Solution Of Population Balance Research Articles

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

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.

Solution of population balances with growth and nucleation by high order moment-conserving method of classes

The high order method of classes, developed in our earlier work [Alopaeus, V., Laakkonen, M., Aittamaa J., 2006a. Solution of population balances with breakage and agglomeration by high order moment-conserving method of classes. Chemical Engineering Science 61, 6732–6752] for solution of population balances (PBs), is extended to problems with growth and primary nucleation. The growth problem leads to a hyperbolic partial differential equation with fundamentally different numerical characteristics than the PB with breakage and agglomeration only. However, we show that the principle of moment conservation in the numerical solution can also be applied to this advection-type problem, leading to extremely accurate numerical solutions. The method is tested for two numerical cases. The first one is mass transfer induced particle growth, and the second one is primary nucleation with constant growth (similar to the Riemann advection problem). For mass transfer induced growth, we first analyze functional form of the growth rate from mass transfer correlation viewpoint, and derive a general analytical solution for the power-law growth. The numerical results from the moment conserving method are also compared to one well established high resolution numerical method for advection problems, namely the Lax–Wendroff method with van Leer flux limiter. It was shown that the present method is far superior by predicting the distribution moments with several order of magnitudes lower numerical error. For the Riemann problem with constant growth rate, the present method predicts the shock front location exactly without any numerical diffusion.

Modeling Particle Formation at Spray Drying Using Population Balances

There have been several efforts to simulate the physical processes in a single droplet during spray drying in the last several years, but most of the models do not describe the solid formation in detail. In this work, the development of the microscopic solid structure in a droplet during spray drying is simulated. A radial-symmetric model of the droplet is used to simulate the mass and heat transport. The solid formation at every radial discretization point is obtained by the solution of population balances. This way, the distribution of the particle number density in the droplet depending on the macroscopic process parameters can be predicted. The model equations are solved in a normalized coordinate system to be able to describe the shrinkage of the droplet. The suitability of these population balances will also be discussed. For the validation, monodisperse single droplets consisting of a solution or suspension are dried with constant boundary conditions.

Integrated Framework for the Numerical Solution of Multicomponent Population Balances. 1. Formulation, Representation, and Growth Mechanisms

Current numerical solutions of population balance problems including growth are limited by the Courant condition time step constraint. As a result, particle size ranges must be restricted and limited growth rate mechanisms can be solved. A thorough analysis of growth rate mechanisms and dynamics is presented that relates the growth rate mechanisms and particle size ranges to the time step limit. These relationships reveal a method for producing the optimal time step for any growth rate model, which can increase the time step by a factor of 107. When these methods are combined with previous approaches, new hybrid methods are developed that allow the user to flexibly choose the optimal time step given the resolution requirements of the problem, producing solutions that are 104−107 times faster than conventional solutions. The numerical accuracy of these solutions compares favorably against analytical solutions, demonstrating their applicability across a wide range of problems.