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Jn, tot itot s ̇ n iFV idVl Jn, tot itot CV i − 1 CV i CV i + 1 y Figure 4.3: Layout of the ith control volume (CV), illustrating the source terms and fluxesforchemicalspecies,i.e.mass(s ̇n andJn,tot),andelectrons,i.e.charge (iFV, idVl, and itot). Finally, Dn, eff denotes the effective diffusion coefficient, which is defined as D =D ·ε /τ2 , (4.4) n, eff n elyte elyte correcting Fick’s diffusion coefficient Dn for the volume fraction εelyte and the tortuos- ity τelyte of the liquid electrolyte phase, which is discussed e.g. in Ref. [210]. Of course, Eq. (4.1) needs to be solved for each species n. By postulating charge conservation and electroneutrality (at the scale of the CVs), the number of equations to be solved is reduced by one, since the condition V ∂Jn,tot 0=∑znFAms ̇n,m−∑znF ∂y (4.5) n,m n always holds and can be used to eliminate Eq. (4.1) for one (charged) species. In words, the flux of the nth species is determined by the flux of the (n − 1) other species so that the total flux (in units of charge) is equal to the amount of charge generated within the CV. Fig. 4.3 summarizes the source terms in and fluxes across the boundaries of a CV. 4.2.3 Electrochemistry The derivation of the (electro-)chemical rate laws follows Ref. [58], which in turn is based on Ref. [211, sec. 9.3]. The following assumptions are required for a more com- pact formulation of the problem: pressure and temperature shall be constant and all electrochemical reactions occur only at interfaces (whereas chemical reactions may also occur within a bulk phase). For the species continuity equation, Eq. (4.1), a gen- eralized source term s ̇n (or s ̇n) was already defined which in turn is calculated from 74PDF Image | Lithium-Sulfur Battery: Design, Characterization, and Physically-based Modeling
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