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Papers by Wolfram Heineken
Journal of Computational and Applied Mathematics - J COMPUT APPL MATH, 2008
A finite volume method with grid adaption is applied to two hyperbolic problems: the ultra-relati... more A finite volume method with grid adaption is applied to two hyperbolic problems: the ultra-relativistic Euler equations, and a scalar conservation law. Both problems are considered in two space dimensions and share the common feature of moving shock waves. In contrast to the classical Euler equations, the derivation of appropriate initial conditions for the ultra-relativistic Euler equations is a non-trivial problem that is solved using one-dimensional shock conditions and the Lorentz invariance of the system. The discretization of both problems is based on a finite volume method of second order in both space and time on a triangular grid. We introduce a variant of the min-mod limiter that avoids unphysical states for the Euler system. The grid is adapted during the integration process. The frequency of grid adaption is controlled automatically in order to guarantee a fine resolution of the moving shock fronts. We introduce the concept of “width refinement” which enlarges the width ...
Applied Numerical Mathematics, Jul 1, 2006
We consider the numerical solution of reaction-diffusion systems using linear finite elements on ... more We consider the numerical solution of reaction-diffusion systems using linear finite elements on a space grid changing in time. For the integration with respect to the time variable a W-method with several variants of implicit/explicit partitioning is used. For grid adaption an algorithm featuring a flexible refinement and coarsening control is proposed. The partitioned W-methods keep the stability of implicit schemes but reduce the size of the linear systems to be solved. We combine local partitioning with partitioning between the diffusion and reaction terms, leading to a large variety of methods. The efficiency of several partitioning methods is compared in numerical tests. The calculations show an increase of efficiency if partitioned schemes are used instead of a fully implicit W-method. We include a numerical comparison of three linear solvers. Optimal truncation of the iteration process is discussed.
Chemical Engineering Science, Sep 1, 2007
We investigate the stability of barium sulphate precipitation in an ideally mixed reactor under s... more We investigate the stability of barium sulphate precipitation in an ideally mixed reactor under stoichiometric conditions. The process is modelled by a population balance equation describing the crystal size distribution coupled with a system of integro-differential equations for the substances in solution. A reduction to a closed moment model is used to analyse the stability of equilibrium solutions. In a certain range of parameters an unstable equilibrium is detected by a numerical calculation of the eigenvalues of the moment system. In this case sustained oscillations of the crystal size distribution are observed in numerical simulations. The numerical results are compared to data obtained from chemical experiments.
Computer Aided Chemical Engineering, 2008
ABSTRACT Crystallization models that take into account direction-dependent growth rates give rise... more ABSTRACT Crystallization models that take into account direction-dependent growth rates give rise to multi-dimensional population balances that require a high computational cost. We propose a model reduction technique based on the quadrature method of moments (QMOM) that simplifies a two-dimensional population balance to a one-dimensional advection system. Our method returns the crystal volume distribution and other volume dependent moments of the crystal size distribution, in contrast to many other QMOM based reduction methods that lose all the volume dependent information. The method is applied to the direction-dependent growth of barium sulphate crystals, showing a close agreement with the solution of the full two-dimensional population balance.
We consider numerical methods for reaction-diffusion problems in the plane as well as on curved s... more We consider numerical methods for reaction-diffusion problems in the plane as well as on curved surfaces. The spatial discretization is carried out with linear finite elements on unstructured triangular grids. Several algorithms for an adaptive grid refinement are presented. A projection technique is proposed for the construction of grids on curved surfaces. The semi-discrete problems are often stiff, either due to strong grid refinement in the presence of diffusion or due to steep gradients of the reaction function. Since in many cases the problem is only locally stiff local partitioning methods might increase the efficiency of numerical solving. We present several partitioning methods based on a W-method. A comparison of numerous partitioning strategies is carried out on three test problems. The numerical methods presented are used for the simulation of excitable media on curved surfaces. These reaction-diffusion problems can have solutions with rotating spiral waves that show an ...
