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(为期六年)的目标是为多变量人口平衡系统的数值解开发准确和有效的方法。作为应用问题，我们考虑了一种创新的择形结晶过程。最终目的是通过使用所开发的数值方法来优化工艺设计和操作。为了实现这一目标，计划开发和系统地比较种群平衡方程中出现的微分和积分算符的数值处理技术。这两种类型的算子在其数学性质以及求解的适当的数值技术方面是完全不同的。我们建立了基准问题，这些问题展示了粒子群体的期望行为。这些基准也是通过实验实现的。通过实验，获得了可靠的测量数据，为数值技术的评价提供了参考值。在第一个项目期间，我们填补了在单变量情况下的数值技术评估方面存在的空白。在第二个项目阶段，我们将重点放在双变量情况(具有两个内部性质的晶体)的实验实现以及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中。在第三个项目阶段，我们计划在实验上实现一个由多个耦合过程单元组成的集成择形结晶过程，并利用开发的数值技术对该过程进行模拟。通过这个过程，我们想要控制晶体产品的大小和形状分布。对于增长控制的多变量过程，将发展改进的代数镇定格式并用于化工问题中的种群平衡系统的模拟。第三阶段的另一个目标是对多变量聚集过程进行数值处理。对于均匀网格，可以使用快速傅立叶变换(FFT)以线性代价来计算聚集积分。这使得可以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所特有的特性。

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.