Probability backflow for a Dirac particle

The phenomenon of probability backflow, previously quantified for a free nonrelativistic particle, is considered for a free particle obeying Dirac's equation. It is known that probability backflow can occur in the opposite direction to the momentum; that is to say, there exist positive-energy states in which the particle certainly has a positive momentum in a given direction, but for which the component of the probability flux vector in that direction is negative. It is shown thar the maximum possible amount of probability that can flow backwards, over a given time interval of duration T, depends on the dimensionless parameter epsilon = root 4h/mc(2)T, where m is the mass of the particle and c is the speed of light. At epsilon = 0, the nonrelativistic value of approximately 0.039 for this maximum is recovered. Numerical studies suggest that the maximum decreases monotonically as epsilon increases from 0, and show that it depends on the size of m, h, and T, unlike the nonrelativistic case.

Model for gas-liquid flow through wet porous medium

A method involving bubbling of air through a fibrous filter immersed in water has recently been investigated (Agranovski et al. [1]). Experimental results showed that the removal efficiency for ultra-fine aerosols by such filters was greatly increased compared to dry filters. Nuclear Magnetic Resonance (NMR) imaging was used to examine the wet filter and to determine the nature of the gas flow inside the filter (Agranovski et al. [2]). It was found that tortuous preferential pathways (or flow tubes) develop within the filter through which the air flows and the distribution of air and water inside the porous medium has been investigated. The aim of this paper is to investigate the geometry of the pathways and to make estimates of the flow velocities and particle removal efficiency in such pathways. A mathematical model of the flow of air along the preferred pathways has been developed and verified experimentally. Even for the highest realistic gas velocity the flow field was essentially laminar (Re approximate to 250). We solved Laplace's equation for stream function to map trajectories of particles and gas molecules to investigate the possibility of their removal from the carrier.

Simulation of mixing in unsteady flow through a periodically obstructed channel

Fluid mixing in steady and unsteady Bow through a channel containing periodic square obstructions has been studied using a finite-difference simulation to determine fluid velocities, followed by the use of passive marker particle advection to look at fluid transport out of the cavities formed between each of the obstructions. The geometry and Bow conditions were chosen from the work by Perkins (1989, M.S. Thesis, Lehigh University; 1992, Ph.D. Thesis, Lehigh University); who investigated heat transfer enhancement due to unsteady flow through such an obstructed channel. Particle advection shows that Bow regimes which are predicted to give good mixing based on snapshots of instantaneous streamline contour plots were not necessarily able to efficiently mix fluid which started in the cavity regions throughout the channel. The use of Poincare sections shows regular regions existing under these conditions which inhibit efficient fluid transport. These regular regions are found to disappear when the unsteady Bow velocity is increased. (C) 1997 Elsevier Science Ltd.

Experimental investigation of liquid flow shift due to gas cross flow in non-wetted packed beds

An experimental study has been carried out for the gas-liquid two-phase flow in a packed bed simulating conditions of the gas and liquid flows in the lower part of blast furnace. The localised liquid flow phenomenon in presence of gas cross flow, which usually occurs around the cohesive zone and raceway in blast furnace, was investigated in detail. Such liquid flow is characterised in terms of liquid shift distance or liquid shift angle that can effectively be measured by the experiments involved in the current study. It is found that liquid shift angle does not significantly increase or decrease with different packing depth. This finding supports the hypothesis of the force balance model where a vectorial relationship among acting forces, i.e. gas drag force, gravitational force and solid-liquid friction force, and liquid shift angle does exist. Liquid shift angle is inversely proportional to particle size and liquid density, and proportional to square of gas superficial velocity, but is almost independent on liquid flowrate and liquid viscosity. The gas-liquid drag coefficient, an important aspect for quantifying the interaction between gas and liquid flows, was conceptually modified based on the discrete feature of liquid flow through a packed bed and evaluated by the combined theoretical and experimental investigation. Experimental measurements suggest that the gas-liquid drag coefficient is approximately a constant (C-DG(')=5.4+/-1.0) and is independent on liquid properties, gas velocity and packing structure. The result shows a good agreement with previous experimental data and prediction of the existing liquid flow model.

Numerical investigations of flow and energy fields near a thermoacoustic couple

The flow field and the energy transport near thermoacoustic couples are simulated using a 2D full Navier-Stokes solver. The thermoacoustic couple plate is maintained at a constant temperature; plate lengths, which are short and long compared with the particle displacement lengths of the acoustic standing waves, are tested. Also investigated are the effects of plate spacing and the amplitude of the standing wave. Results are examined in the form of energy vectors, particle paths, and overall entropy generation rates. These show that a net heat-pumping effect appears only near the edges of thermoacoustic couple plates, within about a particle displacement distance from the ends. A heat-pumping effect can be seen even on the shortest plates tested when the plate spacing exceeds the thermal penetration depth. It is observed that energy dissipation near the plate increases quadratically as the plate spacing is reduced. The results also indicate that there may be a larger scale vortical motion outside the plates which disappears as the plate spacing is reduced. (C) 2002 Acoustical Society of America.