Biomass and Bioenergy, 2014
In this work a novel concept for the decentralized conversion of biogas to electricity is introdu... more In this work a novel concept for the decentralized conversion of biogas to electricity is introduced. It consists of five segments: gas supply, gas treatment, gas reforming, gas usage and post-combustion. The system was designed in a regional project called GREEN-FC. The project is dealing with a design study for the conversion of 1 m 3 h À1 biogas to electricity, based on equilibrium calculations for steam reforming and wateregas shift reaction in combination with CFD simulations. The simulation results revealed that the system converts methane fully and delivers a maximum yield of hydrogen with a low concentration of carbon monoxide, thus making it suitable for a high-temperature polymer eelectrolyte membrane (HT-PEM) fuel cell. The calculated electrical efficiency of the novel process is approximately 40%. Another important result of this work is the modular prototype design, because the individual components of the prototype can be replaced. For example alternative reactors that convert biogas into hydrogen and other technologies that use hydrogen can be included.
Journal of Computational and Applied Mathematics, 2008
A finite volume method with grid adaption is applied to two hyperbolic problems: the ultra-relati... more A finite volume method with grid adaption is applied to two hyperbolic problems: the ultra-relativistic Euler equations, and a scalar conservation law. Both problems are considered in two space dimensions and share the common feature of moving shock waves. In contrast to the classical Euler equations, the derivation of appropriate initial conditions for the ultra-relativistic Euler equations is a non-trivial problem that is solved using one-dimensional shock conditions and the Lorentz invariance of the system. The discretization of both problems is based on a finite volume method of second order in both space and time on a triangular grid. We introduce a variant of the min-mod limiter that avoids unphysical states for the Euler system. The grid is adapted during the integration process. The frequency of grid adaption is controlled automatically in order to guarantee a fine resolution of the moving shock fronts. We introduce the concept of "width refinement" which enlarges the width of strongly refined regions around the shock fronts; the optimal width is found by a numerical study. As a result we are able to improve efficiency by decreasing the number of adaption steps. The performance of the finite volume scheme is compared with several lower order methods.
Computers & Chemical Engineering, 2011
This work aims at the development of a dynamic model for the mathematical description of facilita... more This work aims at the development of a dynamic model for the mathematical description of facilitated transport separation processes carried out in membrane contactors where mass transport phenomena are coupled with chemical reactions. A general model that takes into account the description of all possible mass transport steps and interfacial chemical reactions is initially presented, allowing its application to a
Computers & Chemical Engineering, 2011
ABSTRACT Crystallization models with direction-dependent growth rates give rise to multi-dimensio... more ABSTRACT Crystallization models with direction-dependent growth rates give rise to multi-dimensional population balance equations (PBE) that require a high computational cost. We propose a model reduction based on the quadrature method of moments (QMOM). Using this method a two-dimensional population balance is reduced to a system of one-dimensional advection equations. Despite the dimension reduction the method keeps important volume dependent information of the crystal size distribution (CSD). It returns the crystal volume distribution as well as other volume dependent moments of the two-dimensional CSD. The method is applied to a model problem with direction-dependent growth of barium sulphate crystals, and shows good performance and convergence in these examples. We also compare it on numerical examples to another model reduction using a normal distribution ansatz approach. We can show that our method still gives satisfactory results where the other approach is not suitable.
Chemie Ingenieur Technik, 2007
Chemical Engineering Science, 2007
We investigate the stability of barium sulphate precipitation in an ideally mixed reactor under s... more We investigate the stability of barium sulphate precipitation in an ideally mixed reactor under stoichiometric conditions. The process is modelled by a population balance equation describing the crystal size distribution coupled with a system of integro-differential equations for the substances in solution. A reduction to a closed moment model is used to analyse the stability of equilibrium solutions. In a certain range of parameters an unstable equilibrium is detected by a numerical calculation of the eigenvalues of the moment system. In this case sustained oscillations of the crystal size distribution are observed in numerical simulations. The numerical results are compared to data obtained from chemical experiments.
