Effect of particle size on nonlinear wave propagation in a fluidized bed

The effect of particle size (Archimedes number) on the propagation of a kinematic particle concentration wave in a fluidized bed is investigated. The dependence of the characteristic wave velocity on the porosity of the bed (particle concentration) and the Archimedes number (or the Reynolds number for flow past individual particles of the dispersed phase) is determined. The evolution of a nonlinear perturbation of the bed porosity is investigated and the formation of discontinuities in the concentration of the dispersed phase is studied in relation to the particle size (Archimedes number). It is shown, in particular, that, as distinct from a bed of small particles, in a bed of large particles with quadratic interphase interaction only compression discontinuities can be formed. The results obtained can be used to analyze the formation of inhomogeneities (slugs and bubbles) in a fluidized bed in relation to the particle size.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 96–100, May–June, 1987.     

Steady concentration waves and dispersion effects in a gas-particle two-phase medium with weak particle interaction

The model of a concentrated two-phase medium constructed in [1–3] with allowance for the random small-scale motion of the dispersed phase due to particle interaction is used for analyzing steady particle concentration waves and dispersion effects in the case of negligibly small rates of generation and dissipation of the energy of small-scale motion. The propagation of one-dimensional disturbances in a direction parallel or antiparallel to the force of gravity is investigated. The structure of the steady wave front is found and the conditions of formation of internal concentration discontinuities at the front are determined. Dispersion effects are investigated for weakly nonlinear waves. The results can be used for analyzing wave phenomena in a gas-fluidized bed, a falling bed, pneumatic transport and fast fluidization systems, etc. The model proposed in [1–3] was developed in [4] in order to investigate steady waves and dispersion effects at high rates of generation and dissipation of the energy of small-scale motion of the dispersed phase and in [5] in order to analyze the propagation of particle concentration discontinuities for finite rates of generation and dissipation of the energy of random motion and linear stability in the presence of weak particle interaction.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 42–48, January–February, 1990.     

The propagation of nonlinear waves in a bidisperse fluidized bed

A study is made of the propagation of nonlinear kinematic waves of concentrations of solid particles in a fluidized bed of particles of two different sizes. A hyperbolic system of quasilinear equations is obtained which describes the propagation of the waves. A dependence of the characteristic velocities on the concentrations of the phases and the ratio of the sizes of the particles is found. The influence of an admixture of fine particles on the propagation of porosity waves in the fluidized bed is analyzed. The nature of the formation of jumps in the porosity depending on the concentration of the admixture is studied, as is the process of the transfer of the admixture of fine particles in the bed. The nature of the propagation of nonlinear waves in a fluidized bed of identical particles is clarified. A characteristic velocity is found and conditions are determined for the formation of discontinuities of concentration of the dispserse phase in rarefaction compression waves.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 49–58, January–February, 1985.     

Motion of inhomogeneities of a developed fluidized bed at small reynolds numbers

The instability of a fluidized system in which the particles are uniformly distributed in space [1–3] leads to the development of local inhomogeneities in the internal structure, these taking the form of more or less stable formations of packets of particles [4]. In accordance with the existing ideas based on experimental data [5–8, 13], the particle concentration within a packet may vary in a wide range from very small values (10^–2–10^–3 [8]) for bubbles to the concentration of the unfluidized bed for bunches of particles in a nearly closely packed state. The paper considers the steady disturbed motion of the fluid and solid phases near an ascending or descending packet of particles in a developed fluidized bed. It is assumed that the motion of the solid phase corresponds to a creeping flow of viscous fluid, and the viscosity of the fluidizing agent is taken into account only in the terms that describe the interphase interaction. The velocity fields and pressure distributions of the phases inside and outside a packet are determined. If the particle concentration within a packet tends to zero, the solution describes the slow motion of a bubble in a fluidized bed. The results of the paper are compared with results obtained earlier for the model of ideal fluids [9] and Batchelor's model [10], in which the fluidized bed is treated in a simplified form as a viscous quasihomogeneous continuum.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 57–65, July–August, 1984.     

