n 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
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