DISCRETE PARTICLE SIMULATION OF SIZE SEGREGATION OF PARTICLE MIXTURES IN A GAS FLUIDIZED BED

This paper presents a study of the mixing/segregation behaviour of particle mixtures in a gas fluidized bed by use of the discrete particle simulation. Spherical particles with diameters 2 mm （jetsam） and 1 mm （flotsam） and density 2 500 kg.m^-3 are used as solid mixtures with different volume fractions. The particles are initially packed uniformly in a rectangular bed and then fluidized by gas uniformly injected at the bottom of the bed. The gas injection velocities vary to cover fixed, partially and fully fluidized bed conditions. Segregation/mixing behaviour is discussed in terms of flow patterns, solid concentration profile and mixing kinetics. The results show that segregation, as a transient fluidization process, is strongly affected by gas injection velocities for a given particle mixture. With the increase of the volume fraction of flotsam, size segregation appears at lower velocities.     

THE BOUNDARY INTEGRAL-VARIATIONAL THEOREMS OF AN ARBITRARY ELEMENT IN THE SOLID—COMPUTE THE ENERGY RELEASE RATE OF AN ARBITRARY CRACK EXTENSION

Based on the solid mechanics of the discrete form and its variational principles proposed by Niu, this paper puts forward four kinds of boundary integral-variational theorems of an arbitrarg element.In the course of the fracture analysis, they can be used to compute the energy release rate along the normal direction of the crack boundary. When there is a hole in the solid, and whether there are given surface forces on the hole boundary or not they can be used to compute the variation of the energy along the normal direction of the hole boundary. In the course of the discrete analysis, they can be used to establish the discrete equations, so that the values of the unknown functions are solved. At the same time, from this paper we know that the J-integral proposed by Rice represents an integral to be independent of a path imperfectly.     

EFFECTS OF POISSON’S RATIO ON SCALING LAW IN HERTZIAN FRACTURE

In this paper the Auerbach＇s scaling law of Hertzian fracture induced by a spherical indenter pressing on a brittle solid is studied. In the analysis, the singular integral equation method is used to analyze the fracture behavior of the Hertzian contact problem. The results show that the Auerbach＇s constant sensitively depends on the Poisson＇s ratio, and the effective Auerbach＇s domain is also determined for a given value of the Poisson＇s ratio.     

TENSILE STRENGTH FOR SPLITTING FAILURE OF BRITTLE PARTICLES WITH CONSIDERATION OF POISSON''''S RATIO

1. Introduction The core mechanism of comminution could be reduced to the breakage of individual particles that occurs through contact with other particles or with the grinding media, or with the solid walls of the mill (Potapov & Campbell, 1994). When a particle is subjected to a load, the nature of the stress field around and within the brittle particle, its material properties, and the size and distribution of micro-flaws within the particle govern the size and shape distribution of the fra…     

FUZZY ARITHMETIC AND SOLVING OF THE STATIC GOVERNING EQUATIONS OF FUZZY FINITE ELEMENT METHOD

The key component of finite element analysis of structures with fuzzy parameters, which is associated with handling of some fuzzy information and arithmetic relation of fuzzy variables , was the solving of the governing equations of fuzzy finite element method. Based on a given interval representation of fuzzy numbers, some arithmetic rules of fuzzy numbers and fuzzy variables were developed in terms of the properties of interval arithmetic. According to the rules and by the theory of interval finite element method, procedures for solving the static governing equations of fuzzy finite element method of structures were presented. By the proposed procedure, the possibility distributions of responses of fuzzy structures can be generated in terms of the membership functions of the input fuzzy numbers. It is shown by a numerical example that the computational burden of the presented procedures is low and easy to implement. The effectiveness and usefulness of the presented procedures are also illustrated.     

