Modelling and diagnostics of batch processes and analogous kinetic experiments

In chemical kinetics and batch processes K variables are measured on the batches at regular time intervals. This gives a J×K matrix for each batch (J time points times K variables). Consequently, a set of N normal batches gives a three-way matrix of dimension (N×J×K). The case when batches have different length is also discussed. In a typical industrial application of batch modelling, the purpose is to diagnose an evolving batch as normal or not, and to obtain indications of variables that together behave abnormally in batch process upsets. Other applications giving the same form of data include pharmaco-kinetics, clinical and pharmacological trials where patients (or mice) are followed over time, material stability testing and other kinetic investigations. A new approach to the multivariate modelling of three-way kinetic and batch process data is presented. This approach is based on an initial PLS analysis of the ((N×J)×K) unfolded matrix ((batch×time)×variables) with `local timeʹ used as a single y-variable. This is followed by a simple statistical analysis of the resulting scores and results in multivariate control charts suitable for monitoring the kinetics of new experiments or batches. `Upsetsʹ are effectively diagnosed in these charts, and variables contributing to the upsets are indicated in contribution plots. In addition, the degree of `maturityʹ of the batch can be as predicted vs. observed local time. The analysis of batch data with respect to various questions is discussed with respect to typical objectives, overview and summary, classification, and quantitative modelling. This is illustrated by an industrial example of yeast production.