Electrochemical-Thermal Modeling of Large-Format, Thin-Film, Lithium-Ion Batteries with Cocurrent and Countercurrent Tab Connections Using a Reduced-Order Model

We derive and implement a new reduced-order model for the simulation of large-format, thin-film batteries with cocurrent and countercurrent tab connections. We employ the multi-site, multi-reaction (MSMR) framework to describe the solid phase thermodynamics as well as irreversible phenomena associated with diffusion and electrochemical reactions for a graphite negative and a spinel manganese oxide positive. The calculations are streamlined by using the reduced-order electrochemical model for a porous electrode derived by means of a perturbation analysis, which we term ROM1. For discharge rates less than 1 C, where the 1 C rate corresponds to the current needed to fully discharge the cell in 1 h, ROM1 yields accurate results for traction-battery electrodes. We employ ROM1 in the cell energy balance, with the overall results allowing one to clarify the current and temperature distributions within the cell during discharge and isolate and identify the different heat sources. The governing partial differential equations are coupled and nonlinear in part due to the temperature dependence of the physicochemical properties. We show how cocurrent tab locations yield higher cell energy densities, while countercurrent tab locations yield more uniform current and temperature distributions. Sensitivity analyses underscore the flexibility of the approach. Overall, the equation system and open-source (Python) software enables an efficient and rational tool for cell design and integration.

The topic of thermal analysis of battery systems has a long history and has grown in importance as the energy density of batteries, particularly lithium-ion batteries, has increased, and microporous separators, whose function is to keep the negative and positive electrodes from electrically shorting, have been reduced in thickness to decrease impedance and increase energy density.Putting larger amounts of energy into smaller volumes with less separation between the electrodes places more emphasis on understanding and managing the thermal behavior and stability of battery systems.Early history and context on the thermal modeling of batteries can be found in Refs.1-6.Mathematical models are useful for deconvoluting phenomena governing the electrochemical and thermal processes within a cell, providing foundational underpinnings for cell design and integration within modules, packs (i.e., cellto-pack configurations that bypass the need for battery modules), or devices.
For this work, we examine a cell consisting of a graphite negative and a spinel manganese oxide positive.These active materials are used in battery systems today, and, as will be referenced when various model parameter values are provided (e.g., see Table II and Appendix A), prior publications provide parameter values.We shall examine discharge events and compare and contrast cocurrent and countercurrent tab connections.The model formulation can also be used to examine charge events as well by changing the sign of the cell current density or increasing the cell voltage above that of the initial cell voltage (cf Eq. 15).We do not address degradation phenomena in this work.
][10] The impact of air-cooling methodologies on the high-rate discharge performance has been investigated with three-dimensional thermal-electrochemical models. 11,12A comparison of air vs liquid cooling of battery packs is provided in Ref. 13.The recent models mentioned are computationally intensive, which has given impetus to developing reduced-order models (ROMs) for electrochemicalthermal modeling, and which have proven useful, for example, in the clarification of accelerated aging due to cell heating. 14,15A suite of models ranging from that of a three-dimensional electrochemicalthermal models to various ROMs derived from asymptotic analyses is presented in Ref. 18; the authors are part of the team that developed PyBaMM, 3 a convenient open-source software tool for mathematically modeling battery cells, and which allows for the additional utility of Python routines.][24][25][26][27][28][29][30] In this work, we describe a ROM for the treatment of the cell shown schematically in Fig. 1.The concentration and temperature effects on the physicochemical parameters are described in Appendix A. For the full system of equations, Appendix B is provided.A central theme of this work is that countercurrent tab locations lead to a more uniform current and temperature distributions; however, as will be described in the context of Fig. 3, cocurrent tab locations can lead to more efficient packaging and higher energy density.

