Inverse Problems in Population Balances. Determination of Aggregation Kernel by Weighted Residuals

Abstract

The phenomenology of particulate systems are embedded into the population balance model through breakage, aggregation, or growth kernels. Aggregation kernel is a difficult kernel to predict from first principle because of its nonlinear nature. In this work we demonstrate a new methodology to obtain the aggregation kernel through inverse problem approach. This new approach is based on the method of weighted residuals and does not rely on specific traits of the system like selfsimilarity. The residual approach reduces the inverse problem into solution of a set of linear equations. However, the system of equations is badly conditioned and therefore requires regularization for accurate solution. In this work Tikhonov regularization technique has been used. The new method has been demonstrated successfully for constant, sum, and product kernel. ■ INTRODUCTIONIn the hope of providing a reasonable explanation for contributing to a special issue for one's own felicitation, this paper is an attempt to bring to the forefront the subject of inverse problems that we believe to be an essential aspect of identifying population balance models. While there have been a few publications in the past demonstrating the importance of inverse problems, the overwhelming approach to the use of population balances has been through force-fitting parameters in models that are either empirical or based on mechanisms of uncertain validity. Satisfactory fits often lend unwarranted support to the model as they are frequently under restricted conditions. The paper by Mahoney et al. 1 demonstrates the risks of such fitting. Yet, we wish not to be disparaging of the ever increasing effort on the use of population balance models, as their presence in engineering analysis, design, and control is a reflection of their high relevance to many dispersed phase processes encountered in engineering technology. Because of the ill-posed nature of inverse problems, without appropriate numerical armory to counter its effect, their solution is often difficult. It is our objective in this paper to show some success in this regard and possible directions for the future. Specifically, this work introduces a new technique for extraction of aggregation kinetics using the inverse problem approach.Modeling of aggregation is one of the major areas 2,3 in population balances. Aggregation is the process whereby two or more particles come together and remain bound for long enough to be considered a unique new particle. Two subprocesses are considered important: First, the frequency at which particles come near enough to one another so that they can interact with each other. This is generally termed the collision frequency, and is a well explored area. Smoluchowski 4 predicted the collision frequencies for particles moving by Brownian motion and in laminar shear fields. In another major study, Drake 5 investigated collisions in turbulent fields (collisions due to inertial effects and diffusion).Once the particles have come into contact, a second subprocess can be modeled t...

Modelling simultaneous chain‐end and random scissions using the fixed pivot technique

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“…Having this deficiency would certainly impair the accuracy of the resulting combined predictions. In cases where the evolution of the complete chain‐length distribution is important, the capability of the method of moments and its variants to produce such information is “at best small.”…”

“…[15] Having this deficiency would certainly impair the accuracy of the resulting combined predictions. In cases where the evolution of the complete chain-length distribution is important, [3][4][5][6]10,16,17] the capability of the method of moments and its variants to produce such information is "at best small." [18,19] Another commonly reported approach for solving the continuous PBEs involving simultaneous chain-end and random scissions is by direct discretization of the continuous PBEs.…”

Convergence analysis of finite volume scheme for nonlinear aggregation population balance equation

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