23 May 2020
© 2020 Elsevier Ltd In Fig. 9 of , a comparison of the terminal voltage predicted by the SPMe from Ref. , DFN, SP, and cSP models is presented. In this figure, the SPMe appears to perform poorly at high C-rates. However, there is an implementation error in the code used to generate the figure, which results in an overprediction of the reaction overpotential in the SPMe. The error is in the evaluation of the reaction current density j, which is defined in Ref.  as [Formula presented] but in Ref.  as [Formula presented] so that the reaction rate constants are related by [Formula presented]. The factor of 2 was erroneously omitted in the calculations of . Fig. 1 shows the corrected comparison. We make one further comment on the results. In Ref. , the SPMe is formally derived in the limit of fast diffusion in the electrolyte, with no ad-hoc assumptions. This results in a linear diffusion equation (with a piecewise-constant source term [Formula presented]) for the electrolyte concentration of the form [Formula presented] During the discussion in Ref.  it is suggested that, although it includes formally higher-order terms in the asymptotic expansion, including nonlinear electrolyte diffusion, [Formula presented] may in some situations give better results in practice. The example in Ref.  in which [Formula presented] varies by an order of magnitude, with a discharge rate of 7.5C, is one such situation. At a discharge rate of 7.5C the SPMe (Linear) predicts that the electrolyte concentration goes negative close to the current collector at around 40s. At this point, the linear model is operating outside of its range of validity, and gives results that are non-physical (we therefore terminate the simulation of the SPMe (Linear) at 40s in Fig. 1). This non-physical behaviour is remedied if the linear diffusion (3) is switched to the nonlinear diffusion (4). Then, even at 7.5C, both the SPMe (Nonlinear)  and the cSP  accurately recover the terminal voltage as predicted by the DFN (see Figs. 1 and 2, and Table 1). The code to produce the results shown is available at https://github.com/rtimms/spme-comparison. [Figure presented] Fig. 1. Comparison between the DFN model, the SPM, the cSP , and the SPMe  at different discharge rates. Results for the SPMe with both linear and nonlinear diffusion in the electrolyte are included. [Figure presented] Fig. 2. Absolute errors of the SPM, the cSP , and the SPMe  vs. the DFN at different discharge rates. Results for the SPMe with both linear and nonlinear diffusion in the electrolyte are included. Table 1. Root-mean square error (RMSE) for each reduced-order model compared to the DFN at different C-rates. Note that the error for the linear SPMe at 7.5C is omitted as the model breaks down before the discharge is complete. [Table presented] The authors would like to apologise for any inconvenience caused.
Submitted to ORA: