The Lorentz forces are the only interactions between flowing particles and the lattice. The force balance applies to a continuum, i.e., to the macroscopic, averaged, description of the lattice itself through which ions, electrons, and/or neutral particles flow. The principle of virtual power leads to the usual balance of forces and symmetry of stress tensor σ. If the interference between the electric and magnetic phenomena is restricted to capacitive effects only (i.e., noninductive), the time-dependent hyperbolic of Maxwell’s equations can be replaced by parabolic equations that can be solved in a more simple way.Ī coupling arises between the mass conservation and Maxwell’s equations through Faraday’s law of electrolysis, being ionic species charge carriers.Īssuming small displacements and strains, the conjugated pairs within the internal expenditure of virtual power W int are the Cauchy stress σ and the infinitesimal strain tensor ɛ. For the EQS approximation to hold, the velocity of traveling electromagnetic waves in the material has to be small compared to the ratio between the characteristic time scale and the length characterizing the system, a condition that is generally fulfilled for battery cells (see Salvadori et al., 2015). The EQS model ( Larsson, 2007) is here assumed as an approximation of the full Maxwell’s equations’ set. Maxwell’s equations describe electric and magnetic fields reciprocal interactions. Mass balance equation governs the transport of neutral and ionic lithium (Li and Li + respectively) and counterions (X −) in any battery cell models. Grazioli, in Advances in Battery Technologies for Electric Vehicles, 2015 16.3 Essentials of the multiscale modeling approach (7) is considered before discussing extended versions that include hyporheic exchange flows.Ī. However, the volumetric rates of hyporheic exchange, groundwater exchange, solute concentration in groundwater, and the reaction rate constant are all spatially uniform. The model therefore allows nonuniform flow resulting from exchange with groundwater. (7) states that the solute mass flux declines or increases downstream according to the balance between mass gains and losses resulting from groundwater exchange and mass gains or losses due to reactive production or uptake of the solute. Note that hyporheic fluxes representing recharge to and discharge from the hyporheic zone were collected into one term by recognizing that at steady state, Q hz in = Q hz out and thus each hyporheic term could be replaced by Q hz.Įq. Where C( x) is the concentration of the reactive constituent, Q( x) is volumetric flow rate at distance x along the reach’s centerline from x = 0 upstream to x = L downstream, L is reach length, C gw in is average concentration in discharging groundwater, C hz is concentration in the hyporheic zone, Q gw and Q hz are volumetric exchange fluxes with groundwater and hyporheic, with superscripts denoting discharge flux from groundwater to stream, in, and recharge fluxes from stream to groundwater, out, λ is a first-order rate coefficient for reactive loss of solute, and A is average stream cross-sectional area. (7) d Q C d x = 1 L Q gw in C gw in − Q gw out C + Q hz C hz − C − λ A C
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