# Dr. Paul Weber

Work

## Modeling and Numerical Simulation of Multi-Component Two-Phase Fluid Systems with Ionic Species

### Research Motivation

During the last decades the numerical treatment of complex processes greatly evolved. Since analytical solutions of mathematical models describing technical systems with sufficient accuracy are not available in most cases, numerical simulations become ever more important. As a generalization of classical fluid dynamics, the analysis of two-phase fluid systems utilizes this knowledge, but asks additionally for understanding and research of the processes taking place at and near the interface between both phases. Furthermore, most technical processes involve fluids comprised of multiple components, which induce additional transport mechanisms. Examples for applications are gas scrubbing or gas/liquid reactions in general, which are very common in process engineering or pharmacy. In polar solvents, e.g. aqueous solutions, also ions can be present due to dissolution of molecules. This again increases the complexity due to electric effects, such as coupling of the hydrodynamics to the electric field and transport due to electromigration.

### Scientific Project

Within the context of sharp interface modeling and Direct Numerical Simulation (DNS) we consider a single bubble in a surrounding medium on the basis of continuum mechanics. This requires a thorough modeling and numerical treatment of the underlying physics, which can be naturally divided into three aspects or levels of complexity. In this framework we consider first a class zero model, second a class one model, and third a class one model coupled to the electric field. The class zero model consists of the Navier-Stokes equations with proper jump conditions at the interface, as well as an additional species equation for the transport of a single component. Due to the diluteness assumption the species equation decouples and can be treated separately. Using a class one model allows for multiple and non-dilute components, which interact by multicomponent diffusion and possibly by chemical reactions. In contrast to the class zero model a standard Navier-Stokes solver cannot be used. Extending model and software to be capable of ions, i.e. handling the third level of complexity, requires proper treatment of the generated electric field, it's coupling to hydrodynamics and the corresponding jump conditions. Depending on the system the influence of electromigration and presence of electric double layer at the interface may become significant.

For multiphase flows several numerical methods and libraries are available. As boundary and jump conditions play an important role in this context, the interface tracking method is considered to be a suitable choice. We develop the numerical software capable of treating the mathematical models appropriately based on the advanced C++ library OpenFOAM-Extend. This library provides a solver package named interTrackFoam [1], which is a Navier-Stokes solver for pure systems, forming the starting points of our software development.

### Preliminary Results

The first level of complexity, simulating mass transfer across the fluidic interface, is treated by extending the top-level solver interTrackFoam to be capable of simulating the transport of a dilute species. Stability and convergence issues could be resolved by a detailed analysis of the employed Dirichlet-Neumann coupling at the interface, which yielded also method for significant speed up of the iteration.

The second level of complexity, modeling and simulating multi-component mixtures, involves cross-effects and the adequate treatment thereof. A thermodynamically consistent approach to model diffusion in non-dilute mixtures is by the Maxwell-Stefan approach, which has, however, the drawback of rendering the transport equations non-linear. The linearized theory for the Maxwell-Stefan equations is a severe simplification, but many studies support a high prediction quality nevertheless. A thorough study of the linearized theory was performed, revealing that it is not unconditionally applicable. Indeed, in most of the relevant situations it fails and yields unphysical results. Aiming at high quality simulations for multi-component mixtures, one has therefore to resort to the full (non-linear) Maxwell-Stefan equations.

Currently the main focus is on the third level of complexity, the numerical treatment of ionic species and their influence on mass transfer intensities due to electric effects.

### Acknowledgements

The work of Paul Weber is supported by the “Excellence Initiative” of the German Federal and State Governments and the Graduate School of Computational Engineering at Technische Universität Darmstadt.

### Project References

[1] Z. Tukovic and H. Jasak. A moving mesh Finite Volume Interface Tracking method for surface tension dominated interfacial fluid flow. Computers & Fluids, 55:70-84, 2012.

### Publications

- Weber, PS, and Bothe, D. Applicability of the Linearized Theory of the Maxwell–Stefan Equations. Research article in AIChE Journal. Accepted, 2016.
- Weber, PS, Marschall, H, and Bothe, D. Highly Accurate Two-Phase Species Transfer Based on ALE Interface Tracking. Research Article. Submitted, 2015.