Computational validation of the Generalized Sutherland Equation for bubble–particle encounter efficiency in flotation

Bubble–particle encounter during flotation is governed by liquid flow relative to the rising bubble, which is a function of the adsorbed frothers, collectors, and other surfactants and surface contaminants. Due to surface contamination, the bubble surface in flotation has been considered as immobile (rigid). However, surface contamination can be swept to the backside of the rising bubble due to the relative liquid flow, leaving the front surface of the rising bubble mobile with a non-zero tangential component of the liquid velocity. The bubble with a mobile surface was considered by Sutherland who applied the potential flow condition and analyzed the bubble–particle encounter using a simplified particle motion equation without inertia. The Sutherland model was found to over-predict the encounter efficiency and has been improved by incorporating inertial forces which are amplified at the mobile surface with a non-zero tangential velocity component of the liquid phase. An analytical solution was obtained for the encounter efficiency using approximate equations and is called the Generalized Sutherland Equation (GSE). In this paper, the bubble–particle encounter interaction with the potential flow condition has been analyzed by solving the full motion equation for the particle employing a numerical computational approach. The GSE model was compared with the exact numerical results for the encounter efficiency. The comparison only shows good agreement between the GSE prediction and the numerical data for ultrafine particles (< 10 μm in diameter), the inertial forces of which are vanishingly small. For non-ultrafine particles, a significant deviation of the GSE model from the numerical data has been observed. Details of the numerical methodology and solutions for the (collision) angle of tangency and encounter efficiency are described.     

Analysis of water seepage in a pavement system using the particulate approach

A particulate model has been developed to analyze the effects of transient and steady state seepage of water through a randomly-packed coarse-grained soil as an alternative to conventional seepage analysis based on continuum models. In this model, the soil skeleton and the pore water are volumetrically coupled in the transient and steady-state conditions. The concept of relative density has been used to define different compaction levels of the soil layers forming a pavement filter system and observe the seepage response to compaction. First, Monte–Carlo simulation is used to randomly pack discrete spherical particles from a specified particle size distribution (PSD) to achieve a desired relative density based on the theoretical minimum and maximum void ratios. Then, a water pressure gradient is applied across one two-layer unit to trigger water seepage. The interstitial pore water motion is idealized using Navier–Stokes (NS) equations with provision to incorporate the drag forces acting between the pore fluid and soil particles. The NS equations are discretized using finite differences and applied to discrete elements in a staggered, structured grid. The model predicted hydraulic conductivities are validated using widely used equations.     

An integral assessment of the role of critical process parameters on jigging

In jigging, amplitude and frequency of pulsation, and feed characteristics are the most important process parameters. Effects of these parameters on particle segregation during jigging are studied and explained through experimental as well as numerical means by drawing parallel to liquid/solid fluidization process. This is permissible because jigging could also be viewed as a repetitive process of fluidization and defluidization. Unlike a normal fluidization process, in jigging, particle segregation takes place under rapid rate of change of water velocity. The rate of change of water velocity is decided by the maximum water velocity of the jig cycle, which in turn is controlled primarily by the amplitude. Experimental evidence of the role of maximum water velocity on jigging is given. Next, the feed to the jig that varies in both size and density is conceived as a mixture of many density variant binary systems. This way, the effect of feed characteristic is conveniently analyzed through several indicators of the binary feed namely size ratio, volume fraction, and particle size. Finally, the effect of frequency is studied by considering the particles in the jig bed analogous to tuned mass dampers. This concept is used to explain the preferential segregation of particles at certain jig frequency.     

