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Non-isothermal model. Adding a description of heat sources and transport to the model is straightforward from a theoretical point of view [280]. However, since almost all effects in the cell depend on the temperature one way or the other, this measure would effectively double the number of parameters required. Depending on the phe- nomenon, entropies, activation energies, temperature coefficients, and so on would need to be determined in addition to the present parameters. That issue put aside, this extension is desirable, because it would enable the use of this model as a back-end for higher level, system-scale simulations, which rely on a correct energy balance. A full implementation of heat generation and transfer is available within the mod- eling framework Denis, see e.g. Ref. [281]. The approach taken is to include an addi- tional conservation equation for energy in addition to the conservation of mass and charge: d (wel + wchem + wheat) = div (w⃗ heat) . (5.1) dt Herewel istheelectrical,wchem thechemicallystoredenergydensity,andw⃗heat,the heat flux. The total heat Q tot equals wheat times the volume of the CV; it is composed of several contributions including, among others, heat from exothermic reactions Qelchem, polarization heating Qpol, and resistive heating Qres: Qtot = Qelchem +Qpol +Qres +... . (5.2) These sources need to be implemented one by one. In addition, one more equation is needed, describing the transport of heat. ∂T ∂t where Qj are the heat sources as defined above, ρ is the density, and cp the effective heat capacity at constant pressure. Finally, boundary conditions are needed, describing the flux in and out of the computational domain. A detailed description of the theory and implementation can be found in Ref. [282, chap. 3.4]. To conclude, there are many possible extensions to the model; some exist only as concepts, some ready to use. The only limitation is the number of parameters that can be determined faithfully and reliably. 136 ρcp· =−div(w⃗heat)+∑Qj , (5.3) jPDF Image | Lithium-Sulfur Battery: Design, Characterization, and Physically-based Modeling
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