fast and robust computational tool for predicting battery discharge curves. It provides a useful and quick diagnostic for determining if a battery requires more complicated physics to be properly modelled or if parameters need to be corrected. It can be readily used for real-time SOC and SOH monitoring and in fast methods for parameter optimisation. Despite the computational drawbacks, full simulations of large-scale models may appear attractive because of their inclusion of many physical and chemical effects. However, as these can take several minutes or even hours to run, they are impractical when many cells are stacked. Furthermore, these are completely infeasible for real-time analysis and management. The speed and simplicity of our reduced model combined with its rigorous derivation means that we can easily upscale into battery management systems. Each added cell would require two more differential equations along with appropriate boundary conditions that connect the cells together. Incorporating such a battery management system is an avenue of future work.

Acknowledgements IRM would like to acknowledge financial support from an Irish Research Council New Foundations Grant, a Charlemont Fellowship from the Royal Irish Academy, and from Science Foundation Ireland under grant number SFI/13/IA/1923. MGH has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 707658 and acknowledges travel support from the Mathematics Applications Consortium for Science and Industry. TGM acknowledges financial support from the Ministerio de Ciencia e Innovaci´ on grant MTM2017-82317-P. MGH and TGM have been partially funded by the CERCA Programme of the Generalitat de Catalunya. BRW acknowledges support from an NSERC 19

Canada grant. [1] B. Scrosati, J. Garche, Lithium batteries: Status, prospects and future, Journal of Power Sources 195 (9) (2010) 2419–2430. [2] J.-M. Tarascon, M. Armand, Issues and challenges facing rechargeable lithium batteries, Nature 414 (6861) (2001) 359–367. [3] L. Lu, X. Han, J. Li, J. Hua, M. Ouyang, A review on the key issues for lithium-ion battery management in electric vehicles, Journal of Power Sources 226 (2013) 272–288. [4] K. Smith, C.-Y. Wang, Solid-state diffusion limitations on pulse operation of a lithium ion cell for hybrid electric vehicles, Journal of Power Sources 161 (1) (2006) 628–639. [5] M. A. Hannan, M. H. Lipu, A. Hussain, A. Mohamed, A review of lithium-ion battery state of charge estimation and management system in electric vehicle applications: Challenges and recommendations, Renewable and Sustainable Energy Reviews 78 (2017) 834–854. [6] A. A. Franco, Multiscale modelling and numerical simulation of rechargeable lithium ion batteries: concepts, methods and challenges, RSC Advances 3 (32) (2013) 13027–13058. [7] V. Ramadesigan, P. W. Northrop, S. De, S. Santhanagopalan, R. D. Braatz, V. R. Subramanian, Modeling and simulation of lithium-ion batteries from a systems engineering perspective, Journal of the Electrochemical Society 159 (3) (2012) R31–R45. [8] A. Seaman, T.-S. Dao, J. McPhee, A survey of mathematics-based equivalent-circuit and electrochemical battery models for hybrid and electric vehicle simulation, Journal of Power Sources 256 (2014) 410–423. [9] P. W. Northrop, B. Suthar, V. Ramadesigan, S. Santhanagopalan, R. D. Braatz, V. R. Subramanian, Efficient simulation and reformulation of lithium-ion battery models for enabling electric transportation, Journal of the Electrochemical Society 161 (8) (2014) E3149–E3157. [10] T. Huria, M. Ceraolo, J. Gazzarri, R. Jackey, High fidelity electrical model with thermal dependence for characterization and simulation of high power lithium battery cells, in: Proceedings of the 2012 IEEE International Electric Vehicle Conference, IEEE, 2012, pp. 1–8. [11] J. Newman, K. E. Thomas-Alyea, Electrochemical Systems, John Wiley & Sons, 2004. [12] M. Doyle, T. F. Fuller, J. Newman, Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell, Journal of the Electrochemical Society 140 (6) (1993) 1526–1533. [13] M. Doyle, J. Newman, A. S. Gozdz, C. N. Schmutz, J.-M. Tarascon, Comparison of modeling predictions with experimental data from plastic lithium ion cells, Journal of the Electrochemical Society 143 (6) (1996) 1890–1903. [14] K. Kumaresan, G. Sikha, R. E. White, Thermal model for a Li-ion cell, Journal of the Electrochemical Society 155 (2) (2008) A164–A171. [15] D. Zhang, B. N. Popov, R. E. White, Modeling lithium intercalation of a single spinel particle under potentiodynamic control, Journal of the Electrochemical Society 147 (3) (2000) 831–838. [16] C. Wang, W. Gu, B. Liaw, Micro-macroscopic coupled modeling of batteries and fuel cells I. Model development, Journal of the Electrochemical Society 145 (10) (1998) 3407–3417. [17] V. R. Subramanian, V. D. Diwakar, D. Tapriyal, Efficient macro-micro scale coupled modeling of batteries, Journal of the Electrochemical Society 152 (10) (2005) A2002–A2008. [18] V. R. Subramanian, V. Boovaragavan, V. D. Diwakar, Toward real-time simulation of physics based lithium-ion battery models, Electrochemical and Solid-State Letters 10 (11) (2007) A255–A260. [19] G. Fan, X. Li, M. Canova, A reduced-order electrochemical model of Li-ion batteries for control and estimation applications, IEEE Transactions on Vehicular Technology 67 (1) (2018) 76–91. [20] T.-S. Dao, C. P. Vyasarayani, J. McPhee, Simplification and order reduction of lithium-ion battery model based on porous-electrode theory, Journal of Power Sources 198 (2012) 329–337. [21] V. R. Subramanian, V. Boovaragavan, V. Ramadesigan, M. Arabandi, Mathematical model reformulation for lithium-ion

