A quadrature-based moment method for the evolution of the joint size-velocity number density function of a particle population

doi:10.1016/j.cpc.2021.108072

Computer Physics Communications

Volume 267, October 2021, 108072

https://doi.org/10.1016/j.cpc.2021.108072 Get rights and content

Abstract

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 developed for velocity distribution transport. This approach is combined with the conditional quadrature method of moments to incorporate the dependency of the NDF on particle size, leading to an efficient, stable, quadrature method that uses an analytical solution to determine the size-conditioned velocity moments. The incorporation of source terms accounting for aggregation, breakup, and collisions, as well as acceleration terms such as gravity and drag, is performed using a realizable ODE solver. The approach is then demonstrated by considering zero-dimensional cases to verify the correct integration of the source terms. A set of one-dimensional cases involving droplet evaporation and coalescence is used to validate the velocity-dependent source terms. A two-dimensional case of crossing jets of particles with different sizes is used to demonstrate the proposed method in the case of a polydisperse flow with inertial particles. All work has been implemented in the open-source framework OpenQBMM, based on OpenFOAM®.

Introduction

The approximate solution of kinetic equations describing the spatio-temporal evolution of a particle population has a wide range of applications, from non-equilibrium flows and rarefied gases to multiphase disperse flows. In cases where the population of particles under consideration is polydisperse, and the particle size can evolve due to phenomena such as aggregation, breakage, and growth, the distribution characterizing such a population is a joint size-velocity number density function (NDF). The assumptions regarding the velocity distribution of a population of particles fall into three categories: a) the zero-Stokes-number case in which particles immediately respond to changes in the velocity flow field; b) the case when the Stokes number is small, but not zero, creating a situation where particles of a given size locally move at the same velocity, while particles characterized by different sizes may have different velocities. This case can be modeled under the monokinetic assumption [1] by only considering the mean velocity conditioned on size; c) the case in which particles have significant inertia ( $St ≫ 1$ ), where some of these particles may not follow the fluid streamlines due to non-negligible inertia, leading to particle trajectory crossing [2]. In this situation, in the dense limit, intended as the condition when the particle volume fraction is sufficiently high to make collisions the dominant mechanism of momentum transfer, the velocity distribution function (VDF) tends towards the Maxwellian, and, in the dilute case, it shows non-equilibrium effects. In the latter condition, moments of the VDF of order higher than two, which represents the Navier-Stokes-Fourier limit, must be considered to satisfactorily describe the physics of the flow. A brief review of methods for these three ranges of Stokes number is presented below.

The evolution of a particle population with negligible inertia was studied in [3]. Several methods have been developed to describe flows under these conditions, including the method of classes [4] and the method of moments [5], [6], of which the quadrature method of moments (QMOM) is a well-known example [7], [8]. Example applications for which the assumption of zero Stokes number is acceptable when describing the evolution of particles with a tiny size are flash nano-precipitation [9] and soot formation processes [10].

The second case of a disperse phase with small but finite St, for which the monokinetic assumption holds, is often employed to describe flows with disperse entities made of low-density materials, such as dispersed bubbles, or small particles and droplets. A notable example of a mono-kinetic approach is the multi-fluid model [11], [12], [13], which was used by several authors in the context of the method of classes [14], [15], [16]. Monokinetic quadrature-based moment methods (QBMM), in which the size-velocity moments are used to approximate the underlying distribution, have also been developed [17], [18], [19], [20]. The first two of these methods using QBMM [17], [20] have been formulated using the direct quadrature method of moments (DQMOM), focusing on applications to sprays, while the other two [18], [19] use an adaptive inversion procedure to evolve the moments of a joint size-velocity distribution of a population of bubbles.

