Free Settling Velocity Research Articles

The Hindered Settling Velocity of Particles of Any Shape in Low Reynolds Number Flow

This article takes insights from a previously derived mathematical framework for the free settling velocity of particles of any shape to model analytical constructs to solve the hindered settling velocity of hard particles of any shape. Because the geometry of the physical environment and continuity can be strictly enforced in the construct model, the relative velocity of the fluid front pumped upward by the settling particles can be found, thus allowing for calculation by subtracting the front velocity from the calculated velocity.

Sedimentation behavior of suspensions in milliflow reactors

The increasing industrial interest in performing heterogeneous solid–liquid reactions in tubular reactors requires control over sedimentation and accurate data on the solid velocity in order to optimize residence time distribution. In this research, concentration profiles, velocity profiles and the minimum suspension speed of non-interacting solids in milliflow reactors are determined via image analysis and particle image velocimetry. First, the five different flow regimes are identified. Then, the minimum suspension speed, ensuring a homogeneous/pseudo-heterogeneous flow regime is observed to be between 0.7 and 1 m/s depending on liquid and solid properties. Reducing the liquid viscosity and density difference, decreases the minimum suspension speed. Based on the free settling velocity, the Reynolds number to avoid sedimenting of particles can be calculated. The identification of minimum suspension speed and Reynolds number were found to be the determining factors enabling broader application of heterogeneous solid–liquid reactions in milliflow reactors, in academic and industrial environments.

Settling characteristics of nonspherical porous sludge flocs with nonhomogeneous mass distribution

The paper reports on the development of an advanced Lagrangian particle tracking model of sludge flocs that takes into account its nonspherical shape, the internal porosity and permeability, as well as the nonhomogenous mass distribution. The floc shapes, sizes and free settling velocities are determined based on the experimental measurement of settling sludge flocs originating from a wastewater treatment plant. Based on the floc shape characterization, a prolate axisymmetric ellipsoid is selected as the modelled sludge particle. In order to determine the main particle characteristics, e.g. the internal porosity, the density and the flow permeability, a Lagrangian particle tracking model is developed based on Brenner's drag model for a prolate axisymmetric ellipsoid and a buoyancy force model for a porous particle. The model is implemented for numerical simulations of the free settling process. The obtained floc characteristics are presented in the form of a two-part polynomial fitting curve, which can be used to model floc characteristics. The values of settling velocities of flocs computed by the model show very good agreement with experimental results. Futhermore, as the internal structure of a floc is seldom uniform, the nonhomogeneous mass distribution is considered, influencing the rotational and translational motions of the settling flocs. The nonhomogeneous mass distribution is introduced into the floc settling model. The parametric analyses of different barycentre offsets and shear rates are performed, and their influences on the free settling velocity are evaluated. The presented modelling approach can also be applied to flocculent settling of alum and other flocs in drinking water treatment plants. The developed Lagrangian model is suitable for use as a point source within the framework of Eulerian flow computations, and is solved as a two-phase flow model with a suitable Computational Fluid Dynamics code.

An experimental investigation into modeling solids friction for fluidized dense-phase pneumatic transport of powders

Results are presented of an ongoing investigation into modeling friction in fluidized dense-phase pneumatic transport of bulk solids. Many popular modeling methods of the solids friction use the dimensionless solids loading ratio and Froude number. When evaluated under proper scale-up conditions of pipe diameter and length, many of these models have resulted in significant inaccuracy. A technique for modeling solids friction has been developed using a new combination of dimensionless numbers, volumetric loading ratio and the ratio of particle free settling velocity to superficial conveying air velocity, to replace the solids loading ratio and Froude number. The models developed using the new formalism were evaluated for accuracy and stability under significant scale-up conditions for four different products conveyed through four different test rigs (subject to diameter and length scale-up conditions). The new model considerably improves predictions compared with those obtained using the existing model, especially in the dense-phase region. Whereas the latter yields absolute average relative errors varying between 10% and 86%, the former yielded results with errors from 4% to 20% for a wide range of scale-up conditions. This represents a more reliable and narrower range of prediction that is suitable for industrial scale-up requirements.

