Optimal Moment Sets for Multivariate Direct Quadrature Method of Moments

The direct quadrature method of moments (DQMOM) can be employed to close population balance equations (PBEs) governing a wide class of multivariate number density functions (NDFs). Such equations occur over a vast range of scientific applications, including aerosol science, kinetic theory, multiphase flows, turbulence modeling, and control theory, to name just a few. As the name implies, DQMOM uses quadrature weights and abscissas to approximate the moments of the NDF, and the number of quadrature nodes determines the accuracy of the closure. For nondegenerate univariate cases (i.e., a sufficiently smooth NDF), the N weights and N abscissas are uniquely determined by the first 2N non-negative integer moments of the NDF. Moreover, an efficient product-difference algorithm exists to compute the weights and abscissas from the moments. In contrast, for a d-dimensional NDF, a total of (1 + d)N multivariate moments are required to determine the weights and abscissas, and poor choices for the moment set can lead to nonunique abscissas and even negative weights. In this work, it is demonstrated that optimal moment sets exist for multivariate DQMOM when N ) nd quadrature nodes are employed to represent a d-dimensional NDF with n ) 1-3 and d ) 1-3. Moreover, this choice is independent of the source terms in the PBE governing the time evolution of the NDF. A multivariate Fokker-Planck equation is used to illustrate the numerical properties of the method for d ) 3 with n ) 2 and 3

Quadrature-Based Moment Model for Moderately Dense Polydisperse Gas-Particle Flows

A quadrature-based moment model is derived for moderately dense polydisperse gas-particle flows starting from the inelastic Boltzmann-Enskog kinetic equation including terms for particle acceleration (e.g., gravity and fluid drag). The derivation is carried out for the joint number density function, f(t,x,m,u), of particle mass and velocity, and thus, the model can describe the transport of polydisperse particles with size and density differences. The transport equations for the integer moments of the velocity distribution function are derived in exact form for all values of the coefficient of restitution for particle-particle collisions. For particular limiting cases, the moment model is shown to be consistent with hydrodynamic models for gas-particle flows. However, the moment model is more general than the hydrodynamic models because its derivation does not require that the particle Knudsen number (and Mach number) be small

Validation of Two-Fluid Simulations of a Pseudo-Two-Dimensional Bubble Column with Uniform and Nonuniform Aeration

The multiphase flow simulated in this work corresponds to the pseudo-2D bubble-column experiments at Delft University of Technology (Harteveld et al. Can. J. Chem. Eng. 2003, 81, 389-394). As in the work of Monahan and Fox (AIChE J. 2007, 53, 9-18), the complete set of interphase force models includes drag, added-mass, lift, rotation, and strain. The simulation results are presented in the form of comparisons with the experimental data for the time-averaged gas holdup and the instantaneous and time-averaged air and liquid velocity fields. Overall, the qualitative and quantitative comparisons between experiments and simulations are satisfactory for both uniform and nonuniform aeration. In particular, the model predicts the flow patterns observed in the experiments, but in some cases at slightly different values of the amount of aeration. In the latter cases, it is shown that changing the model parameters does not improve the agreement with experiments. However, changing the liquid-velocity boundary condition from zero stress to zero slip leads to a small improvement

Numerical simulation of turbulent gas-particle flow in a riser using a quadrature-based moment method

Gas-particle flows are used in many industrial applications in the energy, oil and gas fields, such as coal gasification, production of light hydrocarbons by fluid catalytic cracking, catalytic combustion and different treatments aiming to reduce or eliminate pollutants. The particle phase of a gas-particle flow is described by analogy to a granular gas, by finding an approximate solution of the kinetic equation in the velocity-based number density function. In the recent past, many studies have been published on the mathematical modeling of gas-particle flows using hydrodynamic models (e.g. Enwald et al. 1996), where Navier-Stokes-type equations are solved to describe the particle phase as a continuum, computing its stress tensor using moment closures from kinetic theory (Gidaspow 1994). These closures, however, are obtained assuming that the flow is dominated by collisions and near equilibrium, which corresponds to considering a very small particle-phase Knudsen number. This assumption leads to inconsistencies and erroneous predictions of physical phenomena when these models are applied to dilute fluid-particle flows, where rarefaction effects are not negligible. In these flows, the wall Knudsen layers extend inside the bulk of the fluid, and cannot be accounted for with the simple addition of partial-slip boundary conditions. Recently Desjardin et al. (2008) showed that two-fluid models are unable to correctly capture particle trajectory crossing, seriously compromising their ability to correctly describe any velocity moment for finite Stokes numbers. These authors clarified that the particle segregation captured by two-fluid models for finite Knudsen numbers is artificially high due to their mathematical formulation, which leads to the formation of delta-shocks. In order to overcome these shortcomings, Fox (2008) developed a third-order quadrature-based moment method for dilute gas-particle flows, which has been successfully coupled to a fluid solver to compute dilute and moderately dilute gas-particle flows by Passalacqua et al. (2010) in two dimensions. These authors validated their model against Euler-Lagrange and two-fluid simulations. In this work, the fully coupled quadrature-based fluid-particle code described in Passalacqua et al. (2010) is applied to simulate turbulent gas-particle flow in the riser described by He et al. (2009), using a three-dimensional configuration. This application shows the predictive capabilities and the robustness of the quadrature-based moment method to predict the behavior of gas-particle flows in accordance with experiments (He et al. 2009)

Confocal imaging of laminar and turbulent mixing in a microscale multi-inlet vortex nanoprecipitation reactor

