An Analytical Model for Predicting the Rate Performance of Battery Cells Under Mixed Kinetic Control
Authors: Wang Hongxuan , Wang Fan , Tang Ming 
Published: August 2024

The prediction of battery performance conventionally relies on resource-intensive numerical simulations based on the porous electrode theory. For computational speedup, simplified physics-based models have proven attractive as they still capture the underlying reaction mechanisms to a certain degree. In this class of models, physical complexity is usually reduced by assuming a single rate-limiting reaction. For example, the single particle model describes battery cells limited only by the solid diffusion pathway [1]. On the flipside, our previous analytical model exploited inefficiency of electrolyte transport, particularly in thick electrodes [2]. With advancement in both electrode and electrolyte technologies, however, it becomes challenging for these models to handle more realistic batteries.
In this work, we propose an improved physics-based analytical model that enables mixed control of both kinetic processes, reflecting both salt depletion and particle size effect. An extension to our uniform reaction (UR) model, the new model uses salt distribution in electrolyte to probe the accessible lithium in the electrode level. The added solid diffusion module calculates the equilibrium potential at particle surface, lithium concentration at particle surface (Cs), and in turn the particle-level depth of discharge. We refer to the model as the URCs model, recognizing its two components. The model features higher tunability as it now accepts function-valued electrode and electrolyte properties. It also allows for prediction of discharge voltage curves and energy output of the cell.
The model is compared with numerical simulation over a wide range of cell parameters. It shows good agreement with simulated results in rate performance test, specific energy output test, and discharge voltage curves. Better alignment is observed for the NMC/Li half cell than the NMC/Gr full cell, likely due to mixed reaction type in the graphite anode. Figure 1 shows the comparison of voltage curves across various C rates for a half cell with 70um NMC electrode and particles of 4um radius. Additionally, the cells are optimized for their specific capacity against electrode thickness and porosity. The optimal parameters predicted by the model lie closely to the global optimum from numerical simulation. Figure 2 shows the optimization results for a full cell with particles of 4um radius, overlaid on contour lines generated from the URCs model.
The model offers a speed up of over 500 times compared with state-of-art P2D solvers [3]. Furthermore, its compatibility with gradient-based optimization algorithms opens possibilities for more efficient global optimization schemes, in which the number of function calls can be reduced. The high efficiency and the analytical nature of the URCs model render it a powerful tool for both on-board applications and battery cell design.
[1] Doyle, M. and Newman, J., 1997. Analysis of capacity–rate data for lithium batteries using simplified models of the discharge process. Journal of Applied Electrochemistry, 27, pp.846-856.
[2] Wang, F. and Tang, M., 2020. A quantitative analytical model for predicting and optimizing the rate performance of battery cells. Cell Reports Physical Science, 1(9).
[3] Sulzer, V., Marquis, S.G., Timms, R., Robinson, M. and Chapman, S.J., 2021. Python battery mathematical modelling (PyBaMM). Journal of Open Research Software, 9(1).

Figure 1