A PHYSICAL MODEL OF SEPARATION PROCESS BY MEANS OF JIGS

In the case of ideal separation of minerals, partition into products is conducted according to a specific partition feature which is, for instance, the density of raw material. Usually, enrichment in a jig is described by means of the particle density as a partition feature. However, the degree of particle loosening in the jig's bed is influenced by, among others, the particle free settling velocity. After some time of the pulsating movement duration, particle segregation along the vertical axis according to the settling velocity will occur. It can be said that the particle free settling velocity constitutes a feature characteristic of the feed heterogeneous in terms of physical and geometric properties in the jigging process. In the article on the basis of heuristic considerations, a physical model of the partition function (recovery of the i -th fraction), in which interactions between particles in the working bed of the jig are taken into account, is derived. A cause of the formation of the mechanism of particle dispersion around equilibrium layers is given and the accuracy of particle partition for a narrow size fraction in two variations, i.e. in conditions when a partition feature is, accordingly, the particle density and settling velocity, is calculated. These calculations allowed for the analysis of causes of process and inherent dispersion formation which takes place during the jigging process.

The Effects of Ore Properties on the Characterization of Suspension in Settling and Compression

Many studies have considered the effects of suspension properties on the dewatering process but fewhave focused on ore properties. Thus, the present work studied the effects of ore properties (density,particle size, mineralogy) on the dewatering process based on lab and pilot experiments. A hydrocyclonewas used to prepare the required samples for the experiments. To study the effects of mineralogicalproperties, the sedimentation behaviour of hydrocyclone feed and underflow samples were compared. Itwas observed that the free-settling velocity of feed (2 to 6mm/sec) was less than in the underflow sample(2 to 7mm/sec) and the final concentration of underflow sample (0.45 to 0.48t/m3) was more than thefeed sample (0.44 to 0.47t/m3). Additionally, to study the effects of particle size and density, thesedimentation behaviour of hydrocyclone overflow and feed samples were compared. The settlingvelocity and final concentration of overflow sample were obtained at 0.15 to 0.4mm/sec and 0.32t/m3,respectively, which was significantly less than the feed sample. This was due to the amount of clayreduction in the underflow sample and particle size and density reduction in the overflow sample.Following on, the pilot experiments were carried out. It was observed that the bed formation of the feedsample tended to overflow in the sample at low flux (10t/m2/day) and tended to underflow in the sampleat high flux (28.5t/m2/day). This meant that the long time at lower flux created an opportunity for fineparticles to settle easily, similar to coarser particles and as such, ore properties did not play a decisiverole in bed formation, but their effects appeared instead at higher flux. Furthermore, it was observed thatthe underflow concentration increased by decreasing the flux from 28.5 to 10t/m2/day. These increasingamounts were 0.05t/m3 and 0.12t/m3 in hydrocyclone overflow and underflow samples, respectively, at aheight of 2.5 metres. This meant that the compressibility and permeability of the hydrocycloneunderflow sample was much better than in the hydrocyclone overflow sample, which was clearly a resultof the ore properties (density, particle size, mineralogy).

THE ROLE OF ORE PROPERTIES IN THICKENING PROCESS

The role of ore properties (density, particle size, and mineralogy) in thickening process was studied in this research. The shaking table was used to prepare the sample for the tests. The tailings were continuously fed on the table by gravity to separate the tailings in three products as slime, middling and coarse particles. The solid density and particle size of the samples were different. To study the effect of mineralogical properties, the sedimentation behavior of the feed and middling samples were tested. The results showed that the free settling velocity of the feed (2-6 mm/s) was less than that of the middling sample (18-23 mm/s), and the compressibility of middling (density: 0.63-0.86 Mg/m 3 ) was more than that of feed (density: 0.33-0.47 Mg/m 3 ). This was due to the amount of clay reduction in the middling sample. The sedimentation behavior of the slime and the coarse samples were also compared in order to study the effect of particle size and density. The settling velocity of the slime and the coarse particles was obtained as 0.1-0.4 and 26 mm/s, respectively, and the maximum underflow density were obtained as 0.35 and 1.57 Mg/m 3 , respectively. Therefore, the particle size and density reduction reduced the thickener performance. In order to study the effect of particle size, the sedimentation behavior of the slime and coarse samples were compared, and it was obtained that the settling velocity and underflow density increased with the increasing in the particle size.

