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Modeling, Analysis and Computation of heterogeneous catalysis in microchannels
Název práce v češtině: Modelování, analýza a počítačové simulace heterogenní katalýzy v mikroreaktorech Modeling, Analysis and Computation of heterogeneous catalysis in microchannels heterogenní katalýza, spárovaný reakce-difuze a konvekce-difuze systém, teorie nelineárních semigrup, bio-diesel mikroreaktor heterogeneous catalysis, coupled reaction-diffusion/convection-diffusion system, nonlinear semigroup theory, bio-diesel microreactors 2011/2012 diplomová práce angličtina Matematický ústav UK (32-MUUK) prof. RNDr. Josef Málek, CSc., DSc. skrytý - zadáno a potvrzeno stud. odd. 05.02.2012 14.02.2012 20.02.2012 18.09.2013 00:00 01.08.2013 02.08.2013 18.09.2013 doc. RNDr. Tomáš Bárta, Ph.D. Prof. Dr. Dieter Bothe
 Zásady pro vypracování 1. Formulate the mathematical model (see Ref. [1]) and search for and survey related publications. 2. Using the framework of interating continua (theory of mixtures), develop a thermodynamically consistant modelsthat include not only chemica but also mechanical and thermal effects (cf. Heida/Malek/Rajagopal 1 a 2) 3. Introduce concept of solution (strong, weak) and investigate its mathematical properties (for example, prove existence and uniqueness of strong solutions using standard methods; extend the L^2 technique (see papers Pierre and Bothe/Pierre) to this system and use bootstrap to obtain global existence, etc.) 4. Formulate a parabolized form of the system following the ideas in the Ref. [4] (Bothe/Lojewski/Warnecke) 5. Implement a standard numerical scheme to solve this parabolized form 6. Perform numerical experiments to see how the system behaves in the limit of fast reaction/sorption 7. If time allows, prove a rigorous result about the fast limit, following the approach of Ref. [2] (Bothe/Pierre)
 Seznam odborné literatury [1] D. Bothe: Periodic solutions of a nonlinear evolution problem from heterogeneous catalysis, Differential and Integral Equations, Vol. 14, pp. 641-670 (2001). [2] D. Bothe, M. Pierre: Quasi-steady-state approximation for a reaction–diffusion system with fast intermediate, J. Math. Anal. Appl., Vol. 368, pp. 120-132 (2010). [3] D. Bothe, M. Pierre: The instanteneous limit for reaction-diffusion systems with a fast irreversible reaction, Discrete Cont. Dynamical Systems Se. S, Vol. 5, pp. 49-59 (2012). [4] D. Bothe, A. Lojewski, H.-J. Warnecke: Fully resolved numerical simulation of reactive mixing in a T-shaped micromixer using parabolized species equations, Chemical Engineering Science, Vol. 66, pp.6424–6440 (2011). [5] M. Heida, J. Málek, K. R. Rajagopal: On the development and generalizations of Cahn–Hilliard equations within a thermodynamic framework, Z. Angew. Math. Physik, Vol. 63, pp. 145-169 (2012) [6] M. Heida, J. Málek, K. R. Rajagopal: On the development and generalizations of Allen-Cahn and Stefan equations within a thermodynamic framework, Z. Angew. Math. Physik (in print) (2012) and other recommended literature
 Předběžná náplň práce One important recent means for intensification of chemical processes employs microreactors. This is due to the fact that a smaller length scale leads to an acceleration of transport processes, since fluxes scale with the area and the area per volume ratio increases with decreasing spatial dimensions. Chemical reactions often involve catalytic substances and the latter can be immobilized as a coating of the microreactor wall. Here, again, the higher area per volume ratio leads to better performance. But, despite the small sub-millimeter dimensions, diffusion is still slow especially in liquids where Schmidt numbers are in the range of 1000. Therefore, a better understanding of the strong interplay between transport processes, sorption as exchange mechanism between bulk and boundary as well as chemical transformations is strongly required. This can be achieved based on mathematical modeling and numerical simulation. The present master thesis project aims at the development of a thermodynamically consistent mathematical model based on continuum mechanics, its mathematical analysis concerning basic questions of existence, uniqueness and global existence, and the numerical solution for a given simple rectangular channel geometry and strictly laminar flow conditions. A further analytical issue is the rigorous limit for fast sorption kinetics and/or fast boundary reaction kinetics.
 Předběžná náplň práce v anglickém jazyce One important recent means for intensification of chemical processes employs microreactors. This is due to the fact that a smaller length scale leads to an acceleration of transport processes, since fluxes scale with the area and the area per volume ratio increases with decreasing spatial dimensions. Chemical reactions often involve catalytic substances and the latter can be immobilized as a coating of the microreactor wall. Here, again, the higher area per volume ratio leads to better performance. But, despite the small sub-millimeter dimensions, diffusion is still slow especially in liquids where Schmidt numbers are in the range of 1000. Therefore, a better understanding of the strong interplay between transport processes, sorption as exchange mechanism between bulk and boundary as well as chemical transformations is strongly required. This can be achieved based on mathematical modeling and numerical simulation. The present master thesis project aims at the development of a thermodynamically consistent mathematical model based on continuum mechanics, its mathematical analysis concerning basic questions of existence, uniqueness and global existence, and the numerical solution for a given simple rectangular channel geometry and strictly laminar flow conditions. A further analytical issue is the rigorous limit for fast sorption kinetics and/or fast boundary reaction kinetics.