Stochastic Sensitivity Analysis of Population Balance Models

该项目将提高对化学工程中某些过程的数学建模和预测能力。洗衣粉生产中颗粒的形成和烟尘颗粒的生长是特别令人感兴趣的例子。这一过程在任何给定时间的状态由存在的粒子类型和每种类型的数量的详细列表给出。在刚才提到的洗衣粉情况下，颗粒类型将有许多成分，包括大小、重量和化学成分。在所有要考虑的过程中，粒子类型中有几个组件，因此粒子类型的集合非常大。随着这些过程的进行，会发生大量的粒子变换，每次变换的速度最初都是未知的。这个项目的基本焦点是从过程的实验观测中推导出这些过程速率的问题。提出的解决这个问题的方法是对某些微分方程进行数值研究，称为种群平衡方程，可以用来对过程进行建模。大量的粒子类型和变换使得这些方程很难数值求解。因此，我们的方法是使用蒙特卡罗方法，其中真实的粒子由(小得多的)计算粒子集合建模，这些粒子受到随机过渡和变换的影响。我们建议的新方面是从数值上调查转换率的微小变化将如何影响测量的输出。这将允许我们调整模型中的变化率，使数值结果与实验观察相匹配，并应导致新过程的预测模型。它还将使我们了解并最终避免输出随时间变化不稳定的过程。这两个发展都具有重要的商业意义。计算对变化率的敏感度是一项微妙的数值任务，可能涉及两个相似的量的减法，两者都可能出现错误。为了达到更高的精度，我们将设计出尽可能匹配两个要减去的量的误差的数值方案，以便在减去之后误差减小，从而产生更准确的结果。我们将通过在每次计算中耦合粒子的随机行为来实现这一点。该项目将建立在马库斯·克拉夫特博士(DEP.剑桥大学化学工程系)和之前得到EPSRC支持的James Norris博士(剑桥大学数学系)(GR/R85662/01)。该项目将导致化学和计算工程以及数学方面的进步。

The project will improve capabilities for mathematical modelling and prediction of certain processes in chemical engineering. The formation of granules in the production of washing powder and the growth of soot particles are examples of particular interest. The state of such a process at any given time is given by a detailed list of the types of particle present and numbers of each type. In the washing powder case just mentioned the particle types would have a number of components including size, weight, and chemical composition. In all the processes that will be considered, there are several components in the particle type, so that the set of particle types is very large. As these processes proceed, a large number of particle transformations take place, each at a rate which is, initially, unknown. The basic focus of this project is the problem of deducing these process rates from experimental observations of the process.The method proposed to attack this problem is a numerical investigation of certain differential equations, called population balance equations, which can be used to model the processes. The large numbers of particle types and transformations make these equations difficult to solve numerically. Therefore our approach is to use Monte Carlo methods, where the real particles are modelled by a (much smaller) ensemble of computational particles, which are subject to random transitions and transformations. The new aspect of our proposal is to investigate numerically how a small change in the transformation rates will affect the measured outputs. This will allow us to tune the transformation rates in the model to match the numerical results to the experimental observations and should lead to predictive models for new processes. It will also allow us to understand and eventually avoid processes where the output is unstable over time. Both developments are of commercial importance.The calculation of sensitivities to transformation rates is a delicate numerical task, potentially involving the subtraction of two similar quantities, both subject to error. In order to achieve greater accuracy we will devise numerical schemes which match, as far as possible, the errors in the two quantities to be subtracted so that after subtraction the error is decreased thus producing more accurate results. We will do this by coupling the random behaviour of the particles in each calculation.The project will build upon the collaboration of Dr Markus Kraft (Dept. of Chemical Engineering, University of Cambridge) and Dr James Norris (Faculty of Mathematics, University of Cambridge) which has previously been supported by the EPSRC (GR/R85662/01). The project will lead to advances in chemical and computational engineering, and in mathematics.