Review-Mathematical Formulations of Electrochemically Gas-Evolving Systems

Abstract

Electrochemically gas-evolving systems are utilized in alkaline water electrolysis, hydrogen production, and many other applications. To design and optimize these systems, high-fidelity models must account for electron-transfer, chemical reactions, thermodynamics, electrode porosity, and hydrodynamics as well as the interconnectedness of these phenomena. Further complicating these models is the production and presence of bubbles. Bubble nucleation naturally occurs due to the chemical reactions and impacts the reaction rate. Modeling bubble growth requires an accurate accounting of interfacial mass transfer. When the bubble becomes large, detachment occurs and the system is modeled as a two-phase flow where the bubbles can then impact material transport in the bulk. In this paper, we review the governing mathematical models of the physicochemical life cycle of a bubble in an electrolytic medium from a multiscale, multiphysics viewpoint. For each phase of the bubble life cycle, the prevailing mathematical formulations are reviewed and compared with particular attention paid to physicochemical processes and the impact the bubble. Through the review of a broad range of models, we provide a compilation of the current state of bubble modeling in electrochemically gas-evolving systems.

Figure 1.

Schematic of relevant multiscale computational physics and the corresponding length and time scales pertinent to electrochemical gas-evolving systems.

Figure 2.

Schematic of multiscale coupling strategies. (A) Concurrent coupling method requires solving all models at the same time step in a parallel mode then moving to the next time step. (B) Sequential coupling method requires solving models over different time lengths sequentially then passing parameters from one model to another.

Figure 3.

Gibbs free energy change during bubble nucleation. The surface free energy ΔG_{sur f} (dashed red line), bulk free energy ΔG_bulk (dashed black line), homogeneous nucleation free energy ΔG_Hom (blue line), and the heterogeneous nucleation free energy ΔG_Het (green line) are presented as a function of the radius R. The nucleation stages are also sketched for reference. The embryo and cluster states are reversible and the nuclei state is irreversible. Beyond the nuclei phase the bubble starts growing.

Figure 4.

A sketch of the stages of bubble growth before detachment from a substrate: nuclei (orange line), under critical growth (blue line), critical growth (green line), and necking (red line).

Figure 5.

Variation of the current density as function of the overpotential: Butler-Volmer β = 0.5 (red line), Marcus-Hush-Chidsey λ = 30 (green line), Marcus λ = 30 (blue line), Marcus-Hush-Chidsey λ = 15 (yellow line), Marcus λ = 15 (black line).

Figure 6.

Hydrodynamics forces acting on a bubble at the electrode surface. The expressions of each force are defined in Table I. This figure is reproduced from Taqieddin et al., 2017.

Figure 7.

Ionic surface tension increments of aqueous ions at ambient conditions as a function of the excess molar refractivity.^,

Figure 8.

Reduction of electrolyte conductivity relations based on gas bubble volume fraction: Linear (red), Maxwell (green), Meredith/Tobias (blue), Prager (yellow), Bruggeman (black).

Figure 9.

(A) Different shapes of a bubble rising in viscous liquid. The characteristic shape is set by the flow dynamics and fluid properties. (B) Instantaneous streamlines revealing the wakes associated with different bubbles. (From left to right) The rising spherical cap is characterized by a trailing ring wake. The spherical bubble has a smooth wake. Rising elliptical bubbles have a helical vortex wake, and skirted bubbles feature vortex ring wakes. The figure is reproduced from Gumulya et al., 2016.