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additional “global” resistance may be included so that E = φCA, elde − φAN, elde − ρ⋆itot , (4.18) with ρ⋆ representing the combined electric resistivity of the bulk lithium metal, carbon, and both current collectors. While ρ⋆ is typically constant, it may also be influenced by the microstructure of the cell, see below. 4.2.5 Microstructure As previously mentioned, this one-dimensional model cannot explicitly describe the electrodes’ microstructure. Instead, each control volume is assigned properties which are representative of its morphology, namely the volume fractions εp of each phase p present in the control volume. From these, the tortuosities τp, effective diffusion coef- ficients Dn, eff, and specific surface areas AVm are derived. Volume fractions may change because of precipitation and dissolution of material, phase transitions, or (convective) phase transport. For this work, only the first option is taken into account so that the evolution of the volume fractions is completely described by the continuity equation ∂ρpεp ∂t =RpMp , (4.19) which has to be evaluated for each phase p. Here, ρp is the density and Mp the (average) molar mass of the phase, while Rp denotes its net production rate, which in turn is calculated from the (electro-)chemical source terms as defined in Eq. (4.6). Assuming that phases grow and shrink at their interfaces only, Rp can be obtained from the relation Rp = ∑ s ̇n,mAVm , (4.20) m,n∈p where the s ̇n,m is the net production rate of species n in phase p due to all reactions occurring at interface m. Since in the implementation used for this work the space occupied by each CV is fixed, the following relation needs to be fulfilled at all times: ∑εp =1. (4.21) p While there are several more rigorous options for ensuring consistency (see e.g. [215, pp. 53ff]), a simple, yet straight-forward approach was chosen for this work: Using Eq. (4.21), Eq. (4.19) can be eliminated for one phase. This is done by setting the volume fraction of the gaseous phase to fill the entire space not occupied by the solid phases. The flow or pressure increase caused by the volume change is neglected, 78PDF Image | Lithium-Sulfur Battery: Design, Characterization, and Physically-based Modeling
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