Calculation of the Velocity of Free Settling of Solid Particles in a Newtonian Liquid

A cubic equation for the velocity of free settling of solid particles in a Newtonian liquid is derived on the basis of a simple approximate expression for the drag coefficient of spherical particles as a function of the Reynolds number using the concept of the effective diameter of a particle of arbitrary shape, which characterizes the surface, volume, and midsection of the particle. The exact and approximate solutions are compared, and some experimental data are analyzed.     

Comparison of formulas for drag coefficient and settling velocity of spherical particles

Two formulas are proposed for explicitly evaluating drag coefficient and settling velocity of spherical particles, respectively, in the entire subcritical region. Comparisons with fourteen previously-developed formulas show that the present study gives the best representation of a complete set of historical data reported in the literature for Reynolds numbers up to 2 × 10⁵.     

Motion of a magnetic flow follower in two‐phase flow — application to the study of airlift reactor hydrodynamics

A low‐cost and simple magnetic particle tracer method was adapted to characterize the hydrodynamic behavior of an internal‐ and an external‐loop airlift reactor (ALR). The residence time distribution of three magnetic particles differing in diameter (5.5, 11.0 and 21.2 mm) and with a density very close to that of water was measured in individual reactor sections. The measured data were analyzed and used to determine the velocity of the liquid phase. Validation of the experimental results for liquid velocity was done by means of the data obtained by an independent reference method. Furthermore, analysis of the differences found in the settling velocity of the particle in single‐liquid and gas‐liquid phases was carried out, using a simplified 3D momentum transfer model. The model considering particle‐bubble interaction forces resulting from changes in the liquid velocity field due to bubble motion was able to predict satisfactorily the increase in the particle settling velocity in the homogeneous bubbly regime. The effective drag coefficient in two‐phase flow was found to be directly dependent on particle Reynolds number to the power of ? 2 but independent of gas flow‐rate for all particle diameters studied. Based on the experimental and theoretical investigations, the valid exact formulation of the effective buoyancy force necessary for the calculation of the correct particle settling velocity in two‐phase flow was done. In addition, recommendations concerning the use of flow‐following particles in internal‐loop ALRs for liquid velocity measurements are presented. Copyright © 2006 Society of Chemical Industry     

Experimental estimation of the settling velocity and drag coefficient of the hollow cylindrical particles settling in non-Newtonian fluids in an annular channel

The drag coefficient data of particles settling in an annular channel is very much essential for designing different solid–fluid handling equipment, such as the fluidized bed. Experimental settling velocity, wall factor, and drag coefficient data of the hollow-cylinder particle are presented. Carboxymethyl cellulose solution has been used as the working fluid with a flow index of 0.64 ≤ n ≤ 0.91 and a consistency index of 0.31 ≤ K ≤ 1.81. The experimental results covered a wide diameter ratio range (0.14 ≤ d_eq/L ≤ 0.46), hollow cylinder inner to outer diameter ratio (0.2 ≤ d_i/d_o ≤0.8), and Reynolds number (0.05 ≤ Re ≤ 51 and 0.09 ≤ Re_∞ ≤ 55). d_eq, d_i, and d_o are the equivalent inner and outer diameters of the particle, L is the annular gap, and Re and Re_∞ are the Reynolds numbers in the presence and absence of the wall effect, respectively. The wall factor decreased, and the drag coefficient increased with d_eq/L and d_i/d_o ratios. The above parameters declined with the Reynolds number. The hollow cylinder experienced a lesser wall effect than the spherical particles settling in a non-annular channel. In some cases, the wall factor of the hollow cylinder is found to be equal to the spherical particles settling in an annular channel. The developed correlations have successfully predicted the drag coefficients of the hollow cylinder.     

