Scaling and Analytical Approximation of Porous Electrode Theory for Reaction-limited Batteries

TL;DR

This paper introduces a simplified, physics-based framework for porous electrode theory that uses four key dimensionless numbers to enable fast, accurate analysis of battery behavior and scaling relations.

Contribution

It derives a reduced-order 'lean model' with analytical solutions for standard protocols, validated against detailed simulations, facilitating rapid battery analysis.

Findings

Excellent agreement with numerical simulations at negligible computational cost.

The framework unifies modeling of batteries, supercapacitors, and fuel cells.

Analytical solutions enable real-time state estimation and design.

Abstract

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.