Gerald Warnecke

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.

Modeling phase transition for compressible two-phase flows applied to metastable liquids

The seven-equation model for two-phase flows is a full non-equilibrium model, each phase has its own pressure, velocity, temperature, etc. A single value for each property, an equilibrium value, can be achieved by relaxation methods. This model has better features than other reduced models of equilibrium pressure for the numerical approximations in the presence of non-conservative terms. In this paper we modify this model to include the heat and mass transfer. We insert the heat and mass transfer through temperature and Gibbs free energy relaxation effects. New relaxation terms are modeled and new procedures for the instantaneous temperature and Gibbs free energy relaxation toward equilibrium is proposed. For modeling such relaxation terms, our idea is to make use of the assumptions that the mechanical properties, the pressure and the velocity, relax much faster than the thermal properties, the temperature and the Gibbs free energy, and the ratio of the Gibbs free energy relaxation time to the temperature relaxation time is extremely high. All relaxation processes are assumed to be instantaneous, i.e. the relaxation times are very close to zero. The temperature and the Gibbs free energy relaxation are used only at the interfaces. By these modifications we get a new model which is able to deal with transition fronts, evaporation fronts, where heat and mass transfer occur. These fronts appear as extra waves in the system. We use the same test problems on metastable liquids as in Saurel et al. [R. Saurel, F. Petitpas, R. Abgrall, Modeling phase transition in metastable liquids: application to cavitating and flashing flows, J. Fluid Mech. 607 (2008) 313-350]. We have almost similar results. Computed results are compared to the experimental ones of Simoes-Moreira and Shepherd [J.R. Simoes-Moreira, J.E. Shepherd, Evaporation waves in superheated dodecane, J. Fluid Mech. 382 (1999) 63-86]. A reasonable agreement is achieved. In addition we consider the six-equation model with a single velocity which is obtained from the seven-equation model in the asymptotic limit of zero velocity relaxation time. The same procedure for the heat and mass transfer is used with the six-equation model and a comparison is made between the results of this model with the results of the seven-equation model.

A posteriori error analysis for numerical approximations of Friedrichs systems

Abstract. The global error of numerical approximations for symmetric positive systems in the sense of Friedrichs is decomposed into a locally created part and a propagating component. Residual-based two-sided local a posteriori error bounds are derived for the locally created part of the global error. These suggest taking the

Efficient and accurate numerical simulation of nonlinear chromatographic processes

L^2

Numerical solution of population balance equations for nucleation, growth and aggregation processes

-norm as well as weaker, dual norms of the computable residual as local error indicators. The dual graph norm of the residual

A Modified Fractional Step Method for the Accurate Approximation of Detonation Waves

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Evolution Galerkin methods for hyperbolic systems in two space dimensions

is further bounded from above and below in terms of the

A direct Eulerian GRP scheme for compressible fluid flows

L^2

ON THE SOLUTION TO THE RIEMANN PROBLEM FOR THE COMPRESSIBLE DUCT FLOW

norm of

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

h {\vec r}_h