Solution of population balances by high order moment-conserving method of classes: reconstruction of a non-negative density distribution

Abstract

A method is proposed for reconstruction of a non-negative piecewise continuous density distribution from a solution obtained by the high order moment conserving method of classes (HMMC), presented in our earlier papers [Alopaeus, V., Laakkonen, M., Aittamaa, J., 2006]. Solution of population balances with breakage and agglomeration by high order moment-conserving method of classes. Chemical Engineering Science 61, 6732–6752; Alopaeus, V., Laakkonen, M., Aittamaa, J., 2007. Solution of population balances with growth and nucleation by high order moment-conserving method of classes. Chemical Engineering Science 62, 2277–2289.]. The resulting distribution is strictly non-negative, and the overall distribution moments obtained by HMMC are conserved very accurately. The distribution is reconstructed by using the positive region of HMMC as an initial estimate for the continuous density distribution. This estimate is then refined by transforming the abscissas and scaling the ordinates of the density distribution so that the distribution moments are as close to the original solution as possible. The method is tested with a number of numerical examples. These cases are chosen due to their tendency to produce oscillations in HMMC, or due to their nature of producing multimodal distributions. It is shown that when the reconstruction algorithm is applied to the HMMC solution, the analytical density distribution is captured extremely well. This is true even in cases where the width of the analytical distribution is only a fraction of the original HMMC grid size. Multimodal population density distributions can also be reconstructed accurately from a HMMC solution with a reasonably small number of grid points, without a need to assume any basis functions for the reconstruction.