Pore‐scale modeling of surface erosion in a particle bed

A fully coupled transient two‐dimensional model was employed to study fundamentals of flood‐induced surface erosion in a particle bed. The interaction of the liquid and solid phases is the key mechanism related to surface erosion. The solid phase was idealized at a particle scale by using the discrete element method. The fluid phase was modeled at a mesoscale level and solved using the lattice Boltzmann method. The fluid forces applied on the particles were calculated on the basis of the momentum the fluid exchanges with the particle. The proposed approach was used to model both single particles and particle beds subjected to Couette flow conditions. The behavior of both the single particle and the particle bed depended on particle diameter and surface shear fluid velocity. The conducted simulations show that the fluid flow profile penetrates the bed for a small distance. This penetration initiates sheet‐flow and surface erosion as the fluid interacts with particles. The effect of suppressing particle rotation on the fluid‐induced forces on the particle was also examined. Suppressing particle spinning may lead to underestimated erosion rate. Results of fluid and particle velocities were compared against experimental results and appeared to agree with the observed trends.Copyright © 2013 John Wiley & Sons, Ltd.     

The motion of sediment-water mixtures during intense bedload transport: computer simulations

A new model, which couples fluid and particle dynamics, has been developed to study the motion of the sediment-water mixture during intense bedload transport, including the velocity profiles of both sediment and water, the roughness length of an upper plane bed and the thickness of moving sediment layers. Standard mixing length theory is used to model the motion of water above the boundary between the overlying water and the sediment-water mixture. The turbulent flow within the moving sediment layers is described by a shear stress model, in which the effective viscosity of the flowing water is proportional to the velocity difference between the fluid and the sediment. The particle dynamics method, in which the equations of motion of each of many particles are solved directly, is applied to model the movement of sediment particles. The particle-fluid interaction is expressed by a velocity-squared fluid drag force exerted on each sediment particle. Both computer simulation results and theoretical analysis have shown that the velocities of both sediment and fluid during intense sediment transport decrease exponentially with depth in the top layers of a fast-moving sediment—water mixture. The thickness of the moving sediment layers, obtained from the computer simulation results, is proportional to the shear stress, which agrees with previous experimental observations.     

层流区Shields曲线分布规律试验

为探明层流条件下Shields曲线的分布规律,运用电荷耦合组件(Charge Coupled Device,CCD)成像技术结合激光扫描及计算机图像处理技术,在水和甘油混合液中对无黏性均匀玻璃颗粒的起动拖曳力进行测量,并根据层流绕流理论建立了无黏性均匀颗粒的滚动起动模型,提出层流条件下无黏性均匀颗粒的量纲一起动拖曳力公式.试验观测发现在层流范围内希尔兹数呈规则的带状分布,流体作用引起颗粒床表面粗化并导致床面颗粒突起减小,使颗粒起动拖曳力增大1倍以上.结果表明颗粒床表面结构性状直接影响着颗粒起动拖曳力的大小,在层流区Shields曲线具有带状分布特性.     

不同粒度颗粒在非恒定管流中跟随性试验

管道中固液两相流水击对管道和输送系统可能产生严重的破坏,而固液两相在这种非恒定流中的运动特性是计算最大水击压力变化的重要依据。采用粒子图像测速技术（Particle Image Velocimetry,PIV）,通过试验研究水击发生时,水平圆管中不同平均流速、颗粒粒径条件下,流体介质和粗颗粒在管道断面的速度分布以及粗颗粒跟随性的变化规律。研究结果表明:①在水击发生的不同时刻,圆管流中粗颗粒的流速在管道断面分布呈不规则的抛物线型分布,主要表现为靠近管道壁面底部的颗粒流速略小于靠近管道顶部流速,当颗粒粒径大于1.5 mm,平均流速小于2.5 m/s时,粗颗粒表现出明显的沉降特性;②粗颗粒的跟随性与颗粒受力有密切关系,其中颗粒速度与流体速度的变化量是影响颗粒受力的重要参数;③基于试验数据拟合得到了水击条件下粗颗粒跟随性系数k的经验公式,并分析了颗粒粒径、管道直径、两相流平均流速以及水击发生时间等不同参数对粗颗粒跟随性系数的影响,公式计算值与实测值之间的误差在5%以内。     

Multiphase SPH modeling of free surface flow in porous media with variable porosity