MULTI-SCALE NUMERICAL MODEL FOR SIMULATING CONCRETE MATERIAL BASED ON FRACTAL THEORY

A new multi-scale numerical model is presented using the fractal theory and adopting FEM to simulate the failure of concrete.The relation between the fractal box dimension in large scale and the damage to concrete in small scale is deduced.And the evolutionary process of elastic modulus and strength in small scale is given.Consequently,the multi-scale numerical model is proposed to describe the constitutive relation of concrete between small scale and large scale.A two-dimensional static analysis of a concrete block is performed by using this model and the calculation result is discussed.The propagation of cracks of the concrete block is also studied.     

Effect of solid particle loading on nucleophilic addition of epoxy-resin to isophorone diisocyanate

In the preparation of surface coatings made of conductive composites consisting of conductive particulate fillers in a soft matrix, cracks will develop with increase of the particulate loading, which is believed to be related to the nucleophilic addition reaction between glycidyl end-capped poly （bisphenol A-co-epichlorohydrin） and isophorone diisocyanate molecules. This curing reaction is responsible for the generation of crosslinking network throughout the coatings. The influence of solid particle loading on the chemical reaction may be described as a volume-excluded effect, that is, the high solid particle loading will occupy the space between the functional groups thus preventing the chemical reaction to take place. As a direct consequence, the cross-linking network cannot develop properly due to the insufficiency of curing reaction. This will lead to the generation of cracks, as was supported by FT-IR analysis in this work.     

ANALYSIS OF FINANCIAL DERIVATIVES BY MECHANICAL METHOD (Ⅰ)-BASIC EQUATION OF PRICE OF INDEX FUTURES

Similar to the method of continuum mechanics, the variation of the price of index futures is viewed to be continuous and regular. According to the characteristic of index futures, a basic equation of price of index futures was established. It is a differential equation, its solution shows that the relation between time and price forms a logarithmic circle. If the time is thought of as the probability of its corresponding price, then such a relation is perfectly coincided with the main assumption of the famous formula of option pricing, based on statistical theory, established by Black and Scholes, winner of 1997 Nobel’ prize on economy. In that formula, the probability of price of basic assets (they stand for index futures here) is assummed to be a logarithmic normal distribution. This agreement shows that the same result may be obtained by two analytic methods with different bases. However, the result, given by assumption by Black-Scholes, is derived from the solution of the differential equation.     

Clustering behavior of solid particles in two-dimensional liquid-solid fluidized-beds

In this paper, the clustering behavior of solid particles in a two-dimensional （2D） liquid-solid fluidized-bed was studied by using the charge coupled devices （CCD） imaging measuring and processing technique and was characterized by fractal analysis. CCD images show that the distribution of solid particles in the 2D liquid-solid fluidised-bed is not uniform and self-organization behavior of solid particles was observed under the present experimental conditions. The solid particles move up in the 2D fluidized-bed in groups or clusters whose configurations are often in the form of horizontal strands. The box fractal dimension of the cluster images in the 2D liquid-solid fluidized-bed increases with the rising of solid holdup and reduces with the increment of solid particle diameter and superficial liquid velocity. At given solid holdup and solid particle size, the lighter particles show smaller fractal dimensions.     

NUMERICAL CALCULATION OF LONDON-VAN DER WAALS ADHESION FORCE DISTRIBUTIONS FOR DIFFERENT SHAPED PARTICLES

A basic method to calculate van der Waals dispersion force distributions for submicron superquadric particles in particle-wall systems is presented. The force distribution is achieved by rotating particles through a large number of arbitrary spatial orientations, each time keeping constant the contact distance to the wall surface while calculating the dispersion force. To accomplish this, the use of 2D particle shape suffices, that is, through using an inter-dimensional function, which has been determined previously. A further development of the method within digital image analysis may lead to possible applications to forecasting the macroscopic properties of particle systems, for example, flowability, agglomeration behavior or dispersibility. For small ranges of superquadric particle shapes, each with a different size, the way from determining the inter-dimensional function up to applying image analysis is shown in an example.     