Applied Numerical Mathematics, 2006
We consider the numerical solution of reaction-diffusion systems using linear finite elements on ... more We consider the numerical solution of reaction-diffusion systems using linear finite elements on a space grid changing in time. For the integration with respect to the time variable a W-method with several variants of implicit/explicit partitioning is used. For grid adaption an algorithm featuring a flexible refinement and coarsening control is proposed. The partitioned W-methods keep the stability of implicit schemes but reduce the size of the linear systems to be solved. We combine local partitioning with partitioning between the diffusion and reaction terms, leading to a large variety of methods. The efficiency of several partitioning methods is compared in numerical tests. The calculations show an increase of efficiency if partitioned schemes are used instead of a fully implicit W-method. We include a numerical comparison of three linear solvers. Optimal truncation of the iteration process is discussed.
Chaos, 2006
It is claimed by Rhodes, Morari, and Wiggins [Chaos 9, 108-123 (1999)] that the projection algori... more It is claimed by Rhodes, Morari, and Wiggins [Chaos 9, 108-123 (1999)] that the projection algorithm of Maas and Pope [Combust. Flame 88, 239-264 (1992)] identifies the slow invariant manifold of a system of ordinary differential equations with time-scale separation. A transformation to Fenichel normal form serves as a tool to prove this statement. Furthermore, Rhodes, Morari, and Wiggins [Chaos 9, 108-123 (1999)] conjectured that away from a slow manifold, the criterion of Maas and Pope will never be fulfilled. We present two examples that refute the assertions of Rhodes, Morari, and Wiggins. In the first example, the algorithm of Maas and Pope leads to a manifold that is not invariant but close to a slow invariant manifold. The claim of Rhodes, Morari, and Wiggins that the Maas and Pope projection algorithm is invariant under a coordinate transformation to Fenichel normal form is shown to be not correct in this case. In the second example, the projection algorithm of Maas and Pope...
Journal of Computational and Applied Mathematics - J COMPUT APPL MATH, 2008
A finite volume method with grid adaption is applied to two hyperbolic problems: the ultra-relati... more A finite volume method with grid adaption is applied to two hyperbolic problems: the ultra-relativistic Euler equations, and a scalar conservation law. Both problems are considered in two space dimensions and share the common feature of moving shock waves. In contrast to the classical Euler equations, the derivation of appropriate initial conditions for the ultra-relativistic Euler equations is a non-trivial problem that is solved using one-dimensional shock conditions and the Lorentz invariance of the system. The discretization of both problems is based on a finite volume method of second order in both space and time on a triangular grid. We introduce a variant of the min-mod limiter that avoids unphysical states for the Euler system. The grid is adapted during the integration process. The frequency of grid adaption is controlled automatically in order to guarantee a fine resolution of the moving shock fronts. We introduce the concept of “width refinement” which enlarges the width ...
Applied Numerical Mathematics, Jul 1, 2006
We consider the numerical solution of reaction-diffusion systems using linear finite elements on ... more We consider the numerical solution of reaction-diffusion systems using linear finite elements on a space grid changing in time. For the integration with respect to the time variable a W-method with several variants of implicit/explicit partitioning is used. For grid adaption an algorithm featuring a flexible refinement and coarsening control is proposed. The partitioned W-methods keep the stability of implicit schemes but reduce the size of the linear systems to be solved. We combine local partitioning with partitioning between the diffusion and reaction terms, leading to a large variety of methods. The efficiency of several partitioning methods is compared in numerical tests. The calculations show an increase of efficiency if partitioned schemes are used instead of a fully implicit W-method. We include a numerical comparison of three linear solvers. Optimal truncation of the iteration process is discussed.
