An incompressible three-dimensional multiphase particle-in-cell model for dense particle flows

The motion of instantaneous and maintained releases of buoyant fluid through shallow permeable layers of large horizontal extent is described by a nonlinear advectiondiffision equation. This equation admits similarity solutions which describe the release of one fluid into a horizontal porous layer initially saturated with a second immiscible fluid of different density. Asymptotically, a finite volume of fluid spreads as tl/". On an inclined surface, in a layer of uniform permeability, a finite volume of fluid propagates steadily alongslope under gravity, and spreads diffusively owing to the gravitational acceleration normal to the boundary, as on a horizontal boundary. However, if the permeability varies in this cross-slope direction, then, in the moving frame, the spreading of the current eventually becomes dominated by the variation in speed with depth, and the current length increases as t'l'. Shocks develop either at the leading or trailing edge of the flows depending upon whether the permeability increases or decreases away from the sloping boundary. Finally we consider the transient and steady exchange of fluids of different densities between reservoirs connected by a shallow long porous channel. Similarity solutions in a steadily migrating frame describe the initial stages of the exchange process. In the final steady state, there is a continuum of possible solutions, which may include flow in either one or both layers of fluid. The maximal exchange flow between the reservoirs involves motion in one layer only. We confirm some of our analysis with analogue laboratory experiments using a Hele-Shaw cell.

/pdf/gravity-driven-flows-in-porous-layers-m1uen1lr05.pdf

Gravity-driven flows in porous layers

In this paper, we consider the evolution of an initially small voidage disturbance in a gas-fluidized bed. Using a one-dimensional model proposed by Needham & Merkin (1983), Crighton (1991) has shown that weakly nonlinear waves of voidage propagate according to the Korteweg–de Vries equation with perturbation terms which can be either amplifying or dissipative, depending on the sign of a coefficient. Here, we investigate the unstable side of the threshold and examine the growth of a single KdV voidage soliton, following its development through several different regimes. As the size of the soliton increases, KdV remains the leading-order equation for some time, but the perturbation terms change, thereby altering the dependence of the amplitude on time. Eventually the disturbance attains a finite amplitude and corresponds to a fully nonlinear solitary wave solution. This matches back directly onto the KdV soliton and tends exponentially to a limiting size. We interpret the series of large-amplitude localized pulses of voidage formed in this way from initial disturbances as corresponding to the ‘voidage slugs’ observed in gas fluidization in narrow tubes.

Solitons, solitary waves, and voidage disturbances in gas-fluidized beds

Wastewater treatment processes are inherently dynamic because of the large variations in the influent wastewater flow rate, concentration and composition. Moreover, these variations are to a large extent not possible to control. The adaptive behaviour of the involved microorganisms imposes further difficulties in terms of time-varying process parameters. Mathematical models and computer simulations are essential to describe, predict and control the complicated interactions of the processes. The number of reactions and organism species that are involved in the system may be very large. An accurate description of such systems can therefore result in highly complex models, which may not be very useful from a practical, operational point of view. A reduced order dynamic model, describing an activated sludge process performing carbonaceous removal, nitrification and denitrification with reasonable accuracy, is presented. The main objective is to combine knowledge of the process dynamics with mathematical methods for estimation and identification. The identifiability of the model is investigated using both off-line and on-line methods, and its dynamic behaviour is validated by simulations of a recognized model. The information required by the identification algorithms is based on directly measurable real-time data. The simplified model may serve as a tool for predicting the dynamic behaviour of an activated sludge process, since the parameters can be tracked on-line during varying operating conditions. The model is aimed for operation and control purposes as an integral part of a hierarchical control structure. The main objective of the work on settler modelling is to enlighten recent theoretical results. A new one-dimensional settler model is compared to a traditional layer model by means of numerical simulations. Emphasis is put on the numerical solution¹s ability to approximate the analytical solution of the conservation law written as a non-linear partial differential equation. The new settler model is consistent in this respect. Several problems that occur when integrating a model of the biological reactor with a model of the settler are also discussed. In particular, the concentrations of the biological components of the particulate material are of importance for an accurate description of the sludge that is recycled to the biological reactor. Two one-dimensional algorithms have been evaluated. The first algorithm is commonly used and some of its inherent problems are discussed. The second algorithm is a new analytically derived method. Few attempts have been made to take into account the influence of higher order organisms in biofilm systems when developing or applying mathematical models. This work describes a simplified modelling approach to include some possible effects of higher order organisms on nitrification, based on a proposed hypothesis of their oxygen consumption in the biofilm. Three different models are developed and investigated. Model simulations are validated using data from a laboratory experiment using continuous-flow suspended-carrier biofilm reactors, where the predators were selectively inhibited. The proposed models are capable of reproducing several of the observed effects. They are primarily aimed at capturing the steady-state behaviour of the biofilm but may also prove to be a useful basis for describing the dynamics. (Less)

