(424a) Effective Parameter Estimation within a Multi-Dimensional and Multi-Scale Population Balance Model Framework

Multi-dimensional and multi-scale population balance models are often required for the modelling and analysis of particulate systems that are distributed with respect to two or more of their internal coordinates (e.g. granulation) [1]. The population balance model is essentially a meso-scale framework that utilizes microscopic information (e.g. material properties, kinetic and thermodynamic parameters) in the form of kernels, to predict macroscopic particle traits (e.g. size distribution of crystals, porosity distribution of granules etc.). These kernels can be mechanistic (i.e., formulated based on the physics and chemistry of the process) or empirical (i.e., formulated based on power-law expressions with unknown fitting parameters). Given that most particulate processes are complex and intricate, most of the kernels in the literature are of the latter form. An important part of the formulation of the model of a given process is the estimation of the values for model parameters. The estimation of these model parameters is crucial to the applicability of the formulated model for process design, control and optimization. Parameter estimation can sometimes be straightforward, for instance when the process and formulated model are relatively simple and sufficient data is available. However, these conditions are not always met and this can result in difficulties in determining optimal parameter values. The focus of this study is on systems described by population balances and the nature of these systems is such that conventional optimization routines often fail when attempting to determine the optimal set of parameters. In many instances, due to the complexity of these model structures, optimization techniques are not used and parameters are estimated based on trial and error, often resulting in sub-optimal parameter values. The multiple dimensions, nonlinearity, non-convexity, hyperbolic and integral terms associated with the population balance partial-differential equation are potential causes amongst others, for convergence failure [2].

In this work, via two-dimensional crystallization and three-dimensional granulation applications, we aim to investigate and circumvent these failures systematically. A multi-dimensional population balance model is formulated for each application within which an objective function is defined. The objective function is formulated as a nonlinear least squares problem, minimizing the difference between experimental data and simulated model output. Results show that derivative-free methods, such as the Nelder-Mead simplex method, fail to converge to an optimal solution. A similar result was obtained with gradient-based methods such as BFGS, quasi-Newton, Newton, Gauss-Newton, Levenberg-Marquardt and SQP, and with a stochastic genetic algorithm. It was hypothesized that two main issues could contribute to the convergence failure: 1) gradients were calculated based on finite differences; as a result of improper step size determination, the numerical error could be prohibitive resulting in inaccurate derivative information and 2) optimal parameters may not be identifiable. To circumvent these issues, this work addresses the calculation of derivative information based on automatic differentiation to ensure increased accuracy. Issues such as parameter identifiability are also dealt with by analysing an accurate Fisher information matrix. Given the computational burden in calculating accurate Jacobians and Hessians, compounded by the discontinuities introduced into the objective function as a result of crystal/granule nucleation, a global optimization strategy may be warranted and this work addresses that accordingly. Overall, by systematically assessing the problem formulation and mechanisms, results will show that improvements in convergence can be achieved by utilizing appropriate optimization techniques, thus leading to more successful and optimal parameter estimation.

References

1. Iveson, S. M., 2002. Limitations of one-dimensional population balance models of wet granulation processes. Powder Technology 124, 219-229.

2. Wang, F. Y. and Cameron, I. T., 2007. A multi-form modelling approach to the dynamics and control of drum granulation processes. Powder Technology 179, 2-11.