R. O. Fox

Iowa State University | ISU · Department of Chemical and Biological Engineering

When considering moment methods for the resolution of the free-transport term of the 1-D kinetic equation, the hyperbolic quadrature method of moments (HyQMOM) closure introduced in [12] leads to a globally hyperbolic system of conservation equations. Here, the HLL scheme used for its resolution is first proved to be realizable, i.e., allows comput...

Multiphase turbulence driven by coupling between the phases can arise in the absence of mean shear. For example, due to gravity (or other body forces) in disperse multiphase flows (e.g., particle-laden and bubbly flows), the mean-slip velocity between phases and spontaneous cluster formation generate strong turbulence that is nearly one dimensional...

We investigate drop break-up morphology, occurrence, time and size distribution, through large ensembles of high-fidelity direct-numerical simulations of drops in homogeneous isotropic turbulence, spanning a wide range of parameters in terms of the Weber number $We$ , viscosity ratio between the drop and the carrier flow $\mu _r=\mu _d/\mu _l$ , wh...

The effect of biomass particle shape on the conversion of beech wood during pyrolysis in a bubbling fluidized bed (BFB) was experimentally quantified. A lab-scale BFB installed on a high-precision scale was used to characterize the mass loss of the biomass particles immersed in the bed. The scale could monitor the mass loss of the beech wood partic...

Confined Impinging Jet Mixers (CIJMs) are widely used due to their small-scale-mixing efficiency. This property makes them especially useful for nanoprecipitation. The CIJM process parameters are computationally expensive to predict when using direct-numerical simulation (DNS) and large-eddy simulation (LES). In this work, binary nonreactive mixing...

The dynamics of cavitation bubbles are important in many flows, but their small sizes and high number densities often preclude direct numerical simulation. We present a computational model that averages their effect on the flow over larger spatiotemporal scales. The model is based on solving a generalized population balance equation (PBE) for nonli...

The quadrature method of moments (QMOM) for a one-dimensional (1-D) population balance equation was introduced by R. McGraw (Aerosol Science and Technology, 27, 255-265, 1997) to close the moment source terms. QMOM is defined based on the properties of the monic orthogonal polynomials Qi of degrees i=0,1,…,n that are uniquely defined by the set of...

The quadrature method of moments (QMOM) for a one-dimensional (1-D) population balance equation was introduced by R. McGraw (Aerosol Science and Technology, 27, 255-265, 1997) to close the moment source terms. QMOM is defined based on the properties of the monic orthogonal polynomials Q i of degrees i = 0, 1,. .. , n that are uniquely defined by th...

Droplet dispersion in liquid–liquid systems is a crucial step in many unit operations throughout the chemical, food, and pharmaceutic industries, where improper operation causes billions of dollars of loss annually. A theoretical background for the description of droplet breakup has been established, but many assumptions are still unconfirmed by ex...

The multigrain model (MGM) is the most widely used single-particle model, giving a detailed description of phenomena during olefin polymerization. However, the detailed modeling of the diffusion-reaction process inside the catalyst particles requires solving a large number of ordinary differential equations, which is unsuitable for coupling with mu...

The dynamics of cavitation bubbles are important in many flows, but their small sizes and high number densities often preclude direct numerical simulation. We present a computational model that averages their effect on the flow over larger spatiotemporal scales. The model is based on solving a generalized population balance equation (PBE) for nonli...

Eulerian sub-grid models for cavitation like ensemble-averaging are an increasingly viable route for simulating engineering-scale bubbly flow problems. We identify two primary concerns towards enabling physically-faithful simulations: sub-grid model fidelity and computational cost. Previous Euler-Euler models considered the sub-grid bubble radius R...

Improving the efficiency of the production of uniformly sized functional nanoparticles for pharmaceutical and agricultural applications has been a problem of great interest. The macroscale multi-inlet vortex Reactor (macro-MIVR) could potentially be used for this purpose due to its ability to achieve the rapid mixing necessary for the flash nanopre...

A quadrature-based moment method for the approximate solution of the generalized population balance equation (GPBE) governing the evolution of the joint size-velocity number density func-tion (NDF) of a particle population is formulated and tested. The proposed method relies on the third-order hyperbolic conditional quadrature method of moments dev...

An original dynamic model for substrate uptake under transient conditions is established and used to simulate a variety of biological responses to external perturbations. The actual uptake and growth rates, treated as cell properties, are part of the model variables as well as the substrate concentration at the cell–liquid interface. Several regula...

Modeling particle-laden turbulent flows at high volume fractions requires accounting for the coupling between phases. The latter is often a sensitive point, and proper closure of the exchange and production terms due to the presence of particles is not straightforward. In the present work, a Lagrangian probability-density-function model developed f...