Model Formulation
Introductory remarks.-Aschematic of the cell geometry is provided in Fig. 1, which also defines much of the notation used for geometric quantities.Quantities and values for the base-case calculations are provided in Table II.A list of nomenclature appears in Table III.We shall use subscript = k n p or for the negative or positive electrode, respectively.Porous electrode theory 31,32 applied to a lithium ion cell yields differential equations for the solutionphase variables, including the salt concentration, current density, and electric potential, as well as the solid-phase variables, including the Li intercalate concentration, current density, and electric potential, consistent with Fig. 1b; 33,34 the full system of equations are provided in this document with the inclusion of Appendix B. Figure 2 provides problem schematics for the different equation systems that will be referenced in this work.Figure 2a refers to the Full-  Order Model (FOM) that treats a cell using porous electrode theory (cf Appendix B) along with the multi-site, multi-reaction (MSMR) model [22][23][24][25] and = L 0, where L is the length of the current collectors.Equation 2 through 20 of Ref. 36 provide a model for a porous (alloy or intercalation) electrode adjacent to a separator and with a Li counter electrode (often referred to as a half cell), along with the MSMR formulation.For FOM, we replace the Li counter electrode with an alloy or intercalation electrode.Figure 2b refers to a reduced-order model (ROM1).Figures 2c and 2d refer to a cell of finite length L and cocurrent (tabs on the same side of the cell) and countercurrent (tabs on opposing sides of the cell) cell designs, respectively.
The geometry provided in Figs.1a and 1b was examined by the authors 3 in the context of a thin-film battery (TFB) and analytic solutions; the TFB is also the subject of this work, and is relevant to most lithium ion prismatic and pouch cell batteries in use today and for the foreseeable future.It is assumed that the current collecting tabs extend along the full z-dimension of the TFB and that no heat transfer takes place at the z-ends of the TFB, so only variations in x y r , , and are considered, where r is the radial coordinate in the particles.This can be referred to as a P3D (pseudo three dimensional) model, wherein solid-phase Li concentrations that vary with the r coordinate couple to the x y and coordinates only at the particle surfaces.In Ref. 3 we analyzed the secondary current distribution subject to linear charge-transfer kinetics. 32It was shown in Ref.
, with R sheet representing the sheet resistance between the current collectors, l k the current collector thickness, and σ k the current collector electronic conductivity, then an asymptotic analysis of the transport equations in the limit of small μ k provides lines of current flowing normal to the current collectors and separator depicted in Fig. 1; this current density corresponds to ( ) i t x , y in the equations to be presented, where t refers to time.To summarize, there are no x or z components to the current flowing between the two current collectors.
The lines of normal current ( ) i t x , y that flow between the current collectors can be accommodated in a one-dimensional (y-direction only) porous electrode model for the porous electrodes and the separator, as outlined in Appendix E of Ref. 3 and made explicit in this work with the inclusion of Appendix B. Instead of solving the full equation system, we shall instead utilize ROM1 mentioned above to treat a cell unit comprising a negative electrode, a separator, and a positive electrode.This asymptotic approach is valid when transport losses in the electrolyte and ohmic losses in the liquid and solid phases are of less importance.The leading order solution for ROM1 is a single-particle model (SPM) for the active materials of the electrodes, corresponding to the limit of these resistances are defined in Eq. 29.Additional algebraic terms are included to provide the first-order correction associated with small but finite values of R R R R , , , and , as will be discussed in the context of Eqs.35 and 36.The validity of ROM1 for the conditions of this work is the subject of Fig. 7 and the associated discussion.
The result of the two asymptotic approximations outlined above is a reduction in the equation system from a P3D to a P2D model, as local variations in the y-coordinate have been removed, and the independent variables to consider are t x r , , and .We find that for traction battery applications (e.g., electric vehicle batteries) employing lithium-ion battery technologies, both asymptotic approximations are valid over common use conditions, which compels this analysis.
We employ the P2D model in an energy balance to determine the temperature distribution in the thin-film battery.When the cell thickness H is small enough, temperature will be approximately constant in this direction, and it can be calculated to high accuracy using a one-dimensional heat equation (variations in time t and position x only, see Eq. 38 of Ref. 3 and Eq.40 in this work).
Thermodynamics.-We model the thermodynamics of the intercalation electrode with the multi-site, multi-reaction (MSMR) model, [22][23][24][25] which is extended here to treat both the cell negative and positive and the influence of temperature variations.For the active material of each electrode, there are lithium mole fractions x j k with ⩽ ⩽ j N 1 ; k k hence, each electrode k has N k galleries.Each gallery of each species j k is involved in an insertion reaction at the interface between the host particles and the electrolyte of the form Thus, for each gallery j , k a vacant host site H j k can accommodate one Li leading to a filled host site ( − ) Li H . j k We find that there is a small effect of temperature on the OCV (open-circuit voltage, represented by U ).As noted in Appendix A, the entropy of reaction ΔS, is about 10 J/mol-K for the LMO-graphite system at ambient temperature (T = 298 K); this is an average value over the full capacity of the cell, and we treat ΔS as being independent of composition.We can expand the OCV U about the ambient temperature: where the subscripts p and n refer to the positive (LMO) and negative (graphite) electrodes, respectively.We find that mV and is not a negligible term in the energy balance, Eq. 26.We represent the OCV for reaction (1), relative to a Li reference electrode, with the MSMR formulation: [22][23][24][25] where X j k represents the total fraction of available host sites that can be occupied in gallery j , k can be viewed as a standard electrode potential for reaction involving gallery j , k and ω j k is a measure of the material's disorder 23 (smaller values of ω j k yield broader peaks for reaction j k in differential voltage spectroscopy plots).The OCV, = − U U U p n at any position x (cf Fig. 1), is given by Eq. 4, and ∂ ∂ U T as calculated using Eq. 4 is proportional to R F which, as noted previously, is close to for the LMO-graphite cell.
We shall assume galleries k j within each electrode are equilibrated: where the common host-particle potential ( ) U x k k is the OCV of the host material with respect to a lithium reference.Equation 5 is a mathematical relation that can also be applied within the host particles; i.e., if x k is known, we can identify ( ) locally, and this will be important for our formulation of the lithium diffusion and charge-transfer equations.Last, the specific capacity q k of each electrode is given by (e.g., units of C/m 2 ): where L e k , is the thickness of electrode k (see Fig. 1a).
Electrochemical kinetics.-Usingthe MSMR model, we express the kinetics of the electrochemical reaction (1) as [22][23][24][25] where β j k are the symmetry factors for each reaction j .k R f k , is a film resistance due to a surface layer on the electrode.For this work, we shall ignore the film resistance.Note that because of the equilibrium assumption of Eq. 5, the surface overpotential η s k , for each reaction within a host particle k is the same for all j k reactions.In Appendix A, we provide equations for ( ) i T .
Last, as explained above, we treat the electrodes with a perturbation solution based on an SPM; hence, V k represents the potential of any one of the identical electrode particles within an electrode of the differential element depicted in Fig. 1 relative to a Li reference electrode located at the solution-side of the particle-electrolyte interface.
Problem formulation for a thin-film battery.-Theequations for the potential and current density in the current collectors of the TFB are given by 3 sign n 1 and 1 10 In Eq. 9, l , k i , k and Φ k ( = k n p or ) represent the current-collector k thickness (cf Fig. 1), current density, and the associated electric potential, respectively; i k and Φ k change with time and position x, but they do not vary in y; the same is true for i , y the current density normal to the current collectors.The electronic conductivities of the current collectors σ k are temperature dependent (see Appendix A).
For co-current tab connections (cf Fig. 2c), the boundary conditions for Eq. 9 include insulating conditions = x L for = k n p and , and we ground the potential at the negative electrode tab position, = x 0: For counter-current tab connections (cf Fig. 2d), the boundary conditions for Eq. 9 include insulating conditions for the current collectors at the end of the cell opposing the tab connections: x x 0 and 0 13 x L 0 and we ground the potential at the negative electrode tab position, = x L: For both co-current and counter-current tab connections, at the positive electrode tab position, = x 0, It is helpful to battery engineers to identify the cell's 1C-rate, which is the current needed to charge or discharge fully the cell in 1 h in the absence of any potential losses.With example units, we can define the current density at the positive electrode tab connection (at = x 0 per Eqs. 1 and 15) in terms of the cell's C rate as Single particle model for the porous electrodes.-References24, 25, 35, 36 provide the differential equations using the MSMR model for porous electrodes.As mentioned previously, we use the results of Ref. 36 to generate a reduced-order model, ROM1, which was shown in Ref. 36 to agree well with the solution to the full equation system for power cycling consistent with EV applications, our area of interest.The leading order solution for ROM1 is an SPM, which we now provide for a full cell.For any particle in the differential element of Fig. 1, the particle will be at uniform temperature, and, for equimolar counter diffusion of filled and unfilled sites in the host material, the flux of Li-filled sites N is expressed as 24,25,35,36 where μ c D , , and refer to the total site concentration, diffusion coefficient (which is temperature dependent as described in Appendix A), and chemical potential of Li, and r is the radial coordinate of the spherical particle.The material balance for the Li in the host material particles is given by Initial and boundary conditions for Eq.18 are where a k is the particle radius of electrode k.The interfacial current J k is given in Eq. 7.
The voltage across the cell in the y-direction of Fig. 1, for all values of time t and position x, is given by We shall also make use of an earlier ROM developed by the authors, ROM0. 35Whereas the leading order solution for ROM1 is a SPM, Eqs.17-19, the leading order solution for ROM0 is dynamic equilibrium, wherein all resistances vanish, and, upon including the first order correction, a form of Ohm's Law again results:

, avg avg
The resistances and κ ˆin Eq. 22 are provided in Eqs.29 and 30.A superscript avg indicates the average value of the lithium filled sites at position x, which can be determined by the following charge balance for a time-varying current: Equation 22 is an algebraic equation, and Eq.23 a straightforward integration to obtain U , k avg from which x k avg can be obtained via Eq. 4. In contract, ROM1 requires the numerical solution of two nonlinear partial differential equations, that is, Eqs.17-19 for = k n p and .ROM0 is generally less accurate than ROM1; for low rates of operation, typically below about C/5, ROM0 and ROM1 and FOM return the same answer from a practical perspective for cells of the type studied in this work.
Energy balance and temperature distribution.-Keyto formulating the energy balance is to realize that for the thin-film battery depicted in Fig. 1, ≪ H L, and we can neglect temperature variations in the y-direction.Hence, the differential element shown in Fig. 1 is taken to be at uniform temperature, and the cell temperature is a function of time and position x.For the differential element of volume Δ WH x in the cell schematic of Fig. 1, we can construct the following energy balance: where represents the accumulation of energy within the differential element (with example units being J/s), leading to temperature changes, and Q refers to various power contributions to the differential element: , , , and   c h e s n p are associated with conductive heat transfer, convective heat transfer, electrochemical reaction, entropy of reaction, resistance in the negative current collector, and resistance in the positive current collector, respectively.These are given by For Eq. 25, each I refers to a current (example units of A), and each R refers to a resistance (example units of Ω).The total overpotential between the current collectors, A superscript avg indicates that the thermodynamic quantity is evaluated at the average value of the lithium filled sites at position x by means of Eq. 23.To form the differential equation for conservation of energy, we substitute the Q quantities from Eq. 25 into Eq.24, divide the resulting equation by the differential volume Δ WH x, and let Δ → x 0. The result is where we have used the constitutive equations Eq. 8).Each term in this energy balance has dimensions of power per unit volume.
For an initial condition, we assume the cell is at equilibrium with the surrounding temperature, a For boundary conditions, we impose convective heat transfer conditions: The fluid at the end regions may differ from T a if, for example, liquid cooling or cold plates are employed.
Nondimensionalization and scaling of the equations.-Forlisting the equations needed for ROM1, following, for the most part, the nomenclature of Refs.24, 25, 35, 36 we introduce characteristic resistance parameters, with the k-dependent quantities placed to the left of the vertical bar, which is subscripted k.For example, the characteristic electrolyte resistance in the cell electrodes is rendered as which reflects the fact that τ ε L , , and 2, can be different for the negative ( = k n) and positive ( = k p) electrodes, whereas the solution-phase conductivity is independent of the porous electrode.The following resistance definitions shall be used.The electrolyte resistances R R R , , are temperature dependent.
2, 2 0 , 1 note that R m is dimensionless, and, because we are considering temperature variations, we employ = f a a in Eq. 29.In addition, the following definitions allow us to nondimensionalize and scale the problem.
We note that a different dimensionless radial coordinate ̅ r k is used for the negative and positive electrodes, but both are bounded by ⩽ ̅ ⩽ r 0 1. k The same dimensionless time τ is used for all equations.The value of q used in = t q i 0 p 0 can be identified with the limiting electrode, which is commonly that of the positive electrode, as < 1 q q p n to avoid lithium plating at the end of charge (see Table II).
For the characteristic current density i , 0 it is good practice to use the average current density magnitude over the duration t 0 being modeled, as we have done in this work.Last, all dimensionless current densities at the particle surfaces īs j , k are normalized by the exchange current density of the = j 1 k gallery.The dimensionless MSMR equations are Equation 18 can now be written in dimensionless form as The quantity shows on the left side of this equation as τ = , ti q p 0 noted in Eq. 30.The initial and boundary conditions for Eq.32 can be written as U k rk 1 is determined, it can be used to determine ̅ V k from the dimensionless form of Eq. 7: x U 34 The dimensionless for of Eq. 20 for the voltage across the cell in the y-direction of Fig. 1a for all values of time τ and position ̅ x is given by where and Eq.23 becomes We now address the nondimensionalization of the current collector equations.The field Eq. 9 can be written as where l k is the thickness of the current collector k in the y direction.
The boundary conditions become x x 0 and 0 cocurrent tabs x x 0 and 0 countercurrent tabs Last, the energy balance ( 26) can be written in dimensionless and scaled form as The initial and boundary conditions in dimensionless and scaled form are 1 41