The case of particles with $St ≫ 1$ is typical of gas-particle flows and, more generally, of flows with inertial disperse entities. Such flows can be modeled either with Lagrangian methods [21], [22] or in a continuum framework, using the kinetic theory of granular flows (KTGF) [6], [23], [24], [25], [26], [27], [28], [29], [30] to describe the behavior of the disperse phase. Several formulations of KTGF exist, ranging from that used to derive models in the collisional limit [23], [24], [25], [26], [30], [31], where particle properties are transported and change due to collisions ( $Kn ≪ 1$ ), leading the velocity distribution to be Maxwellian, to methods developed explicitly to describe flows where collisions are not the principal method of property transport and evolution, such as in dilute particulate flows. Derivations of the former have primarily been done for monodisperse particle populations [23], [24] or fixed sizes [25], [26], [30], [32], [33]. Several of these derivations assume an isotropic velocity distribution function (VDF) where the mean velocity and granular energy completely describe the particle VDF. There have also been extensions of the KTGF where a polydisperse system of particles is evolved in space and time [31], [32], [34], but the evolution of the size distribution is neglected. Kong and Fox [35] have accounted for the evolution of the particle size distribution using quadrature methods to approximate the size distribution. More recent formulations, such as the QBMM in [36], have included the transport of higher-order moments to more accurately represent the VDF but still neglect changes in the size distribution. An entropic quadrature method of moments was proposed in [37] to combine the features of QBMM and entropy maximization, matching the moments under consideration. Such an entropic quadrature approach has been applied to the solution of the solution of the Boltzmann-BGK equation [37]. While these methods allow for improved predictions compared to the commonly used KTGF description with moments up to second-order [11], they do not consider the local evolution of the particle size.

Methods for describing the evolution of the joint size-velocity NDF and its coupling to a fluid phase, in the context of moment methods, were presented in [8], [18], [19]. However, these approaches were developed making simplifying assumptions, as in the case of the mono-kinetic form of the velocity NDF. For this reason, the present work focuses on the formulation and on the numerical aspects of a robust QBMM to approximate the joint size-velocity NDF lifting such restrictions and without imposing constraints on the NDF, aside from assuming it can be discretized as a weighted sum of multi-variate Dirac distributions, as typically done in the context of QBMM.

The remaining sections of this manuscript are organized as follows: in Sec. 2, the moment transport equations derived from the generalized population balance equation are presented. Sec. 3 describes the quadrature-based method of moment and the inversion procedure used to obtain an approximate NDF from the transported moments. The closures to both advection and source terms are derived, and their approximations are presented in Sec. 4. Sec. 5 describe the numerical procedure for the full solution to the moment transport equations. Finally, in Sec. 6, 0-D, 1-D, and 2-D test cases with two- and three-dimensional velocity distribution test cases are presented to demonstrate the capabilities and robustness of the moment inversion algorithm and solution procedure.

Section snippets

Governing equations

The generalized population balance equation (GPBE) used to describe the evolution of the joint size-velocity NDF of a polydisperse particulate system can be written as [8]: $\frac{\partial f}{\partial t} + \nabla_{x} \cdot (f v) + \nabla_{ξ} \cdot [G (ξ) f] + \nabla_{v} \cdot {[A (ξ, v) + g] f} = C,$ where $f = f (t, x, ξ, v)$ is the joint size-velocity NDF, which depends on time t, position x, particle size ξ, and particle velocity v. The term $\nabla_{x} \cdot (f v)$ represents the advection of the NDF in physical space, $\nabla_{ξ} \cdot [G (ξ) f]$ is the advection term with respect to the internal coordinate ξ, and $G ($

The quadrature method of moments

In order to close advection and source terms of Eq. (2.3), the size-velocity NDF is reconstructed using the quadrature method of moment (QMOM), in which $f (t, x, ξ, v)$ is approximated using a summation of N Dirac delta distributions: $f (ξ, v) \approx \sum_{α = 1}^{N} w_{α} δ (ξ - ξ_{α}) δ (v - v_{α}),$ where N is set to achieve the required accuracy of the method, $w_{α}$ are the quadrature weights, $ξ_{α}$ the size abscissae, and $v_{α}$ the velocity abscissae. Using the approximation in Eq. (3.1), the moments of the NDF are written in terms of

Closures to the moment transport equations

This section describes the details of the advection and source terms' closures, based on the quadrature definition in Sec. 3.

Numerical procedure

The approach described in the previous sections was implemented in the OpenQBMM framework [51], a derivative work of the OpenFOAM® toolbox [58], [59]. The solution to the moment transport equations is performed using operator splitting for advection, acceleration terms, and source terms. By doing this, moment realizability can be ensured at each operator split. In addition, it also allows independent treatment of terms that pose numerical challenges. Specifically, when time scales associated

Results

The verification of the inversion procedure and the numerical method will be done in four parts. First, the time integration of source terms is validated using 0-D test cases for population balance source [51], [60], [61] and polydisperse collisional source terms [36], [62]. Second, the method will be applied to a collection of polydisperse droplet cases [40]. Third, the ability to handle crossing jets with particles of multiple sizes is examined, showing that multiple particle velocities can

Conclusions

A novel method for obtaining the solution to the evolution of the four-dimensional, size-velocity NDF transport has been presented and validated using simple but realistic test cases. The presented model has been derived from the number density function evolution equation, and the relevant source terms have been modified to allow for the volume fraction-based definition of the moments. The source terms were validated using 0-D cases for both PBE [51] and bidisperse collisions [62] and match

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors would like to gratefully acknowledge the support of the U.S. National Science Foundation under the SI²–SSE award NSF – ACI 1440443. Simulations were performed using the HPC cluster funded by the NSF-MRI award no. 1726447.

