An open-source quadrature-based population balance solver for OpenFOAM

Hybrid quadrature moment method for accurate and stable representation of non-Gaussian processes applied to bubble dynamics

Solving the population balance equation (PBE) for the dynamics of a dispersed phase coupled to a continuous fluid is expensive. Still, one can reduce the cost by representing the evolving particle density function in terms of its moments. In particular, quadrature-based moment methods (QBMMs) invert these moments with a quadrature rule, approximating the required statistics. QBMMs have been shown to accurately model sprays and soot with a relatively compact set of moments. However, significantly non-Gaussian processes such as bubble dynamics lead to numerical instabilities when extending their moment sets accordingly. We solve this problem by training a recurrent neural network (RNN) that adjusts the QBMM quadrature to evaluate unclosed moments with higher accuracy. The proposed method is tested on a simple model of bubbles oscillating in response to a temporally fluctuating pressure field. The approach decreases model-form error by a factor of 10 when compared with traditional QBMMs. It is both numerically stable and computationally efficient since it does not expand the baseline moment set. Additional quadrature points are also assessed, optimally placed and weighted according to an additional RNN. These points further decrease the error at low cost since the moment set is again unchanged. This article is part of the theme issue 'Data-driven prediction in dynamical systems'.

A quasi-Monte Carlo based flocculation model for fine-grained cohesive sediments in aquatic environments

The quasi-Monte Carlo (QMC) method was enhanced to solve the population balance model (PBM) including aggregation and fragmentation processes for simulating the temporal evolutions of characteristic sizes and floc size distributions (FSDs) of cohesive sediments. Ideal cases with analytical solutions were firstly adopted to validate this QMC model to illustrate selected pure aggregation, pure fragmentation, and combined aggregation and fragmentation systems. Two available laboratory data sets, one with suspended kaolinite and the other with a mixture of kaolinite and montmorillonite, were further used to monitor the FSDs of cohesive sediments in controlled shear conditions. The model results show reasonable agreements with both analytical solutions and laboratory experiments. Moreover, different QMC schemes were tested and compared with the standard Monte Carlo scheme and a Latin Hypercube Sampling scheme to optimize the model performance. It shows that all QMC schemes perform better in both accuracy and time consumption than standard Monte Carlo scheme. In particular, compared with other schemes, the QMC scheme using Halton sequence requires the least particle numbers in the simulated system to reach reasonable accuracy. In the sensitivity tests, we also show that the fractal dimension and the fragmentation distribution function have large impacts on the predicted FSDs. This study indicates a great advance in employing QMC schemes to solve PBM for simulating the flocculation of cohesive sediments.

Parameter Estimation of the Stochastic Primary Nucleation Kinetics by Stochastic Integrals Using Focused-Beam Reflectance Measurements

The kinetic parameters of stochastic primary nucleation were estimated for the batch-cooling crystallization of L-arginine. It is difficult for process analytical tools to detect the first nucleus. In this study, the latent period for the total number of crystals to be increased to a predetermined threshold was repeatedly measured with focused-beam reflectance measurements. Consequently, the latent periods were different in each measurement due to the stochastic behavior of both primary and secondary nucleation. Therefore, at first, the distribution of the latent periods was estimated by a Monte Carlo simulation for some combinations of the kinetic parameters of primary nucleation. In the simulation, stochastic integrals of the population and mass balance equations were solved. Then, the parameters of the distribution of latent periods were estimated and correlated with the kinetic parameters of primary nucleation. The resulting correlation was represented by a mapping. Finally, the parameters of the actual distribution were input into the inverse mapping, and the kinetic parameters were estimated as the outputs. The estimated kinetic parameters were validated using statistical techniques, which implied that the observed distribution function of the latent periods for the thresholds used in the estimation coincided reasonably with the simulated one based on the estimated parameters.

CFD data of unsteady cavitation around a hydrofoil, based on an extended Schnerr-Sauer model coupled with a nucleation model

The data presented in this article were the basis for the study reported in the research articles entitled “Characterization of unsteady cavitating flow regimes around a hydrofoil, based on an extended Schnerr-Sauer model coupled with a nucleation model” (De Giorgi et al., 2018)[1]. The reference study presented a spatio-temporal characterization of different cavitating flow regimes using Computational Fluid Dynamics (CFD). The authors evaluated the accuracy of an extended Schnerr-Sauer cavitation model. The accuracy of the numerical model has been improved by means of the introduction of a Density Correction Model of the turbulent viscosity, and a simplified Population Balance Modeling (PBM).

Mathematical modelling of expanded bed adsorption - a perspective on in silico process design

Expanded bed adsorption (EBA) emerged in the early 1990s in an attempt to integrate the clarification, capture and initial product concentration/purification process. Several mathematical models have been put forward to describe its operation. However, none of the models developed specifically for EBA allows simultaneous prediction of bed hydrodynamics, mass transfer/adsorption and (unwanted) interactions and fouling. This currently limits the development and early optimization of EBA-based separation processes. In multiphase reactor engineering, the use of multiphase computational fluid dynamics has been shown to improve fundamental understanding of fluidized beds. To advance EBA technology, a combination of particle, equipment and process scale models should be used. By employing a cascade of multiscale simulations, the various challenges EBA currently faces can be addressed. This allows for optimal design and selection of equipment, materials and process conditions, and reduces risks and development times of downstream processes involving EBA. © 2018 The Authors. Journal of Chemical Technology & Biotechnology published by John Wiley & Sons Ltd on behalf of Society of Chemical Industry.