This paper presents a numerical model for simulating free surface flow in porous media with spatially varying porosity. The governing equations are based on the mixture theory. The resistance forces between solid and fluid is assumed to be nonlinear. A multiphase SPH approach is presented to solve the governing equations. In the multiphase SPH, water is modeled as a weakly compressible fluid, and solid phase is discretized by fixed solid particles carrying information of porosity. The model is validated by several numerical examples including seepage through specimen, fast flow through rockfill dam and wave interaction with porous structure. Good agreements between numerical results and experimental data are obtained in terms of flow rate and evolution of free surface. Parameter study shows that (1) the nonlinear resistance law provides more accurate results; (2) particle size and porosity have significant influence on the porous flow.     

Sensitivity of incipient particle motion to fluid flow penetration depth within a packed bed

A discrete element method is applied to a three‐dimensional analysis related to sediment entrainment on a micro‐scale. Sediment entrainment is the process by which a fluid medium accelerates particles from rest and advects them upward until they are either transported as bedload or suspended by the flow. Modelling of the entrainment process is a critically important aspect for studies of erosion, pollutant resuspension and transport, and formation of bedforms in environmental flows. Previous discrete element method studies of sediment entrainment have assumed the flow within the particle bed to be negligible and have only allowed for the motion of the topmost particles. At the same time, micro‐scale experimental studies indicate that there is a small slip of the fluid flow at the top of the bed, indicating the presence of non‐vanishing fluid velocity within the topmost bed layers. The current study demonstrates that the onset of particle incipient motion, which immediately precedes particle entrainment, is highly sensitive to this small fluid flow within the topmost bed layers. Using an exponential decay profile for the inner‐bed fluid flow, the discrete element method calculations are repeated with different fluid penetration depths within the bed for several small particle Reynolds numbers. For cases with slip velocity corresponding to that observed in previous experiments with natural sediment, the predicted particle velocity is found to be a few percent of the fluid velocity at the top of the viscous wall layer, which is a reasonable range of velocities for observation of incipient particle motion. This method for prescribing the fluid flow within the particle bed allows for the current discrete element method to be extended in future studies to the analysis of sediment entrainment under the influence of events such as turbulent bursting. Additionally, predictions for the slip velocities and fluid flow profile within the bed suggest the need for further experimental studies to provide the data necessary for additional improvement of the discrete element method models.     

Extended CFD–DEM for free‐surface flow with multi‐size granules

Computational fluid dynamics and discrete element method (CFD–DEM) is extended with the volume of fluid (VOF) method to model free‐surface flows. The fluid is described on coarse CFD grids by solving locally averaged Navier–Stokes equations, and particles are modelled individually in DEM. Fluid–particle interactions are achieved by exchanging information between DEM and CFD. An advection equation is applied to solve the phase fraction of liquid, in the spirit of VOF, to capture the dynamics of free fluid surface. It also allows inter‐phase volume replacements between the fluid and solid particles. Further, as the size ratio (SR) of fluid cell to particle diameter is limited (i.e. no less than 4) in coarse‐grid CFD–DEM, a porous sphere method is adopted to permit a wider range of particle size without sacrificing the resolution of fluid grids. It makes use of more fluid cells to calculate local porosities. The developed solver (cfdemSolverVOF) is validated in different cases. A dam break case validates the CFD‐component and VOF‐component. Particle sedimentation tests validate the CFD–DEM interaction at various Reynolds numbers. Water‐level rising tests validate the volume exchange among phases. The porous sphere model is validated in both static and dynamic situations. Sensitivity analyses show that the SR can be reduced to 1 using the porous sphere approach, with the accuracy of analyses maintained. This allows more details of the fluid phase to be revealed in the analyses and enhances the applicability of the proposed model to geotechnical problems, where a highly dynamic fluid velocity and a wide range of particle sizes are encountered. Copyright © 2015 John Wiley & Sons, Ltd.     

