Numerical solution methods for coupled population balance systems for the dynamic simulation of multivariate particle processes at the example of shape-selective crystallization

该项目的目标（六年期）是发展准确和有效的方法，用于多变量人口平衡系统的数值解。作为应用问题，我们考虑一种创新的择形结晶过程。最终目标是通过使用开发的数值方法进行最佳工艺设计和操作。为了实现这一目标，计划发展和系统地比较人口平衡方程中出现的微分和积分算子的数值处理技术。这两种类型的运营商是完全不同的，在他们的数学性质，以及在适当的数值技术，为他们的解决方案。 我们建立基准问题，表现出所需的行为的粒子种群。这些基准也实现了实验。通过实验，可靠的测量数据成为可用的参考值，并支持评估的数值技术。在第一个项目期间，我们填补了单变量情况下的数值技术评价方面存在的差距。在第二个项目期间，我们专注于双变量的情况下（两个内部属性的晶体），无论是在其实验实现，以及在合适的数值方法的发展和随后的实验和模拟结果的系统比较。在第三个项目期间，我们计划实验实现一个集成的形状选择性结晶过程与几个耦合的过程单元，并通过使用开发的数值技术来模拟这个过程。通过这种方法，我们希望控制晶体产品的尺寸和形状分布。对于增长主导的多变量过程，将发展改进的代数稳定化方案，并用于模拟化学工程问题中的种群平衡系统。第三阶段的另一个目标是多变量聚集过程的数值处理。它将被证明为均匀的网格，聚合积分可以通过使用快速傅立叶变换（FFT）的线性成本进行评估。这允许在比传统方法更精细的网格上进行模拟。这种方法的先决条件是聚合核的可分离近似-这是许多具有实际意义的核函数的特性。

The goal of this project (six-year period) is the development of accurate and efficient methods for the numerical solution of multivariate population balance systems. As application problem, we consider an innovative shape-selective crystallization process. The final aim is the optimal process design and operation through the use of the developed numerical methods. In order to reach this goal, it is planned to develop and systematically compare techniques for the numerical treatment of the differential and integral operators appearing in population balance equations. These two types of operators are entirely different in their mathematical properties as well as in the appropriate numerical techniques for their solution. We establish benchmark problems which exhibit the desired behaviour of particle populations. These benchmarks are also realized experimentally. Through the experiments, reliable measured data become available which serve as reference values and support the evaluation of the numerical techniques. In the first project period, we closed gaps which existed concerning the evaluation of numerical techniques for the univariate case. In the second project period, we focus on the bivariate case (crystals with two internal properties), both in its experimental realization as well as in the development of suitable numerical methods and a subsequent systematic comparison of experimental and simulation results. In the third project period we plan to realize experimentally an integrated shape-selective crystallization process with several coupled process units and to simulate this process by using the developed numerical techniques. With this process we want to control the size and shape distribution of the crystal product. With respect to growth-dominated multivariate processes, improved algebraic stabilization schemes will be developed and used for the simulation of population balance systems in chemical engineering problems.Another aim of the third period is the numerical treatment of multivariate aggregation processes. It will be shown for uniform grids that aggregation integrals can be evaluated with linear costs by use of the Fast Fourier Transformation (FFT). This allows simulations on much finer grids than with conventional methods. The prerequisite for this approach is a separable approximation of the aggregation kernel - a property featured by many kernel functions of practical relevance.