Compartmental multivariate population balances based on Monte Carlo methods

颗粒的尺寸演化与许多科学和工程领域有关，例如气体到颗粒的转化过程、燃烧过程、材料合成、烟气净化技术、过程工程等。颗粒尺寸分布的动态解可以通过群体平衡方程（PBE）来执行。蒙特卡罗（MC）方法是一种求解PBE的随机模拟方法。其随机性和离散性使其非常适合处理离散事件。与其他方法（如矩量法）相比，MC方法不需要PSD的任何先验信息，如分布的形状。MC方法允许包括其他颗粒性质，如直径、电荷、组成、形态等（多变量性质），而无需额外的计算成本-与分段方法相反。这项工作的目标是开发随机数值方法模拟的多变量分散相的框架中面向流程的模拟系统的基础上的顺序模块化的方法。 将建立一个隔室系统，其中每个隔室包含大量的模拟粒子，并且其中MC隔室间的粒子交换将以概率的方式进行，使用跳跃约束。在动态流程模拟中改变颗粒性质的机制是成核、生长/溶解、凝聚和破碎。运输以及设备功能，如筛分、混合、分选等，可以简单地转换为概率函数，模拟颗粒从一个隔室到另一个隔室的流动。这些功能的MC描述导致模拟粒子数量的变化，因此需要重新调整模拟粒子数量以保证所需精度的策略。在第一阶段，我们集中在MC模拟成核与凝聚相结合，而在第二阶段，我们还包括生长/溶解耦合到连续相。此外，已经开发了用于破损的MC配方并进行了基准测试。平行合并和分裂算法的发展将是必不可少的处理车厢之间的运输，计划为第三期。同时，基于MC对同时凝固、成核、破碎和生长（分别为结晶）的描述，建立了搅拌介质磨和结晶过程的分区模型。蒸发）。将为这些过程建立示例性流程图模拟。

The size evolution of particles is relevant for many fields of science and engineering such as gas-to-particle conversion processes, combustion processes, material synthesis, flue gas cleaning technology, process engineering etc. The dynamic solution of the particle size distribution can be performed by means of Population Balance Equations (PBEs). The Monte Carlo (MC) method is a stochastic simulation method which solves the PBEs. Its stochastic and discrete nature makes it ideally suited to handle discrete events. In contrast to other methods (like moments method), the MC methods do not require any a priori information of the PSD, such as the shape of the distribution. The MC methods allow the inclusion of other particle properties such as diameter, charge, composition, morphology, etc. (multivariate properties) without additional computational costs - in contrast to sectional methods. The goal of this work is to develop stochastic numerical methods for the simulation of a multivariate disperse phase in the framework of a flowsheet-oriented simulation system based on the sequential modular approach. A compartmental system will be set up in which each compartment contains a large number of simulation particles and in which the MC intercompartmental particle exchange will be done in a probabilistic way, using hopping constraints. Mechanisms which change particle properties in a dynamic flowsheet simulation are nucleation, growth/dissolution, coagulation and breakage. Transport as well as equipment functions such as sieving, mixing, sorting etc. can be simply converted in probability functions modeling the particle flow from one compartment to the other. The MC description of these functions leads to a change in the number of simulation particles, so that strategies are required which readjust the number of simulation particles guaranteeing the desired accuracy. In the first period we concentrated on MC simulation of nucleation in combination with coagulation, whereas in the second period we also included growth/dissolution coupled to the continuous phase. Furthermore, a MC formulation for breakage has been developed and benchmarked. The parallel merging and splitting algorithms developed will be essential for dealing with transport between compartments, planned for the 3rd period. Also a compartmental model of a stirred media mill and crystallization processes will be set up, based on the MC description of the simultaneous coagulation, nucleation, breakage and growth (resp. evaporation). Exemplary flowsheet simulations will be set up for these processes.