激光测速粒子对复杂流动的响应研究——Ⅰ颗粒非恒定运动数学模型及其数值方法

流体激光测速的精度与示踪粒子的跟随特性即流体中异质粒子的非恒定运动特性密切相关。首先对粒子非恒定运动方程进行了探讨,着重考虑了在高颗粒雷诺数时该方程的修正问题,简要分析了该方程的数学属性,并构造了这类方程的数值计算方法。分析表明,高颗粒雷诺数下的粒子非恒定运动方程为非线性奇异积分方程,而当颗粒雷诺数小于1时,则线性化为第二类渥尔特拉(Volterra)积分方程。以几种均匀流中球形小颗粒的非恒定运动为算例,计算结果与其解析解及有关实验数据的比较表明,数值方法具有良好的计算精度。     

粗粒料强度及变形特性的细观模拟

粗粒料是一定级配的岩石颗粒集合体,具有独特的物理力学特性。以粗粒料室内三轴固结排水试验成果为基础,基于离散元颗粒流理论,从细观角度出发,以PFC3D为工具,通过自编程及二次开发,得到按级配生成的粗粒料三轴试验数值模型。引入clump颗粒考虑颗粒形状对粗粒料强度及变形的影响,分析剪胀、颗粒形状、颗粒重排的关系。结果表明：颗粒形状是影响粗粒料强度与变形的主要因素,在其他细观参数一定的情况下,改变颗粒形状,可以显著影响粗粒料的力学行为;BPM模型的应力-应变关系只在低围压下与试验值吻合,随着围压的增大,偏差越来越大;而引入clump颗粒的PFC3D数值模型能很好地模拟粗粒料室内三轴固结排水试验的应力-应变特性,但由于BPM及clump都是刚性颗粒,没有考虑颗粒变形及破碎,造成应变剪胀偏大。     

基于物理模型试验的碎屑流流态化运动特征分析

流态化运动是高速远程滑坡的主要运动形式,是揭示高速远程滑坡运动机理的重要基础。基于粒子图像测速（PIV）分析方法,采用物理模型试验对不同粒径组成条件下的颗粒流内部的速度分布、剪切变形及流态特征进行了研究,并对高速远程滑坡流态化运动特征进行了讨论分析。结果表明:碎屑流流态化运动特征与颗粒粒径呈显著的相关性,随着粒径的减小或细颗粒含量的增加,颗粒流底部相对于边界的滑动速度以及整体的运动速度均呈逐渐减小的趋势,颗粒流内部剪切变形程度增加,颗粒的运动形式由“滑动”向“流动”转变;当颗粒粒径较小或细颗粒含量较高时,颗粒流内部剪切速率增大的趋势在颗粒流底部更加显著,反映了粒径减小有助于促进颗粒流内部剪切向底部的集中;在同一颗粒流的不同运动阶段及不同纵向深度,其流态特征具有显著差别,颗粒流前缘及尾部主要呈惯性态,颗粒间以碰撞作用为主,而主体部分则主要呈密集态,颗粒间以摩擦接触作用为主;在颗粒流表面及底部,颗粒间相互作用方式主要是碰撞作用,中间部分则以摩擦作用为主;对于不同粒径的颗粒流,随着粒径的增大或粗颗粒含量的增加,颗粒流内部颗粒的碰撞作用加强,颗粒流整体趋于向惯性态转变。     

The influence of the void fraction on the particle migration: A coupled computational fluid dynamics–discrete element method study about drag force correlations

Granular soils subjected to flow through their soil skeleton can show a behaviour in which fine particles migrate through the pore space between coarser particles. This process is called internal instability or suffusion. This contribution deals with the numerical analysis of the migration of fine particles in a soil column subjected to fluid flow with unresolved coupled computational fluid dynamics–discrete element method (CFD–DEM) with special regards to the used drag force correlation. The contribution investigates the influence of the Schiller–Naumann model and its extension with a voidage term on the migration behaviour of fine particles. The voidage term is further varied with a parameter, which controls the impact of the change of the void fraction on the drag force. It could be observed that the Schiller–Naumann model does not yield in a suffusive behaviour while the extended models show significant particle migration. Thereby, increasing the impact of the void fraction on the drag force results in stronger particle migration. These results reveal the need for good validation techniques. They indicate how the drag force correlation can be adapted to depict the correct particle migration behaviour.     

