Drag Coefficient Of Particles Research Articles

Effects of acceleration, deceleration and particle shape on single‐particle drag coefficients in still air

AbstractMeasurements were made of the drag coefficients of smooth solid particles moving singly through still air at Reynolds numbers in the range of 103 to 104. Spheres were observed during acceleration, deceleration and at terminal velocity. Discs were tested during acceleration and at steady velocity. Other particles in the shapes of discs, circular cylinders and rectangular parallelepipeds were tested at steady velocity in their preferred and, in some cases, non‐preferred orientations. The effects of non‐steady velocity and of particle form were determined. An unexpected influence of particle density was observed for some of the particles.

BED PARTICLE REYNOLDS MODELLING FOR FLUID DRAG

A scaled-up Reynolds model of a sediment particle on a streambed has been used to study the drag force exerted by streamflow. The results have been presented as a data trend relating particle drag coefficient to particle Reynolds number. The modified drag coefficient and Reynolds number used take into account the presence of bounded shear flow by using a characteristic velocity determined from the relation between the approach flow velocity and the shear velocity on the streambed.

A Study of Airlift for the Feeder of Solids

Some experimental and theoretical studies of airlift, which is used for a feeder of particles in the pneumatic conveyance system are described. The feeding performances are obtained experimentally using several kinds of particles, and pressure losses in airlift are analysed to estimate the power required; which are compared with experimental ones. The drag coefficients of particles in a fully developed turbulent pipe flow are determined experimentally to make an accurate calculation of the pressure losses. The feeding performance of airlift can be represented by formulae using the results of a simple discharge test of particles from a two-dimensional port. The agreement of the calculated pressure losses with experimental ones is quite good when the calculation is made using the drag coefficient of particles obtained in the test.

Reply by Author to M. J. Walsh

Henderson, C. B., Coefficients of Spheres in Continuum and Rarefied Flows, AIAA Journal, Vol. 14, June 1966, pp. 707708. Bailey, A. B. and Hiatt, J., Free-Flight Measurements of Sphere Drag at Subsonic, Transonic, Supersonic, and Hypersonic Speeds for Continuum, Transition, and Near-Free-Molecular Flow Conditions, Arnold Engineering Development Center, Tullahoma, Tenn., AEDCTR-70-291, March 1971. Walsh, M. J., Coefficient Equations for Small Particles in High Speed Flows, AIAA Journal, Vol. 13, Nov. 1975, pp. 15261528. Walsh, M. J., Influence of Particle Drag Coefficient on Particle Motion in High-Speed Flow with Typical Laser Velocimetry Applications, NASATN D-8120, Feb. 1976. Zarin, N. A., Measurement of Non-Continuum and Turbulence Effects on Subsonic Sphere Drag, NASA CR-1585, June 1970. Emmons, H. W., Ed., Fundamentals of Gas Dynamics, Vol. HI. High Speed Aerodynamics and Jet Propulsion, Princeton University Press, Princeton, N.J., 1958, p. 689.

Kinetic Theoretical Studies of Shock Waves in Gas-Particle Mixtures

By the moment method for the Boltzmann equation with a collision term between particle and gas molecules, the normal shock wave structures are investigated without any assumptions on initial gas jump conditions, and on the particle drag and heat transfer coefficient. In the profiles for high gas Mach number, a sudden transition from frozen to relaxation region is found. For low Mach number and small particle size, such a transition disappears, and the shock wave is dispersed. The particle drag and heat transfer rate are calculated within the shock transition process. There are two types of transport of momentum and energy, i.e., the one is the Stokes type, and the other is not.

