(Invited) Mathematical Modeling of Electrochemical Cells: Promises, Problems, and Progress
Authors: Zhang Shidong , Hess Steffen , Kadyk Thomas , Beale Steven B. 
Published: December 2024

Fuel cells and electrolyzers are key components in the energy transformation from non-renewable fossil fuels, to technologies based on renewable energy, such as solar and wind, which utilize or synthesize hydrogen (and oxygen) rich fuels/feedstocks in place of hydrocarbons. Following a brief introduction, a description of modeling activities, past, present and future, developed by the authors over many years, are described.
Electrochemical conversion cells may be considered as natural problems in combined heat and mass transfer with the electrochemical reactions functioning as the main driving force for the interconversion of chemical and electrical energy, and heat. Ionic and electronic charge transfer within the electrodes and electrolyte must be considered, and multi-phase flow (liquid-gas) is also frequently present. The authors have led development of a suite of object-oriented open source models for both polymer electrolyte membrane cells (PEMCs) and solid oxide cells (SOCs), the emphasis being on generic (rather than specific) algorithms. The implementation is based around the popular open-source finite-volume-based software library, OpenFOAM. This allows the user access to a flexible framework/library of C++ classes, thereby allowing for further model research, as well as development appropriate to practical real-life applications. The main features of the methodology are described and explored both, with the benefit of hindsight from past experiences, and foresight to the problems to be tackled in the future.
In addition to the above cell-level macro-homogeneous analysis, substantial work is being devoted to micro-scale analysis; for example, detailed studies of flow phenomena within the porous transport layers and electrodes of electrochemical cells as well as construction of digital twins of physical morphologies. These are necessary, not only for basic qualitative understanding and control of the underlying 3-D processes, but also to provide quantitative closure parameters for quantifying the dynamics of macro-scale equations (multi-scale models).
The state-of-the art of CFD-based modeling in electrochemical science is presented with examples of PEMC (and SOC) technologies together with a discussion of the important issues and limitations of the present generation of models as well as examples of success stories. Some of the issues for concern include degradation and evolution of material properties, code stability and numerical issues, suitability of present-generation existing two-phase models (mixture, Euler-Euler, volume of fluid methods, etc.) to electrochemical applications, and computational issues such as parallel efficiency. In spite of numerous unexpected problems encountered over the years, great progress was, is, and will continue to be made in this quest for mathematical descriptions and prototypes of complex electrochemical processes and products, based on physical modeling.
Artificial neural network techniques have found applications in tackling scientific and industrial challenges involving the concept of digital twins. In the domain of electrochemical cells, data-driven models or physically informed data-driven methods have been shown to provide valuable insight into issues that were traditionally explored using physical models. Areas for investigation include; material properties, performance prediction, and control strategies. By integrating physical/mathematical models with data-driven counterparts, a more comprehensive suite of methodologies should become capable to address ever more complex issues in the field.