Stochastic Sensitivity Analysis of Population Balance Models

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.