Multivariate Population Balances via Moment and Monte Carlo Simulation Methods: An Important Sol Reaction Engineering Bivariate Example and “Mixed” Moments for the Estimation of Deposition, Scavenging, and Optical Properties for Populations of Nonspherical Suspended Particles

Abstract

Reactors or crystallizers synthesizing valuable particles can be formally described by combining the laws of continuum transport theory with a population balance equation governing evolution of the "dispersed" (suspended) particle population. Early examples necessarily focused on highly idealized device configurations and populations described locally using only one particle state variable, i.e., "size" (length or volume). However, in almost every application of current/future importance, a multivariate description is required, for which the existing literature offers little guidance. We describe here our recent research on an instructive bivariate prototype of physical interest (coagulating, sintering nanoparticles of prescribed composition, etc.) that will, hopefully, motivate a broader attack on important multivariate population balance problems, including those describing continuous molecular mixtures. We illustrate the use of both physically based bivariate "mixed" moment and Monte Carlo simulation methods amenable to future generalization. Multivariate extensions, improved and fully coupled rate laws, and consideration of interacting (coexisting) populations will be required to deal with future sol reaction engineering problems using similar strategies/techniques.

Population balance modelling of polydispersed particles in reactive flows

“…These methods are more suitable for the simulation of multidimensional PBEs, as they do not scale exponentially with dimensionality. They will not be covered here, but the reader can refer to Rosner et al [211] for more detail.…”

Measurement, modelling, and closed-loop control of crystal shape distribution: Literature review and future perspectives

“…Therefore, various numerical algorithms have been developed for solving PB equations such as method of moment [114][115][116][117], method of characteristics [108,[118][119][120][121], Monte Carlo techniques [122,123], and discretization methods including finite element technique [119,124,125], cell average methods [107], hierarchical solution strategy based on multilevel discretization [126], method of classes [82,95], fixed and moving pivot method [127,128], and finite difference/volume methods [90,119,[129][130][131]. Table 1 summarises these numerical solution methods with the further reviews below.…”

“…While the method is highly efficient when the physics are simple, the approach does not generalize to complex physics. Monte Carlo simulations track the histories of individual particles, each of which exhibits random behaviour according to a probabilistic model [122,123]. Monte Carlo simulations are most suitable for stochastic PB equations, especially for complex systems, but typically very computationally expensive.…”

Application of the Lagrangian Meshfree Approach to Modelling of Batch Crystallisation: Part II—An Efficient Solution of Integrated CFD and Population Balance Equations

“…These approaches can be categorised in five main classes: a) Standard method of moments (SMOM) (Randolph and Larson, 1971), b) Numerical non-linear model reduction approaches: method of characteristics (MOCH) (Hounslow and Reynolds, 2006), quadrature method of moments (QMOM) (McGraw, 1997;Marchisio et al, 2003), fixed quadrature method of moments (FQMOM) (Alopaeus, 2006), Jacobian matrix transformation (JMT) (McGraw and Wright, 2003) and direct quadrature method of moments (DQMOM) (Fan et al, 2004), c) Direct numerical solution approaches involving finite-element or finite-volume discretisation of the partial differential equation (discretised population balances, DPB) (LeVeque, 2002;Gunawan et al, 2004;Costa et al, 2007), d) various methods in the weighted residuals framework such as the least squares, orthogonal collocation and Galerkin methods (Singh and Ramkrishna, 1977;Sporleder et al, 2011;van den Bosch and Padmanabhan, 1974;Nigam and Nigam, 1980;Roussos et al, 2005), and e) Dynamic Monte Carlo simulation (DMC) (Haseltine and Rawlings, 2005;Rosner et al, 2003). The two most often used techniques are the standard method of moments and the quadrature method of moments.…”