Particle velocities in vertical pneumatic conveying systems

The calculation of pressure gradients in solid particle conveying systems requires a knowledge of particle velocities. In the following discussion it is shown that, in the case of large particles in vertical pipes, a general equation for particle velocities can be written in terms of the known particle and system properties and the empirical coefficients, Co, Co, and As, which relate the drag force and the solids force to the flow parameters. It is further shown that, for a given rate of solids flow, the excess pressure gradient due to the presence of the solids passes through a minimum as the airflow is changed and that at this minimum the particle velocity is independent of the solids factor A,. Moreover, if the multi-particle drag coefficient CD is approximately equal to the single particle drag coefficient Co,, then at the minimum, the relative velocity between the air and the solids is approximately 41% greater than the single particle terminal velocity in air. The flow of dilute suspensions of solid particles in air has been studied extensively and satisfactory explanations of the effect of a change in the airflow on the total pressure gradient and the stability of the suspension have been given (Zenz & Othmer 1960; Wen & Galli 1971; Kunii & Levenspiel 1969). In these texts the phenomenon of saltation in horizontal flow is described together with the sudden increase in the system pressure gradient which occurs when, at constant solids flow, the airflow falls below the saltation value. The equivalent choking velocity in vertical flow has also been described and the more gradual increase in pressure gradient as the airflow reduces beyond a pressure gradient minimum is recognised as being due to a fall in particle friction and to a rise in static head as the mixture density increases at lower air and particle velocities. A knowledge of particle velocities is necessary for predicting pressure drops accurately in such systems but no generally accepted analytical method exists which enables these velocities to be calculated. Some investigators have concluded that for vertical flow the relative velocity between the air and the solids is equal to the single particle terminal velocity. Others have refuted this assumption and produced data to show that the relative velocity is not a constant but increases with increasing air velocity. Basic analyses relating air/particle drag forces and frictional forces due to particle collisions have been published by Hinkle (1953) and by Wen & Galli (1971). In the Hinkle thesis the equation developed requires experimental data relating air and particle velocities before a solution for pressure drop can be found. The relationship which was found to correlate the measured particle and gas velocities is of an empirical nature and is of uncertain value outside the conditions in the test. Wen & Galli derive a general equation for horizontal flow which relates particle and gas velocities but its solution requires data on the solids frictions factor As together with an iterative or graphical solution for the solid particle velocity in terms of the air velocity. In the following analysis it is shown that in the case of vertical flow a general solution for particle velocities can be written in terms which include the solids factor As and the multiparticle drag coefficient CD. By considering only the excess pressure gradient due to the

Drag coefficient equations for small particles in high speed flows

The paper determines the effect of various available drag coefficient equations on particle velocity calculations for typical two phase flows encountered in supersonic and turbulent laser velocimeter applications. The predictions of the particle drag coefficient equations are compared with experimental sphere drag data. For the laser velocimeter applications, the relative Mach number less than 2 and the relative Reynolds number less than 200 are of particular importance.

Shock Wave Relations in Lunar Ash Flow

AbstractA detailed analysis of steady normal shock waves in a layer of lunar ash by the theory of two phase flow of a mixture of a gas and small solid particles is presented. New terms of pressure gradient and virtual mass forces in the particle momentum equation and the particle kinetic energy and work done in the particle energy equation are included in the fundamental equations and their influences are investigated. It is found that the virtual mass force is only important at low density ratio of the solid particles and the gas in the free stream, hence it may be neglected in ash flow calculations. On the other hand, the rest new terms must not be neglected in the normal shock wave calculation as was done by previous authors. The flow variables as functions of the free stream Much number M0, initial particle volume fraction Z0 and the density ratio G are presented. The thickness of the relaxation zone is found to increase with decreasing Z0 and almost independent of G for any given values of M0. A new empirical relation of the drag coefficient for the spherical particles in the mixture to fit experimental data for Z up to 0.55 and Rep up to 1,000 is proposed in the calculation.

Maximum Thrust Nozzles for Gas Particle Flows

An optimization analysis is presented for the design of maximum thrust nozzles which contain liquid or solid particles in the flowfield. A design procedure was developed wherein an assumed nozzle contour is analyzed to determine if it is a maximum thrust nozzle, and a relaxation scheme was employed to modify the assumed contour until the maximum thrust contour is obtained. Parametric studies were conducted to illustrate the influence of the particle size, the nozzle inlet angle, the throat radius of curvature, the particle drag and heat-transfer coefficients, the particle to gas mass flow ratio, and the nozzle size.

Bubble‐free expansion of gas‐fluidized beds of fine particles

AbstractFive different fine powders with average particle size of about 100 microns or smaller were fluidized with air over their respective bubblefree ranges of fluidization. Photographs taken through the front transparent wall of the column showed that the beds contained cavities and microchannels whose sizes were of the same order of magnitude as the particle size. It is shown that interparticle van der Waals adn capillary forces play an important role in the bubble‐free expansion of small particles. Measurements of bed expansion and average particle drag coefficients are consistent with the proposed role of interparticle forces.

