Sixth-order compact finite difference method for singularly perturbed 1D reaction diffusion problems

AbstractIn this paper, the sixth-order compact finite difference method is presented for solving singularly perturbed 1D reaction–diffusion problems. The derivative of the given differential equation is replaced by finite difference approximations. Then, the given difference equation is transformed to linear systems of algebraic equations in the form of a three-term recurrence relation, which can easily be solved using a discrete invariant imbedding algorithm. To validate the applicability of the proposed method, some model examples have been solved for different values of the perturbation parameter and mesh size. Both the theoretical error bounds and the numerical rate of convergence have been established for the method. The numerical results presented in the tables and graphs show that the present method approximates the exact solution very well

Kinetic Modeling using the Single-Event Methodology: Application to the Isomerization of Light Paraffins Modélisation cinétique basée sur la méthodologie des événements constitutifs : application à l’isomérisation des paraffines légères

The construction of kinetic models is an important step in the development of refining processes. Indeed, these models can be used to predict the performances of the process, to optimize the operation of a unit and to optimally design a reactor and the associated process (choice of the reactor configuration, determination of optimal operating conditions, use of recycles, etc.). The current demand for models is particularly focused towards the development of detailed predictive models that are able to account for significant changes in process operation. Hence, they need to allow for the extrapolation to considerably different operating conditions or to very diverse feedstocks. Kinetic modeling based on the single-event theory meets these requirements, since it allows detailed prediction of the composition, and hence the yield structure from the reactors. The single-event theory was initially developed for the radical chemistry occurring during thermal cracking [Clymans et Froment (1984) Comput. Chem. Eng. 8, 2, 137-142; Hillewaert et al. (1988) AIChE J. 34, 1, 17-24; Willems et Froment (1988a) Ind. Eng. Chem. Res. 27, 11, 1959-1966; Willems et Froment (1988b) Ind. Eng. Chem. Res. 27, 11, 1966-1971], later extended to acid catalysis [Baltanas et Froment (1985) Comput. Chem. Eng. 9, 1, 71-81; Baltanas et al. (1989) Ind. Eng. Chem. Res. 28, 899-910; Vynckier et Froment (1991) Modeling of the kinetics of complex processes upon elementary steps, in Kinetic and thermodynamic lumping of multicomponent mixtures, Astarita G., Sandler S.I. (eds), Elsevier Science Publishers BV, Amsterdam, pp. 131-161], and recently adapted to metal catalyzed processes [Lozano-Blanco et al. (2006) Oil Gas Sci. Technol. 61, 4, 489-496; Lozano- Blanco et al. (2008) Ind. Eng. Chem. Res. 47, 16, 5879-5891]. This methodology developed in the “Laboratorium voor Petrochemische Techniek” at Ghent University consists in constructing a reaction network which, although exhaustive, is described by a limited number of independent kinetic parameters. The behavior of complex feedstocks can therefore be predicted based on studies conducted on model molecules. This method is applied at IFP Energies nouvelles to several refining processes. The purpose of this article is to describe this methodology in a detailed manner and to present its application to the isomerization of C5-C6 paraffins. L’établissement de modèles cinétiques est une étape importante du développement de procédés de raffinage. En effet, ces modèles peuvent être utilisés pour prédire les performances du procédé, pour optimiser l’opération de l’unité et pour concevoir de façon optimale le réacteur et le procédé associé (choix du réacteur, détermination des conditions de marche optimale, utilisation de recyclages, etc.). La demande actuelle de modèles se focalise particulièrement sur la mise au point de modèles détaillés et prédictifs, qui sont capables de prendre en compte des variations importantes dans la conduite du procédé. De ce fait, ils doivent être extrapolables à des conditions opératoires très différentes ou à des charges très diverses. La modélisation cinétique par événements constitutifs satisfait ces critères car elle permet d’obtenir une prédiction détaillée des effluents des réacteurs. La théorie des événements constitutifs a été initialement développée pour la chimie radicalaire [Clymans et Froment (1984) Comput. Chem. Eng. 8, 2, 137-142; Hillewaert et al. (1988) AIChE J. 34, 1, 17-24; Willems et Froment (1988a) Ind. Eng. Chem. Res. 27, 11, 1959-1966; Willems et Froment (1988b) Ind. Eng. Chem. Res. 27, 11, 1966-1971] et a été étendue ultérieurement à la catalyse acide [Baltanas et Froment (1985) Comput. Chem. Eng. 9, 1, 71-81; Baltanas et al. (1989) Ind. Eng. Chem. Res. 28, 899-910; Vynckier et Froment (1991) Modeling of the kinetics of complex processes upon elementary steps, in Kinetic and thermodynamic lumping of multicomponent mixtures, Astarita G., Sandler S.I. (eds), Elsevier Science Publishers BV, Amsterdam, pp. 131-161], ainsi qu’à la catalyse métallique [Lozano-Blanco et al. (2006) Oil Gas Sci. Technol. 61, 4, 489-496; Lozano- Blanco et al. (2008) Ind. Eng. Chem. Res. 47, 16, 5879-5891]. Cette méthodologie développée au “Laboratorium voor Petrochemische Techniek” à l’Université de Gand consiste à construire un réseau réactionnel exhaustif mais décrit par un nombre limité de paramètres cinétiques indépendants. Le comportement de charges complexes peut alors être prédit sur la base d’études effectuées sur des molécules modèles. Cette méthodologie est appliquée par IFP Energies nouvelles à plusieurs procédés de raffinage. L’objet de cet article est de décrire cette méthodologie de façon détaillée et de présenter son application à l’isomérisation des paraffines C5-C6

