Mathematical Modeling and Advanced Parameter Estimation for Polymerization Processes

在开发数学模型时，建模师对所需的复杂性和细节做出相互关联的决定，简化将做出的假设，以及将用于估计模型参数的技术。适当的决定不仅取决于正在建模的系统的物理和化学，还取决于已知的实验信息、进行进一步实验的成本以及模型的预期用途(例如，过程控制、监测、放大或优化)。我的研究计划的长期目标是：i)为特定的聚合体系建立新的基本模型，以及ii)开发模型构建工具，以解决研究人员在开发和应用机械模型时面临的关键问题。*建议书中描述了三个项目。首先，将开发一个模型来描述半间歇反应器中4-(2-氯异丙基)苯乙烯的碳阳离子均聚合。更重要的是，将对异丁烯与这种亚聚体的共聚(以生产用于生物医学的定制橡胶分子)进行建模。虽然这些模型将预测支化程度和分子量分布随温度的变化，初始浓度和进料速度。来自J.E.Puskas(美国阿克伦大学)实验室的数据将用于参数估计和模型测试。建议的模型将比目前的模型有显著的进步，因为它们将考虑Lewis酸的消耗和产生准分子的交换反应。*在第二个项目中，将对随机微分方程模型中的参数估计的极大似然方法进行推广和测试。这些方法和模型对工业反应堆很有吸引力，因为它们考虑了未知的初始条件、以不规律的时间间隔收集的数据以及影响未来过程操作的扰动。将开发新的目标函数来说明关于模型参数的先验知识。*将开发诊断方法来确定哪些随机微分方程应该包括非平稳随机误差项，以便获得可能的最佳模型预测。*使用可估计性和分析方法来决定是否应该使用有限的实验数据来估计复杂模型中的哪些参数，以及哪些参数应该保留在名义值上，这样就可以避免数值问题，获得可靠的预测。第三个项目将扩展流行的可估计性分析技术，以说明需要良好预测的操作条件，特别是在费舍尔信息矩阵不可逆的困难情况下。此外，还将探索在某些参数无法估计时进行试验序贯设计的新方法。学生将在所有三个项目中接受培训，并将发展加拿大工业所需的技能。*

When developing a mathematical model, modelers make interconnected decisions about the complexity and detail required, simplifying assumptions that will be made, and techniques that will be used to estimate model parameters. Appropriate decisions depend not only on the physics and chemistry of the system being modeled, but also on experimental information that is already known, costs of performing further experiments, and the  intended use of the model (e.g., process control, monitoring, scale-up or optimization). The long-term goals of my research program are: i) to build novel fundamental models for specific polymerization systems, and ii) to develop model-building tools that address key problems faced by researchers when developing and applying mechanistic models. ******Three projects are described in the proposal. In the first, a model will be developed to describe carbocationic homopolymerization of 4-(2-chloroisopropyl) styrene in semi-batch reactors. More importantly, copolymerization of isobutylene with this inimer (to produce tailored rubber molecules for biomedical applications) will be modeled.  The models will predict evolution of branching and molecular weight distribution in response to temperature, initial concentrations and feed rates. Data from the lab of J.E. Puskas (U. of Akron) will be used for parameter estimation and model testing. The proposed models will be a significant advance over current models because they will account for consumption of Lewis acid and for the exchange reaction that generates inimer. ****In the second project, maximum-likelihood methods for parameter estimation in stochastic differential equation models will be extended and tested. These methods and models are attractive for industrial reactors because they account for unknown initial conditions, data collected at irregular time intervals, and disturbances that influence future process operation. New objective functions will be developed to account for prior knowledge about model parameters.*Diagnostic methods will be developed to determine which stochastic differential equations should include nonstationary random error terms so that the best possible model predictions can be obtained. *** Estimability analysis methods are used to decide which parameters should be estimated in complex models using limited experimental data and which should be left at nominal values, so numerical problems can be avoided and reliable predictions obtained. The third project will extend popular estimability analysis techniques to account for operating conditions where good predictions are required, especially in the difficult case where the Fisher Information Matrix is non-invertible.  New methods for sequential design of experiments when some parameters cannot be estimated will also be explored. Students will be trained in all three projects and will develop skills desired in Canadian industry.***