A solution is proposed to a longstanding open problem in kinetic theory, namely, given any set of realizable velocity moments up to order 2n, a closure for the moment of order 2n+1 is constructed for which the moment system found from the free-transport term in the one-dimensional (1-D) kinetic equation is globally hyperbolic and in conservative fo...

This work reports an experimental investigation of a liquid-solid fluidized bed involving inertial particles at a large Reynolds number. Owing to optical techniques and index matching, the statistics of the velocity fluctuations of both the particles and the liquid are measured for a wide range of the particle volume fraction αp. The dynamics of th...

An open-source CFD software OpenFOAM ® is used to simulate two multiphase stirred-tank reactors relevant to industrial processes such as slurry polymerization and fuel production. Gas-liquid simulations are first performed in a single-impeller stirred-tank reactor, studied experimentally by Ford, J. J., T. J. Heindel, T. C. Jensen, and J. B. Drake....

The quadrature-based semi-analytical solution for the conditional moment closure (SA-CMC) given in (A. D. Ilgun, A. Passalacqua, and R. O. Fox, “A quadrature-based conditional moment closure for mixing-sensitive reactions,” Chem. Eng. Sci., 226, 2020) eliminates the additional conditioning-space discretization in CMC applications by assuming that t...

A quadrature-based moment method for the approximate solution of the generalized population balance equation (GPBE) governing the evolution of the joint size-velocity number density function (NDF) of a particle population is formulated and tested. The proposed method relies on the third-order hyperbolic conditional quadrature method of moments deve...

Multiphase turbulence arises in disperse multiphase flows due to (i) gravity, (ii) buoyancy, and (iii) mean shear. In contrast to pseudoturbulence at length scales on the order of the particle diameter, multiphase turbulence occurs on larger scales and is strongly associated with fluctuations in the disperse-phase volume fraction. In this chapter,...

In the context of modeling turbulent scalar mixing using probability density function (PDF) methods, the treatment of molecular mixing is of paramount importance. The conditional moment closure (CMC) offers a high-fidelity description for molecular mixing in nonpremixed flows. Recent work has demonstrated that first-order CMC can be implemented num...

Two solution algorithms are developed for the conditional moment closure (CMC) using quadrature-based moment methods (QBMM). Their primary purpose is to eliminate the necessity of the additional grid for the conditioning variable (e.g., mixture fraction). As in almost every probability-density-function (PDF)-based description, the main technical hu...

QBMMlib is an open source Mathematica package of quadrature-based moment methods and their algorithms. Such methods are commonly used to solve fully-coupled disperse flow and combustion problems, though formulating and closing the corresponding governing equations can be complex. QBMMlib aims to make analyzing these techniques simple and more acces...

QBMMlib is an open source Mathematica package of quadrature-based moment methods and their algorithms. Such methods are commonly used to solve fully-coupled disperse flow and combustion problems, though formulating and closing the corresponding governing equations can be complex. QBMMlib aims to make analyzing these techniques simple and more acces...

Particle-laden flow in a vertical channel was simulated using a Reynolds-averaged Navier-Stokes two-fluid model including a Reynolds-stress model (RSM). Two sets of cases varying the overall mass loading were done using particle sizes corresponding to either a large or small Stokes number. Primary and turbulent statistics extracted from counterpart...

A numerical investigation is performed on buoyancy-driven homogeneous and heterogeneous bubbly flows to compare the bulk gas-liquid heat transfer effectiveness for Prandtl (Pr) numbers 0.2-20 and void fractions 〈 α g 〉 0.3-0.5. For this purpose, transient two-fluid model simulations of bubbles rising in a stagnant pool of liquid are conducted...

Particle‐laden flows in a vertical channel were simulated using an Eulerian–Eulerian, Anisotropic Gaussian (EE‐AG) model. Two sets of cases varying the overall mass loading were done using particle sizes corresponding to either a large or small Stokes number. Primary and turbulent statistics were extracted from these results and compared with count...

A novel algorithm consisting of a quadrature-based semi-analytical solution to the conditional moment closure (CMC) is developed for mixing-sensitive reactions in turbulent flows. When applying the proposed algorithm, the additional grid in mixture-fraction phase space used in CMC codes is eliminated, and at most ten quadrature nodes are needed to...

In this work, model closures of the multiphase Reynolds-Average Navier-Stokes (RANS) equations are developed for homogeneous, fully-developed gas--particle flows. To date, the majority of RANS closures are based on extensions of single-phase turbulence models, which fail to capture complex two-phase flow dynamics across dilute and dense regimes, es...

Buoyancy driven turbulence due to heterogeneous bubble swarms are typically encountered in bubble columns operating in the dense heterogeneous regime. The integral scales of such turbulence are much larger than single bubble produced pseudo-turbulence. Accurate computational fluid dynamics predictions in this regime require correctly formulated ani...

