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5 Single fibre electrode measurements - a versatile strategy for assessing the non-uniform kinetics Simulations Numerical models were used to simulate the cyclic voltammetry at single carbon fibre and carbon felt electrodes. For the single carbon fibre electrodes, a 1D partial differential equation model was developed in cylindrical coordinates, with the surface of the fibre assumed to be homogenous, so that a single standard heterogeneous rate constant (k0) could be used. The partial derivatives were solved via a system of ODEs using the method of lines and the ode15s solver in Matlab, with the electron transfer rate following Butler-Volmer kinetics. The change in concentration of the oxidised and reduced species with respect to time is governed by the cylindrical-PDE: ππΆπΌ =π·[π2πΆπΌ +1ππΆπΌ] (5.1) ππ‘ ππ2 π ππ where πΌ can be either the oxidised species π or the reduced species π , r is the radial position relative to the centre of the carbon fibre. To solve the system, two boundary conditions were defined at the bulk and at the surface. At a large and arbitrary distance from the electrode (r*) the bulk concentration was assumed to be constant and equal to the initial solution concentration, and thus the bulk boundary condition was defined as: π = πβ: ππΆπΌ| =0 (5.2) ππ π=πβ The surface boundary condition was defined by equating the change in concentration at the surface to the reaction rate at the surface: ππΆ π0 (πΆπ,π exp (β πΌππΉπ) β πΆπ ,π exp ((1 β πΌ)ππΉπ)) π π π π (5.3) ππ π=ππ π· π=π: π | = π 55PDF Image | Electron Transfer Kinetics in Redox Flow Batteries
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