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UnderstandininggththeeVVaannaaddiuimumRRedeodxoFxloFwlowBaBttaetrtiesries 335 2.1 Equilibrium potential The stack voltage Ustack depends on the equilibrium voltage Ueq and on the internal losses Uloss; the equilibrium conditions are met when no current is flowing through the stack. In that case, there is no internal loss and Ustack equals Ueq; otherwise, the internal losses modify Ustack. The internal losses1 Uloss will be discussed in section 3.3. Hence Ustack is given by: Ustack(t) = Ueq(t) − Uloss(t) [V] (2) The equilibrium voltage Ueq corresponds to the sum of the equilibrium potential E of the individual cells composing the stack. This potential is given by the Nernst equation and depends on the vanadium species concentrations and on the protons concentrations (Blanc, 2009): E = E′ + RT ln F c + ·c2 VO2 H+ cVO2+ cV2+ cV3+ [V] (3) where R is the gas constant, T the temperature, F the Faraday constant, ci the concentration of thespeciesiandE′ theformalpotential.Ifweassumethattheproduct/ratiooftheactivity coefficients is equal to 1, the formal potential E′ , an experimental value often not available, can be replaced by the standard potential E. 2.1.1 Standard potential from the thermodynamics The standard potential E is an ideal state where the battery is at standard conditions: vanadium species at a concentration of 1 M, all activity coefficients γi equal to one and a temperature of 25◦C . The standard potential is an important parameter in the Nernst equation because it expresses the reaction potential at standard conditions; the second term in the Nernst equation is an expression of the deviation from these standard conditions. Together, they determine the equilibrium cell voltage under any conditions. The standard potential E can be found from thermodynamical principles, namely the Gibbs free enthalpy ΔG and the conservation of energy, and empirical parameters found in electrochemical tables. We introduce here the standard Gibbs free enthalpy of reaction ΔG which represents the change of free energy that accompanies the formation of 1 M of a substance from its component elements at their standard states: 25◦C , 100 kPa and 1 M (Van herle, 2002): ΔG = ΔHr − TΔSr [kJ/mol] (4) where the standard reaction enthalpy ΔHr is the difference of molar formation enthalpies between the products ΔHf ,product and the reagents ΔHf ,reagent: ΔHr = ∑ ΔHf ,product − ∑ ΔHf ,reagent [kJ/mol] (5) products reagents and the standard reaction entropy ΔSr is the difference of molar formation entropies between the products Sf ,product and the reagents Sf ,reagent: ΔSr = ∑ Sf ,product − ∑ Sf ,reagent [J/mol · K] (6) products reagents 1Note that the sign of Uloss depends on the operating mode (charge or discharge).PDF Image | Understanding the Vanadium Redox Flow Batteries
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