Mathematical model of the entry length in the flow of suspensions of solid particles in a gas

A mathematical model consisting of equations of mass and momentum and for the velocity field has been used for computing the entry length of the flow of non-Newtonian fluids in laminar, transition and turbulent regions. Experimental data measured in a vertical flow of a suspension of solid particles in air have been used for verifying the predictions. n flow index for laminar flow - Re Reynolds number defined for the flow of the carrier medium - q exponent for turbulent flow - ratio of core radius with a flat velocity profile to pipe radius - _c ratio of the axial component of local velocity in the core to mean velocity - w mean flow velocity - ratio of axial distance from the pipe entrance to the pipe radius - ratio of the entrance length to the pipe radius - relative mass fraction of particles - ratio of the distance from the pipe wall to the pipe radius - coefficient of pressure loss due to friction     

Pipe flow with solid particles fixed in space

A group of solid particles were hung by slender rods in a pipe to make a model of two-phase flow of coarse particles. Pressure gradient and velocities were measured for different types of the models. The drag on the particles (spheres) were obtained from measurements of pressure gradient with some assumptions. The results are summarized as follows. (1) Mean velocities of fluid are lower in the central part of the pipe than in the circumferential part. Turbulence is remarkably increased by particles. The spectrum distribution of turbulent velocity becomes flatter. These results are similar to the gas-solid flow of coarse particles in a vertical pipe. (2) At a large Reynolds number, the drag coefficient per one sphere in the group is larger than that of a single isolated sphere in a uniform flow. When the spheres are arranged along the same line in the longitudinal direction, the drag coefficient becomes smaller as the longitudinal distance between the spheres is shortened.     

A model for tracking inertial particles in a lattice Boltzmann turbulent flow simulation

A computationally inexpensive model for tracking inertial particles through a turbulent flow is presented and applied to the turbulent flow through a square duct having a friction Reynolds number of Re_τ = 300. Prior to introducing particles into the model, the flow is simulated using a lattice Boltzmann computation, which is allowed to evolve until a steady state turbulent flow is achieved. A snapshot of the flow is then stored, and the trajectories of particles are computed through the flow domain under the influence of this static probability field. Although the flow is not computationally evolving during the particle tracking simulation, the local velocity is obtained stochastically from the local probability function, thus allowing the dynamics of the turbulent flow to be resolved from the point of view of the suspended particles. Particle inertia is modeled by using a relaxation parameter based on the particle Stokes number that allows for a particle velocity history to be incorporated during each time step. Wall deposition rates and deposition patterns are obtained and exhibit a high level of agreement with previously obtained DNS computational results and experimental results for a wide range of particle inertia. These results suggest that accurate particle tracking through complex turbulent flows may be feasible given a suitable probability field, such as one obtained from a lattice Boltzmann simulation. This in turn presents a new paradigm for the rapid acquisition of particle transport statistics without the need for concurrent computations of fluid flow evolution.     

The distribution of solid particles suspended in a turbulent flow: A stochastic approach

Basic fluid mechanics and stochastic theories are applied to show that the concentration distribution of suspended solid particles in a direction normal to the mean streamlines of a two-dimensional turbulent flow is greatly influenced by the lift force exerted on them in the vicinity of the wall. Analytic solution shows that, when the direction of the mean flow is horizontal, the probability density functionp (y, t) for random displacements of the particles will have a maximum value at a point from the wall where the perpendicular component of the lift force precisely balances particle gravity. Interpretation of experimental observations is presented using this theory.     