Interaction of local inhomogeneities of a fluidized bed

In connection with an analysis of transport processes in fluidized beds a study is made of the steady motion of a system of local inhomogeneities of the bed porosity, which are modeled by packets of particles. The interaction of the inhomogeneities is taken into account on the basis of a cell model. The velocity fields and the pressure distributions of the solid and gas phases together with the rising and sinking velocities of the system of packets are determined in the approximation of a double continuum. The flow regimes of the fluidizing agent are investigated. It is shown that with increasing concentration of packets in the bed the velocity of their motion decreases, and the circulation region of the dispersion medium surrounding a packet or trapped within it contracts. The dependence of the rate of flow of the fluidizing agent through the transverse section of the reactor on the concentration of packets in the bed is found.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 42–49, January–February, 1991.     

Propagation of disturbances in a two-phase magnetic particle dispersion

The propagation of one-dimensional vertical perturbations of the magnetic particle concentration in a liquid-particle or gas-particle disperse medium in an external uniform magnetic field, oriented parallel or antiparallel to the force of gravity, is considered. The propagation of linear disturbances is analyzed. Linear stability criteria for layers of magnetized particles fluidized with a gas or liquid are found. The propagation of nonlinear long (kinematic) concentration waves in a medium containing dispersed magnetized particles is investigated. It is shown that in the kinematic wave approximation the propagation of the nonlinear concentration perturbations can be described by the Burgers equation. The formation of regions with a sharp change in concentration (shock fronts) is analyzed. The structure of the shock front, in particular its width, is found. It is shown that a magnetic field leads to broadening of the shock front, thus preventing the formation of concentration discontinuities and hence leading to the smearing of the boundaries of the inhomogeneities formed in the fluidized bed. This provides a basis for a qualitative explanation of the stabilization of developed nonuniform fluidized beds of magnetic particles in a magnetic field.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 76–83, July–August, 1992.     

Thermomechanical oscillations of a fluidized bed

Within the framework of the simplest one-dimensional model of a fluidized bed as a medium consisting of two interpenetrating interacting fluids, it is shown in the paper that linear thermomechanical oscillations can occur if the gas viscosity depends on the temperature and a nonisothermic chemical reaction takes place in the fluidized bed. The conditions under which oscillations occur and their frequencies are determined and their dependence on the fluidization parameters, the heat loss (or supply) rate, the rate and heat of the reaction, and quantities characterizing the dependence of the gas viscosity on the temperature and the force of the interphase interaction on the concentration of the solid particles is found.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 75–80, September–October, 1984.     

Mass transfer between bubbles and the continuous phase in a fluidized bed of variable cross section

A study is made of mass transfer in an inhomogeneous fluidized bed whose cross section varies over the height. The field of the liquid phase around a bubble is constructed, conditions are obtained for the existence of a region of closed circulation of the liquid phase, and the boundaries of this region are determined. A solution is given to the problem of convective diffusion of the substance to the region of closed circulation, and the mass-transfer coefficient between a bubble and the continuous phase is determined as a function of the flow parameters. By the same token, the results of [1] are generalized to a fluidized bed of variable cross section. It is shown that in this case the mass transfer is improved.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 20–29, November–December, 1974.     

Influence of an underwater ridge on a nonlinear wave expanding from a point on the surface of a liquid

A study is made of the influence of an underwater ridge on a solitary wave that prior to the interaction with the ridge has the form of a circle situated outside the ridge. It is shown that the nonlinear effects lead to a concentration of the wave energy above the ridge. As they move away from the source, the waves propagating above the ridge are not damped in the considered approximation but are damped everywhere away from the ridge. An analogy is pointed out between the propagation of the wave and two-dimensional steady flows of a fluid, and this makes it possible to use hydrodynamic intuition for qualitative predictions about the nature of the wave propagation in various cases. All the results of the paper can be extended to the case of waves that are periodic in time.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 100–105, July–August, 1982.     