ON FREE WAVE PROPAGATION IN ANISOTROPIC LAYERED MEDIA

The method of reverberation-ray matrix （MRRM） is extended and modified for the analysis of free wave propagation in anisotropic layered elastic media. A general, numerically stable formulation is established within the state space framework. The compatibility of physical variables in local dual coordinates gives the phase relation, from which exponentially growing functions are excluded. The interface and boundary conditions lead to the scattering relation, which avoids matrix inversion operation. Numerical examples are given to show the high accuracy of the present MRRM.     

BURNING OF SINGLE CARBON PARTICLE APPROACHED BY ACTIVATION ENERGY ASYMPTOTICS

The method of activation energy asymptotics is used to treat the combustion of a singlecarbon particle in quiescent gas mixture with high temperature.Both heterogeneousreactions 2C+O_2→2CO,C+CO_2→2CO and homogeneous reaction2CO+O_2 (?) 2CO_2 are considered.It is shown that the burning of the particle principallyis carried out during a diffusion-limited period.Four brief and complex periods throughwhich the history of the particle evolves from a heat-up period to the diffusion-limited periodare described.A comparison between results of activation energy asymptotics and e(?)numerical solutions is given.The agreement is considered satisfactory.     

TENSILE STRENGTH FOR SPLITTING FAILURE OF BRITTLE PARTICLES WITH CONSIDERATION OF POISSON＇S RATIO

The core mechanism of comminution could be reduced to the breakage of individual particles that occurs through contact with other particles or with the grinding media, or with the solid walls of the mill. When brittle particles are loaded in compression or by impact, substantial tensile stresses are induced within the particles. These tensile stresses are responsible for splitting failure of brittle particles. Since many engineering materials have Poisson‘s ratios very close to 0.3, the influence of Poisson‘s ratio on the tensile strength is neglected in many studies. In this paper, the state of stress in a spherical particle due to two diametrically opposed forces is analyzed theoretically. A simple equation for the tensile stress at the centre of the particle is obtained. It is found reasonable to propose this tensile stress at the instant of failure as the tensile strength of the particle. Moreover, this tensile strength is a function of the Poisson‘s ratio of the material. As the state of stress along the z-axis in an irregular specimen tends to be similar to that in a spherical particle compressed diametrically with the same force, this tensile strength has some validity for irregular particles as well. Therefore, it could be used as the tensile strength for brittle particles in general. The effect of Poisson‘s ratio on the tensile strength is discussed.     

RESISTANCE EFFECT OF ELECTRIC DOUBLE LAYER ON LIQUID FLOW IN MICROCHANNEL

Poisson-Boltzrnann equation for EDL (electric double layer) and Navier-Stokes equation for liquid flows were numerically solved to investigate resistance effect of electric double layer on liquid flow in microchannel. The dimension analysis indicates that the resistance effect of electric double layer can be estimated by an electric resistance number, which is proportional to the square of the liquid dielectric constant and the solid surface zeta potential, and inverse-proportional to the liquid dynamic viscosity, electric conductivity and the square of the channel width. An "electric current density balancing" (ECDB) condition was proposed to evaluate the flow-induced streaming potential, instead of conventional "electric current balancing" (ECB) condition which may induce spurious local backflow in neighborhood of the solid wall of the microchannel. The numerical results of the flow rate loss ratio and velocity profile are also given to demonstrate the resistance effect of electric double layer in microchannel.     