Chemical Engineering Science, Sep 1, 2007
We investigate the stability of barium sulphate precipitation in an ideally mixed reactor under s... more We investigate the stability of barium sulphate precipitation in an ideally mixed reactor under stoichiometric conditions. The process is modelled by a population balance equation describing the crystal size distribution coupled with a system of integro-differential equations for the substances in solution. A reduction to a closed moment model is used to analyse the stability of equilibrium solutions. In a certain range of parameters an unstable equilibrium is detected by a numerical calculation of the eigenvalues of the moment system. In this case sustained oscillations of the crystal size distribution are observed in numerical simulations. The numerical results are compared to data obtained from chemical experiments.
Computer Aided Chemical Engineering, 2008
ABSTRACT Crystallization models that take into account direction-dependent growth rates give rise... more ABSTRACT Crystallization models that take into account direction-dependent growth rates give rise to multi-dimensional population balances that require a high computational cost. We propose a model reduction technique based on the quadrature method of moments (QMOM) that simplifies a two-dimensional population balance to a one-dimensional advection system. Our method returns the crystal volume distribution and other volume dependent moments of the crystal size distribution, in contrast to many other QMOM based reduction methods that lose all the volume dependent information. The method is applied to the direction-dependent growth of barium sulphate crystals, showing a close agreement with the solution of the full two-dimensional population balance.
We consider numerical methods for reaction-diffusion problems in the plane as well as on curved s... more We consider numerical methods for reaction-diffusion problems in the plane as well as on curved surfaces. The spatial discretization is carried out with linear finite elements on unstructured triangular grids. Several algorithms for an adaptive grid refinement are presented. A projection technique is proposed for the construction of grids on curved surfaces. The semi-discrete problems are often stiff, either due to strong grid refinement in the presence of diffusion or due to steep gradients of the reaction function. Since in many cases the problem is only locally stiff local partitioning methods might increase the efficiency of numerical solving. We present several partitioning methods based on a W-method. A comparison of numerous partitioning strategies is carried out on three test problems. The numerical methods presented are used for the simulation of excitable media on curved surfaces. These reaction-diffusion problems can have solutions with rotating spiral waves that show an ...
Biomass and Bioenergy, 2014
In this work a novel concept for the decentralized conversion of biogas to electricity is introdu... more In this work a novel concept for the decentralized conversion of biogas to electricity is introduced. It consists of five segments: gas supply, gas treatment, gas reforming, gas usage and post-combustion. The system was designed in a regional project called GREEN-FC. The project is dealing with a design study for the conversion of 1 m 3 h À1 biogas to electricity, based on equilibrium calculations for steam reforming and wateregas shift reaction in combination with CFD simulations. The simulation results revealed that the system converts methane fully and delivers a maximum yield of hydrogen with a low concentration of carbon monoxide, thus making it suitable for a high-temperature polymer eelectrolyte membrane (HT-PEM) fuel cell. The calculated electrical efficiency of the novel process is approximately 40%. Another important result of this work is the modular prototype design, because the individual components of the prototype can be replaced. For example alternative reactors that convert biogas into hydrogen and other technologies that use hydrogen can be included.
Journal of Computational and Applied Mathematics, 2008
A finite volume method with grid adaption is applied to two hyperbolic problems: the ultra-relati... more A finite volume method with grid adaption is applied to two hyperbolic problems: the ultra-relativistic Euler equations, and a scalar conservation law. Both problems are considered in two space dimensions and share the common feature of moving shock waves. In contrast to the classical Euler equations, the derivation of appropriate initial conditions for the ultra-relativistic Euler equations is a non-trivial problem that is solved using one-dimensional shock conditions and the Lorentz invariance of the system. The discretization of both problems is based on a finite volume method of second order in both space and time on a triangular grid. We introduce a variant of the min-mod limiter that avoids unphysical states for the Euler system. The grid is adapted during the integration process. The frequency of grid adaption is controlled automatically in order to guarantee a fine resolution of the moving shock fronts. We introduce the concept of "width refinement" which enlarges the width of strongly refined regions around the shock fronts; the optimal width is found by a numerical study. As a result we are able to improve efficiency by decreasing the number of adaption steps. The performance of the finite volume scheme is compared with several lower order methods.
