M. P. Elsner

A comparative study of high resolution schemes for solving population balances in crystallization

Abstract This article demonstrates the applicability and usefulness of high resolution finite volume schemes for the solution of population balance equations (PBEs) in crystallization processes. The population balance equation is considered to be a statement of continuity. It tracks the change in particle size distribution as particles are born, die, grow or leave a given control volume. In the population balance models, the one independent variable represents the time, the other(s) are “property coordinate(s)”, e.g. the particle size in the present case. They typically describe the temporal evolution of the number density functions and have been used to model various processes. These include crystallization, polymerization, emulsion and cell dynamics. The high resolution schemes were originally developed for compressible fluid dynamics. The schemes resolve sharp peaks and shock discontinuities on coarse girds, as well as avoid numerical diffusion and numerical dispersion. The schemes are derived for general purposes and can be applied to any hyperbolic model. Here, we test the schemes on the one-dimensional population balance models with nucleation and growth. The article mainly concentrates on the re-derivation of a high resolution scheme of Koren (Koren, B. (1993). A robust upwind discretization method for advection, diffusion and source terms. In C. B. Vreugdenhill, & B. Koren (Eds.), Numerical methods for advection–diffusion problems, Braunschweig: Vieweg Verlag, pp. 117–138 [vol. 45 of notes on numerical fluid mechanics, chapter 5]) which is then compared with other high resolution finite volume schemes. The numerical test cases reported in this paper show clear advantages of high resolutions schemes for the solution of population balances.

The Claus process: teaching an old dog new tricks

Abstract The potential of integrating reaction with adsorption in a single piece of equipment to favourably displace chemical equilibria has attracted considerable attention in recent years. In contrast to the most such multifunctional reactor concepts investigated so far, the aim of our work is to study an industrially relevant chemical system with all its peculiarities. The feasibility of enhancing conversion in fixed-bed adsorptive reactors has been evaluated for the Claus reaction used in sulphur recovery, which is usually carried out in a multistage process, to counteract the severe equilibrium limitations at high conversions. Both experimental and simulation results indicate that the adsorption/reaction kinetics and the adsorbent capacity have to exhibit appropriate and compatible values in order to overcome the equilibrium limitation and attain high conversions in a single-step process. A crucial point is the selection of suitable reaction conditions, since the occurrence of undesirable side-reactions (e.g. suppression of COS elimination in the case of the Claus process) may be amplified by the distorted concentration profiles in adsorptive reactor operation. In industrial applications such by-products may be of particular importance and thus exert a decisive influence on the concentration profile modifications possible.

Adaptive high-resolution schemes for multidimensional population balances in crystallization processes

Abstract This article focuses on the application of adaptive high-resolution finite volume schemes for solving multidimensional population balance models (PBM) in crystallization processes. For the mesh redistribution, we use the moving mesh technique of Tang and Tang [Tang, H.-Z. & Tang, T. (2003). Adaptive mesh methods for one- and two-dimensional hyperbolic conservation laws. SIAM Journal of Numerical Analysis , 41 , 487–515] which they have developed for hyperbolic conservation laws in conjuction with finite volume schemes. In this technique, an iterative procedure is used to redistribute the mesh by moving the spatial grid points. The corresponding numerical solution at the new grid points is obtained by solving a linear advection equation. The method avoids the usual unsatisfactory, interpolation procedure for updating the solution. The finite volume schemes were originally derived for compressible fluid dynamics. The schemes have already shown their accuracy and efficiency in resolving sharp peaks and shock discontinuities. The accuracy of these schemes has been improved further by using the adaptive meshing techniques. The application of these high-resolution schemes for multidimensional crystallization processes demonstrates their generality, efficiency, and accuracy. The numerical test cases presented in this article show the clear advantage of finite volume schemes and show further improvements when combined with a moving mesh technique.

Adsorptive reactors for enhancing equilibrium gas-phase reactions—two case studies

The conversion enhancement potential of fixed-bed adsorptive reactors has been evaluated for two heterogeneously catalysed gas-phase equilibrium reactions: the Claus reaction used in sulphur recovery and the direct synthesis of hydrogen cyanide from carbon monoxide and ammonia. It was found that kinetics of both reaction and adsorption as well as adsorbent capacity have to be very compatible to achieve high conversions. The specific parameters of the reaction/catalyst system, such as the deposition of solids (e.g. sulphur, coke) or the formation of undesirable by-products have to be taken into account for the successful application of adsorptive reactor concepts. A crucial point is the selection of the reaction conditions (i.e. temperature), since the occurrence of side reactions may be enhanced in adsorptive reactor operation due to the inherently distorted concentration profiles.

Enzyme-assisted physicochemical enantioseparation processes - Part III: Overcoming yield limitations by dynamic kinetic resolution of asparagine via preferential crystallization and enzymatic racemization

The application of enantioseparation methods alone can only yield up to 50% of the desired chiral product. Thus enantioseparation becomes more attractive when accompanied by the racemization of the counter-enantiomer. Here we present first results of dynamic kinetic resolution of L-asparagine (L-Asn) via preferential crystallization and enzymatic racemization from a racemic, supersaturated solution on a 20 mL scale. An enzyme lyophilisate (WT amino acid racemase from P. putida KT2440 (E.C. 5.1.1.10), overexpressed in E. coli BL21(DE3)) was used for in situ racemization (enzyme concentrations varying from 0 to 1 mg/mL). When preferential crystallization was applied without any enzyme, a total of 31 mg of L-Asn monohydrate could be crystallized, before crystal formation of d-Asn started. Crystallization experiments accompanied by enzymatic racemization led to a significant increase of crystallized L-Asn (198 mg L-Asn monohydrate; >92%ee) giving the first experimental proof for this new process concept of dynamic kinetic resolution via preferential crystallization and enzymatic racemization. Measurements of the racemase activity before and after the crystallization process showed no significant differences, which would allow for enzyme recovery and recycling.

A Worst-Case Observer for Impurities in Enantioseparation by Preferential Crystallization

Abstract In this contribution, we show how a model is used to obtain worst-case estimates for product impurities in preferential crystallization of enantiomers. The estimates can be obtained for either a simple batch process in a single vessel, or for a coupled mode of operation of two vessels. We demonstrate, supported by a case study, how the estimates enable the use of a combined control strategy consisting of continuous feedback control and process stopping when critical impurity constraints are hit.