A simple model for the perfusion of porous hydrogel scaffolds under culture in a sustentation like bioreactor

Over the last decade hydrogels, and more recently porous hydrogels, have shown to be promising biocompatible, resorbable scaffolds for tissue engineering (Gloria et al, 2010). Such porous scaffolds generally show a poor mechanical behavior which makes them difficult to manipulate. Since one key issue of tissue engineering is to properly perfuse the scaffolds in order to promote cell seeding in 3D scaffold, increase mass transport and mechanically condition by shear and compressive forces the cells under culture, the perfusion strategy has to be adapted for such materials. As a matter of fact the use of such scaffolds in perfusion systems in which the scaffolds are clamped is prohibited and only bioreactors in which the scaffolds are freely suspended are possible. In order to quantify the perfusion performance of porous hydrogel scaffolds in such bioreactors we :1) characterized the free settling velocity, measured the permeability of the scaffolds,2) derived a simple model for the perfusion of the scaffolds under free falling suspension,3) compared the obtained perfusion characteristics with literature,4) designed and built a bioreactor system to host the scaffolds under proper flow conditions performed cell culture experiments. The present work relates to the 4 first steps, details of the 2 later ones can be found in Knapp et al (2012).

Segregation in water fluidized beds of sand particles

Abstract Fluidization experiments were conducted on four fractions of filtration (quartz) sand using columns of 144 and 64 mm i.d. The size distribution of the fractions obtained by sieving followed an approximately modified normal (Gaussian) particle size distribution with respect to the projected particle diameter, while the shape factor decreased linearly with increasing projected particle diameter. The expansion characteristics of the sand particles followed a Richardson–Zaki relationship. The intercept velocity, i.e., the extrapolated value of the fluid superficial velocity to e = 1 on the plot log ( U ) vs. log ( e ), agreed quite well with the experimentally determined mean free settling velocity of the cloud of about 30 randomly selected particles. During fluidization, the beds were hydraulically separated into 10 sub-fractions, which are then analyzed. For the mixture d p = 0.75–1.25 mm during fluidization at an overall bed expansion of 35%, the bed partially segregated since, remarkably, the smallest particles were concentrated at the top, while the largest particles were concentrated in the bottom zone. In the remaining part of the bed (about 80% of total mass), the particles were well mixed. Similar experiments were performed with sand mixtures d p = 1.60–2.00 mm, d p = 1.40–2.00 mm and d p = 1.166–2.00 mm. By analyzing the particle size distribution at the bottom and at the top of the bed, it was concluded that the bed was well mixed if the sieving ratio of the largest to smallest sieve opening was less than about 1.5.

Free settling of aggregates with mixed statistics

Abstract The free settling model is presented for aggregates of multilevel structure. Primary aggregates, formed by primary particles, join together to produce larger objects, which are treated as monomers for bigger clusters until the final aggregate is obtained. The objects at each structural level have specific size and the fractal dimension. The obtained aggregate free settling velocity normalized by that for primary particles can be expressed as dependent on the ratio of their sizes and the total number of primary particles. Some experimental data are quoted to support the obtained dependence. They are that for kaolin flocs of two-level structure measured by light-scattering technique and free settling test.

Settling velocity of cubes in Newtonian and power law liquids

New extensive data on the free settling velocity of thirty cubes of various densities and sizes falling in scores of Newtonian and Power law liquids are reported herein to supplement the existing data, for there is very little prior data on cubes in power law liquids. The new data embrace the range of conditions as follows: sphericity of 0.805; power law index, 0.61 to 1 and consistency index, 0.0078–15.31 Pa s n ; Reynolds number, 0.0013 to 860. The new results are shown to be consistent with an existing drag correlation which has been tested extensively using the literature data for spherical and non-spherical particles falling in Newtonian and power law liquids with acceptable levels of accuracy.