Mass production of functional nanoparticles may be realized through flash nanoprecipiation in microscale reactors such as the multi-inlet vortex reactor (MIVR). A comprehensive understanding of mixing in the MIVR is required for process control and reactor design. Mixing in the MIVR is studied using a technique coupling laser induced fluorescence with confocal laser scanning microscopy. It is shown to provide meaningful qualitative and statistical data of the scalar field for analysis and comparison with numerical simulations. Data were collected for four flow rates, showing that mixing is incomplete even at the highest flow rate

A microscale multi-inlet vortex nanoprecipitation reactor: Turbulence measurement and simulation

Microscale reactors capable of generating turbulent flow are used in Flash NanoPrecipitation, an approach to produce functional nanoparticles with unique optical, mechanical and chemical properties. Microreactor design and optimization could be greatly enhanced by developing reliable computational models of the nanoprecipitation process. A microscale multi-inlet vortex nanoprecipitation reactor was investigated using microscopic particle image velocimetry and computational fluid dynamics. Velocity data such as the mean velocity and turbulent kinetic energy displayed good agreement between experiment and simulation over flow conditions ranging from fully laminar to turbulent, demonstrating the accuracy of the simulation model over the entire turbulent transition range

Simulation of Mono- and Bidisperse Gas-Particle Flow in a Riser with a Third-Order Quadrature-Based Moment Method

Gas-particle flows can be described by a kinetic equation for the particle phase coupled with the Navier−Stokes equations for the fluid phase through a momentum exchange term. The direct solution of the kinetic equation is prohibitive for most applications due to the high dimensionality of the space of independent variables. A viable alternative is represented by moment methods, where moments of the velocity distribution function are transported in space and time. In this work, a fully coupled third-order, quadrature-based moment method is applied to the simulation of mono- and bidisperse gas-particle flows in the riser of a circulating fluidized bed. Gaussian quadrature formulas are used to model the unclosed terms in the moment transport equations. A Bhatnagar−Gross−Krook (BGK) collision model is used in the monodisperse case, while the full Boltzmann integral is adopted in the bidisperse case. The predicted values of mean local phase velocities, rms velocities, and particle volume fractions are compared with the Euler−Lagrange simulations and experimental data from the literature. The local values of the time-average Stokes, Mach, and Knudsen numbers predicted by the simulation are reported and analyzed to justify the adoption of high-order moment methods as opposed to models based on hydrodynamic closures

Dynamics of scalar dissipation in isotropic turbulence: a numerical and modelling study

The physical mechanisms underlying the dynamics of the dissipation of passive scalar fluctuations with a uniform mean gradient in stationary isotropic turbulence are studied using data from direct numerical simulations (DNS), at grid resolutions up to 5123. The ensemble-averaged Taylor-scale Reynolds number is up to about 240 and the Schmidt number is from ⅛ to 1. Special attention is given to statistics conditioned upon the energy dissipation rate because of their important role in the Lagrangian spectral relaxation (LSR) model of turbulent mixing. In general, the dominant physical processes are those of nonlinear amplification by strain rate fluctuations, and destruction by molecular diffusivity. Scalar dissipation tends to form elongated structures in space, with only a limited overlap with zones of intense energy dissipation. Scalar gradient fluctuations are preferentially aligned with the direction of most compressive strain rate, especially in regions of high energy dissipation. Both the nature of this alignment and the timescale of the resulting scalar gradient amplification appear to be nearly universal in regard to Reynolds and Schmidt numbers. Most of the terms appearing in the budget equation for conditional scalar dissipation show neutral behaviour at low energy dissipation but increased magnitudes at high energy dissipation. Although homogeneity requires that transport terms have a zero unconditional average, conditional molecular transport is found to be significant, especially at lower Reynolds or Schmidt numbers within the simulation data range. The physical insights obtained from DNS are used for a priori testing and development of the LSR model. In particular, based on the DNS data, improved functional forms are introduced for several model coefficients which were previously taken as constants. Similar improvements including new closure schemes for specific terms are also achieved for the modelling of conditional scalar variance

Turbulence in a microscale planar confined impinging-jets reactor

Confined impinging-jets reactors (CIJR) offer many advantages for rapid chemical processing at the microscale in applications such as precipitation and the production of organic nanoparticles. It has been demonstrated that computational fluid dynamics (CFD) is a promising tool for ‘‘experiment-free’’ design and scale-up of such reactors. However, validation of the CFD model used for the microscale turbulence applications requires detailed experimental data on the unsteady flow, the availability of which has until now been very limited. In this work, microscopic particle-image velocimetry (microPIV) techniques were employed to measure the instantaneous velocity field for various Reynolds numbers in a planar CIJR. In order to illustrate the validation procedure, the performance of a particular CFD model, the two-layer k–3 model, was evaluated by comparing the predicted flow field with the experimental data. To our knowledge, this study represents the first attempt to directly measure and quantify velocity and turbulence in a microreactor and to use the results to validate a CFD model for microscale turbulent flows

Three-dimensional conditional hyperbolic quadrature method of moments

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 spatial flux term in the corresponding moment equations derived from a kinetic equation for the 3-D VDF. Here, CHyQMOM is extended for 3-D velocity phase space using the modified conditional quadrature method of moments with 16 (or 23) trivariate velocity moments up to fourth order. In order to verify the numerical implementation, it is applied to simulate several canonical particle-laden flows including crossing jets, cluster-induced turbulence (CIT), and vertical channel flow. The numerical results are compared with those from Euler–Lagrange simulations and two other quadrature-based moment methods, namely, anisotropic Gaussian (AG) and 8-node tensor-product (TP) quadrature. The relative advantages and disadvantages of each method are discussed. The crossing-jet problem highlights that CHyQMOM handles particle crossing more accurately than AG. For CIT, the results from all methods are similar, but the computational cost of TP is significantly larger than AG and CHyQMOM, both of which have nearly the same cost