Kinetic Modeling using the Single-Event Methodology: Application to the Isomerization of Light Paraffins

The construction of kinetic models is an important step in the development of refining processes. Indeed, these models can be used to predict the performances of the process, to optimize the operation of a unit and to optimally design a reactor and the associated process (choice of the reactor configuration, determination of optimal operating conditions, use of recycles, etc.). The current demand for models is particularly focused towards the development of detailed predictive models that are able to account for significant changes in process operation. Hence, they need to allow for the extrapolation to considerably different operating conditions or to very diverse feedstocks. Kinetic modeling based on the single-event theory meets these requirements, since it allows detailed prediction of the composition, and hence the yield structure from the reactors. The single-event theory was initially developed for the radical chemistry occurring during thermal cracking [Clymans et Froment (1984) Comput. Chem. Eng. 8, 2, 137-142; Hillewaert et al. (1988) AIChE J. 34, 1, 17-24; Willems et Froment (1988a) Ind. Eng. Chem. Res. 27, 11, 1959-1966; Willems et Froment (1988b) Ind. Eng. Chem. Res. 27, 11, 1966-1971], later extended to acid catalysis [Baltanas et Froment (1985) Comput. Chem. Eng. 9, 1, 71-81; Baltanas et al. (1989) Ind. Eng. Chem. Res. 28, 899-910; Vynckier et Froment (1991) Modeling of the kinetics of complex processes upon elementary steps, in Kinetic and thermodynamic lumping of multicomponent mixtures, Astarita G., Sandler S.I. (eds), Elsevier Science Publishers BV, Amsterdam, pp. 131-161], and recently adapted to metal catalyzed processes [Lozano-Blanco et al. (2006) Oil Gas Sci. Technol. 61, 4, 489-496; Lozano- Blanco et al. (2008) Ind. Eng. Chem. Res. 47, 16, 5879-5891]. This methodology developed in the “Laboratorium voor Petrochemische Techniek” at Ghent University consists in constructing a reaction network which, although exhaustive, is described by a limited number of independent kinetic parameters. The behavior of complex feedstocks can therefore be predicted based on studies conducted on model molecules. This method is applied at IFP Energies nouvelles to several refining processes. The purpose of this article is to describe this methodology in a detailed manner and to present its application to the isomerization of C5-C6 paraffins

Single-event microkinetic model for Fischer-Tropsch synthesis on iron-based catalysts

A single-event microkinetic (SEMK) model was developed for Fischer-Tropsch synthesis and applied to experimental data obtained on an iron-based catalyst in a temperature range from 523 to 623 K, total pressures from 0.6 to 2.1 MPa, and H-2/CO inlet ratios from 2 to 6 mol/mol. The use of the single-event concept allowed a significant reduction of the number of adjustable parameters. The single-event pre-exponential factors were calculated based on statistical thermodynamics. The reaction enthalpies, as well as initial guesses for the activation energies, were obtained through the unity bond index-quadratic exponential potential (UBI-QEP) method. Ten activation energies of the kinetically relevant reaction families and four atomic chemisorption enthalpies remained as adjustable parameters, the latter corresponding to so-called catalyst descriptors. The SEMK model describes well the product distribution over a wide range of operating conditions with physically sound kinetic parameters. The reductive elimination toward n-alkanes and the beta-hydride elimination involved in the formation of 1-alkenes with activation energies amounting to 117.8 and 96.3 kJ/mol are the two most kinetically significant steps determining the product distribution. In particular, the symmetry effects specifically accounted for by the single-event concept appeared critical in the interpretation of the deviations from the Anderson-Schulz-Flory distribution at low carbon numbers. Because of its fundamental character, the SEMK model developed here can be easily applied to describe Fischer-Tropsch synthesis over other catalysts

Single Event Kinetic Modelling without Explicit Generation of Large Networks: Application to Hydrocracking of Long Paraffins

The single event modelling concept allows developing kinetic models for the simulation of refinery processes. For reaction networks with several hundreds of thousands of species, as is the case for catalytic reforming, rigorous relumping by carbon atom number and branching degree were efficiently employed by assuming chemical equilibrium in each lump. This relumping technique yields a compact lumped model without any loss of information, but requires the full detail of an explicitly generated reaction network. Classic network generation techniques become impractical when the hydrocarbon species contain more than approximately 20 carbon atoms, because of the extremely rapid growth of reaction network. Hence, implicit relumping techniques were developed in order to compute lumping coefficients without generating the detailed reaction network. Two alternative and equivalent approaches are presented, based either on structural classes or on lateral chain decomposition. These two methods are discussed and the lateral chain decomposition method is applied to the kinetic modelling of long chain paraffin hydroisomerization and hydrocracking. The lateral chain decomposition technique is exactly equivalent to the original calculation method based on the explicitly generated detailed reaction network, as long as Benson’s group contribution method is used to calculate the necessary thermodynamic data in both approaches