Couette flow with particle injection

The steady flow of a viscous fluid between two infinite parallel flat walls with injection of particles from one wall and removal from the other is analyzed. The velocity and temperature distributions of the fluid and the particles are solved in analytical form for Stokes particle drag coefficient and in numerical form for constant coefficient. Arbitrary values of axial pressure gradient, viscous dissipation, particle injection angle and speed, and initial particle temperature are allowed. For the case of vertical injection and in the absence of axial pressure gradient, it is found that the wall shear stress is always decreased by particle injection whereas heat transfer can be increased or decreased. The dimensionless parameters which characterize the effects of particle injection on particle-free Couette flow are identified and discussed.

The effect of fluid-drag forces on the discharge of free-flowing solids from hoppers

By carrying out a force balance on the particles in the orifice region of a hopper, an equation has been derived for predicting free-discharge flowrates. The fluid-drag force was estimate from single particle drag coefficients. The equation gives values which agree with experimental results for free-flowing materials, except for hoppers with small orifices where wall-friction forces become significant.

PARTICLE SIZE CLASSIFICATION BY DEPOSITION ANGLE IN A GAS CENTRIFUGE AT REDUCED PRESSURE

Centrifugal particle size classification by a gas centrifuge which produces a forced vortex was investigated at reduced pressures. In this method, different particle trajectories cause the formation of a continuous gradation of particle size on the rotor wall. Here, particles in the subsieve and submicron ranges were classified with good resolution because the ''Cunningham correction'' increases as pressure decreases. Also, the computed solutions gave excellent agreement with the experimental results. The numerical solutions with and without the integral term for non-uniform motion have been compared, the approximate equation of best fit for the drag coefficient of spherical particles being used in both cases.

NEW APPROXIMATE EQUATION OF DRAG COEFFICIENT FOR SPHERICAL PARTICLES

An approximate equation for the drag coefficient of spherical particles is proposed in the form of a quadratic equation in logarithmic co-ordinates. The constants of the equation have been determined by the least square method from experimental values5). This equation gives better agreement with experimental values than the equations proposed previously by other authors. The constants of the equation and the comparison of its accuracy are summarized in Tables I and 2. This equation is valid even at Reynolds numbers between O and 70, 000, the standard deviation of the relative error being 5%.

Particle Dynamics in Solids-Gas Flows in Vertical Pipe

Abstract An investigation was carried out to study the particle dynamics of a known mixture of two narrowly sized particles for the first time. The flow characteristics determined by means of a special photographic technique (dot-streak-dot) were particle velocities, turbulence intensities, pressure drop, slip velocities and drag coefficients. The results are discussed with respect to the loading ratio, pipe Reynolds number, and composition of the mixtures. The particle velocity profile and the particle density distribution were analyzed in light of recent theoretical treatments on transport of solids in pipes. Moreover, the concept of volume/surface mean diameter as the representative diameter of the mixed-particle flow system is evaluated in correlating these flow parameters.