The conditional velocity time averages (⟨Ui|ξ⟩) and conditional mixture fraction time averages (⟨Φ|ωi⟩) were computed based on the Eulerian approach from the experimental data measured in a macroscale multi-inlet vortex chemical reactor. The conditioning events were determined by equally sized intervals of the sample space variable for the mixture...

Starting from coupled Boltzmann–Enskog (BE) kinetic equations for a two-particle system consisting of hard spheres, a hyperbolic two-fluid model for binary, hard-sphere mixtures is derived with separate mean velocities and energies for each phase. In addition to spatial transport, the BE kinetic equations account for particle–particle collisions, u...

A theoretical model of liquid and particle random fluctuations is proposed for gravity-driven flows of inertial homogeneous suspensions. It is based on a paradigm assuming that fluctuations of both liquid velocity and particle slip velocity are driven by fluctuations of the phase indicator function. It is shown that this model accurately predicts t...

In this work, we present a rigorous derivation of the volume-filtered viscous compressible Navier–Stokes equations for disperse two-phase flows. Compared to incompressible flows, many new unclosed terms appear. These terms are quantified via a posteriori filtering of two-dimensional direct simulations of shock-particle interactions. We demonstrate...

The Eulerian–Eulerian two-fluid model is widely used for computational fluid dynamics simulations of gas–solid flows. For non-isothermal flows, the averaged conservation equations solved in the two-fluid model require closures for drag, gas–solid heat transfer, pseudo-turbulent velocity fluctuations and the pseudo-turbulent heat flux (PTHF). Howeve...

A computational algorithm for polydisperse bubbly flow is developed by combining quadrature-based moment methods (QBMM) with an existing two-fluid solver for gas-liquid flows. Care is taken to ensure that the two-fluid model equations are hyperbolic by generalizing the kinetic model for the bubble phase proposed by Bieseuvel and Gorissen (1990). Th...

In an attempt to validate our Monte-Carlo simulations of a multi-variate population balance model for cell growth in a chemostat, we observed that the mean age at rupture of a population is less than the so called doubling time. A confirmation of that result was found in Powell's work published in the 50's. However an opposite conclusion was found...

Starting from a generalized population balance equation and the Boltzmann–Enskog collision model for hard spheres, a kinetic theory model for polydisperse gas–particle flows is presented. Here, polydispersity results from spherical particles with the same material density but different diameters. The particle size distribution (PSD)of the particles...

Swarm-driven turbulent bubbly flows are typically encountered in bubble columns operating in the dense heterogeneous regime. The momentum phase coupling between gas and liquid leads to bubble swarm formation resulting in mesoscale turbulence. Accurate computational fluid dynamics predictions in this regime require correctly formulated anisotropic t...

The second-order realizable ζ moment advection scheme developed in Laurent and Nguyen, (2017) is extended to the case of unstructured grids with cells of arbitrary shape. The necessary modifications to the scheme and the conditions under which the scheme ensures the realizability of the advected moment set are presented. The implementation of the s...

Velocity fluctuations in the carrier phase and dispersed phase of a dispersed multiphase flow are studied using particle-resolved direct numerical simulation. The simulations correspond to a statistically homogeneous problem with an imposed mean pressure gradient and are presented for $\Re_m=20$ and a wide range of dispersed phase volume fractions...

The conditional hyperbolic quadrature method of moments (CHyQMOM) was introduced by Fox et al. [19] to reconstruct 1- and 2-D velocity distribution functions (VDF) from a finite set of integer moments. The reconstructed VDF takes the form of a sum of weighted Dirac delta functions in velocity phase space, and provides a hyperbolic closure for the s...

Simultaneous measurements of velocity and concentration using stereoscopic particle image velocimetry (stereo-PIV) and planar laser-induced fluorescence (PLIF) were used to investigate the mixing performance of a scaled-up multi-inlet vortex reactor (MIVR). Data were collected in three measurement planes located at different heights from the reacto...

Inertial particles in turbulent flows are characterised by preferential concentration and segregation and, at sufficient mass loading, dense particle clusters may spontaneously arise due to momentum coupling between the phases. These clusters, in turn, can generate and sustain turbulence in the fluid phase, which we refer to as cluster-induced turb...

The objective of the presented work is to verify a delayed detached eddy simulation (DDES) model for simulating transitional swirling flow in a micro-scale multi-inlet vortex reactor (MIVR). The DDES model is a k-ω based turbulence model with a low Reynolds number correction applied to the standard k-ω model such that the Reynolds-averaged Navier-S...

Turbulent wall-bounded flows exhibit a wide range of regimes with significant interaction between scales. The fluid dynamics associated with single-phase channel flows is predominantly characterized by the Reynolds number. Meanwhile, vastly different behaviour exists in particle-laden channel flows, even at a fixed Reynolds number. Vertical turbule...

