Hi all! Hope you had a great vacation.
I hope I am not off-topic by asking math & model derivation questions here, but I am trying to get to the crux of Marquis et al 2019’s paper which derives the SPMe model that is used in PyBaMM, and my brain alone just doesn’t cut it.
The SPMe model in PyBaMM yields results almost on par with the DFN model, and is significantly lighter to run. However, in the paper, I am struggling to understand exactly how they do the derivation.
I am getting lost at the timescale C_e (which is a ratio of Li migration timescale and discharge timescale), and in the limit C_e->0 (right after equation 12, page A3697). They way I understand it, it just means that the migration effects are so fast that they have no time-dependence, but as far as I understand the derivation, they just count it out? Further, they do a leading- and first-order asymptotic expansion of e.g. equation 3d. Here it seems like they assume that dc_e/dt = 0. Do I misunderstand this? If not, how can it be justified?
A bit later they also state that at this limit, all the particles in DFN isn’t just reduced to a single particle, but all the particles behave exactly the same, and thus it is enough to calculate one representative particle. I assume the reason this works here and not in the regular SPMe culminates into equation 36, but is there any more intuitive way to understand why it is acceptable to just calculate one representative particle?
Thank you to anyone willing to discuss/help me out
Best,
Amund