Solid Particle Velocity Research Articles

Terminal Velocities of Single Spherical Particles Falling in a Water-Filled Vertical Pipe.

Terminal velocities of single solid particles or bubbles are one of the most fundamental knowledges required for modeling and simulation of dispersed two-phase flows in vertical pipes. However, there are few available correlations which can cover a wide range of parameters, i.e., the ratio of a particle diameter to a pipe diameter, particle density as well as the falling motions of particles such as rectilinear and rocking. In this study, we proposed a semi-empirical model of terminal settling velocity which can cover a wide range of parameters, based on a singular pressure loss about a particle. Experiments were also carried out to develop an empirical correlation based on the proposed semi-empirical model of terminal settling velocity of a particle falling through a liluid-filled pipe. It was confirmed through comparisons of terminal velocities between the measurements and the developed correlation that the proposed model can give good estimations of terminal velocities under various terminal settling conditions.

Acceleration of solid particles during cumulation of detonation products in vacuum

The possibility of increasing the velocity of solid particles accelerated by an explosion of a long tubular charge of a high explosive (HE) in vacuum is analyzed. The experimental results obtained indicate that the acceleration velocity cannot be considerably increased. The probable causes are erosion of the material from the inner surface of the HE tube and a significant decrease in the mass flow velocity when the length of the HE tube exceeds the optimum length.

Flow development in a gas‐solids downer fluidized bed

AbstractThe development of gas and solids flow structure was studied in a 9.5 m high and 0.10 m diameter, gas‐solids cocurrent downflow circulating fluidized bed (downer). Local solids concentration and particle velocity were measured using two separate optical fibre probes at different radial positions on several axial levels along the downer. The results show that the flow development is significantly influenced by the operating conditions. For most of the conditions under which the experiments were conducted, the gas‐solids flow reaches its fully developed zone within 3 to 8 m away from the entrance. On the other hand, the development zone can extend as long as the downer itself, under certain conditions. When the solids circulation rate is over 100 kg/m2s, an increasing solids circulation rate largely extends the length of radial flow development. It is found that the flow developments in the core and at the wall are not quite simultaneous. For solids concentration, the core develops more quickly at low gas velocities and the wall region develops faster at high gas velocities. For particle velocity, higher gas velocity speeds up the development of the wall region but does not significantly affect the development of the core region. The wall region is much more sensitive to the change of superficial gas velocity than the core region. At high superficial gas velocities (> 7 m/s), a “semi‐dead” region is observed in the fully developed zone adjacent to the wall where the dilute solids are moving at a very low velocity.

A theoretical investigation of gas–solid flow exiting a fluidized bed into a standpipe

A theoretical solution for a two-dimensional steady flow of a gas–solid mixture exiting a fluidized bed into a vertical standpipe is presented. Initially, analytical expressions for the particle velocity and gas pressure are obtained inside the fluidized bed and standpipe, assuming that changes in the mixture density are negligible. For inertial dominated flow regimes it is shown that the gas pressure at the standpipe entry region is very sensitive to the magnitude of inertial and gravitational forces, which could explain the observed particle and gas flow instabilities in industrial fluidized bed/standpipe systems. When density changes in the gas–solid mixture are considered, it is shown that the granular `fluid' density varies considerably in the standpipe entry region with the solid fraction increasing near the top edge of the standpipe walls. The rapid density gradients at the standpipe entry are compensated by gas pressure gradients of similar order, yielding gas velocities comparable in magnitude with solid particle velocities. This first approximation solution suggests that the flow of a granular fluid from a fluidized bed into a standpipe may be governed primarily by the particle phase flow behavior, and rapid changes in particle velocity and density at the standpipe entry could potentially propagate downwards into the standpipe, rendering the flow of the granular fluid erratic and difficult to control.

Experimental study of heat transfer in a co-current downflow fluidized bed (downer)

Local heat transfer was studied in a 9.3 m tall, 100 mm diameter gas–solid concurrent downflow fluidized bed (downer) with FCC particles. A miniature cylindrical heat transfer probe (6.4 mm in diameter and 38 mm in length) was inserted into the bed to measure the local heat transfer coefficients. The radial and axial distribution of heat transfer coefficients between the suspended surface and the gas–particle flow suspension was obtained using a miniature heat transfer probe, under different operating conditions. The solids concentration and particle velocity, which are considered as the two most important parameters, are measured by two optical fibre probes. The results show that the radial and axial distributions of heat transfer in the downer are not the same as those in the riser. The heat transfer rate in the downer is closely related to the hydrodynamics, with bed suspension density being the most influential factor. The average heat transfer coefficient decreases with decreasing solids circulating rate due to the decreased solids holdup. The heat transfer coefficient does not always decrease with increasing gas velocity given the increased importance of gas convective heat transfer under high gas velocities and low solids holdups in the downer.

Hydrodynamics Characteristics of High Speed Multi-Phase Flow in AWJ Nozzles. 2nd Report, Measurement of Exit Phase-Velocities and Velocity Change in Nozzles.

The experimental results on the abrasive water jet (AWJ) are described in this paper. AWJ is the high-speed water jet containing solid particles and is used for cutting of various industrial materials. In order to obtain the velocities of solid, gas and liquid phases at the mixing nozzle exit, the momentum of jet was measured. Phase velocity distribution in the axial direction of jet was also estimated based on the measured pressure distribution in the AWJ nozzle with the aid of the continuous and the momentum equations. The result shows that the decrease of the liquid phase velocity in the mixing nozzle is small in the case of the gas liquid two phase flow, and the velocity of liquid phase is remarkably reduced on the contrary in the gas liquid-solid three-phase flow. In the latter case, the total momentum of jet is reduced comparing with two phase flow. It is also found that the mean velocity of solid particles at the mixing nozzle exit is about 30% of the velocity of liquid phase.

Finite element analysis of the flow and heat transfer of solid particles in moving beds

A numerical analysis for the flow and heat transfer of solid particles in moving beds of heat exchangers is presented. The solid particles pass through a bundle of heat source tubes as the result of the gravitational force. Heat energy is transferred through direct contact of particles with the heat source tubes. A viscous-plastic fluid model and a convective heat transfer model are employed in the analysis. The flow field dominantly determines the total heat transfer in the heat exchanger. As the velocities of solid particles around the heat source tubes increase, the heat transfer from the tubes also increases. Examples are presented to show the performance of the numerical model. The effect of flow on heat transfer has also been studied. © 1998 John Wiley & Sons, Ltd.

Finite element analysis of flow and heat transfer in solid particle moving beds of heat exchanger

Numerical analysis for the flow and heat transfer in solid particle moving beds of heat exchangers is presented. The solid particle flow through the bundle of heat source tubes by the gravitational force. The heat energy is transferred through the direct contact of particles with the heat source tubes. The viscous-plastic fluid model and the convective heat transfer model are employed in the analysis. The flow field dominantly influences the total heat transfer in a heat exchanger. As the velocities of solid particles around the heat source tubes increase, the amount of heat transfer from the tubes increases. Some examples are presented to show the performance of the numerical model. The flow effect on the heat transfer is also studied through the examples.

Mathematical modelling of steady state updraft gasification of jute stick particles of definite sizes packed randomly — An analytical approach

An experimental study was carried out on an insulated MS reactor of total length 1650 mm and constant diameter 76 mm for updraft gasification of jute stick particles with air as the gasifying agent. The results and performances were evaluated for different sizes of jute stick particles at various superficial air inlet velocities ranging from 0·0337 to 0·1011 m s −1 at STP. Accordingly, the superficial mass velocities of solid particles were changed from 0·0225 to 0·0950 kg/(m 2 s 1). The calorific value of the producer gas was found to be in the range of 4306–4912 kJ Sm −3. An attempt was made to develop a predictive model of the gasification system. Gas diffusion through the solid pores was postulated to be controlling the rate processes in the developed non-equilibrium, mechanistic model. Approach to solve the governing equations was analytical and the total process was found to be controlled by a few dimensionless numbers containing the process variables. Comparison of experimental data and the model predictions indicated the validity of the model under the given circumstances.

An improved formula for terminal velocity of rigid spheres

An asymptotic analysis for terminal velocity of a rigid sphere has been made employing the Oseen law for drag force. On the basis of the asymptotic analysis, our previous result (a four-parameter approximation) for terminal velocity of rigid spheres (Nguyen et al., 1994) has been improved. The terminal velocity can be adequately described by a three-parameter model given by Eq. (21) in our present paper. The improved formula: (1) does not numerically differ from our previous model; (2) is simpler than the previous one since it is only a three-parameter approximation; and (3) provides a unified and useful approximation for terminal velocity of solid particles in the modelling of the flotation process.

Calculation of settling velocity of small solid particles in stratified suspension flows

Calculation of settling velocity of small solid particles in stratified suspension flows

Electrostatic, inductive ring probe bandwidth

Electrostatic, inductive, non-intrusive probes are employed for measuring the electric charge as well as the mass flow rate, volume loading, and mean flow velocity of solid particles especially while travelling through pipes of pneumatic transport or in ambient air when a flux of those particles forms a rather compact column-like jet of a nearly constant cross sectional area. The bandwidth of such measuring devices is a very important factor which should be determined carefully, or even planned in advance, before designing the probe itself intended for the use under well defined conditions. In the paper the bandwidth of the inductive ring probes is discussed. To find the probe bandwidth the derivation of a general formula is presented along with the accepted assumptions and an original way of formulation and consideration of that problem. The general formula obtained is an inductive ring probe frequency response function and permits the bandwidth of any inductive, non-intrusive probe, sensor, transducer, etc, to be easily determined for given mean flow velocities of particles and probe widths.

Take-off threshold velocity of solid particles lying under a turbulent boundary layer

The take-off of free solid particles by wind erosion has been investigated. A preliminary bibliographic study (Foucaut and Stanislas 1995) has enabled an analysis of the Bagnold criterion (1941) to be made and its accordance with the results obtained by White (1982) to be shown, thus leading to a semi-empirical model. This initial study led to a more judicious choice of the primary parameters, allowing a more physical representation of the threshold velocity. In the present study, the criterion validation was carried out in a specific boundary layer wind tunnel, by means of a direct measurement of the threshold velocity. The basic idea was to increase the wind tunnel velocity slowly and linearly and to perform an optical detection of the first take-offs.

Slip velocities in Solid-Water Mixture Flows with Nearly Equal Density in Pipe

Solid particle velocities are observed experimentally in water slurries of solid particles slightly thinner or denser than water, and the dynamics of the mixture flows are investigated. Slip velocities given by the difference between the mean velocities of solid particles and water in a circular pipe are reported in the present study.

A Semianalytical Analysis of Compressible Electrophoretic Cake Formation

Leaks in geomembrane liners of waste landfills and liquid impoundments cause chemical contaminants to leak into the subsurface environment. A mathematical model is presented to simulate electrophoretic sealing of impoundment leaks. The model describes the formation of a compressible clay cake because of electrical and gravitational forces. The model includes mass balance equations for the solid particles and liquid phase, modified Darcy's law in an electrical field, and Terzaghi's definition of effective stress. The formulation is presented in the Eulerian coordinates. The resulting second‐order, nonlinear partial differential equation and the lower boundary condition are linearized to obtain an analytical solution for time‐dependent settlement. After discretizing in time the analytical solution is applied to simulate compression of an accreting sediment. In the simulation of an accreting sediment, solid fluxes on either side of suspension/sediment interface are coupled using a no‐jump condition. The velocity of a discrete particle in the suspension zone is assumed to be equal to the algebraic sum of electrophoretic and Stoke's settling velocities. An empirical relationship available in the literature is used to account for the effect of concentration on the velocity of solid particles in the suspension zone. The validity of the semianalytical approach is partially verified using an exact steady state solution for self‐weight consolidation. The simulation results obtained for a set of material parameters are presented graphically. It is noted that the electrokinetic consolidation of sediment continues even after the completion of electrophoretic settling of all clay particles. An analysis reveals that the electrophoretic cake formation process is quite sensitive to voltage gradient and the coefficient of compressibility.

Particle behavior in the turbulent boundary layer. II. Velocity and distribution profiles

The velocity, flux, and concentration distribution of solid particles in a turbulent boundary layer of a horizontal water flume were investigated experimentally by means of LDA and visualization techniques. The particles were of polystyrene (specific density ∼1.05). Results show that coherent wall structures are responsible for most of the characteristics of particle behavior throughout the boundary layer. Particles are often concentrated in regions of low velocity, associated with wall structures, and as a result the average particle velocity is lower than the fluid’s. This was also noted previously by Rashidi et al., but not explained. The actual relative velocity between the particles and the surrounding fluid is often small. In addition, the data suggest that as the shear rate increases, the particle flux profiles asymptotically approach a shape where a strong gradient of particle flux exists in the lower part of the boundary layer (y+≤250), while it is relatively constant at higher elevations. This phenomenon may also be attributable to interactions with the wall structures.

Wave propagation in heterogeneous, porous media: A velocity‐stress, finite‐difference method

A particle velocity‐stress, finite‐difference method is developed for the simulation of wave propagation in 2-D heterogeneous poroelastic media. Instead of the prevailing second‐order differential equations, we consider a first‐order hyperbolic system that is equivalent to Biot’s equations. The vector of unknowns in this system consists of the solid and fluid particle velocity components, the solid stress components, and the fluid pressure. A MacCormack finite‐difference scheme that is fourth‐order accurate in space and second‐order accurate in time forms the basis of the numerical solutions for Biot’s hyperbolic system. An original analytic solution for a P‐wave line source in a uniform poroelastic medium is derived for the purposes of source implementation and algorithm testing. In simulations with a two‐layer model, additional “slow” compressional incident, transmitted, and reflected phases are recorded when the damping coefficient is small. This “slow” compressional wave is highly attenuated in porous media saturated by a viscous fluid. From the simulation we also verified that the attenuation mechanism introduced in Biot’s theory is of secondary importance for “fast” compressional and rotational waves. The existence of seismically observable differences caused by the presence of pores has been examined through synthetic experiments that indicate that amplitude variation with offset may be observed on receivers and could be diagnostic of the matrix and fluid parameters. This method was applied in simulating seismic wave propagation over an expanded steam‐heated zone in Cold Lake, Alberta in an area of enhanced oil recovery (EOR) processing. The results indicate that a seismic surface survey can be used to monitor thermal fronts.

Expansion of the detonation products of lead azide via a supersonic nozzle

The cloud of products formed following the detonation of lead azide (LA) contains gaseous species and solid particles. The dynamics of the detonation products expanding freely or through a supersonic nozzle into vacuum is unraveled via the temporal profiles of the pressure, the emission from Pb atoms and the attenuation of a He-Ne beam. The velocity of the fastest gaseous species is found from the onset of the pressure rise and the emission at a given distance from the LA sample, and the velocity of the fastest solid particles from the attenuation. In the free expansion, the respective velocities are 4.5±0.1 and 3.8±0.2 km/s and in the nozzle expansion 5.1±0.2 and 1.4±0.2 km/s. The expansion into atmospheric pressure air is also monitored and found to be much slower than that into vacuum. The utilization of nozzles as a means for obtaining a particle free, transparent medium of detonation products is stressed in the context of exploiting explosives for achieving chemical lasers in the visible wavelength region.

A simple algorithm for the calculation of the terminal velocity of a single solid sphere in water

A method for predicting the terminal settling velocity of a single solid spherical particle in water is described. The terminal velocity is a function of, among others, particle size and the Archimedes number of 0 to 512000. The prediction is in agreement with that by using Stokes law or Allen law or Newton law for drag force and is useful for the modeling of the flotation elementary processes of the particle-bubble interaction.

Modeling of Compressible Cake Filtration

The transport of suspended solid particles in a liquid through porous media has importance from the viewpoint of engineering practice and industrial applications. Deposition of solid particles on a filter cloth or on a pervious porous medium forms the filter cakes. Following a literature survey, a governing equation for the cake thickness is obtained by considering an instantaneous material balance. In addition to the conservation of mass equations for the liquid, and for suspended and captured solid particles, functional relations among porosity, permeability, and pressure are obtained from literature and solved simultaneously. Later, numerical solutions for cake porosity, pore pressure, cake permeability, velocity of solid particles, concentration of suspended solid particles, and net rate of deposition are obtained. At each instant of time, the porosity decreases throughout the cake from the surface to the filter septum where it has the smallest value. As the cake thickness increases, the trends in pressure variation are similar to data obtained by other researchers. This comparison shows the validity of the theory and the associated solution presented. A sensitivity analysis shows higher pressure values at the filter septum for a less pervious membrane. Finally, a reduction in compressibility parameter provides a thicker cake, causes more particles to be captured inside the cake, and reduces the volumetric filtrate rate. The increase of solid velocity with the reduction in compressibility parameter shows that more rigid cakes compress less.