Drag on a sphere in a spherical dispersion containing Carreau fluid

The drag on a rigid sphere in a spherical dispersion containing Carreau fluid is investigated theoretically based on a free surface cell model for Reynolds number in the range [0.1,100], Carreau number in the range [0,10], the power-law index in the range [0.3,1], and the void fraction in the range [0.271,0.999]. The influences of the particle concentration, the nature of the Carreau fluid, and Reynolds number, on the drag coefficient are examined. We show that the drag coefficient declines with the decreasing particle concentration, and the reversal of the flow field in the rear region of a sphere is enhanced by the shear-thinning nature of the fluid. An empirical relation, which correlates the drag coefficient with the void fraction (= 1 − particle concentration), the nature of the Carreau fluid, and Reynolds number, is proposed.     

Finite element simulation of deep bed filtration

In this paper, the numerical model for separation efficiency and transport in periodic porous media is studied. Finite element method was used to simulate the development of a predictive model of behavior of porous media during injection of particles. This paper describes the effects of injected particle size, Reynolds number and particle drag coefficient. The numerical results show that the separation efficiency increased with injected particle size increase. The separation efficiency is found to increase with increasing Reynolds number. For the effect of drag force, C_D, in porous media, numerical results show that for C_D<10 and C_D>100, the separation efficiency is not affected by drag coefficient in the range of drag coefficient from 10 to 100, and the separation efficiency significantly depends on the Reynolds number.     

THE STOKES HYDRODYNAMIC RESISTANCE OF NONSPHERICAL PARTICLES

A review is presented of the motion of an isolated, nonspherical particle of general shape settling at small Reynolds numbers through an unbounded quiescent fluid—with a view towards establishing whether or not all particles ultimately attain a unique, time-independent terminal state, independently of their initial orientation and state of motion. Effects of inhomogeneities in internal mass distribution are incorporated into the analysis. Differences are pointed out between gravity and centrifugal settling rates for nonspherical particles. These arise from the tendency of such particles to adopt preferential orientations in a centrifugal field of force owing to variations in field strength over the length of the particle, ft is pointed out that Coriolis forces acting on both the fluid and particle in a centrifuge cause the particles to settle more slowly. Moreover, in the case of spherical particles, the particle path deviates from a purely radial trajectory. Effects of both translational and rotational Brownian motions on the mean settling velocities of submicron particles is discussed, again for generally-shaped particle. A detailed summary of the contents of this paper is provided at its conclusion.     

Lift and drag forces on a sphere attached to a wall in a Blasius boundary layer

Lift and drag forces on a sphere attached to a planar wall, over which a laminar flat plat boundary layer flows, are examined numerically in this study. Particle Reynolds number ranged from 0.1–250, which represents steady, laminar flow about the sphere, and the plate Reynolds number was held constant at 32 400. A finite-volume computational fluid dynamics program was utilised. Simulation results were validated against analytical results for drag and lift in creeping flow and against experimental results available in the literature for lift at higher particle Reynolds number. The model results were curve-fitted and interpolating drag and lift coefficient functions are reported. The lift and drag results are shown to be weakly dependent upon plate Reynolds number. The resulting correlations are expected to be useful in the development of particle impending motion and aerosol entrainment predictions of particles adhering to planar walls.     

Numerical evaluation of virtual mass force coefficient of single solid particles in acceleration

Virtual mass force is an indispensable component in the momentum balance involved with dispersed particles in a multiphase system.In this work the accelerating motion of a single solid particle is math-ematically formulated and solved using the vorticity-stream function formulation in an orthogonal curvi-linear coordinate system.The total drag coefficient was evaluated from the numerical simulation in a range of the Reynolds number (Re) from 10 to 200 and the dimensionless acceleration (A) between-2.0 to 2.0.The simulation demonstrates that the total drag is heavily correlated with A,and large decel-eration even drops the drag force to a negative value.It is found that the value of virtual mass force coef-ficient (Cv) of a spherical particle is a variable in a wide range and difficult to be correlated with A and Re.However,the total drag coefficient (Coy) is successfully correlated as a function of Re and A,and it increases as A is increased.The proposed correlation of total drag coefficient may be used for simulation of solid-liquid flow with better accuracy.     