L
By using ROM1, the reduced-order model of Ref. 36 we can reduce the P3D battery model to a P2D model, as local variations in the y-coordinate have been removed, and the independent variables to consider are the dimensionless forms of t r x , , and .A mapping of the equations and variables is provided in Table I.

Results and Discussion
A central theme of this work is to clarify the pros and cons (strengths and weaknesses) of cocurrent and countercurrent tab connections depicted in Fig. 1.Figures 3a and 3b provide schematics of the comparison made in this work, wherein we examine cocurrent and countercurrent tab connections for cells of the same length L. In actual designs of modules and packs using such cells, countercurrent tab connections require a space of length δ T in the x-direction for the tab connection on both sides of the cell, versus the much smaller length δ P for the packaging to ensure hermetic sealing of the cell end opposite the tabs for cocurrent tab connections.For example, if 3 cm are needed on any side that has a current-collector tab and attendant bus-bar paraphernalia, then for the 1-m cell we investigate in this work, about a 3 percent decrease in energy density results for countercurrent tabbing vs that of cocurrent.It can be argued that from a cell volume perspective, a countercurrent cell of length L 2 , Fig. 3e, should be compared to two cocurrent cells butted up against one another, as shown in Fig. 3f, as the total length of each cell system is δ ( + ) L 2 .
T Such a comparison does not consider that the cocurrent cell of Fig. 3f would be more expensive to manufacture due to (1) the need for four tab connections (verses two in Fig. 3e) and ( 2) shorter cells also lead to increased manufacturing costs per unit energy (e.g., $/kWh).For this work, we consider δ δ = = 0, P T as our base-case cell is = L 1 m in length, and we ignore the much smaller lengths δ δ and .

P T
An advantage of countercurrent cell designs is a more uniform current distribution, as will be made clear in the ensuing analysis.The simple equivalent circuit in Fig. 3c (cocurrent) and Fig. 3d (countercurrent) shows that while the countercurrent design yields a more uniform current distribution, i.e., half the total current flows through each impedance element, the overall impedance is higher.This is because for the simple cocurrent circuit of Fig. 3c, threefourths of current flows through only the first vertical impedance element, and the remaining one-fourth of the current flows through the other three impedance elements of the circuit.Hence, at the beginning of discharge of a battery cell, before active materials are discharged, we expect lower polarization for the cocurrent configure, as will be discussed in the context of Fig. 12.
Base-case analysis, assessment of assumptions.-Wementioned in the Introduction that if the parameter , then lines of current can be treated as flowing normal to the current collectors, 3 giving rise to ( ) i t x , y in this analysis.A good measure of R sheet is R ohmic depicted in Fig. 10  Thus, the values of μ n and μ p are much less than 1 for all calculations associated with the base case, and we can assume lines of current flow normal to the current collectors.The same is true for all calculations in this work, and we would expect this assumption to hold generally for large-format, thin-film cells.If we were to consider the additional impact of resistance associated with diffusion within the particles and reaction at the particle surfaces, which is difficult to extract  Table II.Quantities and values for the base case, and relationships employed in the calculations.See appendix A for temperature and concentration dependence of additional parameters not listed.LMO values are taken from 25 and graphite values from, 49 including the MSMR parameters.

Quantity
ECS Advances, 2023 2 040505 with ROM1, μ n and μ p would be further reduced.(We could employ R y 0 of Eq. 22, which would be larger than R , ohmic but that quantity is dependent on the electrodes' Li content, and the point is made clear using the smaller R ohmic ).
A second assumption associated with the thin-film battery is that temperature variations in y can be ignored.The perturbation analysis in 3 showed that a key parameter to assess this approximation is which is equal to × − 4.24 10 8 for the base case; the smallness of this parameter supports the assumption that temperature variations in y can be ignored.For the analysis of a cell with = L 0.5 m (cf Fig. 13), the shortest cell we examine, the value increases to × − 1.70 10 . 7Hence, we expect this assumption to be valid for this work and likely for large-format, thin-film cells of commercial interest at present and in the foreseeable future.The General Motors Bolt EV multilayer pouch cells have 32 negative-electrode coatings and 33 positive-electrode coatings, and the separator wraps around the entire assembly, yielding 34 separator layers.For such a design, hence, for such large-format, thin-film, multilayer cells, the same approach used in this work can be employed, and temperature variations in y can be ignored for expected use conditions.We now examine the validity of using ROM1 in place of a P2D model to connect the electrodes, as shown schematically in Fig. 1b.The leading order solution for ROM1 is a SPM, which implies that all particles in the subject porous electrode are identical, and the first-order correction leads to the term ( )