Rheological behaviour of two-phase flow of solid particles in a gas

The present investigation was concerned with the rheological behaviour of dilute suspensions of solid particles in a gas in a vertical cocurrent flow moving upwards. Starting from the experimentally determined dependence of the pressure drop on the concentration of solid particles and the Reynolds number of the carrier medium in the steady flow region, the rheological parameters were estimated using pseudo-shear diagrams. Air was the carrier medium and the dispersed phase was one of six fractions of polypropylene powder and five fractions of glass ballotini. The results show that the investigated two-phase systems have pseudoplastic character which becomes more pronounced with increases in concentration, equivalent diameter and density of solid particles in the flowing suspension. C _d coefficient of particle resistance - d _e equivalent diameter of particles - D column diameter - Fr Froude number - g gravitational acceleration - K rheological parameter - L length - n rheological parameter - p _t pressure drop due to friction - p _m total pressure drop - p _ag pressure drop due to acceleration of the gas phase - p _as pressure drop due to acceleration of the solid phase - p _g hydrostatic pressure of the gas phase - p _s specific effective weight of the dispersed phase - r radius - Re Reynolds number - Re _p Reynolds number of a particle - Re _G generalized Reynolds number - Re _G1 generalized Reynolds number relating to the end of the laminar flow region - Re _G2 generalized Reynolds number relating to the beginning of the turbulent flow region - w _z axial component of velocity - u _t steady free-fall velocity of a single particle - w average velocity - w _g average velocity of the gas phase - w _s average velocity of the dispersed phase of solid particles - relative mass fraction of solid particles - x _s volume fraction of solid particles - _g coefficient of pressure drop due to friction - µ dynamic viscosity - _g density of the gas phase - _m density of the suspension - _s density of solid particles - _ds density of the dispersed phase - _w shear stress at the wall     

Drag models for simulating gas–solid flow in the turbulent fluidization of FCC particles

This paper examines the suitability of various drag models for predicting the hydrodynamics of the turbulent fluidization of FCC particles on the Fluent V6.2 platform. The drag models included those of Syamlal–O’Brien, Gidaspow, modified Syamlal–O’Brien, and McKeen. Comparison between experimental data and simulated results showed that the Syamlal–O’Brien, Gidaspow, and modified Syamlal–O’Brien drag models highly overestimated gas–solid momentum exchange and could not predict the formation of dense phase in the fluidized bed, while the McKeen drag model could not capture the dilute characteristics due to underestimation of drag force. The standard Gidaspow drag model was then modified by adopting the effective particle cluster diameter to account for particle clusters, which was, however, proved inapplicable for FCC particle turbulent fluidization. A four-zone drag model (dense phase, sub-dense phase, sub-dilute phase and dilute phase) was finally proposed to calculate the gas–solid exchange coefficient in the turbulent fluidization of FCC particles, and was validated by satisfactory agreement between prediction and experiment.     

A model to describe the flow curves of concentrated suspensions of spherical particles

Summary The explanation for the very common phenomenon of linearity of double logarithmic plots of stress and shear-rate for shear-thinning concentrated suspensions, proposed in earlier papers, has been much extended. A definite model is proposed which, while not claiming to describe real systems in detail, leads to the linearity of the log-log curves and probably represents fairly well the actual behaviour of such systems as thick fresh dairy creams at low temperatures.     

A concurrent micromechanical model for predicting nonlinear viscoelastic responses of composites reinforced with solid spherical particles

A concurrent micromechanical model for predicting nonlinear viscoelastic responses of particle reinforced polymers is developed. Particles are in the form of solid spheres having micro-scale diameters. The composite microstructures are idealized by periodically distributed cubic particles in a matrix medium. Each particle is assumed to be fully surrounded by polymeric matrix such that contact between particles can be avoided. A representative volume element (RVE) is then defined by a single particle embedded in the cubic matrix. A spatial periodicity boundary condition is imposed to the RVE. One eighth unit-cell model with four particle and polymer subcells is generated due to the three-plane symmetry of the RVE. The solid spherical particle is modeled as a linear elastic material. The polymeric matrix follows nonlinear viscoelastic behaviors of thermorheologically simple materials. The homogenized micromechanical relation is developed in terms of the average strains and stresses in the subcells and traction continuity and displacement compatibility at the subcells’ interfaces are imposed. A stress–strain correction scheme is also formulated to satisfy the linearized micromechanical and the nonlinear constitutive relations. The micromechanical model provides three-dimensional (3D) effective properties of homogeneous composite responses, while recognizing microstructural geometries and in situ material properties of the heterogeneous medium. The micromechanical formulation is designed to be compatible with general displacement based finite element (FE) analyses. Experimental data and analytical micromechanical models available in the literature are used to verify the capability of the above micromechanical model for predicting the overall composite behaviors. The proposed micromodel is also examined in terms of computational efficiency and accuracy.     

