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UnderstandininggththeeVVaannaaddiuimumRRedeodxoFxloFwlowBaBttaetrtiesries 3379 3.1.2 Input, output and average concentrations of vanadium ions We know now that the vanadium concentrations change within the cells when the battery is operating. Therefore, the concentrations are not uniformly distributed through the electrolyte circuit (Fig. 4). Indeed, four concentrations are located in the VRB: the tank concentration ctank, the concentration at the cell input cin, the concentration inside the cell ccell and the concentration at the cell output cout. Usually, the size of the reservoir is large compared to the electrolyte flowrate; thus the change in concentrations due to the flow of used electrolyte is so small that the tank concentrations are considered homogeneous. And therefore, the input concentrations cin correspond exactly to ctank. The tank concentration ctank reflects the past history of the battery; indeed the change in ctank is proportional to the quantity of vanadium that has been transformed in the stack: this value corresponds to the quantity of electrons involves in the reaction. Therefore, ctank is defined by the initial ion concentrations cinitial, the size of the reservoir Vtank and the total molar flowrate of electrons N ̇ e− : tot tanki initial cini(t)=ctanki(t)=ctanki 1 +V tank (15) where b is a sign factor that reflects the direction of the reaction in accordance with Tab. 2: −1 for V2+ and V5+ ions b= 1forV3+ andV4+ ions [−] (16) The description of the output concentration cout is difficult because it depends on the electrolyte flowrate Q, the length of the electrolyte circuit and on the current i that the electrolyte encounters during the cell crossing. Since the distribution of the vanadium ions inside the cell is unknown, we consider that the model has no memory and reacts instantly to a change in the operating conditions. In that case, cout is related to the electrons molar flowrate N ̇ e− , the electrolyte flowrate Q and on the input concentration cin: tot N ̇ e − ( t ) couti (t) = cin (t) + b tot b N i ( t ) = cin (t) + cell where: ci Q(t) = = i i F Q(t) initial = ctanki 1 b + Vtank F i(t)dt [mol/l] bN ̇e− (t)dt tot [mol/l] concentration of the different vanadium ions (17) [mol/l] [l/s] Q(t) flowrate of the electrolyte For a quasi steady state, where the current and the flowrate are almost constant, the model predicts accurately the output concentrations. Unfortunately, it is not able to predict the transient behaviour when the system encounters extreme conditions such as the combination of a low flowrate, few active species and sudden current change. But when these conditions are avoided, (17) offers a very good insight of the battery behaviour. We still have to establish the most important concentration: the concentration inside the cell ccell that is necessary to solve the Nernst equation (3). Because the ion concentrations are not uniformly distributed inside the cell, we will make an approximation to determine ccell from the mean value of cin and cout: ccelli (t) = cini (t) + couti (t) [mol/l] (18) 2PDF Image | Understanding the Vanadium Redox Flow Batteries
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