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mass-action kinetics. For each reaction q, the term s ̇n,q is defined as s ̇ =ν k an′ −kq,rev ∏an′′ , (4.6) n,q n,q q,fwd ∏ νn′,q νn′′,q n′ ∈q, fwd n′′ ∈q, rev where kfwd and krev are the forward and backward rate constants, νn the stoichiometric coefficients of species n in the corresponding reaction q, and an the activities. The products run over all reactants n′ or n′′ participating in the forward or reverse reaction, respectively. The total production rate of species n at interface m is simply the sum of the contributions of all reactions q at that interface: s ̇n,m = ∑ s ̇n,q . (4.7) q∈m The equation for bulk reactions has exactly the same form, except that the units of activities and rate constants differ, so that s ̇n is a volume-specific production/ consumption rate instead of an area-specific one. The forward rate constant is given by a general modified Arrhenius expression (index q omitted in the following equations): ̃ Eact (1−α)zF kfwd = k0,fwdTβ exp − RT exp − RT ∆φ . (4.8) Since there is no reason to assume otherwise, the symmetry factor α is chosen to be 1/2 for all reactions. Also, because the model is assumed strictly isothermal, both the temperature factor Tβ and the activation energy Eact may be eliminated so that Eq. (4.8) is reduced to zF kfwd = k0,fwd exp −2RT∆φ , (4.9) with z being the number of electrons transferred, F Faraday’s constant, R the uni- versal gas constant, T the (now fixed) temperature, and ∆φ the potential step at the corresponding interface: ∆φ = φelde − φelyte. The pre-exponential factor k0, fwd now incorporates the contributions of the two temperature-dependent terms in Eq. (4.8), which are reduced to a fixed factor in the case of constant temperature. For mere chemical reactions (i.e. z = 0), kfwd is equal to k0, fwd. While reverse reactions rates may be specified analogously, i.e. in the form zF krev = k0, rev exp 2RT ∆φ , (4.10) this is not the case for most reactions in this model. Instead, reverse rates are calculated 75PDF Image | Lithium-Sulfur Battery: Design, Characterization, and Physically-based Modeling
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