Danny van den Eertwegh

Although alkaline water electrolysis is a promising technique for the production of green hydrogen, it faces limitations due to the gaseous products that are formed. The bubbles significantly limit the efficiency of the process: they actively decrease the electrode surface area available for reaction, they change the local flow field and thereby influence the local mass transfer phenomena, and their general presence decreases the space for ions to move, increasing the overall Ohmic resistance. Alkaline water electrolysis is thus a complex interplay of different physics, ranging from the transport of charged ionic species to multiphase fluid dynamics. In this work, this complex system is studied using numerical simulation methodologies. This involves incorporation of conventional methods, but also the development and introduction of novel computational approaches.

Chapter 2 focusses on modelling of charged ionic species in a single-phase system. In these types of systems, a species transport equation in combination with the Nernst-Planck flux is employed. In addition, a transport equation for the electrical potential is required. For the latter, multiple conventional approaches exist: using the Poisson equation or assuming electroneutrality. The former is most versatile from a physical point of view as it allows for the separation of charge. However, from a numerical point of view it is more stringent, as it requires resolving the length and time scales of charge separation. In case of the electroneutrality assumption, no separation of charge in the system is assumed. In addition to these two conventional approaches, a new numerical approach is introduced, which effectively functions as a continuous numerical switch between the Poisson equation and the transport equation for the electrical potential obtained in case of electroneutrality. This charge conservation equation is compared with the two conventional models for three different types of systems: a small-scale system where charge separation is expected in most of the domain, a large-scale system where charge separation is significantly less important, and a multi-ion liquid junction system where the importance of charge separation depends on the chosen parameters. The results generated using the charge conservation equation are compared to the results obtained with the Poisson equation and/or the electroneutrality equation, depending on the system. It is found that usage of the charge conservation equation produces accurate results for each of the systems, while both the Poisson equation and the electroneutrality equation fail to produce accurate results in all cases. In addition, the method can capture the charge separation when necessary while resorting to electroneutrality when possible.

In Chapter 3, the transport of charged ionic species in a multiphase system is studied. The gas bubbles are represented using static spheres, allowing for a comparison of the simulation results with theoretical models. The presence of bubbles in the system obstructs the transport of ions and, as a result, affects the effective electrical conductivity of the electrolyte. Two descriptions for ion transport are used: the first based on the charge conservation equation developed in chapter 2 and the other one based on the assumption of electroneutrality. Note that each model applies slightly different boundary conditions at the gas-liquid interface. To apply the boundary conditions at the interface, a (coupled) Immersed Boundary Method (IBM) is used. To determine the effect of bubbles on the conductivity, a system consisting of two electrodes opposite to each other and periodic boundaries in the other two coordinate directions is considered. At the electrodes, a current density is applied, leading to a current in the system. Due to the bubbles in the system, the ions need to move around them. This results in changes in the current density distribution in the system. By directly comparing the current densities generated using both models, it is found that the results generated by the method applying the electroneutrality assumption significantly differ from the charge conservation equation, most notably for higher gas fractions. These differences occur especially near bubble clusters. To determine the reason for this difference, the results generated for a single bubble using both models are compared. These results show that in the case of a single bubble, the results differ by approximately 1% in terms of the applied current density. This effect becomes more pronounced when there are multiple bubbles in close proximity, which explains the mismatch in the results. Finally, the effective conductivity as a function of the gas fraction generated using both models is compared to theoretical models. There is a good match for the model using the charge conservation equation, with either a slight underprediction or a slight overprediction of the effective conductivity depending on the exact method used for determining the effective conductivity. For the model assuming electroneutrality, an underprediction of the effective conductivity is obtained, especially for higher gas fractions, showing almost no conductivity at a gas fraction of 40%.

Chapter 4 concerns the simulation of small bubbles in multiphase flows. Using a conventional one-fluid approach, spurious currents are formed at the interface, which scale with the magnitude of the surface tension. Due to the high curvature associated with small bubbles, the spurious currents will be in the same order of magnitude or even higher than the bubble rise velocity. Therefore, this chapter focuses on a sharp interface approach using a Front Tracking (FT) method to describe the gas-liquid interface. Two different approaches were considered, namely an approach based on IBM and one based on a Ghost Fluid Method (GFM). Both approaches aim to include an interfacial pressure jump at the interface, but differ regarding the exact inclusion of this jump. Based on the simulations for different bubbles sizes with an analytical curvature value based on a perfect sphere, it can be concluded that the spurious currents are negligible as the capillary numbers are close to machine precision and there are negligible errors in the Laplace pressure. Subsequently, a parameter study is performed adding artificial noise to the analytical curvature. This reveals that both the spurious currents and the error in the Laplace pressure increase with an increase in the noise, showing the importance of a proper and consistent local curvature calculation method. In addition, the results reveal that while the results generated using GFM initially showed higher spurious currents than the results generated with IBM, the transient behaviour of GFM is more stable. Based on these conclusions, the curvature calculation algorithm is tested for a representative interface mesh. Tests are performed to increase both the accuracy and consistency of the curvature calculation algorithm. Based on the results, a polynomial fit is chosen to determine the curvature, using a subset of 48 neighbours chosen from a set consisting of minimally 56 neighbours. To increase the consistency of the local values of the curvature, a smoothing is performed that includes 5 rings. Using the results obtained here, static bubble simulations are performed using a computed curvature. The results obtained for GFM and IBM are compared to a traditional one-fluid approach, generally showing spurious currents with a similar order of magnitude. However, for the small radii, the results obtained using IBM are one order of magnitude lower. The error in the Laplace pressure is lower for the one-fluid approach, which can be explained on basis of the smoothing that occurs in the one-fluid approach due to the mass weighing of the body force to the background Eulerian grid. It is concluded that spurious currents can be reduced to machine precision errors when the curvature calculation is exact. However, the currently available methods for calculating the curvature are not able to provide a locally accurate result for the curvature on the meshes used in FT approaches.

Chapter 5 introduces a formulation for a sharp interface approach based on GFM that includes a phase transition to model a rising hydrogen bubble next to an electrode at which hydrogen is produced. The gas-liquid interface is directly tracked by a triangulated surface, which is updated on basis of a combination of the local gas velocity and the velocity due to phase change. The movement of the ions and hydrogen in the system are modelled using species transport equations, where the boundary conditions at the gas-liquid interface are enforced using IBM. At the electrode, the Butler-Volmer equation is included to model the hydrogen producing half-reaction under alkaline conditions. To verify the model, the implementation of the Butler-Volmer equation is first tested, which shows negligible mass losses. Second, the implementation of the velocity probing is verified with a growing bubble with a constant mass flux. In these simulations, the inward pointing probes produce superior results, which is attributed to the relatively low influence of the gas phase velocity for determining the interface velocity. Finally, a static growing hydrogen bubble is considered, in which the bubble grows as a result of a hydrogen flux. The small mass losses observed in this test case can be attributed to the unphysical hydrogen concentration profile at the beginning of the simulation and the explicit nature of the concentration gradient calculation. In addition, this test case shows that the initially dissolved hydrogen is transferred to the gas phase, which is in line with expectations. Unfortunately, the final set-up with a hydrogen bubble rising next to the electrode shows divergence, and thus no results can be obtained. This is most likely caused by the jump in the velocity due to the mass flux.