Settling velocity of Cryptosporidium parvum and Giardia lamblia

Understanding transport behavior of Cryptosporidium parvum oocysts and Giardia lamblia cysts (together referred to as (oo)cysts) in overland flow is important for beneficial uses of receiving water bodies. Like sediment, (oo)cysts are subjected to deposition once they are present in overland flow or low flow environments like reservoirs, wetlands and sedimentation basins. The objectives of this paper are to present the theory and experiment to determine the free settling velocity ( v s) of (oo)cysts and to compare experimental settling velocities to estimates using Stokes’ law. A settling experiment was designed to quantify the v s of (oo)cysts in an aqueous column. C. parvum oocysts used were spherical with average diameter (±1SD) of 6.6±1.1 μm. G. lamblia cysts were oval shaped (average eccentricity =1.48±0.19) with average size of 11.8 ±1.3 μm. Average densities were 1009 kg m −3 for C. parvum oocysts and 1013 kg m −3 for G. lamblia cysts. Observed experimental settling velocities are 0.27 μm s −1 and 0.67 μm s −1 for C. parvum and G. lamblia, respectively. Estimated average settling velocities using Stokes’ law were 0.36 μm s −1 for C. parvum and 0.84 μm s −1 for G. lamblia. R-squared values of the observations from the settling experiments with the Stokes’ law estimation are 0.87 and 0.88 for G. lamblia and C. parvum, respectively. Our results suggest that Stokes’ law can be used to estimate settling velocities of (oo)cysts. Qualitatively, the low settling velocities indicate that (oo)cysts will very slowly settle out of suspension.

Drag on non-spherical particles in power law non-Newtonian media

The free settling velocity of cylinders and disks falling in quiescent Newtonian and power law liquids has been measured over wide ranges of experimental conditions of the particle Reynolds number (10 − 5 –∼300), power law flow behaviour index (0.31–1) and the length-to-diameter ratio, ∼0.4–∼14. The corresponding range of sphericity is 0.62 to 0.86. An existing drag expression which has been tested extensively for spherical particles falling in Newtonian and in power law fluids has been slightly modified here for non-spherical particles. In particular, the use of this drag expression necessitates a knowledge of an equal volume sphere diameter (to evaluate the Reynolds number and drag coefficient) and the ratio of the surface area to the projected area of a non-spherical particle. With these modifications, the approach outlined here reproduces the present and the literature data for a wide range of non-spherical particles including cones, prisms, needles, cylinders settling in both Newtonian and power law fluids with reasonable levels of accuracy.

Mass–radius relation for fractal aggregates of polydisperse particles

Abstract Mass–radius relation for fractal aggregates of polydisperse particles is derived from the Hausdorff measure. The Debye–Brinkman concept of treating polymer coils as uniformly permeable spheres is practically utilized for aggregates composed of not many polydisperse primary particles. The method is based on the calculation of permeability of the system contained inside the sphere circumscribed on the aggregate. The values of normalized hydrodynamic radius for aggregates with log-normal distribution of constituents are close if calculated by the mass–radius relation and by the permeability. Mass–radius relation for aggregates of polydisperse particles is compatible with the aggregation act equation, previously verified by aggregate structure, its dynamics and the aggregation kinetics. It is applicable to describe the free settling velocity of fractal aggregates of polydisperse solids.

Prediction of the settling velocity of irregularly shaped particles

This work presents a new empirical relationship between Reynolds number Re t and Archimedes number Ar. From this relationship it is possible to calculate the free settling (terminal) velocity of irregularly shaped (e.g. crushed) solids in water. Experimental terminal velocities for various size fractions from earlier works were correlated with the new equation. The data refer to crushed quartz and galena particles. The new relationship gives for a very wide range of Archimedes numbers (flow regimes) a good correlation between experimental and calculated terminal velocities. Comparisons with the Ganguly relationship also showed a good agreement for the prediction of the terminal velocities. The current work shows that the terminal velocities in water can be calculated with a single equation for various irregularly shaped solids and for various size fractions.

