Orthogonal Collocation Simulation of the Rotating Ring Disk Electrode: The Use of Exponentially Weighted Polynomials

Abstract

The partial differential equations describing the mass transfer to a rotating ring disk electrode (RRDE) have been solved by the orthogonal collocation technique. The two spatial dimensions (normal and radial) are discretized leaving a continuous dependence in time. There is a semi-infinite boundary condition in the direction normal to the electrode. The space in the radial direction is, however, bounded. Thus, polynomials with a weighting factor ofe -~, with ~ = 1 and ~ = 1, 2, or 3, which are orthogonal to each other in [0, ~], are used in the direction normal to the electrode whereas the "shifted" Jacobi polynomials, which are orthogonal to each other in [0, 1], are used in the radial direction. Two different boundary conditions at the disk are examined, namely, a constant potential in the limiting current region and a constant applied current. Results are presented for an electron transfer at the disk as well as at the ring. The transient and steady-state currents at the disk and the ring electrodes as well as the simulated collection efficiency are in excellent agreement with previous theoretical treatments.

Comparison between CFD calculations of the flow in a rotating disk cell and the Cochran/Levich equations

“…The discussion is centered on the axial flow along the central axis (r = 0) since this is the commonly used velocity profile employed in modeling studies based on the Cochran equation (e.g. [19,23,22,25,24,21].…”

“…When the velocity profile is known, in fact, it is possible to model the system (e.g. [19,[23][24][25]) and determine the kinetic constants from the experimental data (e.g. [21,22]).…”

Modelling and numerical simulation of hydrodynamical processes in a confined rotating electrode configuration

“…Many studies have appeared on the modelling and simulation of hydrodynamical processes in electrochemical cells . Different configurations have been investigated such as stationary or rotating disk electrodes [5][6][7][8][9][10][16][17][18]23,17,[25][26][27][28][29][30][31][35][36][37], ring-disk electrodes [13,17,18] line electrodes [32], paddle cells [33,34], rotating hemispheral electrodes (RHSE) [40,41], wall-jet electrodes [44], etc. We note that usually the authors consider monodimensional flux and that bidimensional full calculations have attracted less attention, mainly in relation to electromigration phenomena during electrolysis processes [15].…”

“…[17,41,42]), current and potential transient responses as well as impedance spectra (e.g. [17]). Mechanistic complications occurring at the electrode vicinity such as bubble formation, chemical solution reactions [16,33,34], etc., can be taken into account and their influence on species transfer and current or potential responses modelised.…”

Minimum grid digital simulation of chronoamperometry at a disk electrode

“…Jacobi polynomial roots require cumbersome computation, and Chebyshev polynomial roots were found to perform equally well. Speiser and Pons applied the method to a two-dimensional simulation [17,18], as did Parikh and Liddel [51][52][53]. Descriptions of OC are normally somewhat heavy on linear algebra but in fact OC, whether in one or more dimensions, can be simply implemented by the derivative approximation formula applied to a stretch of points at x j , j = 0, .…”