Lukassen, Axel Ariaan (2018)
Simulation of chemical systems with fast chemistry.
Technische Universität Darmstadt
Dissertation, Erstveröffentlichung
Kurzbeschreibung (Abstract)
This PhD thesis deals with the numerical simulation of chemical reaction systems. Chemical reaction systems occur in many different areas of our lives. In some areas, such as biology, scientists try to analyze the occurring chemical reactions for better understanding of the underlying mechanisms. In other areas, such as the chemical industry, the focus is on optimization of reactor design in order to increase the productivity. All cases have in common that the temporal development of the chemical reaction systems is described by differential equations. The derivation of the corresponding differential equations is given in chapter 2. The resulting differential equations are generally large nonlinear systems. Therefore, analytical solutions are usually not available, and an approximation of the solution is calculated by numerical integration methods on a computer. The size and the stiffness of the corresponding differential equations often lead to an unfavorably long computing time or an exceeding of the available memory space. Hence, many scientists have developed reduction mechanisms for chemical reaction systems. These reduction mechanisms exploit that there are very slow as well as very fast processes in chemical reaction systems. Depending on the discretization, the timescales of the fast processes are much smaller than the used time step of the numerical integration method. Thus, the fast processes are approximated by their partial equilibrium. The relaxation assumption results in algebraic equations, which can be used to reduce the dimension of the differential equation. Furthermore, the reduced differential equations are often not or less stiff. A description of the partial equilibrium assumption of single reactions or the quasi-stationary state assumption of individual chemical species is given in chapter 3. In addition, frequently used automatic reduction mechanisms of different authors are summarized in the corresponding chapter. Despite the great popularity of reduction mechanisms, the reduction mechanisms listed in chapter 3 are not applicable for all chemical reaction systems. Most of them are based on the existence of a low-dimensional manifold which is approached by the state of the reaction system in a fraction of the considered time step. However, such a low-dimensional manifold does not always exist. Furthermore, the choice of fast processes depends on time and space. Therefore, the dimension of the reduced differential equation can vary. This leads to additional problems. In order to reduce the stiffness of the differential equation, a new approach is introduced in chapter 4. In opposition to many other methods, this new approach is also applicable if the relaxation of the fast processes does not restrict the state of the system onto a low-dimensional manifold, and if the number of fast processes changes in time and space. Reduction of stiffness can decrease the necessary computing time of the numerical solver. The computing time is particularly important in parameter identification. The rate of each chemical reaction is described by at least one parameter. In order to determine unknown parameters, the solution of the differential equation has to be computed for many different parameter sets. Hence, for parameter estimation it is desirable to reduce the computing time of each solution. Since the algebraic equations from chapter 3 also depend on the unknown parameters, it is not possible to replace the differential equation for all possible parameter sets by one reduced differential equation. However, the additional information from the partial equilibrium assumption or the quasi-stationary state assumption can be used to calculate some unknown parameters as a function of all other parameters. This reduces the dimension of the parameter space, and decreases the computing time of parameter identification. The procedure is described in chapter 5. For the sake of simplicity, ordinary differential equations are considered in the chapters 3 to 5. However, many chemical reaction systems that are not homogeneous in space are described by partial differential equations. In the case of splitting methods, a sequence of partial differential equations for the transport and ordinary differential equations for the chemical reaction steps is solved. Thus, if splitting methods are used, a homogeneous chemical reactor is considered for each spatial node in the chemical reaction step. Therefore, the results of the chapters 3 to 5 are applicable. However, an additional splitting error is introduced. Popular splitting methods are the first order Lie-Trotter splitting and the second order Strang splitting. Stiffness of the considered differential equation results in order reduction for the Strang splitting scheme. Therefore, Strang splitting is only a first order scheme for stiff chemical reaction systems. However, the extrapolated Lie-Trotter splitting is a second order scheme for stiff chemical reaction systems. The analysis of the extrapolated Lie-Trotter splitting for chemical reaction systems is presented in chapter 6.
