# The mathematics of redox flow batteries

*By Prof. Michael Vynnycky*

Modern demands for increasingly efficient energy delivery and the anticipated demand for renewable energy have generated considerable interest in redox ow batteries (RFBs) as an energy storage technology. The potential applications of RFBs are numerous, including load levelling and peak shaving, uninterruptible power supplies, emergency backup and facilitation of wind and photovoltaic energy delivery. One of the most popular is the vanadium redox flow battery (VRFB), which was originally pioneered in the 1980s.

**Fig. 1: A schematic of the overall operation of a ****vanadium redox flow battery**

A schematic diagram showing the operation of a VRFB is given in Figure 1. It consists of an assembly of power cells, usually termed a stack, although the figure shows only one cell, in which the two electrolytes are separated by a proton exchange membrane. Both electrolytes are vanadium-based: the electrolyte in the positive electrode contains VO_{2}^{+} and VO^{2+} ions, whilst the electrolyte in the negative electrode contains V^{3+} and V^{2+} ions. In VRFBs, both electrodes, which are typically made of porous carbon felt, are additionally connected to storage tanks and pumps, so that very large volumes of the electrolytes can be circulated through the cell. When the VRFB is being charged, the VO^{2+} ions in the positive electrode are converted to VO_{2}^{+} ions, and electrons are removed from the positive terminal of the cell via a current collector. Similarly, in the negative electrode, electrons are introduced via another current collector, converting the V^{3+ }ions into V^{2+ }; during discharge, the process is reversed. Charging and discharging can be summarized by the electrochemical reactions. Typically, each cell in a VRFB operates at around 30^{o}C at a nominal cell voltage of 1.15-1.55 V; however, a VRFB as a whole will normally consist of tens or hundreds of such cells connected in series in order to provide the required voltage in a given application.

**Fig. 2: Charge-discharge curve at a current density of ****400 A m ^{-2} from a 2D transient and 1- and 2-term **

**asymptotic models**

Mathematical modelling is an obvious tool for helping to describe the operation of the VRB and, indeed, a quick search on ISI Web of Science for “vanadium redox model” indicates around 100 publications over the last fifteen years so. Mathematically, this involves the numerical solution of a coupled system of non-linear partial differential equations (PDEs) and algebraic relations that describe the diffusion, convection, migration and reaction of the ionic species mentioned above. However, in our work, we have used asymptotic methods to reduce a two-dimensional (2D) transient model having 11 PDEs to one consisting just four coupled ordinary differential equations. Most significantly, the model preserves all of the original physical assumptions, as is seen in Fig. 2 from the comparison of the results for the charge-discharge curve, i.e. the cell potential,

*E*, as a function of time,

_{cell}*t*. Furthermore, the solution of the reduced asymptotic model is found to require around 300 times less computational time than that of the original model. The asymptotic model would therefore useful for carrying out rapid parameter studies and can form the basis for including further physical phenomena that were not included in the model earlier.

More details can be found in Ref. [1].

**References**

[1] Vynnycky, M. & Assunção, M.,

*The vanadium redox flow battery – an asymptotic perspective,*accepted for publication in SIAM J. Appl. Maths (2019). DOI: 10.1137/18M1168984

*Biography: Michael Vynnycky is Professor of Applied Mathematics at the University of Limerick and is active within MACSI (Mathematics Applications Consortium for Science and Industry). Aside from vanadium redox flow batteries, some of his industrial mathematics pursuits include the continuous casting of steel, pharmaceutical freeze drying, smectic liquid crystals, brazing of aluminium alloys and high pressure die casting.*

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