Experimental and computational studies on flow behavior of gas–solid fluidized bed with disparately sized binary particles

This paper presents experimental and computational studies on the flow behavior of a gas-solid fluidized bed with disparately sized binary particle mixtures. The mixing/segregation behavior and segregation efficiency of the small and large particles are investigated experimentally. Particle composition and operating conditions that influence the fluidization behavior of mixing/segregation are examined. Based on the granular kinetics theory, a multi-fluid CFD model has been developed and verified against the experimental results. The simulation results are in reasonable agreement with experimental data. The results showed that the smaller particles are found near the bed surface while the larger particles tend to settle down to the bed bottom in turbulent fluidized bed. However, complete segregation of the binary particles does not occur in the gas velocity range of 0.695-0.904 m/s. Segregation efficiency increases with increasing gas velocity and mean residence time of the binary particles, but decreases with increasing the small particle concentration. The calculated results also show that the small particles move downward in the wall region and upward in the core. Due to the effect of large particles on the movement of small particles, the small particles present a more turbulent velocity profile in the dense phase than that in the dilute phase.     

Nonlinear dispersion of long internal waves

The propagation of long weakly nonlinear waves in an atmospheric waveguide is considered. A model system of Kadomtsev-Petviashvili equations [1], which describes the propagation of such waves, is derived. In the case of one excited wave mode the system of model equations goes over into the Kadomtsev-Petviashvili equation, in which, however, the variables x and t are interchanged. The reasons for this are clarified. In the two-dimensional case an approximate solution of the model equations is constructed, and steady nonlinear waves and their interaction in a collision are considered. The results of a numerical verification of the stability of the approximate steady solutions and of the solution to the problem of decay of the wave into quasisolitons are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 151–157, May–June, 1988.     

Mass transfer to particle clusters and bubbles in a fluidized bed at low Reynolds numbers

The process of mass transfer to a particle cluster or bubble rising in a developed fluidized bed rapidly enough for a region of closed circulation of the fluidizing agent (cloud) to be formed is investigated in the Stokes approximation on the basis of a model of the steady-state motion of the fluid and solid phases near the cluster or bubble [1]. Within the cloud surroundinga local inhomogeneity of the fluidized bed intense mixing of the fluid phase takes place and the mass transfer between the cloud and the surrounding medium is determined by diffusion. The method of matched asymptotic expansions is used to obtain an analytic solution of the problem of the concentration field and the diffusion mass flux to the surface of the cloud at small and large values of the Péclet number. The latter is determined from the relative velocity of the cluster, the radius of the cloud, and the effective diffusion coefficient. In the limiting case of zero concentration of the solid phase within the cluster the solution obtained describes the mass transfer to a bubble in the fluidized bed. A comparison is made with the corresponding results previously obtained within the framework of a model of the solid phase as an inviscid fluid [2]. It is shown that the effect of viscosity on the mass transfer to the bubble is most important at large Péclet numbers, and that the correction to the total diffusion flux to the surface of the closed circulation zone due to viscosity effects may reach 40%.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 60–67, July–August, 1986.     

Sudden Blocking of a Subcritical Open–Channel Flow

This paper reports results of experiments in which a steady nonuniform flow in a rectangular channel with an even horizontal bottom was blocked by a rapidly falling shield. Data on the height of the splash–up of water on the shield and the shape, propagation speed, height, and internal structure of the upstream propagating wave of the bore type are obtained for various liquid flow rates at the channel entrance. It is established that the bore produces a strong stratification in the liquid particle velocity, and under particular conditions, the speed of propagation and height of the bore, and the height of the water splash–up on the wall are constant and are determined only by the critical depth for unperturbed flow, i.e., by the specified flow rate.     

Transmission of waves in a liquid through absorbing layers and subsequent interaction of the waves with an obstacle

The propagation of waves in porous media is investigated both experimentally [1, 2] and by numerical simulation [3–5]. The influence of the relaxation properties of porous media on the propagation of waves has been investigated theoretically and compared with experiments [3, 4]. The interaction of a wave in air that passes through a layer of porous medium before interacting with an obstacle has been investigated with allowance for the relaxation properties [5]. In the present paper, in which the relaxation properties are also taken into account, a similar investigation is made into the interaction with an obstacle of a wave in a liquid that passes through a layer of a porous medium before encountering the obstacle.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 53–53, March–April, 1983.     