| Typ des Eintrags: | Dissertation | ||||
|---|---|---|---|---|---|
| Erschienen: | 2018 | ||||
| Autor(en): | Lukassen, Axel Ariaan | ||||
| Art des Eintrags: | Erstveröffentlichung | ||||
| Titel: | Simulation of chemical systems with fast chemistry | ||||
| Sprache: | Englisch | ||||
| Referenten: | Kiehl, Prof. Dr. Martin ; Lang, Prof. Dr. Jens | ||||
| Publikationsjahr: | 7 August 2018 | ||||
| Ort: | Darmstadt | ||||
| Datum der mündlichen Prüfung: | 23 Oktober 2018 | ||||
| URL / URN: | https://tuprints.ulb.tu-darmstadt.de/8131 | ||||
| Kurzbeschreibung (Abstract): | This PhD thesis deals with the numerical simulation of chemical reaction systems. Chemical reaction systems occur in many different areas of our lives. In some areas, such as biology, scientists try to analyze the occurring chemical reactions for better understanding of the underlying mechanisms. In other areas, such as the chemical industry, the focus is on optimization of reactor design in order to increase the productivity. All cases have in common that the temporal development of the chemical reaction systems is described by differential equations. The derivation of the corresponding differential equations is given in chapter 2. The resulting differential equations are generally large nonlinear systems. Therefore, analytical solutions are usually not available, and an approximation of the solution is calculated by numerical integration methods on a computer. The size and the stiffness of the corresponding differential equations often lead to an unfavorably long computing time or an exceeding of the available memory space. Hence, many scientists have developed reduction mechanisms for chemical reaction systems. These reduction mechanisms exploit that there are very slow as well as very fast processes in chemical reaction systems. Depending on the discretization, the timescales of the fast processes are much smaller than the used time step of the numerical integration method. Thus, the fast processes are approximated by their partial equilibrium. The relaxation assumption results in algebraic equations, which can be used to reduce the dimension of the differential equation. Furthermore, the reduced differential equations are often not or less stiff. A description of the partial equilibrium assumption of single reactions or the quasi-stationary state assumption of individual chemical species is given in chapter 3. In addition, frequently used automatic reduction mechanisms of different authors are summarized in the corresponding chapter. Despite the great popularity of reduction mechanisms, the reduction mechanisms listed in chapter 3 are not applicable for all chemical reaction systems. Most of them are based on the existence of a low-dimensional manifold which is approached by the state of the reaction system in a fraction of the considered time step. However, such a low-dimensional manifold does not always exist. Furthermore, the choice of fast processes depends on time and space. Therefore, the dimension of the reduced differential equation can vary. This leads to additional problems. In order to reduce the stiffness of the differential equation, a new approach is introduced in chapter 4. In opposition to many other methods, this new approach is also applicable if the relaxation of the fast processes does not restrict the state of the system onto a low-dimensional manifold, and if the number of fast processes changes in time and space. Reduction of stiffness can decrease the necessary computing time of the numerical solver. The computing time is particularly important in parameter identification. The rate of each chemical reaction is described by at least one parameter. In order to determine unknown parameters, the solution of the differential equation has to be computed for many different parameter sets. Hence, for parameter estimation it is desirable to reduce the computing time of each solution. Since the algebraic equations from chapter 3 also depend on the unknown parameters, it is not possible to replace the differential equation for all possible parameter sets by one reduced differential equation. However, the additional information from the partial equilibrium assumption or the quasi-stationary state assumption can be used to calculate some unknown parameters as a function of all other parameters. This reduces the dimension of the parameter space, and decreases the computing time of parameter identification. The procedure is described in chapter 5. For the sake of simplicity, ordinary differential equations are considered in the chapters 3 to 5. However, many chemical reaction systems that are not homogeneous in space are described by partial differential equations. In the case of splitting methods, a sequence of partial differential equations for the transport and ordinary differential equations for the chemical reaction steps is solved. Thus, if splitting methods are used, a homogeneous chemical reactor is considered for each spatial node in the chemical reaction step. Therefore, the results of the chapters 3 to 5 are applicable. However, an additional splitting error is introduced. Popular splitting methods are the first order Lie-Trotter splitting and the second order Strang splitting. Stiffness of the considered differential equation results in order reduction for the Strang splitting scheme. Therefore, Strang splitting is only a first order scheme for stiff chemical reaction systems. However, the extrapolated Lie-Trotter splitting is a second order scheme for stiff chemical reaction systems. The analysis of the extrapolated Lie-Trotter splitting for chemical reaction systems is presented in chapter 6. |
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| URN: | urn:nbn:de:tuda-tuprints-81316 | ||||
| Sachgruppe der Dewey Dezimalklassifikatin (DDC): | 500 Naturwissenschaften und Mathematik > 510 Mathematik | ||||
| Fachbereich(e)/-gebiet(e): | 04 Fachbereich Mathematik 04 Fachbereich Mathematik > Numerik und wissenschaftliches Rechnen |
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| Hinterlegungsdatum: | 16 Dez 2018 20:55 | ||||
| Letzte Änderung: | 16 Dez 2018 20:55 | ||||
| PPN: | |||||
| Referenten: | Kiehl, Prof. Dr. Martin ; Lang, Prof. Dr. Jens | ||||
| Datum der mündlichen Prüfung / Verteidigung / mdl. Prüfung: | 23 Oktober 2018 | ||||
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| Suche nach Titel in: | TUfind oder in Google | ||||
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