Cluster-based drag coefficient model for simulating gas-solid flow in a fast-fluidized bed

Drag coefficient is of essential importance for simulation of heterogeneous gas-solid flows in fast-fluidized beds, which is greatly affected by their clustering nature. In this paper, a cluster-based drag coefficient model is developed using a hydrodynamic equivalent cluster diameter for calculating Reynolds number of the particle phase. Numerical simulation is carried out in a gas-solid fast-fluidized bed with an Eulerian-Lagrangian approach and the gaseous turbulent flow is simulated using large eddy simulation (LES). A Lagrange approach is used to predict the properties of particle phase from the equation of motion. The collisions between particles are taken into account by means of direct simulation Monte Carlo (DSMC) method. Compared with the drag coefficient model proposed by Wen and Yu, results predicted by the cluster-based drag coefficient model are in good agreement with experimental results, indicating that the cluster-based drag coefficient model is suitable to describe various statuses in fast-fluidized beds.     

Drag coefficient of flow around a sphere: Matching asymptotically the wide trend

An empirical relationship of drag coefficient of flow around a sphere is developed for the entire range of Particle Reynolds numbers reported in the literature from Stokes regime to the condition when turbulent boundary layer prevails. The relationship is obtained using an approach to match asymptotically the wide trend of drag coefficient. The matching approach, which relies on dividing the wide trend into smaller segments that can be combined into an overall relationship, employs regression techniques and thus warrants the best-fit accuracy results. The relationship is calibrated with experimental data available in the literature covering the entire range for Reynolds numbers up to ~ 10⁶. For Reynolds values greater than 10⁶, the relationship renders a drag coefficient of 0.2. The performance of the relationship is tested and compared with other suitable models found in the literature. This relationship is also transformed into an explicit expression for settling velocity calculations.     

Wall effects on the velocities of a single sphere settling in a stagnant and counter-current fluid and rising in a co-current fluid

Experimental results were obtained on the steady settling of spheres in quiescent media in a range of cylindrical tubes to ascertain the wall effects over a relatively wide range of Reynolds number values. For practical considerations, the retardation effect is important when the ratio of the particle diameter to the tube diameter (λ) is higher than about 0.05. A new empirical correlation is presented which covers a Reynolds number range Re = 53-15,100 and a particle to tube diameter ratio λ < 0.88. The absolute mean deviation between the experimental data and the presented correlation was 1.9%. The well-known correlations of Newton, Munroe and Di Felice agree with the presented data reasonably well. For steady settling of spheres in a counter-current water flow, the slip velocity remains practically the same as in quiescent media. However, for rising spheres in a co-current water flow, the slip velocity decreases with increasing co-current water velocity, i.e., the wall factor decreases with increasing co-current water velocity. Consequently, the drag coefficient for rising particles in co-current water flow increases with increasing water velocity.     

Primary Study of a Settling Fiber Particle in a Particle Cloud

Fiber particles have some unique behaviors due to their special shapes, which are important to those related industries. When a single fiber is in a particle cloud, its behavior will be influenced by others around it. Hence, the behavior of an isolated fiber particle will be different from that in a particle cloud, such as aggregation, orientation, and drag coefficient. However, little information is available on these phenomena, especially drag coefficient. Therefore, this article focuses on the settling process of a fiber particle in a particle cloud. Experiments were conducted to observe the settling behavior of fiber particles and to determine the drag coefficients of an isolated fiber particle in a particle cloud. The relationship between drag coefficient, orientation, and Re for different fiber particles is obtained, which is independent of volume concentration. It is further observed that the aspect ratio has little influence on the drag coefficients of fiber particles. By comparison, it is noticed that these relationships are similar to those found for an isolated single-fiber particle. Furthermore, the orientation of a fiber particle in a particle cloud fluctuates around the stable horizontal orientation in the same way as a single-fiber particle, but returns to the steady state more quickly.     

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 sⁿ; 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.     

