ON THE IMPLEMENTATION OF MOMENT TRANSPORT EQUATIONS IN OPENFOAM TO PRESERVE CONSERVATION, BOUNDEDNESS AND REALIZABILITY

Different industrial scale multiphase systems can be successfully described by considering their polydispersity (e.g. particle/droplet/bubble size and velocity distributions) and phase coupling issues are properly overcome only by considering the evolution in space and time of such distributions, dictated by the so-called Generalized Population Balance Equation (GPBE). A computationally efficient approach for solving the GPBE is represented by the quadrature-based moment methods, where the evolution of the entire particle/droplet/bubble population is recovered by tracking some specific moments of the distribution and the quadrature approximation is used to solve the "closure problem" typical of moment-based methods. In this contribution some crucial computational and numerical details concerning the implementation of these methods into the opensource Computational Fluid Dynamics (CFD) code OpenFOAM are discussed. These aspects are in fact very often overlooked, resulting in implementations that do not satisfy the properties of conservation, realizability and boundedness. These constraints have to be satisfied in a consistent way, with respect to what done with the other conserved transported variables (e.g. volume fraction of the disperse phase) also when higher-order discretization schemes are used. These issues are illustrated on examples taken on our work on the simulation of fluid-fluid multiphase system

A Normalized and Extended Correlation Equation for Predicting Single-Collector Efficiency in Physicochemical Filtration in Saturated Porous Media

The colloidal transport and deposition are phenomena involved in different engineering problems. In the environmental engineering field the use of micro- and nano-scale zerovalent iron (M-NZVI) is one of the most promising technologies for groundwater remediation. Colloid deposition is normally studied from a micro scale point of view and the results are then implemented in macro scale models that are used to design field-scale applications. The single collector efficiency concept predicts particles deposition onto a single grain of a complex porous medium in terms of probability that an approaching particle would be retained on the solid grain. Different approaches and models are available in literature to predict it, but most of them fail in some particular conditions (e.g. low fluid velocity and/or very small or very big particle dimension) because they predict efficiency values exceeding unity. By analysing particle fluxes and deposition mechanisms and performing a mass balance on the entire domain, the traditional definition of efficiency was reformulated and a novel total flux normalized correlation equation is proposed for predicting single-collector efficiency under a broad range of parameters. The new equation has been formulated starting from a combination of Eulerian and Lagrangian numerical COMSOL Multiphysics® simulations, performed under Smoluchowski-Levich conditions in a geometry which consists of a sphere enveloped by a cylindrical control volume (Figure 1). The normalization of the deposited flux is performed accounting for all of the particles entering into the control volume through all transport mechanisms (not just the upstream convective flux as conventionally done) to provide efficiency values lower than one under any possible combination of transport mechanisms. The particle fluxes onto the collector and through the control volume have been described mathematically as a summation of terms. In order to guarantee the independence of each term, the correlation equation is derived through a rigorous hierarchical parameter estimation process, accounting for single and mutual interacting transport mechanisms. The new correlation equation provides efficiency values lower than one over a wide range of parameters (Figure 2) and it is valid both for point and finite-size particles. Moreover the correlation equation is extended to include porosity dependence and reduced forms are also proposed by elimination of the less relevant terms without losing the main features of the full equation

An extended and total flux normalized correlation equation for predicting single-collector efficiency

In this study a novel total flux normalized correlation equation is proposed for predicting single-collector efficiency under a broad range of parameters. The correlation equation does not exploit the additivity approach introduced by Yao et al. (1971), but includes mixed terms that account for the mutual interaction of concomitant transport mechanisms (i.e., advection, gravity and Brownian motion) and of finite size of the particles (steric effect). The correlation equation is based on a combination of Eulerian and Lagrangian simulations performed, under Smoluchowski–Levich conditions, in a geometry which consists of a sphere enveloped by a cylindrical control volume. The normalization of the deposited flux is performed accounting for all of the particles entering into the control volume through all transport mechanisms (not just the upstream convective flux as conventionally done) to provide efficiency values lower than one over a wide range of parameters. In order to guarantee the independence of each term, the correlation equation is derived through a rigorous hierarchical parameter estimation process, accounting for single and mutual interacting transport mechanisms. The correlation equation, valid both for point and finite-size particles, is extended to include porosity dependency and it is compared with previous models. Reduced forms are proposed by elimination of the less relevant terms

Quadrature-based moment methods for the simulation of turbulent polydisperse multiphase systems