Euler-Euler CFD modeling of fluidized bed＂ Influence of specularity coefficient on hydrodynamic behavior

Euler-Euler two-fluid model is used to simulate the hydrodynamics of gas-solid flow in a bubbling flu- idized bed with Geldert B particles where the solid property is calculated by applying the kinetic theory of granular flow （KTGF）. Johnson and Jackson wall boundary condition is used for the particle phase, and different amount of slip between particle and wall is given by varying the specularity coefficient （φ） from 0 to 1. The simulated particle velocity, granular temperature and particle volume fraction are compared to investigate the effect of different wall boundary conditions on the hydrodynamic behavior, Some of the results are also compared with the available experimental data from the literature. It was found that the model predictions are sensitive to the specularity coefficient. The hydrodynamic behavior deviated sig- nificantly for φ = 0 and φ = 0.01 with maximum deviation found at φ = 0 i.e. free-slip condition. However, the overall bed height predicted by all the conditions is similar.     

ANALYSIS OF POSTBUCKLING STRENGTH OF STRINGER STIPFENED CYLINDRICAL SHELLS UNDER AXIAL COMPRESSION

An analysis of the postbuckling strength of stringer stiffened cylindrical shells, subject to axial compres-sion is described. The method used in this paper is based on plastic analysis extending Murray's method which was used to analyse postbuckling behaviour of stiffened plates loaded axially and in bendin9. The mechanism tripping of stringer and crumpling of shell plate is described based on how the test specimens are deformed after buckling. Finally the theoretical analyses are compared with the experimental results of steel specimens. The theoretical results coincide quite well with the experimental data. It should therefore be possible to use the method described here to analyse postbuckling strength of stringer stiffened cylindrical shells and to estimate energy absorption capabilities in relation to collision studies.     

Steady rotation of a composite sphere in a concentric spherical cavity

The problem of steady rotation of a compositesphere located at the centre of a spherical container has beeninvestigated.A composite particle referred to in this paperis a spherical solid core covered with a permeable sphericalshell.The Brinkman’s model for the flow inside the composite sphere and the Stokes equation for the flow in the spherical container were used to study the motion.The torque experienced by the porous spherical particle in the presence ofcavity is obtained.The wall correction factor is calculated.In the limiting cases,the analytical solution describing thetorque for a porous sphere and for a solid sphere in an unbounded medium are obtained from the present analysis.     

A FINITE DIFFERENCE METHOD FOR SIMULATING TRANSVERSE VIBRATIONS OF AN AXIALLY MOVING VISCOELASTIC STRING

A finite difference method is presented to simulate transverse vibrations of an axially moving string.By discretizing the governing equation and the equation of stress- strain relation at different frictional knots,two linear sparse finite difference equation systems are obtained.The two explicit difference schemes can be calculated alternatively, which make the computation much more efficient.The numerical method makes the nonlinear model easier to deal with and of truncation errors,O(Δt~2 Δx~2).It also shows quite good stability for small initial values.Numerical examples are presented to demonstrate the efficiency and the stability of the algorithm,and dynamic analysis of a viscoelastic string is given by using the numerical results.     

POROSITY CALCULATION OFMIXTURES OF FIBROUS PARTICLES

The initial forming of fiber blend to high green density, i.e. the packing of fibrous particles, is important to the reinforcement of composite materials. It is very useful to develop a general predictive method for the optimum selection of particle mixtures for the property control of ceramic or composite products. This paper presents such a mathematical model developed on the basis of the similarity analysis between the spherical and non-spherical particle packings and assesses its applicability to the packing of fibrous particles with discrete and/or continuous length distributions. The results indicate that the model can predict this packing system well and hence provide an effective way to solve various packing problems in the composite materials processing.     

ANALYTICAL SOLUTIONS FOR RESPONSE OF COLLISION OF PARTICLE WITH CONICAL ROD CAUSED BY LONGITUDINAL VIBRATION

The objective is to present exact analytical solutions of longitudinal impact analysis for slender conical rods struck by a particle and a new method is proposed for conical rod-particle impact analysis, in which the superposition method is used and the response of the rod is presented. These analytical results are exact and can be used to validate the numerical methods or other analytical results. The numerical example shows that one of the advantages of the present method is that the analytical form is very simple. The result is that mass ratio and some variables describing the geometrical shape of rods such as taper, length and radius play an important role in impact dynamic system.