Unsaturated liquid percolation flow through nonwetted packed beds

The unsaturated flow of liquid through packed beds of large particles was studied using six different liquids, all with contact angles greater than 90degrees on the bed packing (wax spheres of 9, 15 and 19.4 mm diameter). The liquid flow was discrete in nature, as drops for low flow rates and rivulets for high flow rates. For unsaturated liquid flows, the actual percolation velocity, not superficial velocity, should be used to characterize the flow. The percolation velocity did not vary with packed-bed depth, but was a strong function of liquid flow rate, liquid and particle properties. Effects of liquid and particle properties (but not flow rate) are well captured by a simple correlation between the liquid-particle friction factor and Reynolds number based on actual percolation velocities. Liquid dispersion, characterized by the maximum dispersion angle, varies significantly with liquid and particle properties. The tentative correlation suggested here needs further validation for a wider range of conditions.

Flow patterns in granulating systems

Particle flow patterns were investigated for wet granulation and dry powder mixing in ploughshare mixers using Positron Emission Particle Tracking (PEPT). In a 4-1 mixer, calcium carbonate with mean size 45 mum was granulated using a 50 wt.% solution of glycerol and water as binding fluid, and particle movement was followed using a 600-mum calcium hydroxy-phosphate tracer particle. In a 20-1 mixer, dry powder flow was studied using a 600-mum resin bead tracer particle to simulate the bulk polypropylene powder with mean size 600 mum. Important differences were seen between particle flow patterns for wet and dry systems. Particle speed relative to blade speed was lower in the wet system than in the dry system, with the ratios of average particle speed to blade tip speed for all experiments in the range 0.01-015. In the axial plane, the same particle motion was observed around each blade; this provides a significant advance for modelling flow in ploughshare mixers. For the future, a detailed understanding of the local velocity, acceleration and density variations around a plough blade will reveal the effects of flow patterns in granulating systems on the resultant distribution of granular product attributes such as size, density and strength. (C) 2002 Elsevier Science B.V All rights reserved.

Flow regime, hydrodynamics, floc size distribution and sludge properties in activated sludge bubble column, air-lift and aerated stirred reactors

This paper presents a comparative study how reactor configuration, sludge loading and air flowrate affect flow regimes, hydrodynamics, floc size distribution and sludge solids-liquid separation properties. Three reactor configurations were studied in bench scale activated sludge bubble column reactor (BCR), air-lift reactor (ALR) and aerated stirred reactor (ASR). The ASR demonstrated the highest capacity of gas holdup and resistance, and homogeneity in flow regimes and shearing forces, resulting in producing large numbers of small and compact floes. The fluid dynamics in the ALR created regularly directed recirculation forces to enhance the gas holdup and sludge flocculation. The BCR distributed a high turbulent flow regime and non-homogeneity in gas holdup and mixing, and generated large numbers of larger and looser floes. The sludge size distributions, compressibility and settleability were significantly influenced by the reactor configurations associated with the flow regimes and hydrodynamics.

Mine waste beach profile and flow resistance equations

Four mine waste beach longitudinal profile equations are compared theoretically and in statistical analyses of profile data from 64 field and laboratory beaches formed by mine tailings, co-disposed coal mine wastes, and sand. All four equations fit the profile data well. The best performing equation both accounts for particle sorting and satisfies hydraulic constraints, and the combination of assumptions underlying it is considered to best represent the processes occurring on mine waste beaches. Combining these assumptions with the Lacey normal equation leads to a variant of the Manning resistance equation. Features that it is desirable to incorporate in theoretical and numerical models of mine waste beaches are listed.

Effect of particle hydrophobicity on particle and water transport across a flotation froth

Froth recovery measurements have been conducted in both the presence (three-phase froth) and absence (two-phase froth) of particles of different contact angles in a specially modified laboratory flotation column. Increasing the particle hydrophobicity increased the flow rate of particles entering the froth, while the recovery of particles across the froth phase itself also increased for particle contact angles to 63 and at all vertical heights of the froth column. However, a further increase in the contact angle to 69 resulted in lower particle recovery across the froth phase. The reduced froth recovery for particles of 69 contact angle was linked to significant bubble coalescence within the froth phase. The reduced froth recovery occurred uniformly across the entire particle size range, and was, presumably, a result of particle detachment from coalescing bubbles. Water flow rates across the froth phase also varied with particle contact angle. The general trend was a decrease in the concentrate flow rate of water with increasing particle contact angle. An inverse relationship between water flow rate and bubble radius was also observed, possibly allowing prediction of water flow rate from bubble size measurements in the froth. Comparison of the froth structure, defined by bubble size, gas hold-up and bubble layer thickness, for two- and three-phase froths, at the same frother concentration, showed there was a relationship between water flow rate and froth structure. (c) 2005 Elsevier B.V. All rights reserved.

Aquatic passive sampling of herbicides on naked particle loaded membranes: Accelerated measurement and empirical estimation of kinetic parameters

Water-sampler equilibrium partitioning coefficients and aqueous boundary layer mass transfer coefficients for atrazine, diuron, hexazionone and fluometuron onto C18 and SDB-RPS Empore disk-based aquatic passive samplers have been determined experimentally under a laminar flow regime (Re = 5400). The method involved accelerating the time to equilibrium of the samplers by exposing them to three water concentrations, decreasing stepwise to 50% and then 25% of the original concentration. Assuming first-order Fickian kinetics across a rate-limiting aqueous boundary layer, both parameters are determined computationally by unconstrained nonlinear optimization. In addition, a method of estimation of mass transfer coefficients-therefore sampling rates-using the dimensionless Sherwood correlation developed for laminar flow over a flat plate is applied. For each of the herbicides, this correlation is validated to within 40% of the experimental data. The study demonstrates that for trace concentrations (sub 0.1 mu g/L) and these flow conditions, a naked Empore disk performs well as an integrative sampler over short deployments (up to 7 days) for the range of polar herbicides investigated. The SDB-RPS disk allows a longer integrative period than the C18 disk due to its higher sorbent mass and/or its more polar sorbent chemistry. This work also suggests that for certain passive sampler designs, empirical estimation of sampling rates may be possible using correlations that have been available in the chemical engineering literature for some time.

A review of flow modeling for dense medium cyclones

A critical assessment is presented for the existing fluid flow models used for dense medium cyclones (DMCs) and hydrocyclones. As the present discussion indicates, the understanding of dense medium cyclone flow is still far from the complete. However, its similarity to the hydrocyclone provides a basis for improved understanding of fluid flow in DMCs. The complexity of fluid flow in DMCs is basically due to the existence of medium as well as the dominance of turbulent particle size and density effects on separation. Both the theoretical and experimental analysis is done with respect to two-phase motions and solid phase flow in hydrocyclones or DMCs. A detailed discussion is presented on the empirical, semiempirical, and the numerical models based upon both the vorticity-stream function approach and Navier-Stokes equations in their primitive variables and in cylindrical coordinates available in literature. The existing equations describing turbulence and multiphase flows in cyclone are also critically reviewed.

Exploratory modelling of grinding pressure within a compressed particle bed

With increasing industry interest in high pressure roll grinding (HPGR) technology, there is a strong incentive for improved understanding of the nature of grinding pressure that exists in the interior of a compressed particle bed. This corresponds to the crushing region of the HPGR. The relationship between applied pressure (stress) to the particle bed and induced pressure (stress) within particles and at contact points between particles is of particular interest. A detailed parametric investigation is beyond the scope of this exploratory paper. However, this exploratory investigation does suggest some interesting behaviour. The compressed particle bed within an 80 turn diameter piston has been modelled using Particle Flow Code for three dimensions. PFC3D is a discrete element code. The total number of simulated particles was 1225 and 2450 for two beds of different thickness. Particle diameters were uniformly distributed between 4 and 4.5 mm. The results of the simulations show that stress intensity within the simulated particle beds and within the observed particles increased with increase of the applied stress. The intensity of the average vertical stress in the selected particles tended to be comparable with the intensity of the pressure applied to the surface of particle bed and was only occasionally higher. However, the stress at contact points between particles could be several times higher. In a real crusher, such high stress amplification at contacts will quickly decrease due to local crushing and a resultant increase the size of the contact area. Therefore, its significance is likely to be relatively small in an industrial context. The modelling results also suggest that failure within the particle bed will progress from the crushing surface towards the depth of the bed. (c) 2006 Elsevier Ltd. All rights reserved.

Study of flow behaviour in a three-product cyclone using computational fluid dynamics

Simplicity in design and minimal floor space requirements render the hydrocyclone the preferred classifier in mineral processing plants. Empirical models have been developed for design and process optimisation but due to the complexity of the flow behaviour in the hydrocyclone these do not provide information on the internal separation mechanisms. To study the interaction of design variables, the flow behaviour needs to be considered, especially when modelling the new three-product cyclone. Computational fluid dynamics (CFD) was used to model the three-product cyclone, in particular the influence of the dual vortex finder arrangement on flow behaviour. From experimental work performed on the UG2 platinum ore, significant differences in the classification performance of the three-product cyclone were noticed with variations in the inner vortex finder length. Because of this simulations were performed for a range of inner vortex finder lengths. Simulations were also conducted on a conventional hydrocyclone of the same size to enable a direct comparison of the flow behaviour between the two cyclone designs. Significantly, high velocities were observed for the three-product cyclone with an inner vortex finder extended deep into the conical section of the cyclone. CFD studies revealed that in the three-product cyclone, a cylindrical shaped air-core is observed similar to conventional hydrocyclones. A constant diameter air-core was observed throughout the inner vortex finder length, while no air-core was present in the annulus. (c) 2006 Elsevier Ltd. All rights reserved.