Particle size distribution reconstruction: the moment surface method

Numerical simulation of typical chemical engineering processes, such as crystallisation, liquid-liquid extraction, milling and other multi-phase operations in which exist discrete and continuous phases are highly computationally intensive problems. For this reason numerical techniques, such as the Method of Moments (MOM) and Quadrature Method of Moments (QMOM), are utilised to improve the computational efficiency of these simulations. The downside to these approaches is that the simulations only produce the moments of the Particle Size Distribution (PSD), with the actual distribution not preserved. Knowledge of the PSD is very important for many industrial unit operations, particularly in dynamic multi-phase flows in chemical engineering where the composition of the discrete phase(s) evolves in time or space. For example, control of the PSD in crystallisation operations may be required to ensure more efficient downstream operations such as filtration and clarification. Several methods for the reconstruction of a distribution from its respective moments are available in the literature. Typically these techniques are quite computationally expensive. The novel technique presented in this paper involves the pre-calculation of the moments of a pre-defined 2-parameter Probability Density Function (PDF) for a range of values of each parameter. This pre-calculation results in moment surfaces where the surfaces are a function of the two defining parameters. The intersection of constant moment contour lines (termed moment iso-lines) on these surfaces using simulation moment outputs results in the recovery of the defining parameters. Knowledge of the PDF and the total particle count or solids loading allows for the reconstruction of the full PSD. This technique proves to be very efficient which makes it ideal for the reconstruction of large numbers of distributions, for example in transient population balance models or model-based control algorithms, without the need for repeated application of optimisation algorithms.