Electrokinetic sedimentation of contaminated Welland River sediment

An experimental study on electrokinetic sedimentation of contaminated sediment is conducted. The sediment dredged from the Welland River in Ontario is heavily contaminated by metals, oil, and grease as a result of industrial activities over past decades. Characterization of the sediment is presented, followed by descriptions of the sedimentation columns used in the experiments and testing procedure. The application of electrokinetics on the Welland River sediment suspensions significantly increases the free settling velocity (up to 110%) and final solid concentration (up to 33%) and reduces the coefficient of sedimentation (up to 111%) and overall sedimentation time (up to 50%). These changes are functions of the initial solid concentration and applied electric field intensity. Based on the evaluation of all influencing factors, it is concluded that a continuous dc current generates better performance than intermittent current and electrokinetic sedimentation is more effective than chemical coagulation. All other conditions being equal, the combined application of electrokinetics and FeCl3 generated the best dewatering effect. The electrokinetic treatment generates pH changes ranging from pH 6.0 at the anode to pH 9.4 at the cathode which will generate a negligible environmental impact, as most metal contaminants will not be released to water in this pH range.Key words: electrophoresis, electrokinetics, sedimentation, contaminated sediment, waste management.

Wall effects on free-settling velocity of non-spherical particles in viscous media in cylindrical tubes

This work sets out to investigate the effect of confining walls on the settling velocity of variously shaped rigid particles in stationary Newtonian liquids. Cubes, parallelepipeds, cylinders, needles, thin plates and circular discs of different sizes and made of different materials (1200–7800 kg/m 3) have been sedimented in scores of liquids in cyclindrical tubes of different inside diameters (20–100 mm). The retardation effect due to the presence of bounding walls has been quantified in terms of a wall factor which is a function only of particle size and shape, and tube diameter in the low Reynolds number region ( <7). Simple expressions are presented that not only correlate well the present results but also describe satisfactorily the scant literature data.

Drag on chains and agglomerates of spheres in viscous newtonian and power law fluids

AbstractNew experimental data on the free settling velocity of straight chains (up to twenty spheres) and planar clusters of touching spheres in Newionian and power law media are reported. The results embrace the following ranges of conditions: 0.65 ≤ n ≤ 1; Re &lt; ‐ 2.5 and 1.22 &lt; m &lt; 48.87 Pa·sn. The straight chain drag measurements are in line with theoretical predictions for Newtonian fluids. The present results in power law fluids seem to suggest that it is possible to express the drag on a straight chain of spheres in terms of that on a single sphere of equal volume. Limited results with planar clusters are satisfactorily correlated using a volume equivalent sphere diameter.

Applicability of various averaging concepts to predict the average dynamic properties of poly-dispersed liquid-solid fluidized bed systems

To avoid the necessity of having a large number of correlations to predict the dynamic properties of poly-dispersed fluidized bed systems, a statistical approach has been adopted in the present paper to get an expression which can predict such property from a knowledge of similar property of the individual species that constitute such heterogeneous bed. In the present paper, the dynamic property means any property that is associated with, or dependent on a term “velocity”, e.g., beginning elutriation velocity, u be; free settling velocity, u t; onset semifluidization velocity, u osf; expanded bed height, L e, etc. Based on 10,517 computations, embracing 5 different dynamic properties of liquidsolid fluidized bed systems and covering 13 different averaging formulae — similar to those, used to calculate average particle size for a poly-dispersed system — the following expression has been found to give the best fit: P=[ 1 σ ( X i P 3 i )] 1 3

Sedimentation of a sphere along the axis of a long square duet filled with non‐newtonian liquids

AbstractBased on extensive experimental results, it is shown that the retardation effect caused by the confining walls on the free settling velocity of a sphere is smaller with square walls than that with cylindrical boundaries. This is true for both Newtonian and power law fluids, provided the particle Reynolds number is small (&lt; about 5). The values of the wall factor for Newtonian liquids are in excellent agreement with theory (up to R / L ≤ 0.1) while those for power law fluids have been correlated empirically via a linear relationship. The results reported here encompass the following ranges of conditions: 1 ≥ n ≥ 0.7; Re &lt; 15 and 0.024 &lt; R/L &lt; 0.238.