Existence and multiplicity for elliptic problems with quadratic growth in the gradient

We show that a class of divergence-form elliptic problems with quadratic growth in the gradient and non-coercive zero order terms are solvable, under essentially optimal hypotheses on the coefficients in the equation. In addition, we prove that the solutions are in general not unique. The case where the zero order term has the opposite sign was already intensively studied and the uniqueness is the rule.Comment: To appear in Comm. PD

Diffeomorphism-invariant properties for quasi-linear elliptic operators

For quasi-linear elliptic equations we detect relevant properties which remain invariant under the action of a suitable class of diffeomorphisms. This yields a connection between existence theories for equations with degenerate and non-degenerate coerciveness.Comment: 16 page

From Computational Fluid Dynamics to Structure Interpretation via Neural Networks: An Application to Flow and Transport in Porous Media

The modeling of flow and transport in porous media is of the utmost importance in many chemical engineering applications, including catalytic reactors, batteries, and CO2 storage. The aim of this study is to test the use of fully connected (FCNN) and convolutional neural networks (CNN) for the prediction of crucial properties in porous media systems: The permeability and the filtration rate. The data-driven models are trained on a dataset of computational fluid dynamics (CFD) simulations. To this end, the porous media geometries are created in silico by a discrete element method, and a rigorous setup of the CFD simulations is presented. The models trained have as input both geometrical and operating conditions features so that they could find application in multiscale modeling, optimization problems, and in-line control. The average error on the prediction of the permeability is lower than 2.5%, and that on the prediction of the filtration rate is lower than 5% in all the neural networks models. These results are achieved with at least a dataset of ~ 100 CFD simulations

Computational analysis of transport in three-dimensional heterogeneous materials: An OpenFOAM®-based simulation framework

Porous and heterogeneous materials are found in many applications from composites, membranes, chemical reactors, and other engineered materials to biological matter and natural subsurface structures. In this work we propose an integrated approach to generate, study and upscale transport equations in random and periodic porous structures. The geometry generation is based on random algorithms or ballistic deposition. In particular, a new algorithm is proposed to generate random packings of ellipsoids with random orientation and tunable porosity and connectivity. The porous structure is then meshed using locally refined Cartesian-based or unstructured strategies. Transport equations are thus solved in a finite-volume formulation with quasi-periodic boundary conditions to simplify the upscaling problem by solving simple closure problems consistent with the classical theory of homogenisation for linear advection–diffusion–reaction operators. Existing simulation codes are extended with novel developments and integrated to produce a fully open-source simulation pipeline. A showcase of a few interesting three-dimensional applications of these computational approaches is then presented. Firstly, convergence properties and the transport and dispersion properties of a periodic arrangement of spheres are studied. Then, heat transfer problems are considered in a pipe with layers of deposited particles of different heights, and in heterogeneous anisotropic materials

SIMULATION OF FLOW AND PARTICLE TRANSPORT AND DEPOSITION IN POROUS MEDIA WITH COMPUTATIONAL FLUID DYNAMICS

The simulation of transport and deposition of colloidal particles in porous media finds important applications in many engineering and environmental problems, such as particle filtration, catalytic processes carried out in filter beds, chromatographic separation and aquifer remediation. This study focuses in particular on remediation of contaminated groundwater via direct injection of nano-sized zerovalent iron particles, which have been shown to be able to efficiently degrade a large variety of contaminants. Application of this technology on full scale applications poses a number of challenges, the most important of which regards the mobility of the particles and their delivery to the contaminated site in the soil. Particles migration is usually quantitatively expressed by a single parameter: the deposition efficiency in the porous bed, whose theoretical reference lies in the classical colloid filtration theory, which moreover further subdivides the process of deposition in the three mechanisms by which particles can reach the solid grain: Brownian diffusion, steric interception, and gravitational sedimentation. This theory, however, has been developed only for very simple geometrical representations of the porous media and a narrow range of fluid conditions. The difficulties in investigating this kind of systems from the experimental point of view have prevented the development of accurate models able to account for the high degree of complexity which characterizes a porous medium, both in the grain arrangement and in their shape. The aim of this study is therefore to simulate the transport of the nanoparticles and their interaction with the porous media (at the microscopic scale), in order to improve the current understanding of these phenomena and obtain predictive models for the deposition efficiency of the colloids on the surface of the grains constituting the porous medium; moreover, eventually, to evaluate the effectiveness of the zerovalent iron technology. Several two and three dimensional microscale (the order of millimiters) representations of grain packings with different degrees of complexity were analyzed. First, two dimensional random arrangements of spheres were considered. Then, the analysis was extended to domains reconstructed from SEM images of a real porous medium. The work was then expanded in three dimensions, first considering simplified domains constituted by irregular packings of spheres, and finally geometries constituted by grains of realistic shapes. These last geometries were created using an algorithm simulating the grain sedimentation process in porous media (Settledyn). Flow field and particle transport was then investigated using finite volume CFD codes (Fluent and OpenFoam), solving the Navier-Stokes equations for the flow and using an Eulerian approach for the colloid transport, eventually obtaining, for each case, an estimate of the colloidal transport efficiency. After having validated the methodology used in this work by comparing our results with proved analytical results available for simplified cases, new predictive equations for each of the individual contributions of the three deposition mechanisms were derived, highlighting the differences from the theoretical model due to the wider range of operating conditions investigated and/or the different geometrical characteristics of the porous media

CFD-DEM characterization and population balance modelling of a dispersive mixing process

This work investigates the breakup dynamics of solid agglomerates in a polymer compounding operation, by using computational fluid dynamics (CFD) simulations together with discrete element method (DEM) simulations. CFD simulations are used to compute the flow field and the shear stress distribution inside a 2D section of a typical internal mixer for polymer compounding. DEM simulations are instead used to predict the mechanical response of the agglomerates and to detect the critical viscous shear stress needed to induce breakup. DEM breakup data and viscous stress distributions are correlated by a first–time passage–statistics and used to calibrate a population balance model. The work returned detailed insights into the flow field characteristics and into the dispersive mixing kinetics. The simulation strategy herein reported can be adapted to study generic solid–liquid disperse flows in which the breakup of the solid phase is found at the core of the system behaviour

CFD-PBE modelling of continuous Ni-Mn-Co hydroxide co-precipitation for Li-ion batteries

A modelling framework is proposed to simulate the co-precipitation of Ni-Mn-Co hydroxide as precursor of cathode material for lithium-ion batteries. It integrates a population balance equation with computational fluid dynamics to describe the evolution of the particle size in (particularly continuous) co-precipitation processes. The population balance equation is solved by employing the quadrature method of moments. In addition, a multi-environment micromixing model is employed to consider the potential effect of molecular mixing on the fast co-precipitation reaction. The modelling framework is used to investigate the co-precipitation of Ni0.8Mn0.1Co0.1(OH)2 in a multi-inlet vortex micromixer, as a suitable candidate for the study of fast co-precipitation processes in continuous mode. Finally, the simulation results are discussed, and the role of the different phenomena involved in the formation and evolution of particles is identified by inspecting the predicted trends