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Section 2.8 Activation overpotential 2.8.2 Simplified equations For the charge transfer coefficient α, it is widely accepted to assume an equal value of 0.5 for all four different coefficients. This assumption allows for simplifying the Butler- Volmer equation by expressing the double-exponential term as hyperbolic sine, as shown in the Eqs. (2-51) and (2-52). 1 FEAct 1 FEAct iTF FkRC⋅c2C⋅c3C⋅ e 2 GT e 2 GT 2FkRC⋅c2C⋅c3C⋅sinh FEAct 2GT 1 FEAct 1 FEAct iTF=FkRC⋅c4C⋅cC⋅ e 2 GT e 2 GT 2FkRC⋅cC⋅c5C⋅sinh FEAct 2GT (2-51) (2-52) Using Eq. (2-50), we can finally convert the Butler-Volmer equation into a form which only uses the macroscopic quantities, available in a lumped-parameter model, as shown in the Eqs. (2-53) and (2-54). The assumption of transfer coefficients equal to 0.5 allows for a direct computation of the activation overpotential out of the cell current, the concentrations of the vanadium ions, the electrochemical rate constant and the specific surface area of the graphite felt. EAct 2GTarsinh IC F 2sF VE FkRC ⋅c2C ⋅c3C EAct 2GTarsinh IC F 2sF VE FkRC ⋅c4C ⋅c5C 2.8.3 Numeric example and critical assessment (2-53) (2-54) Because of the variety of different values for each input parameter, the derived activation overpotential varies significantly, as shown in Figure 2-13. With an assumed ASR of 1.5 Ωcm2, the ohmic overpotential is 37.5 mV for a current density of 25 mAcm−2 and 112.5 mV for a current density of 75 mAcm−2. With the parameters taken from [18], the activation overpotential exceeds the ohmic overpotential for low and high SoC values. With the parameters presented in [57], the activation overpotential is significantly smaller. Finally, for the parameters given in [22], the activation overpotential practically becomes negligible. The determination of the correct parameters, e.g. the electrochemical rate constant, is still subject to recent research work [58, 59]. 39PDF Image | Model-based Design Vanadium Redox Flow Batteries
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