Intensification of a detonation wave in the zone of secondary reactions in a two-phase medium

A method is proposed for the numerical calculation of one-dimensional nonsteady-state flows of a mixture of a gas with particles, based on the separation of a system of differential equations for a two-phase medium into two subsystems. The problem is solved concerning the propagation of a plane detonation wave in a mixture of a detonating gas with particles, behind the front of which secondary chemical reactions are taking place between the vapors of the particle material and the detonation products. The velocity profiles of the gas and of the thermodynamic functions behind the detonation wave front are determined, and also the dependence of the detonation velocity on the distance to the point of initiation. The conditions for intensification of the detonation wave are obtained in the zone of secondary reactions.Leningrad. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 92–96, September–October, 1972.     

An analysis of the chaotic motion of particles of different sizes in a gas fluidized bed

The dynamic behavior of individual particles during the mixing/segregation process of particle mixtures in a gas fluidized bed is analyzed. The analysis is based on the results generated from discrete particle simulation, with the focus on the trajectory of and forces acting on individual particles. Typical particles are selected representing three kinds of particle motion： a flotsam particle which is initially at the bottom part of the bed and finally fluidized at the top part of the bed; a jetsam particle which is initially at the top part of the bed and finally stays in the bottom de-fluidized layer of the bed; and a jetsam particle which is intermittently joining the top fluidized and bottom de-fluidized layers. The results show that the motion of a particle is chaotic at macroscopic or global scale, but can be well explained at a microscopic scale in terms of its interaction forces and contact conditions with other particles, particle-fluid interaction force, and local flow structure. They also highlight the need for establishing a suitable method to link the information generated and modeled at different time and length scales.     

Circulation instability of steady falling of a flat layer of fine dispersed particles

A numerical investigation has been made of the linear instability of the steady falling under gravity of an infinite horizontal layer of fine dispersed particles in an incompressible atmosphere. The layer has an inhomogeneous vertical distribution of the dispersed phase and a small volume concentration of the particles, the hydrodynamic interaction between which occurs solely through the carrier phase. It is shown that steady falling is unstable and that the layer of particles breaks up into individual convective cells with a characteristic scale of the order of the thickness of the layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 78–83, January–February, 1991.     

Unsteady waves in a liquid containing vapor bubbles

Unsteady wave processes in vapor-liquid media containing bubbles are investigated taking into account the unsteady interphase heat and mass transfer. A single velocity model of the medium with two pressures is used for this, which takes into account the radial inertia of the liquid with a change in volume of the medium and the temperature distribution in it [1]. The system of original differential equations of the model is converted into a form suitable for carrying out numerical integration. The basic principles governing the evolution of unsteady waves are studied. The determining influence of the interphase heat and mass transfer on the wave behavior is demonstrated. It is found that the time and distance at which the waves reach a steady configuration in a vapor-liquid bubble medium are considerably less than the correponding characteristics in a gas-liquid medium. The results of the calculation are compared with experimental data. The propagation of acoustic disturbances in a liquid with vapor bubbles was studied theoretically in [2]. The evolution of waves of small but finite amplitude propagating in one direction in a bubbling vapor-liquid medium is investigated in [3, 4] on the basis of the generalization of the Burgers-Korteweg-de Vries equation obtained by the authors. An experimental investigation of shock waves in such a medium is reported in [5, 6], and the structure of steady shock waves is discussed [7].Translated from Izvestiya Akademii Nauk SSSR, Hekhanika Zhidkosti i Gaza, No. 5, pp. 117–125, September–October, 1984.     

Thermal mechanism of wave damping in a gas-liquid mixture

The effect of interphase heat transfer on the process of propagation of a wave in a monodisperse gas-liquid mixture is investigated. A three-wave equation of the Boussinesq type is derived and formulas for the dependence of the thermal components of the dissipative coefficients on the thermophysical parameters of the mixture are obtained. The limits of applicability of the short-wave method are determined. The theoretical and experimental values of the phase wave velocity are compared and found to be in good agreement.Yerevan. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 75–83, November–December, 1994.