Drag coefficient and settling velocity for particles of cylindrical shape

Solid particles of cylindrical shape play a significant role in many separations processes. Explicit equations for the drag coefficient and the terminal velocity of free-falling cylindrical particles have been developed in this work. The developed equations are based on available experimental data for falling cylindrical particles in all flow regimes. The aspect ratio (i.e., length-over-diameter ratio) has been used to account for the particle shape. Comparisons with correlations proposed by other researchers using different parameters to account for the geometry are presented. Good agreement is found for small aspect ratios, and increasing differences appear when the aspect ratio increases. The aspect ratio of cylindrical particles satisfactorily accounts for the geometrical influence on fluid flow of settling particles.     

Motion of particles entrained in a plasma jet

The motion of small particles (glass microspheres, 30 to 140 microns in diameter) entrained in a free argon plasma jet was studied by means of high-speed cine streak photography. Radial temperature and velocity profiles as well as axial profiles of temperature, velocity, and argon concentration in the jet were experimentally determined by means of a plasma calorimetric probe. The system was found to be characterized by low relative Reynolds numbers (0.2 to 20) and extremely high deceleration rates (about –2,000 g). Under these conditions, an increase of drag coefficient over that predicted by the standard curve was experimentally observed. This increase was attributed to the nonsteady flow field around the particle (the so-called “history term” in the equation of motion). A general computer program has been proposed which predicts the particle velocity, acceleration and temperature along its trajectory.     

Numerical study of cluster and particle rebound effects in a circulating fluidised bed

Gas-particle flows in a vertical two-dimensional configuration appropriate for circulating fluidised bed applications were investigated numerically. In the computational study presented herein the motion of particles was calculated based on a Lagrangian approach and particles were assumed to interact through binary, instantaneous, non-frontal, inelastic collisions including friction. The model for the interstitial gas phase is based on the Navier-Stokes equations for two-phase flows. The numerical study of cluster structures has been validated with experimental results from literature in a previous investigation. Numerical experiments were performed in order to study the effects of different cluster and particle rebound characteristics on the gas-particle flow behaviour.Firstly, we investigated the hard sphere collision model and its effect on gas-particle flow behaviour. The coefficient of restitution in an impact depends not only on the material properties of the colliding objects, but also on their relative impact velocity. We compared the effect of a variable restitution coefficient, dependent on the relative impact velocity, with the classical approach, which supposes the coefficient of restitution to be constant and independent of the relative impact velocity.Secondly, we studied the effects of different cluster properties on the gas-particle flow behaviour. Opposing clustering effects have been observed for different particle concentrations: within a range of low concentrations, groups of particles fall faster than individual particles due to cluster formation, and within a well-defined higher concentration range, return flow predominates and hindered settling characterises the suspension. We propose herein a drag law, which takes into account both opposing effects and have compared the resulting flow behaviour with that predicted by a classical drag law, which takes into account only the hindered settling effect.     

Settling of cylinders in power law liquids

New measurements on drag coefficient and wall effects on the free settling motion of cylinders in two shearthinning polymer solutions are reported. The ranges of conditions covered in this study are: 1 < Re_∞∞ < 40; 0.25 < (LID) < 2; 0.079 < dID < 0.4, and n = 0.65 and 0.74. Both wall correction factor and drag coefficient results are in line with the Newtonian behavior provided a modified Reynolds number is used to represent the results.     

A quantitative model of the motion of particles in the RSM/mintek on-stream particle size analyzer

The average radial velocities of particles in the regions adjacent to the inner and outer walls of the sizer helix have been calculated from measured particle concentrations using limited solutions to the continuity equations for particles and fluid. Reynolds number - drag coefficient curves based on the average velocities have been constructed and it was found that the correlation at the outer wall did not account satisfactorily for the effect of particle and fluid density. This was attributed to additional phenomena affecting particle motion that could not be taken into account in solving the continuity equations, so a separate curve had to be constructed for each particle density. The curves were used to predict the variation in separation size over a range of flow velocities and particle concentrations in the feed.