AGGREGATION AND GELATION .1. ANALYTICAL SOLUTIONS FOR CST AND BATCH OPERATION

This work is part of a study of model discrimination for aggregation processes characterised by continuous particle size distributions. Criteria are sought for the acceptance or rejection of proposed functional forms of the aggregation kernel, of a more fundamental nature than the goodness of fit which they might afford to a perhaps narrow range of experimental data. New analytical results for well-mixed, continuously operated vessels are contrasted with those for batch operation. It is shown that some forms of kernel result in solutions to the population balance that exhibit the mathematical equivalent of a phase-transition phenomenon, manifested as divergence of the sixth moment of the particle size distributions, of a type referred to in the literature as gelation. Kernels that predict this ''mathematical gelation'' for one mode of operation, e.g. continuous, need not do so for another mode, e.g. batch. It is moreover shown that, for a kernel that predicts mathematical gelation for both these modes of operation, the gelation points correspond to different values of the index of aggregation for the two modes. In addition, the gelation points are dependent on the shape of the feed or charge PSD. We propose that gelling kernels-those capable of predicting gelation-be rejected a priori as unsuitable for modelling systems that do not exhibit physical gelation, since whatever their powers for interpolation of experimental data, they are unsafe for extrapolation from one mode of operation to another, from smaller to larger values of the index of aggregation, or from one feed to another.