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A Laplace Transformation Based Technique for Reconstructing Crystal Size Distributions

by Qamar, S., Warnecke,G., Elsner, M.P., Seidel-Morgenstern, A..

Series: 2007-35, Preprints

35L65 Conservation laws
35L45 Initial value problems for first-order hyperbolic systems
35L67 Shocks and singularities

This article introduces a technique for reconstructing crystal size distributions (CSDs)
described by well-established batch crystallization models.
The method requires the knowledge of the initial CSD which can also be used to calculate the initial moments and
initial liquid mass.
The solution of the reduced four-moment system of ordinary differential equations (ODEs) coupled with an algebraic equation for the mass
gives us moments and mass at the discrete points of the given computational time domain. This
information can be used to get the discrete values of growth and nucleation rates. The discrete
values of growth and nucleation rates along with the initial distribution are sufficient to reconstruct the final
CSD. In the derivation of current technique
the Laplace transformation of the population balance
equation (PBE) plays an important role. The proposed technique has dual purposes. Firstly, it can be used as a numerical technique to solve the given
population balance model (PBM) for batch crystallization. Secondly, it can be used to reconstruct the
final CSD from the initial one and also vice versa. The method is very efficient, accurate and easy to implement.
Several numerical test problems of batch crystallization processes are considered here.
For validation, the results of the proposed technique are compared with those from the
high resolution finite volume scheme which solves the given PBM directly.

Population balance models, crystallization pro

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