/pdf/modelling-aspects-of-wastewater-treatment-processes-36vj6gx2tk.pdf

Modelling Aspects of Wastewater Treatment Processes

The consolidation behavior of flocculated alumina suspensions has been analyzed as a function of the interparticle energy. Consolidation was performed by a centrifugal force field or by gravity, and both the time-dependent and equilibrium density profiles were measured by a gamma-ray absorption technique. The interparicle energy at contact was controlled by adsorbing fatty acids of varying molecular weight at the alumina/decalin interface. We found that strongly attractive interactions result in a particle network which resists consolidation and shows compressible behavior over a large stress range. The most weakly flocculated suspension showed an essentially incompressible, homogeneous density profile after consolidation at different centrifugal speeds. We also found a significant variation in the maximum volume fraction, φm, obtained, with φm∼ 0.54 for the most strongly flocculated suspension to φm∼ 0.63 for the most weakly flocculated suspension. The compresive yield stresses show a behavior which can be fitted to a modified power law. In this paper, we discuss possible correlations between the fitting parameters and physical properties of the flocculated suspensions.

/pdf/consolidation-behavior-of-flocculated-alumina-suspensions-4f1setqolm.pdf

Consolidation Behavior of Flocculated Alumina Suspensions

The consolidation or concentration of suspended particulate solids under the influence of gravitational forces is a problem of widespread practical and theoretical interest. The literature, which is scattered over several fields, contains most of the elements necessary for a complete understanding of gravity settling, but considerable controversy and confusion persists about their synthesis. Here we propose to construct a quantitative theory covering the full range of processes from transient settling of large, stable particles to the slow consolidation of flocculated suspensions of submicron particles. Conditions for the existence of shocks are identified and the basic equations describing the phenomena are solved numerically for several Peclet numbers.

The resolution of shocks and the effects of compressible sediments in transient settling

An experimental investigation of the sedimentation of monodisperse colloidal silica spheres with grafted octadecyl chains with three different interaction potentials is presented. Small particles (0.27 μm) behaved as hard spheres in cyclohexane, but larger ones (0.60 and 0.94 μm) are weakly flocculated by van der Waals attractions. The smallest particles (0.08 μm) in hexadecane are strongly flocculated by attractions between the octadecyl layers. A medical computer tomography (CT) scanner provided an accurate and absolute density measurement without disrupting the process. For the hard spheres and the weakly flocculated systems, the kinetics of sedimentation for the dispersed phase could readily be predicted utilizing the flux curve. For flocculated networks, we found a power-law relationship between compressive yield stresses and solids fractions comparable with other experimental systems.

The transient settling of stable and flocculated dispersions

A substantial amount of drilling fluid can invade a permeable bed during the drilling of an oil well. The presence of this fluid, often referred to as filtrate, can greatly influence the performance of instruments lowered into the wellbore for the purpose of locating these permeable beds. The invaded filtrate can also substantially alter the physical properties of the porous rock. For these reasons, it is of great interest to known where the filtrate goes upon entering the bed. The objective of this study is to quantify the influence of the difference in density between the filtrate and the naturally occurring formation fluid on the shape of the filtrate front as the filtrate invades the formation. This type of phenomenon is often referred to as buoyancy or gravity segregation. In this study, Part 1, we determine the behaviour of the filtrate as it accumulates (and spreads out) at a horizontal impermeable barrier within the formation. This is a combined theoretical and experimental study in which an X-ray CT scanner is extensively used to determine the appropriateness and limitations of the simplifying assumptions used in the theory. In Part 2, the flow of the invading filtrate within the entire bed will be presented. The problem addressed in Part 1 may be viewed from the broader, more fundamental, perspective, as a well-defined model fluid mechanics problem for flow in porous media. One fundamental issue infrequently addressed concerns the consequence on the dynamics of the fluids of heterogeneities, always present to some degree, in consolidated porous solids. The X-ray CT scanner enables the assessment of the appropriateness of modelling such porous solids as spatially homogeneous, a very popular assumption. This study also addresses the limitation of the small-slope approximation when a fluid–fluid interface occurs in a porous solid, an approximation which has enjoyed great success in free-surface fluid mechanics problems when no porous media is present.

Buoyancy-induced flow in porous media generated near a drilled oil well. Part 1. The accumulation of filtrate at a horizontal impermeable boundary

The filtration of colloidal gels, consisting of octadecyl-coated silica spheres in hexadecane, is examined experimentally using CAT scanning. A number of interparticle strengths are explored, with significant differences in qualitative behavior relative to the filtration of stable dispersions. For stiff gels, the transmission of stresses from the cell wall to the dispersion results in a fracturing phenomenon during consolidation on a length scale substantially larger than the particle size. For more pliant gels, the radial propagation of stress is more local to the wall region and gravitational effects play a prominent role. The existing theory of network consolidation is compared against our experimental results for pliant gels, independent of the form of the constitutive functions for the sedimentation velocity and yield stress. Though our measurements seem consistent with the theory at early times, deviations at later moments question whether a compressive yield stress is actually present.

François M. Auzerais

Papers

The resolution of shocks and the effects of compressible sediments in transient settling

The transient settling of stable and flocculated dispersions

Buoyancy-induced flow in porous media generated near a drilled oil well. Part 1. The accumulation of filtrate at a horizontal impermeable boundary

Colloidal gel filtration: Experiment and theory