R t x , y
showing in Eq. 20, and its dimensionless analogue R τ ( ̅ ) x , y in Eq. 36.Shown in Fig. 4 are results for the base case and uniform temperature at a 1 C discharge, which can be achieved numerically by employing ̅ → ∞ C for Eq.40, leading to τ 1for all times and locations.The upper row provides the LMO results, and the graphite results show in the lower row.The leftmost column provides the equilibrium potentials using the MSMR parameters provided in Table II.The middle column provides plots for the fraction of Li-filled sites x k vs dimensionless time τ.If there were no diffusion resistance in the solid phases, x k would plot over the line x avg vs τ (red dashed line in each plot) for all times.The LMO electrode is capacity limiting, and the calculation was stopped when τ ( ) x , 1 p neared 1 at τ = 0.81, corresponding to 81 percent utilization of the positive electrode at 2916 s.The rightmost column shows x k vs / r a k for various discharge times τ.
The P2D FOM involves the solution to the porous electrode equations corresponding to the schematic in Fig. 1b (see Appendix B, with → L 0); results using the same conditions as Fig. 4 ( ̅ → 1 are provided in Fig. 5. Figure 6 is developed with the same conditions as Fig. 5, with the exception that the discharge current is increased from 1 C to 2 C. Variations in the liquid-phase salt concentration and electric potential with the scaled y-coordinate normal to the current collectors are provided in Fig. 5 and Fig. 6, along with histories of the fraction of lithium filled sites x k for various locations, including at the electrode/current-collector and electrode-separator interfaces.It is clear that there is a lot of variation in x k throughout the porous electrodes; however, as seen in Fig. 7, ROM1 provides an accurate estimation of FOM for the cell potential during the 1 C discharge.We shall not examine cell currents higher than 1 C in the subsequent thermal analysis, but right left plot of Fig. 7 does show that ROM1 provides a good engineering estimate of FOM for the cell potential during a 2 C discharge, with the max difference in the cell potential between ROM1 and FOM being about 0.5 percent near 1100 s.Base-case dependent variable distributions and cell voltage.-Weutilize the full embodiment of the electrochemical-thermal model, with the equations that are solved summarized in Table I. 4 The current distribution results for the base case are shown in Fig. 8 for both cocurrent (solid lines) and countercurrent (dashed lines) tab locations.If there were no potential drop in the current collectors, ̅ i y (left plot) would be 1 for all ̅ x .It is evident that the countercurrent tab locations provide a more uniform current distribution for the dimensionless normal current ̅ i .y The right plot shows the dimensionless current densities ̅ i n and ̅ i p in the current collectors.Positive currents flow in the positive ̅ x direction.For discharge, current flows out of the positive tabs, in the negative ̅ x direction, and ̅ i n and ̅ i p are both negative at the positive tab for both cocurrent and countercurrent tabs.In addition, for both cocurrent and countercurrent tabs, ̅ i n falls to zero opposite the negative tab, and ̅ i p falls to zero opposite the positive tab.The current density ̅ i p always flows in the negative ̅ x direction for both the positive and negative current collectors for countercurrent tab connection, whereas for cocurrent connection, the sign of the current is reversed, as ̅ i n flows in the opposite direction of ̅ i .p Last, the more uniform the current distribution in ̅ i y (left plot), the   II).Solid lines: cocurrent tabs (see Fig. 2c and Fig. 3a).Dashed lines: countercurrent tabs (see Fig. 2d and Fig. 3b).The legend refers to the colors for both the cocurrent and countercurrent traces.Left: normal current density τ ̅ ( ̅ ) i x , y flowing across the cell.Right: axial current density flowing in the current collectors īk ( = k n or p for negative and positive, respectively).For end of discharge, = t 3060 end s, correspond to τ = 0.85 (yielding about 85 percent capacity utilization of the limiting positive electrode material (see Fig. 9 and associated discussion).More uniform normal and axial current densities result for countercurrent tabs.
b For base-case calculations, we requested 200 uniform time steps per constantcurrent discharge in addition to 50 smaller and uniform time steps for the first 0.1 percent of the final time.(The actual number of time steps is determined by the CasADi solver, https://web.casadi.org/docs/,employed by PyBaMM, which is informed by the user's request.)51 uniformly spaced volume elements were employed for ⩽ ̅ ⩽ r 0.95 1 and 21 uniformly spaced volume elements for ⩽ ̅ < r 0 0.95).51 uniformly spaced volume elements were used for both the electrode and separator in the ̅ x direction.Finer meshes in time and position did not yield noticeably different results as plotted, with the exception of the heat sources (see Fig. 11) near the tabs.The CPU time was less than 30 seconds.To resolve the heat sources, we employed the same time stepping, 151 uniformly spaced volume elements for ⩽ ̅ ⩽ r 0.95 1 and 51 uniformly spaced volume elements for ⩽ ̅ < r 0 0.95), and 201 uniformly spaced volume elements were used for both the electrode and separator in the ̅ x direction, which required 640 seconds CPU time for cocurrent and 393 seconds for countercurrent tab locations.All computations were done on an HP ZBook (2.6 GHz processor and 32 GB RAM), using PyBaMM 21.9 software.more the right side of Eq. 38 becomes constant, and, hence, the slopes in ̅ i n and ̅ i p are nearly constant and show little variation throughout the discharge.In contrast, the more nonuniform current distribution in ̅ i y for the cocurrent tab connection (left plot) leads to more variation in the right side of Eq. 38, which contains ̅ i , y giving rise to greater variations in ̅ i n and ̅ i p throughout the discharge.The distributions in Li-filled sites are depicted in Fig. 9 for the base case.For the upper left plot and cocurrent tabs, at and the calculation is terminated, as the positive electrode is nearly filled with Li, and the equation system becomes numerically singular.We also terminated the countercurrent calculation at this time to allow for a direct comparison.The far left plots correspond to ̅ = x 0 (see Fig. 1); for cocurrent (solid lines) with both tabs at ̅ = x 0, the LMO positive active material fills faster at this location than for the case of countercurrent (dashed lines), and, similarly, the graphite negative active material Li content falls faster for the cocurrent tabs than for countercurrent tabs.The opposite situation takes place at the end of the cell, ̅ = x 1, wherein for cocurrent tabs, the discharge of the active materials lags that seen for countercurrent tabs.For the conditions of the base case, at the middle of the cell, ̅ = x 0.5, and the profiles in x p and x n are nearly equivalent for the active materials for both cocurrent and countercurrent connections.
The distributions in temperature and resistance, R R τ = ( ̅ ) x , ohmic y (cf Eq. 36), are shown in Fig. 10, and the distribution in heat sources are shown in Fig. 11.From the left plot in in Fig. 10, it is clear that the more uniform current distribution leads to a more unform temperature distribution for countercurrent tabs vs cocurrent tabs.The same is true for the heat inputs shown in Fig. 11 and the resistance distribution, in the positive (subscript p, upper row) and negative (subscript n, lower row) active materials for the base case (cf Table II).Solid lines: cocurrent tabs (see Figs. 2c and 3a).Dashed lines: countercurrent tabs (see Figs. 2d and 3b).Far left column: ̅ = x 0, corresponding to the far left of the cell ( = x 0 in Fig. 1).Middle column: ̅ = x 0.5, corresponding to middle of the cell ( = / x L 2 in Fig. 1).Far right column: ̅ = x 1, corresponding to the far left of the cell ( = x L in Fig. 1).For the upper left plot and cocurrent tabs, at and the calculation is terminated, as the positive electrode is filled with Li, and the equation system becomes numerically singular.2c and 3a).Dashed lines: countercurrent tabs (see Figs. 2d and 3b).The more uniform current distributions (see Fig. 8) for the countercurrent tabs yield more uniform temperature and resistance distributions and lower maximum temperatures during discharge.Because the temperatures are lowest at ̅ = x 1 for cocurrent tabs, the resistance is highest at ̅ = x 1.
consistent with the resistances being temperature dependent; see Eq. 29 for resistance definitions, Eq. 36 for the use of R R τ = ( ̅ ) x , , ohmic y and Appendix A for the temperature dependence of the physicochemical parameters that determine the resistances.
The cell voltage for the base case is provided in Fig. 12.The behavior depicted is linked to the various distributions discussed previously.As shorter times (e.g., times less than = t 0.1 30.6 end s, where = t 3060 end s), the temperature has not increased significantly from that of the ambient for both the cocurrent and countercurrent cases.Consistent with the earlier discussion of the simple circuit in Figs.3c and 3d, the cocurrent tab design yields a lower impedance than that of countercurrent, and we see that the discharge voltage is higher for the cocurrent case relative to the countercurrent case for times less than about 1000 s.Throughout discharge, the resistance shown in Fig. 10 is always highest at the cell end location ̅ = x 1 for the cocurrent design, as both ends are cooled, but less current flows through the end of the cell, giving rise to less current-collector heat generation at ̅ = x 1 (cf left plot of Fig. 11) and lower cell temperatures.As a result, near the end of discharge for the cocurrent design, the current that is forced to react the remaining active material near ̅ = x 1 must pass through a higher resistance, giving rise to larger potential losses, which explains why the discharge voltage is lower for the cocurrent case relative to the countercurrent case after about 2200 s.Without a model to sort out these complicated and interacting phenomena that impact the cell voltage, it would be difficult to provide rational cell designs that maximize, for example, cell energy density.
Sensitivity analyses.-Weexamine changes to design variables from those of the base case with the assistance of the model.The trend for large format cell has been to increase the cell length L (cf Fig. 1) to reduce cost per unit energy and increase energy density.Halving the cell length relative to the base case yields a more uniform current density for both cocurrent and countercurrent designs, as is shown in Fig. 13.(Note that the ordinate scales for both plots in Fig. 13 are compressed relative to Fig. 9.) In addition, as seen in Fig. 14, the temperatures in the shorter cell are both lower and more uniform, and the cell voltage curves for the cocurrent and countercurrent designs are nearly identical as plotted.By comparing results for the shorter cell with those of the base case, we underscore the need to be more diligent of thermal management concerns for the longer cell designs coming to market.
Extreme cooling of the ends of the cell ( ̅ ̅ → ∞ h h , L 0 ) forces the temperature at the cell ends to match that of the ambient (25 °C, see Table II), which alters modestly the current distribution that is shown in Fig. 15 relative to that of the base case depicted in Fig. 8.However, the maximum temperature that is observed in the inner portion of the cell (at about ̅ = x 0.4) is less affected, as it nears 50 °C in Fig. 15 vs about 55 °C Fig. 10 (near about ̅ = x 0.2).The effectiveness of cooling the large area portion of the cell, that is, the cell surfaces in the x-z plane (see Fig. 1a) is examined in Fig. 16.By increasing = h h n p from 0 (for the base case) to / 0.15W m s, 2 it is possible to not exceed 40 °C during discharge, thereby achieving improved cell durability.There is minimal impact on the cell voltage, as can be seen by comparing the rightmost plots in Figs. 12 and Fig. 16.This result again underscores the utility of such an electrochemical-thermal mode to facilitate cell design.

