Scaling and Analytical Approximation of Porous Electrode Theory for Reaction-limited Batteries
Porous electrode theory (PET) provides essential insights into electrochemical states, but its computational complexity hinders real-time control and obscures scaling relations. To bridge the gap between high-fidelity simulations and reduced-order models, we present a framework of scaling analysis and analytical approximations. By assuming high-performance electrodes minimize transport limitations and overpotentials, we derive a simplified "lean model" governed by four dimensionless numbers: (i) a traditional Damk\"ohler number, $Da$, scaling the characteristic reaction rate to the diffusion rate in the electrolyte-filled pores; (ii) the "process Damk\"ohler number," $Da_p$, scaling the reaction rate to the applied capacity utilization rate (C-rate); (iii) the "wiring Damk\"ohler number," $Da_w$, scaling the reaction rate to an effective electromigration rate for ions in the pores in series with electrons in the conducting matrix; and (iv) the "capacitive Damk\"ohler number," $Da_c$, comparing the rates of Faradaic reactions and double-layer charging. For batteries, we derive analytical solutions for standard protocols, including galvanostatic discharge, chronoamperometry, and electrochemical impedance spectroscopy. Validated against numerical simulations of a practical NMC half-cell, our formulae show excellent agreement at negligible computational cost. This interpretable, physics-based framework accelerates battery design and state estimation while unifying the modeling of batteries, supercapacitors, fuel cells, and other porous electrode systems.