颗粒沉降的格子Boltzmann模拟与PIV实验验证

在格子Boltzmann方法中引入大涡模拟,对球形颗粒在静水中沉降引起的紊动流场进行了数值模拟。数值模拟沉速与理论值以及粒子图像测速系统(PIV)实验结果吻合,验证了模型的合理性。同时分析比较了颗粒沉降过程中尾部紊动流场分布以及尾流流速值,发现数值模拟结果与实测结果趋势、数值基本一致,进一步说明了利用格子Boltzmann方法与大涡模拟技术相结合可以合理模拟泥沙颗粒在紊流区的沉降。     

固液两相流体中的空泡振荡计算

本文导出了固液两相流体中球空泡的振荡方程,并用数值方法讨论了固体颗粒对它泡运动及泡壁压强的影响,得到了固相浓度、颗粒尺寸等因素与空泡运动之间的定性关系。在分析过程中,考虑了液体与颗粒之间力的双向耦合作用。     

3D numerical modelling of fluid flow in the Val-d’Or orogenic gold district: major crustal shear zones drain fluids from overpressured vein fields

Fluid flow patterns have been determined using oxygen isotope isopleths in the Val-d’Or orogenic gold district. 3D numerical modelling of fluid flow and oxygen isotope exchange in the vein field shows that the fluid flow patterns can be reproduced if the lower boundary of the model is permeable, which represents middle or lower crustal rocks that are infiltrated by a metamorphic fluid generated at deeper levels. This boundary condition implies that the major crustal faults so conspicuous in vein fields do not act as the only major channel for upward fluid flow. The upper model boundary is impermeable except along the trace of major crustal faults where fluids are allowed to drain out of the vein field. This upper impermeable boundary condition represents a low-permeability layer in the crust that separates the overpressured fluid from the overlying hydrostatic fluid pressure regime. We propose that the role of major crustal faults in overpressured vein fields, independent of tectonic setting, is to drain hydrothermal fluids out of the vein field along a breach across an impermeable layer higher in the crust and above the vein field. This breach is crucial to allow flow out of the vein field and accumulation of metals in the fractures, and this breach has major implications for exploration for mineral resources. We propose that tectonic events that cause episodic metamorphic dehydration create a short-lived pulse of metamorphic fluid to rise along zones of transient permeability. This results in a fluid wave that propagates upward carrying metals to the mineralized area. Earthquakes along crustal shear zones cause dilation near jogs that draw fluids and deposit metals in an interconnected network of subsidiary shear zones. Fluid flow is arrested by an impermeable barrier separating the hydrostatic and lithostatic fluid pressure regimes. Fluids flow through the evolving and interconnected network of shear zones and by advection through the rock matrix. Episodic breaches in the impermeable barrier along the crustal shear zones allow fluid flow out of the vein field.     

Modeling of flowing film concentrators: Part 1. Water split behaviour

A large number of mineral processing equipment employs the basic principles of gravity concentration in a flowing fluid of a few millimetres thick in small open channels where the particles are distributed along the flow height based on their physical properties and the fluid flow characteristics. Fluid flow behaviour and slurry transportation characteristics in open channels have been the research topic for many years in many engineering disciplines. However, the open channels used in the mineral processing industries are different in terms of the size of the channel and the flow velocity used. Understanding of water split behaviour is, therefore, essential in modeling flowing film concentrators. In this paper, an attempt has been made to model the water split behaviour in an inclined open rectangular channel, resembling the actual size and the flow velocity used by the mineral processing industries, based on the Prandtl's mixing length approach.