35 research outputs found
A Comparison of Fick and Maxwell-Stefan Diffusion Formulations in PEMFC Cathode Gas Diffusion Layers
This paper explores the mathematical formulations of Fick and Maxwell-Stefan
diffusion in the context of polymer electrolyte membrane fuel cell cathode gas
diffusion layers. Formulations of diffusion combined with mass-averaged Darcy
flow are considered for three component gases. Fick formulations can be
considered as approximations of Maxwell-Stefan in a certain sense. For this
application, the formulations can be compared computationally in a simple, one
dimensional setting. We observe that the predictions of the formulations are
very similar, despite their seemingly different structure. Analytic insight is
given to the result. In addition, it is seen that for both formulations,
diffusion laws are small perturbations from bulk flow. The work is also
intended as a reference to multi-component gas diffusion formulations in the
fuel cell setting.Comment: 12 pages, submitted to the Journal of Power Source
Second-Order Convergence of a Projection Scheme for the Incompressible Navier–Stokes Equations with Boundaries
A rigorous convergence result is given for a projection scheme for the Navies–Stokes equations in the presence of boundaries. The numerical scheme is based on a finite-difference approximation, and the pressure is chosen so that the computed velocity satisfies a discrete divergence-free condition. This choice for the pressure and the particular way that the discrete divergence is calculated near the boundary permit the error in the pressure to be controlled and the second-order convergence in the velocity and the pressure to the exact solution to be shown. Some simplifications in the calculation of the pressure in the case without boundaries are also discussed
Stable Fourth Order Stream-Function Methods for Incompressible Flows with Boundaries
Fourth-order stream-function methods are proposed for the time dependent, incompressible Navier-Stokes and Boussinesq equations. Wide difference stencils are used instead of compact ones and the boundary terms are handled by extrapolating the stream-function values inside the computational domain to grid points outside, up to fourth-order in the noslip condition. Formal error analysis is done for a simple model problem, showing that this extrapolation introduces numerical boundary layers at fifth-order in the stream-function. The fourth-order convergence in velocity of the proposed method for the full problem is shown numerically
Asymptotic reduction of a porous electrode model for lithium-ion batteries
We present a porous electrode model for lithium-ion batteries using
Butler--Volmer reaction kinetics. We model lithium concentration in both the
solid and fluid phase along with solid and liquid electric potential. Through
asymptotic reduction, we show that the electric potentials are spatially
homogeneous which decouples the problem into a series of time-dependent
problems. These problems can be solved on three distinguished time scales, an
early time scale where capacitance effects in the electrode dominate, a
mid-range time scale where a spatial concentration gradient forms in the
electrolyte, and a long-time scale where each of the electrodes saturate and
deplete with lithium respectively. The solid-phase concentration profiles are
linear functions of time and the electrolyte potential is everywhere zero,
which allows the model to be reduced to a system of two uncoupled ordinary
differential equations. Analytic and numerical results are compared with full
numerical simulations and experimental discharge curves demonstrating excellent
agreement.Comment: Accepted in SIAM Journal on Applied Mathematic
Analysis of the Spatial Error for a Class of Finite Difference Methods for Viscous Incompressible Flow
Several first and second order finite difference methods for incompressible flow based on prescribed forms of the discrete gradient and divergence operators are considered. Expansions for the spatial error for these methods are presented. So-called alternating expansions and numerical boundary layers are required to describe the errors arising from schemes with decoupled pressure approximations and regularizing terms, respectively. Alternating expansions in the discrete projection operator can be amplified by the viscous term and lead to a reduction in the accuracy of the computed pressure. These error expansions can be combined with simple stability estimates to show the convergence of the discrete solutions to the nonlinear time-dependent and steady problems when a discrete adjoint condition is satisfied. However, the analysis here does not consider the split-step nature of projection methods. Convergence order predictions are verified in a careful numerical study. Key words: error e..