Summary and Conclusions
Our focus is on large-format thin-film batteries represented schematically in Fig. 1.We begin our analysis with a statement of the governing equations, including the use of a recently published reduced-order model for a cell's electrochemical performance, 36 ROM1, which is then incorporated into an energy balance, Eq. 26.We employ the multi-site, multi-reaction (MSMR) framework [22][23][24][25] to describe the solid phase thermodynamics, Eq. 4, speciation needed for the charge-transfer relations, Eq. 7, and diffusion due to chemical potential gradients in Li-filled sites, Eqs.17-19.The temperature dependence of the physicochemical properties are provided in Appendix A, and Appendix B provides the equations needed for a full-order model and, consequently, simulations that preclude the need for ROM1.
A practical objective of this work is to compare cocurrent vs countercurrent tab designs of large-format, thin-film cells that are relevant to electric vehicle applications.Hence, for the discussion of  II).Solid lines: cocurrent tabs (see Figs. 2c and 3a).Dashed lines: countercurrent tabs (see Figs. 2d and 3b).The more uniform current distributions (see Fig. 8) for the countercurrent tabs yield more uniform heat-source distributions during discharge.II).
ECS Advances, 2023 2 040505 results associated with Fig. 8 through Fig. 16, which provide various sensitivity analyses, on the same plots we show results for both cocurrent and countercurrent tab locations, enabling direct comparisons.A consistent theme in the comparisons is that countercurrent tab locations yield more uniform current and temperature distributions.We note in the discussion of Fig. 3, however, that care must be exercised in such comparisons: while countercurrent cells exhibit more uniform distributions in temperature and current, cocurrent cells allow for more efficient packaging, leading to higher energy density.Overall, the equation system and the use of open-source (Python) software for the numerical solver facilitates efficient and rational cell design and integration, all of which may be a useful addition to the toolkit of a battery engineer.