References (64)

O. Desjardins et al.
J. Comput. Phys.
(2008)
J.C. Cheng et al.
J. Colloid Interface Sci.
(2010)
A. Zucca et al.
Chem. Eng. Sci.
(2006)
R.T. Lahey
Nucl. Eng. Des.
(2005)
R. Bannari et al.
Comput. Chem. Eng.
(2008)
R.O. Fox et al.
J. Comput. Phys.
(2008)
J. Capecelatro et al.
J. Comput. Phys.
(2013)
Z. Chao et al.
Chem. Eng. Sci.
(2011)
N. Böhmer et al.
J. Comput. Phys.
(2020)
R.O. Fox
J. Comput. Phys.
(2008)

C. Yuan et al.

J. Comput. Phys.

(2011)

R.O. Fox et al.

J. Comput. Phys.

(2018)

D.L. Wright

J. Aerosol Sci.

(2007)

A. Passalacqua et al.

Comput. Phys. Commun.

(2020)

V. Vikas et al.

J. Comput. Phys.

(2011)

V. Vikas et al.

J. Comput. Phys.

(2013)

A. Passalacqua et al.

Chem. Eng. Sci.

(2018)

T.T. Nguyen et al.

J. Comput. Phys.

(2016)

M. Vanni

J. Colloid Interface Sci.

(2000)

E. Madadi-Kandjani et al.

Chem. Eng. Sci.

(2015)

F. Laurent et al.

J. Comput. Phys.

(2004)

M. Massot

A.D. Randolph

Can. J. Chem. Eng.

(1964)

D. Ramkrishna

Population Balances Theory and Applications to Particulate Systems in Engineering

(2000)

H. Struchtrup

Macroscopic Transport Equations for Rarefied Gas Flows: Approximation Methods in Kinetic Theory

(2005)

S. Chapman et al.

The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases

(1990)

R. McGraw

Aerosol Sci. Technol.

(1997)

D.L. Marchisio et al.

Computational Models for Polydisperse Particulate and Multiphase Systems

(2013)

D.A. Drew

Annu. Rev. Fluid Mech.

(1983)

R. Jackson

The Dynamics of Fluidized Particles

(2000)

F. Laurent et al.

Combust. Theory Model.

(2001)

T. Frank et al.

Cited by (3)