This talk will present and discuss a class of approaches for the simulation of turbulent multiphase flows, called quadrature-based moment methods (QBMM). These methods are based on an Eulerian description of the multiphase system, where polydispersity is modeled through a number density function (NDF). Since only transport equations for the (pure and mixed) moments of the NDF are solved, the NDF must be reconstructed from the moments. This is done by using basis functions (such as Dirac delta functions) that result in the use of quadrature approximations for overcoming the so-called closure problem. The talk is divided in two parts. In the first part the Generalized Population Balance Equation (GPBE), that dictates the evolution of the NDF, will be presented and its relationship with similar balance equations (e.g., Boltzmann, Williams, particle dynamics and population balance equations) will be highlighted. Moreover, the derivation from the GPBE of the characteristic equations of the Eulerian-Eulerian multifluid models, widely adopted in many commercial computational fluid dynamics (CFD) codes will be analyzed and its limitations discussed. In the second part of the talk some practical cases will be presented. In particular, the use of our implementations of the quadrature method of moment (QMOM), conditional quadrature method of moments (DQMOM) and direct quadrature method of moments (DQMOM) in Ansys/Fluent, Openfoam and TransAT will be illustrated. Particular attention will be devoted to the coupling with turbulence models, both with the Reynolds-average Navier-Stokes equations (RANS) and large-eddy simulation (LES) approaches. The issues related to the implementation of QMOM, CQMOM and DQMOM in CFD codes based on the finite-volume method will also be considered. The problems of moment corruption, realizability and conservation and the relative strategies to overcome them will be presented and discussed. The potentials of QBMM will be eventually demonstrated by presenting data concerning the simulation of mixing and segregation in fluidized beds, bubble coalescence, break-up and mass transfer in bubble columns and gas-liquid stirred tanks and of turbophoresis in turbulent particle-laden flow

Normalization and extension of single-collector efficiency correlation equation

The colloidal transport and deposition are important phenomena involved in many engineering problems. In the environmental engineering field the use of micro- and nano-scale zerovalent iron (M-NZVI) is one of the most promising technologies for groundwater remediation. Colloid deposition is normally studied from a micro scale point of view and the results are then implemented in macro scale models that are used to design field-scale applications. The single collector efficiency concept predicts particles deposition onto a single grain of a complex porous medium in terms of probability that an approaching particle would be retained on the solid grain. In literature, many different approaches and equations exist to predict it, but most of them fail under specific conditions (e.g. very small or very big particle size and very low fluid velocity) because they predict efficiency values exceeding unity. By analysing particle fluxes and deposition mechanisms and performing a mass balance on the entire domain, the traditional definition of efficiency was reformulated and a novel total flux normalized correlation equation is proposed for predicting single-collector efficiency under a broad range of parameters. It has been formulated starting from a combination of Eulerian and Lagrangian numerical simulations, performed under Smoluchowski- Levich conditions, in a geometry which consists of a sphere enveloped by a control volume. In order to guarantee the independence of each term, the correlation equation is derived through a rigorous hierarchical parameter estimation process, accounting for single and mutual interacting transport mechanisms. The correlation equation provides efficiency values lower than one over a wide range of parameters and is valid both for point and finite-size particles. A reduced form is also proposed by elimination of the less relevant terms

Precipitation in turbulent fluids

This thesis concerns the simulation with computational fluid dynamics and population balances of a precipitation proces

SIMULATION OF A REACTIVE GAS-LIQUID SYSTEM WITH QUADRATURE-BASED MOMENTS METHOD

The description of the interaction between fluid dynamics and fast chemical reactions in gas-liquid systems is complicated by the fact that the gas phase is poly-dispersed, namely it is constituted by bubbles characterized by a distribution of velocity, size and composition values. Phase coupling can be successfully described only if the modeling approach acknowledges the existence of this distribution, whose evolution in space and time is governed by the so-called Generalized Population Balance Equation (GPBE). A computationally efficient approach for solving the GPBE is represented by the Quadrature-Based Moment Methods (QBMM), where the evolution of the entire bubble population is recovered by tracking some specific moments of the distribution. In the present work, one of these methods, the Conditional Quadrature Method of Moments (CQMOM) has been implemented in the OpenFOAM two-fluid solver compressibleTwoPhaseEulerFoam , to simulate a chemically reacting gas-liquid system. To reduce the computational time and increase stability, a second-order operator-splitting technique for the solution of the chemically reacting species was also implemented, allowing to solve the different processes involved with their own time-scale. This modeling approach is here validated by comparing predictions with experiments, for the chemical absorption of CO 2 in NaOH solution, performed in a rectangular bubble column