Following a pulse addition of susbtrate in a chemostat, it is often observed that the amount of cell formed does not match the expectations (based on constant yield). This is because the susbtrate uptaken is not transformed into new cells. In order to depict this experimental fact, one has to uncouple uptake from growth. By the way it happens that...

The Lagrangian probability-density-function model, proposed in Part I for dense particle-laden turbulent flows, is validated here against Eulerian-Lagrangian direct numerical simulation (EL) data for different homogeneous flows, namely statistically steady and decaying homogeneous isotropic turbulence, homogeneous-shear flow and cluster-induced tur...

Inertial particles in turbulent flows are characterised by preferential concentration and segregation and, at sufficient mass loading, dense particle clusters may spontaneously arise due to momentum coupling between the phases. These clusters, in turn, can generate and sustain turbulence in the fluid phase, which we refer to as cluster-induced turb...

Cell growth in a chemostat is a well-documented research topic. How cells uptake the available substrate to gain weight and engage cell division is not generally taken into account in the modelling bioreactors. In fact, the growth rate is related to a population doubling time whereas the microorganisms’ growth in mass is due to the mass transfer of...

The conditional quadrature method of moments (CQMOM) was introduced by Yuan and Fox (2011) [4] to reconstruct a velocity distribution function (VDF) from a finite set of its integer moments. The reconstructed VDF takes the form of a sum of weighted Dirac delta functions in velocity phase space, and provides a closure for the spatial flux term in th...

The hyperbolicity condition of the system of partial differential equations (PDEs) of the incompressible two-fluid model, applied to gas–liquid flows, is investigated. It is shown that the addition of a dispersion term, which depends on the drag coefficient and the gradient of the gas volume fraction, ensures the hyperbolicity of the PDEs, and prev...

The numerical simulation of multiphase chemically reacting flows is very challenging due to their multiscale nature. In this chapter, the focus is on disperse multiphase flows with a continuous phase (gas or liquid) surrounding one or more disperse phases (e.g., particle, drops, or bubbles). In such flows, the disperse phase is usually characterize...

The extended quadrature method of moments (EQMOM) for the solution of population balance equations (PBE) is implemented in the open-source computational fluid dynamic (CFD) toolbox OpenFOAM as part of the OpenQBMM project. The moment inversion procedure was designed (Nguyen et al., 2016) to maximize the number of conserved moments in the transporte...

We present a verification study of three simulation techniques for fluid–particle flows, including an Euler–Lagrange approach (EL) inspired by Jackson's seminal work on fluidized particles, a quadrature–based moment method based on the anisotropic Gaussian closure (AG), and the traditional two-fluid model. We perform simulations of two problems: pa...

An extended quadrature method of moments using the β kernel density function (β-EQMOM) is used to approximate solutions to the evolution equation for univariate and bivariate composition probability distribution functions (PDFs) of a passive scalar for binary and ternary mixing. The key element of interest is the molecular mixing term, which is des...

Many fluid–particle flows occurring in nature and in technological applications exhibit large variations in the local particle volume fraction. For example, in circulating fluidized beds there are regions where the particles are close-packed as well as very dilute regions where particle–particle collisions are rare. Thus, in order to simulate such...

An Euler-Euler anisotropic Gaussian approach (EE-AG) for simulating gas-particle flows, in which particle velocities are assumed to follow a multivariate anisotropic Gaussian distribution, is used to perform mesoscale simulations of homogeneous cluster-induced turbulence (CIT). A three-dimensional Gauss–Hermite quadrature formulation is used to cal...

An overview of the fundamental modeling aspects related to disperse multiphase flow is provided. For clarity and accessibility, the discussion is limited to two-phase flows where one phase is disperse (e.g., solid particles) and the other phase is continuous (i.e., a fluid). Because the ultimate goal is to understand the fundamental modeling aspect...

This contribution summarizes the state of the art concerning the simulation with computational fluid dynamics of reacting flows with a particular focus on the interaction between turbulent mixing and chemistry. When dealing with nonisothermal gas-phase reactions (e.g., combustion) and rapid reactions (e.g., acid–base) in the liquid phase, the calcu...

The accurate description and robust simulation, at relatively low cost, of global quantities (e.g. number density or volume fraction) as well as the size distribution of a population of fine particles in a carrier fluid is still a major challenge for many applications. For this purpose, two types of methods are investigated for solving the populati...

Turbulent mixing in the confined swirling flow of a multi-inlet vortex reactor (MIVR) was investigated using planar laser induced fluorescence (PLIF). The investigated Reynolds numbers based on the bulk inlet velocity ranged from 3290 to 8225, and the Schmidt number of the passive scalar was 1250. Measurements were taken in the MIVR at three differ...