Migration of Particles to a Hole Wall in a Drilling Well

Abstract The rate at which a particle will migrate from the rotating drill pipe to the hole wall was calculated as functions of the mud properties, pipe rotational speed, and hole size. The information can be used to estimate the relative carrying capacity of one mud compared to another Then the changes that need to be made when a hole. is not cleaning can be determined. An adaptive control process for field use is suggested Two major simplifying assumptions are that axial flow does not influence the liquid viscosity and that the viscous resistance to particle movement in the radial direction is proportional to the particle's velocity. The calculations assume a power model liquid, spherical cuttings, and concentric pipe in a round hole. A concept of "theoretical zones" where the calculations are valid interspersed with areas where the particles are uniformly redistributed, is used to relate the simplified theory to actual hole conditions. Introduction This theoretical study was conducted to improve the understanding of hole cleaning and how various drilling conditions and muds influence the ability to lift cuttings to the surface. The primary object of the study was to see how fast a particle might move from the drill pipe to the hole wall and how factors of drilling and liquid properties influence the rate of radial movement. The rate of movement may be important as the centrifugal force set up by the rotating drill pipe tends to force the particle toward the hole wall, an area of low vertical mud velocity and, therefore, poor lifting. PRIOR WORK Removal of cutting from a drilling well by circulating a structured liquid such as bentonite-water is a physically complex operation. Explicit prediction of the path of a cutting requires the ability to calculate the velocity of the particle relative to the liquid and the velocity field of the liquid. Various phases have been investigated. Williams and Bruce showed experimentally the path of particles in a laboratory wellbore and developed slip correlations for turbulent flow. Many investigations have been made for drag coefficients of particles settling in stationary Newtonian liquids under conditions of laminar and turbulent flow. The only work with inclined discs is with very low Reynolds numbers, an area of little interest in hole cleaning. Some investigations with drag coefficients in non-Newtonian liquids and the only work with structured liquids indicates the drag coefficient is different in stationary liquid from the drag coefficient in moving liquid. Various methods of predicting slip velocities are available, as well as correlations of hole cleaning based on experimental well tests. Bentonite drilling muds are generally a shear thinning liquid in which stress-rate relation at a fixed time depends on past history. Most rheological work is done under steady-state conditions to show the influence of chemical additives and temperature changes. There has been little work on dynamic response such as has been done with viscoelastic liquids to select time dependent constitutive equations or experiments with shear in two directions. Most pressure drop calculations for field use are based on annular flow without rotating drill pipe, and radial migration of particles and angular liquid velocities are neglected. Savins and Wallick calculated annular pressure drops, including rotating drill pipe, for a liquid described by a three constant Oldroyd model. The calculations were based on a viscosity relation for one-dimensional flow but used the vector sum of the longitudinal and angular components of stress to calculate the viscosity. This procedure was shown correct for a viscoelastic liquid. SPEJ P. 147ˆ

Effects of turbulence on the drag coefficients of spheres in a supercritical flow Ŕegime

AbstractDrag coefficients of aerodynamically smooth spheres varying in diameter from 0.0625 to 1.004 in. and in density from 0.195 to 7.80 g./cc. were obtained at acceleration rates ranging from 120 to −30 ft./sq.sec. The particles were subjected to relative turbulence intensities of 7 to 35% and to ratios of Eulerian macroscale to particle diameter of about 0.4 to 5.Quantitative measurement of particle drag coefficients was made possible by the use of a new particle tracing technique which permits the resolution of time to the nearest tenth of a millisecond. The resulting data extend farther into the supercritical flow regime than any other measurements previously reported.The variation in drag coefficient with Reynolds number indicates a continuous alteration in the flow pattern around a sphere in this region. The effect of turbulence is, essentially, to increase supercritical drag, although this effect was found to diminish with increasing Reynolds number. Possible mechanisms for the effects of Reynolds number and turbulence on the particle drag coefficient are suggested.

Comment on "Drag Coefficient of Small Spherical Particles"

Comment on "Drag Coefficient of Small Spherical Particles"

Drag coefficient of small spherical particles.

An experimental study was conducted to determine the drag coefficient of small, inert, spherical particles accelerating in a laminar, nonreacting, incompressible continuum flow. The Reynolds number range covered in the study was from 200 to 1700, and particle sizes ranged from 150 to 450/1. Glass beads, HP 295 ball powder, and sapphire balls were used. The convective flow behind the shock wave in a shock tube was used to accelerate the particles. The diameter and temporal displacement of a single particle were determined by use of a rotating drum camera in conjunction with an oscillating light source. Using the photographic data, the particle density, the shock speed, the initial temperature and pressure, and the normal shock relations, the particle drag coefficient was calculated. Experiments with HP 295 ball powder produced drag results that were as much as 85% higher than the standard drag coefficient curve. This increase was found to be dependent on the relative Mach number of the flow about the particle even though the Mach number was less than 0.3. The glass beads yielded similar results but to a lesser extent. The sapphire balls produced the best results, yielding the lowest drag values, less scatter, and little sensitivity to Mach number. Many possible influences were considered to explain the foregoing anomalous results. Photomicrographs showed that the ball powder was very and that the sapphire balls were the most smooth. It is concluded that the particles in solid rocket nozzles are rough and hence their drag coefficients are much higher than would normally be the case.

Drag coefficient of particles in a rocket nozzle.

Drag coefficient of particles in a rocket nozzle.