## Introduction

Several industrial processes, such as catalytic polymerization, combustion, and gasification, involve fluidized bed reactors. These are attractive because they maximize the contact area between the phases and guarantee excellent heat and mass transfer. Even so, developing, innovating, and scaling up these processes is still quite challenging, because the dynamics and reactive behavior of fluidized suspensions are extremely difficult to predict and control. The complexity originates from the many physical and chemical phenomena that occur concurrently: chemical reactions take place, which usually affect the properties of the particles; in addition, these can aggregate or break into subelements, whereas others can form through nucleation. The end-product quality strongly depends on all these competing phenomena, which in turn are influenced by the suspension fluid dynamics and, indirectly, by the reactor internals, geometry, and size.

To design these units, process engineers have resorted for many years to experimental correlations and pilot plants. However, since these correlations are valid only for the specific units investigated, they cannot help engineers to innovate or improve design and performance; pilot plants, conversely are expensive and time-consuming, not always leading to adequate scale up.

Thanks to the availability of high-speed computer processors, computational fluid dynamics (CFD) plays nowadays a key role in understanding the behavior of multiphase systems and in particular fluidized beds. The improvement in accuracy of recent fluid dynamic models1–5 has substantially increased the interest of industry in this technique; nevertheless, since many limitations in the predictive capabilities of such models still exist, much theoretical research is required to turn CFD into a fully reliable design tool. One of the assumptions which restricts even the most advanced models is the particles having constant and equal size.6–13 As just pointed out, in industrial processes there exists a particle size distribution (PSD), whose changes in time and space reflect the course of the very physical and chemical phenomena characterizing the processes. These changes in PSDs are associated with the possible occurrence of segregation phenomena, which result into uneven distribution of the particles within the bed. Depending on the application at hand, segregation may be beneficial or detrimental,14–17 but in either case being able to predict its extent and dynamics is key to properly design and operate fluidized bed reactors.

To partially overcome this limitation, research groups have extended to polydisperse suspensions models originally developed for monodisperse. This approach still hinges on the constant-size assumption, but now two or more particle classes differing in size are accounted for, so that powders can segregate.18–27 Even so, variations in size are not allowed for, whereas in reality particles can shrink, aggregate, break, and nucleate, their size distribution varying continuously in time and space. Predicting this evolution, which depends upon the local conditions wherein the system operates, is essential for a reliable description of the suspension behavior, but requires a more powerful modeling strategy.

To account for size-changing phenomena, which characterize the physics and chemistry of the process at hand, and describe how the PSD evolves locally within the reactor, we need to solve, along with, or in place of, the averaged transport equations of conservation of mass, linear momentum and possibly energy, a population balance equation (PBE). Doing so, however, is not trivial, because the dimensionality of this equation depends on the application and on the strategy that the modeler wishes to use (e.g., on how many internal coordinates he uses to characterize the state of the particles). Hence, PBEs are not necessarily three-dimensional (3-D) and cannot be easily integrated within customary CFD codes. In the context of multiphase flows, not so many research groups have used this modeling approach. Olmos et al.28 simulated bubble columns considering 10 different size classes to represent the bubbles, but solved only the dynamical equation for the mixture. The bubbles consequently shared the same velocity. Using similar strategies, other groups have simulated gas–liquid systems.29–31 Dense fluid–solid systems, conversely, in which the phases strongly interact and move with different velocities, have been investigated much less.32–34

Various techniques can solve PBEs numerically; for a comprehensive review we refer to Ramkrishna.35 Here, we focus on the so-called method of moments. Frequently engineers do not really need to know the particle density function, which describes how the population of elements is distributed locally over the properties of interest, but are only interested in some integral properties of the latter. Such properties, called moments, may be important because they control the product quality or because they are easy to measure and monitor. The idea behind the method of moments is to derive transport equations for the moments of interest by integrating out the internal coordinates from the PBE.36 The method is attractive, because the transport equations that govern the moments are 3-D and the number of moments to be tracked is small; however, the transport equations are unclosed, because for any given set of moments that the modeler wishes to consider, the equations normally involve also higher-order moments external to the set.37, 38

The quadrature method of moments (QMOM), which approximates the particle density function using a quadrature formula, overcomes this problem; turning integrals of the density function into summations, the formula eliminates the problem of closure.39, 40 To compute the quadrature nodes and weights, QMOM forces them to agree with a set of independent lower-order moments41 that the model tracks by integrating their transport equations. From this set, QMOM then determines the finite-mode representation of the density function.

For monovariate distributions, that is, distributions with only one internal coordinate, to back-calculate the quadrature nodes and weights from the moments of the density function we can adopt the product-difference (PD) algorithm of Gordon,42 which requires finding the eigenvalues of a real symmetrical tridiagonal matrix, or the algorithm of Wheeler.43 Nevertheless, these algorithms cannot be applied when a higher number of internal coordinates is present. The quadrature approximation must then be determined using multivariate inversion algorithms, such as Brute-Force methods,44 Tensor-Product methods,45 or Conditional QMOM.46 Even if some of these algorithms are very efficient, in this work we let the PBE account only for size dependencies; under this hypothesis, particles with the same size move with the same velocity, the latter being excluded from the set of internal coordinates, and the density function is monovariate. We then solve the PBE with the averaged dynamical equations of multiphase flows, adopting a hybrid approach.

In the present work, we develop and implement a new version of QMOM into the multifluid model of the commercial CFD code Fluent. There are two important novelties: (1) the model is based on a volume, and not on a number, density function, so that it deals with volume fractions instead of number densities, and (2) the particles no longer share the same velocity, so that they can freely mix and segregate. The method is quite general and can treat any type of particulate process, but in what follows we verify and validate it on a simple process in which the particles neither react nor agglomerate nor break. The PSD changes solely because the powders mix. This is a relatively simple problem, but its very simplicity is key to test the method, understand it better and highlight possible issues or limitations. We believe that before tackling more complex problems, involving continuous and discontinuous changes in particle size, this analysis is necessary.

The article is thus structured. First, we introduce the system investigated. Next, we describe the experimental methodology and findings. We then present the mathematical model and the numerics, reporting the predictions of the simulations and showing how these compare with the experimental data. As we shall see, when QMOM is solved with spatial discretization schemes that use higher-order numerical schemes or when it tracks a sufficiently high number of moments (eight in this work), some moments corrupt, this leading to poor results or generating instabilities that eventually make the simulations crash altogether. We thus discuss the problem of moment corruption and present a few strategies that may be able to overcome it. One of these, reported by Wright,47 is implementing a corrective algorithm that replaces invalid moment sets with valid ones in the cells where moments corrupt. To conclude the article, we describe and discuss this method, assessing its potentials and limitations.