20

battery simulations: Galvanostatic boundary conditions, Journal of the Electrochemical Society 156 (4) (2009) A260–A271. [22] D. Di Domenico, G. Fiengo, A. Stefanopoulou, Lithium-ion battery state of charge estimation with a Kalman filter based on a electrochemical model, in: Proceedings of the 17th IEEE International Conference on Control Applications, Ieee, 2008, pp. 702–707. [23] C. Speltino, D. Di Domenico, G. Fiengo, A. Stefanopoulou, Comparison of reduced order lithium-ion battery models for control applications, in: Proceedings of the Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, IEEE, 2009, pp. 3276–3281. [24] L. Cai, R. E. White, Reduction of model order based on proper orthogonal decomposition for lithium-ion battery simulations, Journal of the Electrochemical Society 156 (3) (2009) A154–A161. [25] K. A. Smith, C. D. Rahn, C.-Y. Wang, Model-based electrochemical estimation and constraint management for pulse operation of lithium ion batteries, IEEE Transactions on Control Systems Technology 18 (3) (2010) 654–663. [26] A. M. Bizeray, S. Zhao, S. R. Duncan, D. A. Howey, Lithium-ion battery thermal-electrochemical model-based state estimation using orthogonal collocation and a modified extended Kalman filter, Journal of Power Sources 296 (2015) 400–412. [27] J. Marcicki, M. Canova, A. T. Conlisk, G. Rizzoni, Design and parametrization analysis of a reduced-order electrochemical model of graphite/LiFePO4 cells for SOC/SOH estimation, Journal of Power Sources 237 (2013) 310–324. [28] N. T. Tran, M. Vilathgamuwa, T. Farrell, S. S. Choi, Y. Li, J. Teague, A Pad´ e approximate model of lithium ion batteries, Journal of the Electrochemical Society 165 (7) (2018) A1409–A1421. [29] K. A. Smith, C. D. Rahn, C.-Y. Wang, Model order reduction of 1D diffusion systems via residue grouping, Journal of Dynamic Systems, Measurement, and Control 130 (1) (2008) 011012. [30] A. Jokar, B. Rajabloo, M. D´ esilets, M. Lacroix, Review of simplified Pseudo-two-Dimensional models of lithium-ion batteries, Journal of Power Sources 327 (2016) 44–55. [31] V. Ramadesigan, K. Chen, N. A. Burns, V. Boovaragavan, R. D. Braatz, V. R. Subramanian, Parameter estimation and capacity fade analysis of lithium-ion batteries using reformulated models, Journal of the Electrochemical Society 158 (9) (2011) A1048–A1054. [32] J. C. Forman, S. J. Moura, J. L. Stein, H. K. Fathy, Genetic identification and fisher identifiability analysis of the doyle–fuller–newman model from experimental cycling of a LiFePO4 cell, Journal of Power Sources 210 (2012) 263–275. [33] J. Li, Y. Cheng, M. Jia, Y. Tang, Y. Lin, Z. Zhang, Y. Liu, An electrochemical–thermal model based on dynamic responses for lithium iron phosphate battery, Journal of Power Sources 255 (2014) 130–143. [34] R. Zhao, J. Liu, J. Gu, Simulation and experimental study on lithium ion battery short circuit, Applied Energy 173 (2016) 29–39. [35] M. Safari, C. Delacourt, Modeling of a commercial graphite/lifepo4 cell, Journal of the Electrochemical Society 158 (5) (2011) A562–A571. [36] P. Amiribavandpour, W. Shen, D. Mu, A. Kapoor, An improved theoretical electrochemical-thermal modelling of lithiumion battery packs in electric vehicles, Journal of Power Sources 284 (2015) 328–338. [37] I. R. Moyles, M. G. Hennessy, T. G. Myers, B. R. Wetton, Asymptotic reduction of a porous electrode model for lithium-ion batteries, arXiv:1805.07093v2 (2018). [38] Q. Wang, P. Ping, X. Zhao, G. Chu, J. Sun, C. Chen, Thermal runaway caused fire and explosion of lithium ion battery, Journal of Power sources 208 (2012) 210–224. [39] J. S. Newman, C. W. Tobias, Theoretical analysis of current distribution in porous electrodes, Journal of the Electrochemical Society 109 (12) (1962) 1183–1191. [40] S. Whitaker, Advances in theory of fluid motion in porous media, Industrial & Engineering Chemistry 61 (12) (1969) 14–28.

21

[41] J. Bear, Dynamics of Fluids in Porous Media, Courier Corporation, 1972. [42] A. C. Fowler, Mathematical Models in the Applied Sciences, Vol. 17, Cambridge University Press, 1997. [43] M. Kaviany, Principles of heat transfer in porous media, Springer Science & Business Media, 2012. [44] J. Newman, W. Tiedemann, Porous-electrode theory with battery applications, AIChE Journal 21 (1) (1975) 25–41. [45] T. F. Fuller, M. Doyle, J. Newman, Simulation and optimization of the dual lithium ion insertion cell, Journal of the Electrochemical Society 141 (1) (1994) 1–10. [46] S. A. Krachkovskiy, J. M. Foster, J. D. Bazak, B. J. Balcom, G. R. Goward, Operando mapping of Li concentration profiles and phase transformations in graphite electrodes by MRI and NMR, The Journal of Physical Chemistry C 122 (38) (2018) 21784–21791. [47] A123systems, Nanophosphate high power lithium ion cell ANR26650m1A, https://www.rcgroups.com/forums/show att.php?attachmentid=4155818. Accessed February 26, 2018. (2010). [48] M. P. Dalwadi, Y. Wang, J. R. King, N. P. Minton, Upscaling diffusion through first-order volumetric sinks: a homogenization of bacterial nutrient uptake, SIAM Journal on Applied Mathematics 78 (3) (2018) 1300–1329.

22