Appendix A
For the energy balance of Eq. 26, we can construct the effective heat capacity and thermal conductivity (in the x-direction, parallel to the current collectors shown in Fig. 1 37 shows that their values are close to values reported by others for similar lithium-ion cells, including those of Maleki et al. 38 Lin et al. 39 recently reported similar values for large-format pouch cells, along with ρ = 2.18 where T is in K and ρ represents electrical resistively.These changes with temperature are not insignificant.For example, in going from and all other conditions identical to the base case.Left plot: solid lines for cocurrent tabs and dashed lines for countercurrent tabs.Temperature is maintained at the ambient value throughout discharge and the cell ends.Relative to the base case shown in Fig. 10, the maximum temperature is now kept below 50 °C, but the distribution is more nonuniform.The current distributions are like those of the base case shown in Fig. 8, with differences emerging due to increased polarization in the cooler end regions.20 °C to 30 °C, the electronic conductivities of both Cu and Al decrease by about 4%. Viswanathan et al. 40 have determined the entropy of reaction for various lithium-ion cells, including Li-graphite||LMO.For the 14 cells chemistries, the average entropy of the cell reaction (over the range of ⩽ ⩽ SOC 0 100%) was positive, leading to reversible heating during full cell discharge (i.e., form 100% SOC to 0% SOC).We use their average value, and, relative to Eq. 26, ∆ ̅ = S 10 avg J/mol-K.For the electrolyte properties, we employ the work of Stewart and Newman 41 for p d1 and Landersfeind and Gasteiger 42 for all other coefficients in Eq. 46; specifically, for the LiPF 6 salt in a 3:7 w-w EC-EMC solvent: For the thermodynamic factor = + TDF 1 , dlnf dlnc there is less variation with temperature, and we fit the m coefficients through the approximate average over the temperature range -10 to 50 °C. 42The transference number was taken to be constant, = + t 0.4.0 43 The thermodynamic factor and transference number are used in the quantity κ ˆ2 (see Eqs. 30 and 36).
A recent review of studies associated with the temperature dependence of the lithium diffusion in graphite appears in Schmalstieg et al. 44 The data are well represented by an Arrhenius equation, with the activation energy E a n 1, shown in Eq. 47. Schmalstieg et al. 44 also provide an Arrhenius equation for the exchange current density for graphite, which we employ in this work.We use the same exchange current density for each gallery in graphite (cf Eq. 7); a more complete characterization of graphite is lacking currently.The value for D n ref 1, is taken from. 45Similarly, a recent review of studies associated with the temperature dependence of the lithium diffusion in LMO, along with new results, appears in Kuwata et al., 46,47  A plot of the liquid-phase property variations with salt concentration and temperature in accordance with the relations of this appendix appears in Fig. 17, and the solid-phase property variations with temperature show in Fig. 18.

Appendix B
We present equations to be used in place of Eqs.20 and 21 for the ROM1 approach when a P2D model (i.e., FOM) is employed for the cell schematic depicted in Fig. 1b.As noted previously, the  where i i and 1 Φ c , and 2 2 vary with x throughout the full cell depicted in Fig. 1a, but we have not indicated this dependence explicitly in Eq. 49.The MSMR expressions (4) and ( 5), the chargetransfer relations provided in Eq. 7, and the equations for Li diffusion, Eqs.17-19, apply as written.In the separator, = i 0,

Figure 1 .
Figure 1.Cell schematic.The differential element used in formulating the energy balance is shown with a volume of ∆ HW x, where H is the cell height and W is the cell width in the z-direction, which is perpendicular to the x-y plane.(b) provides more detail about the P2D (FOM) model within the differential element shown in (a).The legend in the upper left identifies tab locations for cocurrent and countercurrent connections.

