We present a mathematical model to study the steady-state performance of a membrane-less reversible redox flow battery formed by two immiscible electrolytes that spontaneously form a liquid-liquid system separated by a well-defined interface [1]. The model assumes a two-dimensional battery with two coflowing electrolytes and flat electrodes at the channel walls. In this configuration, the analysis of the far downstream solution indicates that the interface remains stable in all the parameter range under study. To simplify the description of the problem, we use the dilute solution theory to decouple the calculation of the flow field and the species concentration distributions. Once the velocity field is known, we obtain the distribution of the mobile ionic species along with the current and the electric potential field of the flowing electrolyte solution [2]. The numerical integration of the problem provides the variation of the battery current density I_app with the State of Charge (SoC) for different applied cell voltages V_cell.

A detailed analysis of the concentration density plots indicates that the normal operation of the battery is interrupted when reactant depletion is achieved near the negative electrode during both charge and discharge. The effect of the electrolyte flow on the performance of the system is studied by varying the Reynolds, Re, and Péclet, Pe, numbers. As expected, the flow velocity only affects the polarization curve in the concentration polarization region, when V_cell is well below the equilibrium potential, resulting in limiting current densities that grow with Re as j_lim ~ Re^0.3. In addition, both the single-pass conversion efficiency Ψ and the product Ψ·j_lim decrease with Re. Concerning the later, the decay rate with Re exhibits a power law with an exponent that almost doubles previous theoretical predictions obtained by imposing a prescribed velocity profile for the electrolyte in a membrane-less laminar flow battery with a liquid oxidant and gaseous fuel.

The present work constitutes the first modelling attempt that simultaneously solves the fluid dynamical system formed by the two immiscible electrolytes and the electrochemical problem that determines the response of the membrane-less battery. The proposed model could be used as a valuable tool to optimize future flow battery designs based on immiscible electrolytes.

Acknowledgments

This work has been partially funded by the Agencia Estatal de Investigación (PID2019-106740RB-I00 and PID2019-108592RB-C41/AEI/10.13039/501100011033), by Grant IND2019/AMB-17273 of the Comunidad de Madrid and by project MFreeB from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 726217). D. Ruiz-Martín acknowledges the support of an FPI predoctoral fellowship (BES-2016-078629) under project ENE2015-68703-C2-1-R (MINECO/FEDER, UE).

References

1. Navalpotro, J. Palma, M. Anderson, R. Marcilla, Angewandte Chemie, 129(41), 12634-12639 (2017).

2. Ruiz-Martín, D. Moreno-Boza, R. Marcilla, M. Vera, M. Sánchez-Sanz, Applied Mathematical Modelling, 101, 96-110 (2022).