A Eulerian population balance/Monte Carlo approach for simulating laminar aluminum dust flames
2024, Particuology
Recently, metal powders have been conceptualized as carbon-free recyclable energy carriers that may form a cornerstone of a sustainable energy economy. The combustion of metal dusts in oxidizing atmospheres is exothermal and yields oxide particles that could, potentially, be retrieved and, subsequently, recharged by conversion to pure metals using green primary energy sources. As a step towards a predictive tool for designing metal dust combustors, we present a fully Eulerian modelling approach for laminar particle-laden reactive flows that is, conceptually, based on a population balance description of the dispersed particles and relies on a stochastic Eulerian solution strategy. While the population balance equation (PBE) is formulated for the number-weighted distribution of characteristic properties among all particles near a spatial location, it is kinetically informed by the rates at which mass, momentum and heat are exchanged between the carrier gas and the particulate phase on the single particle level. Within the scope of the Eulerian Monte Carlo solution scheme, the property distribution is discretely represented in terms of the total number density and a finite number of property samples and the computational work is channelled towards the Eulerian estimation of mean particle properties.
For the case of reactive aluminum particles, we combine a kinetic description of the gas-particle heat and mass transfer with a transport-limited continuum formulation to obtain rate expressions that are valid across the entire particle size range from the free molecular through the continuum regime. Besides velocity, the particle properties include only the particle mass, temperature and oxide mass fraction. This set of thermochemical degrees of freedom is retained also as phase transitions due to melting occur, drawing on a smooth blend of the solid and liquid thermodynamic and material properties. The particle-level formulation encompasses aluminum evaporation, surface oxidation, scavenging of oxide smoke, oxide evaporation/dissociation and radiation. After investigating how these effects translate, through the PBE, to the particle population level and affect the combustion in a homogeneous dust reactor, we analyze the combustion of an aluminum dust in a counterflow flame and validate predictions of the particles’ centerline velocity profile and the flame speed by comparison with available experimental data. Concomitantly, nitrogen oxide emissions are investigated along with the particle burnout and outlet size distribution.
The generalized quadrature method of moments
2023, Journal of Aerosol Science
Citation Excerpt :
The same issue is faced when solving the moment source terms numerically (Nguyen et al., 2016), so that, generally speaking, we expect that GQMOM will generate more robust, and more accurate, numerical solvers for the multi-scale, multi-physics codes used in real-world applications (Bryngelson et al., 2020; Heylmun et al., 2021, 2019; Ilgun et al., 2021; Passalacqua et al., 2018; Wick et al., 2017).
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 $i = 0, 1, \dots, n$ that are uniquely defined by the set of $2 n$ moments up to order $2 n - 1$ . The moment of order $2 n$ is fixed to the boundary of moment space such that the distribution function is approximated by a sum of $n$ Dirac delta functions. Using the recursion coefficients of the orthogonal polynomials for $i > n \geq 1$ , the generalized quadrature method of moments (GQMOM) extends the quadrature representation to a sum of $N > n$ terms using the same moments as QMOM. In doing so, the known moments are preserved and higher-order moments correspond to a distribution function in the interior of moment space. Here, GQMOM closures for distributions on $R$ , $R^{+}$ , and $(0, 1)$ are defined and analyzed. Generally speaking, GQMOM provides a more accurate moment closure than QMOM without increasing the number of moments and at nearly the same computational cost.
Quadrature-based moment methods for particle-laden flows
2022, Modeling Approaches and Computational Methods for Particle-laden Turbulent Flows
The method of moments for the solution of a kinetic equation is introduced in this chapter. Generalizations of the kinetic equation for problems involving the evolution of the particle size and strongly coupled heat transfer phenomena are introduced. Linearized forms of the collision operator such as the Bhatnagar–Gross–Krook (BGK) and ellipsoidal statistical BGK models are discussed. The dimensionless form of the Boltzmann kinetic equation is considered to illustrate the role of the Knudsen number on the choice of a computational model for particle-laden flows. Specific forms of the particle velocity distribution are discussed in relation to the Stokes number of the disperse phase. The concept of moment of a multivariate number density function (NDF) is introduced and the physical interpretation of some of the moments of the NDF is discussed. Particular attention is paid to defining the concept of realizability of a moment vector, presenting criteria to determine if a moment vector is realizable on the support where it is defined. Causes of unrealizable moment vectors are discussed and realizable numerical methods to integrate the conservation equations for a moment vector are discussed. The limitations of low-order moment closures are then illustrated, and the need for hyperbolic high-order closures to predict the correct physical phenomena observed in particle-laden flows with inertial particles is explained. A generic formulation of the conditional quadrature method of moments for joint size–velocity distributions is then shown as example of multivariate quadrature-based moment closure. Finally, the second-order anisotropic Gaussian closure is presented.

^☆: The review of this paper was arranged by Prof. Hazel Andrew.

View full text

A quadrature-based moment method for the evolution of the joint size-velocity number density function of a particle population☆

Abstract

Introduction

Section snippets

Governing equations

The quadrature method of moments

Closures to the moment transport equations

Numerical procedure

Results

Conclusions

Declaration of Competing Interest

Acknowledgements

J. Comput. Phys.

J. Colloid Interface Sci.

Chem. Eng. Sci.

Nucl. Eng. Des.

Comput. Chem. Eng.

J. Comput. Phys.

J. Comput. Phys.

Chem. Eng. Sci.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Aerosol Sci.

Comput. Phys. Commun.

J. Comput. Phys.

J. Comput. Phys.

Chem. Eng. Sci.

J. Comput. Phys.

J. Colloid Interface Sci.

Chem. Eng. Sci.

J. Comput. Phys.

Can. J. Chem. Eng.

Population Balances Theory and Applications to Particulate Systems in Engineering

Macroscopic Transport Equations for Rarefied Gas Flows: Approximation Methods in Kinetic Theory

The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases

Aerosol Sci. Technol.

Computational Models for Polydisperse Particulate and Multiphase Systems

Annu. Rev. Fluid Mech.

The Dynamics of Fluidized Particles

Combust. Theory Model.