Figure 2 .
Figure 2. Schematics of various problems addressed in this work.(a): Full-order model (FOM) for a P2D cell, corresponding to the schematic shown in Fig. 1b (no variation in x, and only r and y coordinates are relevant, see Appendix B).(b) Same as (a), but a reduced-order model (ROM1) replaces FOM.(c) P2D cell employing ROM1 to calculate the current flow between the current collectors for cocurrent tab connections (Φ 1 grounded at = x 0).(d) Same as (c), but countercurrent tab connections (Φ 1 grounded at = x L).For (c) and (d), see Eqs. 1 through 23.
is small relative the cell's average OCV of about 3.7 V. Hence, the influence of temperature on the OCV is minimal for the LMO-graphite cell, and, coincidently,

0
For constant current operation, with the incorporation of Eq. 16,

A/m 2 ( 3 .
in its dimensionless form, R = f i R , 58 mA/cm 2 ).The lowest value of R ohmic in Fig.10is 0.77 at the end of discharge (3060 s) and at an axial position = x 0.22, or R ohmic = 0.000648 Ω-m 2 .At 298 K,

Figure 4 . 1 pFigure 5 .
Figure 4. Results employing ROM1 for the LMO|graphite cell for a 1 C discharge.See Fig. 2b for the problem schematic.The cell is isothermal and there are no variations in x for these calculations ( = ) L 0 .Upper row: LMO results.Lower row: graphite results.Leftmost column: equilibrium (MSMR model, Eq. 4).Middle column: fraction of Li-filled sites x k vs dimensionless time τ.The LMO electrode is the capacity limiting, and the calculation was stopped when τ ( ) x , 1 p

Figure 6 .
Figure 6.Results employing FOM for the LMO|graphite cell for a 2 C discharge.These results are analogous to the 1 C discharge shown in Fig. 5.The differences in x , k c , 2 and Φ 2 across the electrodes are larger than those of Fig. 5 due to the larger discharge current.

Figure 7 .
Figure 7.Comparison of various solutions at a uniform temperature for 1 C (left) and 2 C (right) discharge.Even though we see differences in x k across the electrodes for FOM in Fig. 5, ROM1 provides a sufficiently accurate representation of the cell potential at 1 C.The max difference in the cell potential between ROM1 and FOM is about 0.5 percent near 1100 s for the 2 C discharge.The TFB calculations correspond to =L 0 (see Fig.1).The TFB results plot directly over those of ROM1.The good agreement between ROM1 and FOM motivates the use of ROM1 as depicted in the problem schematics shown in Figs.2c and 2d.

Figure 10 .
Figure 10.Base-case (cf Table II) distributions of temperature τ ( ̅ ) T x , , left plot, cf Eq. 40, and resistance R R τ ( ̅ ) = x , , y ohmic right plot, cf Eq. 36.Solid lines: cocurrent tabs (see Figs.2c and 3a).Dashed lines: countercurrent tabs (see Figs.2d and 3b).The more uniform current distributions (see Fig.8) for the countercurrent tabs yield more uniform temperature and resistance distributions and lower maximum temperatures during discharge.Because the temperatures are lowest at ̅ = x 1 for cocurrent tabs, the resistance is highest at ̅ = x 1.

Figure 11 .
Figure 11.Distributions of current-collector heat input, left plot, corresponding to the last two terms on the right side of Eq. 40, and heat input associated with the electrochemical reactions, corresponding to the terms on the right side of Eq. 40 that multiply onto the normal current density ī , y for the base case (cf TableII).Solid lines: cocurrent tabs (see Figs.2c and 3a).Dashed lines: countercurrent tabs (see Figs.2d and 3b).The more uniform current distributions (see Fig.8) for the countercurrent tabs yield more uniform heat-source distributions during discharge.

Figure 12 .
Figure 12. Cell voltage history for the base case (cf TableII).

Figure 13 .
Figure 13.Current distribution results for = L 0.5 m, and all other conditions identical to the base case shown in Fig. 8. Solid lines: cocurrent tabs (see Figs. 2c and 3a).Dashed lines: countercurrent tabs (see Figs. 2d and 3b).Left: normal current density τ ̅ ( ̅ ) i x , y flowing across the cell.Right: axial current density flowing in the current collectors īk ( = k n or p for negative and positive, respectively).For end of discharge, = t 3060 end s, correspond to τ = 0.85.Note that the ordinate scales for both plots are compressed relative to Fig. 8.

Figure 14 .
Figure 14.Temperature distribution (left plot) and cell voltage history (right plot) for = L 0.5 m, and all other conditions identical to the base case.Left plot: solid lines for cocurrent tabs and dashed lines for countercurrent tabs.The results for the shorter cell yield lower maximum temperatures, as compared to Fig. 10, and identical voltage histories as plotted, in contrast to Fig. 12.
3 -K, which we take to be constant.The electronic conductivities in the current collectors along with their temperature dependencies are well known:5

Figure 16 .2
Figure 16.Temperature distribution (left plot) and cell voltage history (right plot) for = = / h h 0.15W m s, n p 2 and all other conditions identical to the base case.Left plot: solid lines for cocurrent tabs and dashed lines for countercurrent tabs.The additional face-plate cooling over the x-z planes of the cell (cf Fig. 1) is shown to keep the maximum temperature below 40 °C.The cell voltage histories are like those of Fig. 12.

1 ,
and the data are well represented by an Arrhenius equation, with the activation energy E a p 1, shown in Eq. 47.The value for D p ref is taken from Ref. 25.For the LMO exchange current density, we use the activation energy of Ref.48 and the two reference exchange current densities are taken from Ref. 25.For both Eqs.47 and 48, the reference temperature * = T 298K.

ECS Advances, 2023 2
040505normal current density i y flows across the cell,

1 2
represent the solid and liquid phase current densities, respectively.For both the negative ( = )k n and positive ( = ) k p electrodes, for the five unknowns ( we have the following five equations:6,35,36 ε

Table 6 of
Werner et al.

Table III .
Additional nomenclature not covered in Table I or Table II.A subscript k refers to electrode active material k, and a subscript j k refers to a gallery j k within electrode active material k (see definitions above Eq.1).eff Effective heat capacity, Eq. 43 R m For each resistance R , m the dimensionless resistance is R = f i R