"Current mathematical and numerical challenges in thermal separation science", Prof. Kai Langenbach, Thermische Verfahrenstechnik, Universität Innsbruck

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ABSTRACT: In thermal separation science, the computation of thermophysical properties of fluids and their interfaces is still one of the biggest challenges. On the one hand side, this is due to the complexity of many-particle systems, which complicate the construction of statistics-based models for these properties. On the other hand, derived models may become ill-conditioned in certain physical situations, and the numerical solution of which is typically computationally expensive.
While, in principle, the physics of multi-particle systems are understood, the systems in thermal separation science are simply too large to be modeled in a bottom-up approach. Therefore, statistical physics is used to derive models that do not require explicit particle trajectories, but rather rely on particular free statistical measures such as density and temperature, thereby introducing dependent variables (e.g. pressure and entropy). The resulting models can be formulated as optimization problems of so-called potentials. In the case of phase equilibria, which are extensively used in thermal separation science to purify substances, these potentials must have at least a double fold structure. The hypersurface of the free energy is the most important mathematical object in these applications and the basis for virtually all other possible simulations since it links the introduced free statistical variables to their dependent counterparts.
Using free energy potentials, it is possible to arrive at equations for simulating interfaces between two or more phases, fluid dynamics, mass transport, and many more. Even though equations for these have been developed independently in the past, introducing free energy models to them will result in a much stronger generality, at the expense of a tighter binding between, e.g., advective and diffusive flux, or at the expense of larger numerical effort.
Some current challenges in this area are highlighted after a brief introduction to free energy surfaces. Mathematically, nonlinear optimization for scalar variables and functions under constraints is the basic challenge. Depending on the solution method, this can, in a first step, lead to stiff systems of strongly coupled nonlinear partial differential equations. Moreover, ideally, these equations have to bridge scales from the molecular scale up to the macroscopic scale in space and time, where fractal structures (e.g. turbulence) or sharp changes of variables (e.g. shock waves, creation of new phases) may occur depending on the choice of initial and boundary conditions and on thermophysical properties. This poses a huge challenge for discretization. The state-of-the-art methods are discussed and their results are illustrated. The main focus of this talk is on the open challenges regarding modeling and numerical approximation.