An improved model for two-phase flow through beds of coarse particles

A one-dimensional model for two-phase flow in packed particle beds is presented. Compared with earlier models, the improved model takes into account the effect of interfacial drag forces between liquid and gas, which are of considerable influence in beds of coarse particles. The model is based on the momentum equations for separated flow, which are closed by empirical relations for the wall shear stress and the interfacial drag. The model is applied to the situation of a one-component two-phase flow in an internally heated bed of uniform spherical particles. An increased dryout heat flux is predicted if liquid enters the bed through the bottom.     

A new drag model for TFM simulation of gas–solid bubbling fluidized beds with Geldart-B particles

In this work, a new drag model for TFM simulation in gas–solid bubbling fluidized beds was proposed, and a set of equations was derived to determine the meso-scale structural parameters to calculate the drag characteristics of Geldart-B particles under low gas velocities. In the new model, the meso-scale structure was characterized while accounting for the bubble and meso-scale structure effects on the drag coefficient. The Fluent software, incorporating the new drag model, was used to simulate the fluidization behavior. Experiments were performed in a Plexiglas cylindrical fluidized bed consisting of quartz sand as the solid phase and ambient air as the gas phase. Comparisons based on the solids hold-up inside the fluidized bed at different superficial gas velocities, were made between the 2D Cartesian simulations, and the experimental data, showing that the results of the new drag model reached much better agreement with experimental data than those of the Gidaspow drag model did.     

The lateral migration of solid particles in a laminar flow near a plane

The lateral inertial migration of a solid spherical particle suspended in a laminar flow over a vertical wall is considered theoretically. Formulae for the migration velocity are obtained for both neutrally buoyant and non-neutrally buoyant particles and also for the case of zero flow over the wall. Situations in which the particle is either free to rotate or prevented from rotating are considered. Such results are found to agree qualitatively with known experimental data.     

A unifying model for fluid flow and elastic solid deformation: A novel approach for fluid–structure interaction

Fluid–structure coupling is addressed through a unified equation for compressible Newtonian fluid flow and elastic solid deformation. This is done by introducing thermodynamics within Cauchy׳s equation through the isothermal compressibility coefficient that is experimentally measurable for both fluids and solids. The vectorial resolution of the governing equation, where every component of velocity vectors and displacement variation vectors is calculated simultaneously in the overall multi-phase system, is characteristic of a monolithic resolution involving no iterative coupling. For system equation closure, mass density and pressure are both re-actualized from velocity vector divergence, when the shear stress tensor within the solid phase is re-actualized from the displacement variation vectors. This novel approach is first validated on a two-phase system, involving a plane fluid–solid interface, through the two following test cases: (i) steady-state compression and (ii) longitudinal and transverse elastic wave propagations. Then the 3D study of compressive fluid injection towards an elastic solid is analyzed from initial time to steady-state evolution.     

Effect of laden solid particles on the turbulent flow structure of a round free jet

The effect of solid particles on the flow structure of a round air jet in a stagnant surrounding was investigated experimentally. Information on the averaged two-component velocities, the kinetic energies, and the u′ v′-properties were obtained for both phases by means of a monochromatic three beam laser Doppler anemometer. The particle number density was also measured by this system. Glass beads of 64 μm and 132 μm diameter were used for a constant mass loading ratio of 0.3 in a jet with a Reynolds number of 20 000. The lateral mean velocity and number density profiles were expressed by best fitting functions and several invariable coefficients were found. The standard drag force coefficient C _D for a single particle was applicable for a dilute particle cloud even in a non-uniform air velocity field.     

A linear viscoelastic model for solid polyethylene

Stress relaxation tests have been carried out on a blue, pipe grade PE 80 medium density polyethylene (BP Chemicals), to provide thermo-viscoelastic rheology for use in calculating thermal stresses in pipe production. Stresses up to 4 MPa were used, with strains up to about 2%, in tests at temperatures from 23° to 90°C. Within this range a linear viscoelastic model was applicable, provided the initial ramp strain rate was less than 7×10^–5 s^–1. The stress relaxation data was fitted directly by a model incorporating an elastic response to volumetric strains, and a generalised linear solid model, consisting of two Maxwell elements and a purely elastic element in parallel, for deviatoric strains. Arrhenius type temperature dependence of relaxation times and shear moduli is found, and within experimental accuracy the temperature dependence of all these model parameters is the same. As a consequence, and provided that the duration of the strain ramp is sufficiently short relative to relaxation times, the model leads to time-temperature superposition of the relaxation moduli, using the same shift factor on both the response magnitude and time axes.