On the solution of bivariate population balance equations for aggregation: Pivotwise expansion of solution domain
- 作者：Abhinandan Chiney ; Sanjeev Kumar ; chesanjeev@gmail.com ; sanjeev@chemeng.iisc.ernet.in
- 关键词：Population balance ; Particulate processes ; Mixing ; Agglomeration ; Mathematical modelling ; Discretization methods
- 刊名：Chemical Engineering Science
- 出版年：2012
- 期刊代码：18_00092509
- 类别：ce
- 出版时间：9 July, 2012
- 卷：76
- 期：Complete
- 页码：14-25
- 文件大小：643 K
摘要
The solution of a bivariate population balance equation (PBE) for aggregation of particles necessitates a large 2-d domain to be covered. A correspondingly large number of discretized equations for particle populations on pivots (representative sizes for bins) are solved, although at the end only a relatively small number of pivots are found to participate in the evolution process. In the present work, we initiate solution of the governing PBE on a small set of pivots that can represent the initial size distribution. New pivots are added to expand the computational domain in directions in which the evolving size distribution advances. A self-sufficient set of rules is developed to automate the addition of pivots, taken from an underlying X-grid formed by intersection of the lines of constant composition and constant particle mass. In order to test the robustness of the rule-set, simulations carried out with pivotwise expansion of X-grid are compared with those obtained using sufficiently large fixed X-grids for a number of composition independent and composition dependent aggregation kernels and initial conditions. The two techniques lead to identical predictions, with the former requiring only a fraction of the computational effort. The rule-set automatically reduces aggregation of particles of same composition to a 1-d problem. A midway change in the direction of expansion of domain, effected by the addition of particles of different mean composition, is captured correctly by the rule-set. The evolving shape of a computational domain carries with it the signature of the aggregation process, which can be insightful in complex and time dependent aggregation conditions.