Abstract
Several efficient analytical methods have been developed to solve the solid-state diffusion problem, for constant diffusion coefficient problems. However, these methods cannot be applied for concentration-dependent diffusion coefficient problems and numerical methods are used instead. Herein, grid-based numerical methods derived from the control volume discretization are presented to resolve the characteristic nonlinear system of partial differential equations. A novel hybrid backward Euler control volume (HBECV) method is presented which requires only one iteration to reach an implicit solution. The HBECV results are shown to be stable and accurate for a moderate number of grid points. The computational speed and accuracy of the HBECV, justify its use in battery simulations, in which the solid-state diffusion coefficient is a strong function of the concentration.
Highlights
[21] While these authors focused on the Crank-Nicolson time domain discretization, which sometimes produces oscillatory solutions, we present a backward Euler control volume method (BECV) to resolve the spherical diffusion problem with a variable diffusion coefficient
C prac max c where Dref is defined as the reference diffusion coefficient of 2 × 10−16 m2·s−1, Ctheo is the theoretical capacity of the electrode material (277.84 mAh·g−1) and Cprac is the practical capacity of the electrode material of 160 mAh·g−1
The backward Euler control volume (BECV) and the hybrid backward Euler control volume (HBECV) methods are presented as efficient numerical methods to resolve the solid-state spherical diffusion problem for a variable diffusion coefficient
Summary
Since the discovery of the electrochemical intercalation reaction of lithium in layered titanium disulfide (TiS2) by Whittingham in 1976, lithium-ion batteries (LIB) have grown in popularity and application to surpass all other electrical. [21] While these authors focused on the Crank-Nicolson time domain discretization, which sometimes produces oscillatory solutions, we present a backward Euler control volume method (BECV) to resolve the spherical diffusion problem with a variable diffusion coefficient This method incorporates all the advantages of the CVM, with the added advantage of being stable and easier to implement. Assume mi (t ) , the amount of electrochemically active species [mol] inside vi at arbitrary time t, can be expressed as a product of the concentration at node point i and volume of corresponding spherical shell. Equations (27)-(29) represent a coupled system of equations since all values at time step j are unknown while values at time step j −1 are unknown
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