A Lagrangian passive scalar solver for mass transport in electrolytes and coupling to the particle-resolved Bluebottle

Abstract

Water electrolysis is a popular energy storage technique used in tandem with many renewable sources to convert the generated energy into storeable hydrogen. The efficiency of water electrolyzers is greatly affected by overpotential losses. Bubble evolution is a unique characteristic of flows in water electrolysis. The evolution of bubbles alter electrokinetics close to the electrodes and vary ionic mass transport. The localized flow features close to a bubble are often attributed to the variation in the current density and subsequently, the electrolyzer efficiency. For the purpose of modelling flows close to a bubble better, a Lagrangian method for the simulation of passive tracers is developed and programmed to be coupled to the flow field of Bluebottle, an open-source particulate multiphase flow solver that uses the Physalis algorithm.
The dynamics of the tracers are modelled using a simplified Langevin equation. In the present work, the migration flux is omitted and priority is given to convection and diffusion with the objective of establishing a foundation for the simulation of ionic mass transport. Brownian motion is described using a random displacement term. The coupling with the flow field is achieved using trilinear interpolation. The domain boundaries in regards to tracer dynamics are modelled as either a rigid wall pair or as a periodic boundary pair. Specular reflection is programmed for the former ensuring elastic collision of a tracer with the domain boundary. For the latter, the tracer position is altered so as to place the tracer in the opposite side of the domain in the axis of intrusion. Particles are assumed to be non-penetrative and hence, specular reflection is implemented at the surface of each particle. Since the tracer module is coupled one-way with the flow field and executed after a Bluebottle time-step, a subroutine is developed to push the tracers out of a particle radially if a tracer is located inside a particle after a Bluebottle time-step. To ensure particle interaction is ensured across periodic boundaries, a subroutine is developed that places the particle in an apparent location that enables particle-tracer interaction. 
The module execution time is found to be linearly proportional to the number of particles and the number of tracers and consumes roughly 10% of a Bluebottle iteration execution time in nominal tests. The module is tested to ensure the Brownian displacement term obeys diffusion statistics and also to ensure that the trilinear interpolation works as intended. The numerically enforced no-penetration boundary at particle surfaces is also tested and observed to prevent intrusion of tracers. 
The tracer module is then used to stochastically simulate mass transport across a particulate suspension in a stagnant and a sheared flow field. The Sherwood number Sh determined from the tracer module is found to agree well with the expected experimental and numerical results of Wang et al. (2009). The tracer module is also compared to a scalar field solver of Bluebottle. The tracer module is observed to capture features of the flow field quite well. However, the transient tracer positions upon conversion to a transient continuous concentration field exhibits noise due to the discrete nature of the tracers. Hence, transient comparisons with a continuous field in terms of absolute magnitude requires a large number of tracers.
Recommendations for improvement of the code is provided. The present work is intended to be followed up with the addition of migration flux to the equations of motion for the tracers through the solution of an additional equation for velocity of the tracer using a force equivalence of Coulomb's law and Stokes' drag law. Future challenges that will be encountered in the development of an accurate ionic mass transport solver is briefly discussed.