Computers & Chemical Engineering, 2011
This work aims at the development of a dynamic model for the mathematical description of facilita... more This work aims at the development of a dynamic model for the mathematical description of facilitated transport separation processes carried out in membrane contactors where mass transport phenomena are coupled with chemical reactions. A general model that takes into account the description of all possible mass transport steps and interfacial chemical reactions is initially presented, allowing its application to a
Computers & Chemical Engineering, 2011
ABSTRACT Crystallization models with direction-dependent growth rates give rise to multi-dimensio... more ABSTRACT Crystallization models with direction-dependent growth rates give rise to multi-dimensional population balance equations (PBE) that require a high computational cost. We propose a model reduction based on the quadrature method of moments (QMOM). Using this method a two-dimensional population balance is reduced to a system of one-dimensional advection equations. Despite the dimension reduction the method keeps important volume dependent information of the crystal size distribution (CSD). It returns the crystal volume distribution as well as other volume dependent moments of the two-dimensional CSD. The method is applied to a model problem with direction-dependent growth of barium sulphate crystals, and shows good performance and convergence in these examples. We also compare it on numerical examples to another model reduction using a normal distribution ansatz approach. We can show that our method still gives satisfactory results where the other approach is not suitable.
Chemie Ingenieur Technik, 2007
Chemical Engineering Science, 2007
We investigate the stability of barium sulphate precipitation in an ideally mixed reactor under s... more We investigate the stability of barium sulphate precipitation in an ideally mixed reactor under stoichiometric conditions. The process is modelled by a population balance equation describing the crystal size distribution coupled with a system of integro-differential equations for the substances in solution. A reduction to a closed moment model is used to analyse the stability of equilibrium solutions. In a certain range of parameters an unstable equilibrium is detected by a numerical calculation of the eigenvalues of the moment system. In this case sustained oscillations of the crystal size distribution are observed in numerical simulations. The numerical results are compared to data obtained from chemical experiments.
Applied Numerical Mathematics, 2006
We consider the numerical solution of reaction-diffusion systems using linear finite elements on ... more We consider the numerical solution of reaction-diffusion systems using linear finite elements on a space grid changing in time. For the integration with respect to the time variable a W-method with several variants of implicit/explicit partitioning is used. For grid adaption an algorithm featuring a flexible refinement and coarsening control is proposed. The partitioned W-methods keep the stability of implicit schemes but reduce the size of the linear systems to be solved. We combine local partitioning with partitioning between the diffusion and reaction terms, leading to a large variety of methods. The efficiency of several partitioning methods is compared in numerical tests. The calculations show an increase of efficiency if partitioned schemes are used instead of a fully implicit W-method. We include a numerical comparison of three linear solvers. Optimal truncation of the iteration process is discussed.
Chaos, 2006
It is claimed by Rhodes, Morari, and Wiggins [Chaos 9, 108-123 (1999)] that the projection algori... more It is claimed by Rhodes, Morari, and Wiggins [Chaos 9, 108-123 (1999)] that the projection algorithm of Maas and Pope [Combust. Flame 88, 239-264 (1992)] identifies the slow invariant manifold of a system of ordinary differential equations with time-scale separation. A transformation to Fenichel normal form serves as a tool to prove this statement. Furthermore, Rhodes, Morari, and Wiggins [Chaos 9, 108-123 (1999)] conjectured that away from a slow manifold, the criterion of Maas and Pope will never be fulfilled. We present two examples that refute the assertions of Rhodes, Morari, and Wiggins. In the first example, the algorithm of Maas and Pope leads to a manifold that is not invariant but close to a slow invariant manifold. The claim of Rhodes, Morari, and Wiggins that the Maas and Pope projection algorithm is invariant under a coordinate transformation to Fenichel normal form is shown to be not correct in this case. In the second example, the projection algorithm of Maas and Pope...