In this paper, a mathematical model for the all-vanadium battery is presented and analytical solutions are derived. The model is based on the principles of mass and charge conservation, incorporating the major resistances, the electrochemical reactions and recirculation of the electrolyte through external reservoirs. Comparisons between the model results and experimental data show good agreement over practical ranges of the vanadium concentrations and the flow rate. The model is designed to provide accurate, rapid solutions at the unit-cell scale, which can be used for control and monitoring purposes. Crucially, the model relates the process time and process conditions to the state of charge via vanadium concentrations. Electrochemical energy storage systems are crucial to the regulation and transmission of intermittent power derived from wind, solar and tidal sources. One promising example of such a storage system is the redox flow battery (RFB), which is suitable for both medium and large scales storage needs.1 Other important applications of RFB technology include power balancing and peak shaving for incumbent power generation methods. Unisearch Limited, University of New South Wales (UNSW) Australia as the applicant. The success of the VRFB is largely attributable to its high energy efficiencies (between 80 and 90% in large installations), the soluble state of the active species (no metal deposition), its potentially low cost per kilo watt hour for large storage capacities, the minimal gas evolution during normal operation, and use of the same element in both half-cells, avoiding problems associated with cross-contamination during long-term use. The main electrode reactions for the VFRB are as followswith further side reactions (notably gas evolution) when the cells are overcharged. The electrolytes for each cell are circulated through the electrochemical cell and external reservoirs/tanks. The half cells are separated by an ion-selective membrane, typically Nafion, to transport protons. In theory, the energy storage capacity increases with the volume of the reservoirs and the concentrations of vanadium species, while the power output depends on the active electrode surface area and number of cells (when placed in a stack). Laboratory-based investigations (considering materials, operating conditions, additives and cell structure) can be highly costly, as well as time-and labour-intensive. In order to reduce costs and timescales, modelling and simulation can be employed during the design and test cycles, and used to control and monitor systems in real time, [10][11][12][13] provided of course that model parameters are available from suitable experimental data. Mathematical models of the VRFB system have been developed by Shah et al. [14][15][16][17] and by Li and Hikihara.18 These models incorporate the fundamental modes of transport, the electrochemical kinetics (including hydrogen and oxygen evolution 16,17 ) and heat losses. 15 It is not feasible, however, to incorporate this level of detail in control/mon...
