Nonhomogeneous Solids Research Articles

Reactive particle-laden flow in industrial-scale supercritical water gasification reactor: Modeling and study on flow patterns

Coal gasification in supercritical water (SCW) is a highly complicated reactive particle-laden flow process, which couples multiphase flow hydrodynamics, conjugate heat transfers, and complex chemical reactions. A quantitative understanding is the key issue for reactor design and optimization. In this study, the Eulerian-Eulerian model is used to describe the particles-SCW flow in a novel industrial-scale thermally integrated supercritical water gasification (TISCWG) reactor with conjugate heat transfer boundary conditions and porous media model. The sophisticated chemical reactions are using a species transport model with a seven-step lumped kinetics. Four zones of the TISCWG reactor and four typical flow patterns in the gasification chamber are firstly revealed as 1) the bottom accumulation region with the nonhomogeneous solids distribution and back-mixing of particles, 2) the upper developed fluidization region with the SCW and coal particles moving in a plug-flow regime in the center and a downward annular gravity-driven flow near the wall, 3) the jet-affected region strongly affected by the vortexes near the nozzle and presented as a trans-critical jet regime and 4) the dilute region where dilute solids slow down and fall back into the gasification zone. Flow in the gasification chamber is strongly affected by the heat transfer with the preheat zone, as reflected by an annular gravity-driven flow film flowing downwards near the gasification wall. A funnel-shaped structure promotes a more uniform distribution of solids in the accumulation region rather than a cylindrical structure. The effect of flow patterns on coal gasification processes is also studied. Volatile pyrohydrolysis reactions depend on the time-varying solids distribution and the gaseous products at the system outlet with a typical H2 mole fraction of 61.42% and CO2 of 34.98%.

Interplay between super and subdiffusive behaviors in photothermal phenomena

We analyze an extension for the Fourier law to describe the heat propagation in non-homogeneous solids by taking into accounts simultaneously different diffusion regimes such as the subdiffusion and superdiffusion. It considers differential operators of non-integer order, i.e., fractional differential operators, which are a simple way, from the formal point of view, of understanding the photothermal phenomena, especially the ones obtained with photoacoustic techniques, for a media with these characteristics. Our analysis considers the heat transfer in solid materials by a pulsed light source by considering superdiffusive elements in a subdiffusive matrix. The photoacoustic signal generated by the rear incidence of light (OPC technique) was determined and the influence of superdiffusive and subdiffusive effects was analyzed.

The behavior of vertically non-homogeneous elastic solids under internal rectangular loads

This paper examines the internal rectangular loading problems for an elastic halfspace where the shear modulus varies exponentially or linearly and the Poisson’s ratio keeps constant or varies linearly with depth. The numerical method is developed through applying the fundamental solution of layered elastic solids and integrating numerically it over the loading area. The adaptive integration of the displacement and traction integrals over the loading area is designed to calculate the nearly singular integral for the source point close to an element. The discretization approach is applied to deal with an arbitrarily depth-heterogeneous elastic solid. OpenMP directives are used to parallelize the internal loop, which controls element iterations so that a high computing speed can be obtained. For an axisymmetric internal loading problem, the displacements obtained with the present formulation are in a very good agreement with existing closed-form solutions. Finally, stresses and displacements in non-homogeneous halfspaces induced by horizontally and vertically uniform rectangular loadings are presented. Results illustrate the effect of non-homogeneous properties on the stress and displacement fields.

Boundary element formulation of axisymmetric problems in vertically non-homogeneous solids subject to normal traction

This paper presents an efficient and accurate boundary element method (BEM) for the elastic analysis of axisymmetric problems in vertically non-homogeneous solids without or with cavity. This BEM uses the fundamental solution of the elastic field in a multilayered elastic solid induced by the body force uniformly concentrated at a circular ring. This solution is also called Yue's solution. The effective integration methods are used for dealing with the integrals in the discretized boundary integral equations. The discretization of the boundary surface uses one-dimensional boundary elements. It also adopts the infinite boundary element to take into account the influence of a far-field region on the boundary surface. Numerical verifications of displacements and stresses for three benchmark problems are conducted, which gives the excellent agreement with previously published results. Case studies are presented to numerically illustrate the influences of both vertically non-homogeneous elastic material properties and spherical cavity on the elastic fields induced by uniform tractions on the boundary surface. These numerical results show that this new BEM is a fast and simple numerical algorithm for accurately computing the axisymmetric elastic fields in vertically non-homogeneous solids with or without cavity induced by normal tractions.

Analysis of Boundary Layer Influence on Effective Shear Modulus of 3-1 Longitudinally Porous Elastic Solid

The evaluation of the effective properties of nonhomogeneous solids using analytical methods is, in general, based on the assumption that these solids have infinite dimensions. Here, we investigate the influence of both the number of holes and the boundary layer of a solid with finite dimensions on the determination of these properties. We use the Asymptotic Homogenization Method (AHM) to determine the effective shear modulus of an elastic solid with infinite dimensions containing a uniform and periodic distribution of circular cylindrical holes arranged on a hexagonal lattice. We also use the Finite Element Method (FEM) to determine this modulus in the case of a solid with finite dimensions containing the same uniform distribution of cylindrical holes away from its boundary. Near the boundary, we consider a layer of material with no holes, which is usually left in the fabrication process of samples. Both solids have the same elastic properties and are subjected to similar anti-plane shear loadings. For the finite medium, we study two sequences of domains discretized by the FEM, which are called the Fixed Layer Sequence (FLS) and the Fixed Domain Sequence (FDS). For the FLS, the layer thickness is kept fixed and both the dimensions of the domain and the number of holes vary. For the FDS, the dimensions of the domain are kept fixed and both the number of holes and the layer thickness vary. Results obtained from numerical simulations are then used to generate graphs of the effective shear modulus versus void volume fraction. It is observed that, in the FLS case, the shear modulus obtained from the numerical simulations converges to the analytical solution obtained via AHM. It is also observed that, in the FDS case, the shear modulus obtained from the numerical simulations converges to a limit function, which is close to the analytical solution obtained via AHM. For comparison purposes, we have also calculated the effective shear modulus of porous elastic solids containing a square array of circular cylindrical holes. We then show graphs of this modulus versus void volume fraction for both hexagonal and square arrangements that are very close to each other up to void volume fraction of 0.5.

One approach to solving the basic equation of magnetostatics in the case of nonhomogeneous magnets

We propose a general approach for solving the basic magnetostatic equation in the case of nonhomogeneous magnetic solids when their magnetic permeability depends on the coordinates. Seeking the strength of the resulting field reduces to finding two scalar functions satisfying a system of one integral and one differential equation with certain boundary conditions. Using this approach, we obtain an analytic solution of the magnetostatic equation for a ball with a model permeability in an arbitrary external field.

Wave processes in the locally nonhomogeneous solids

Wave processes in the locally nonhomogeneous solids

A numerical implementation of the Dynamic Thermal Network method for long time series simulation of conduction in multi-dimensional non-homogeneous solids

The Dynamic Thermal Network (DTN) approach to the modelling of transient conduction was conceived by Claesson (1999,2002,2003) as an extension of the network representation of steady-state conduction processes. The method is well suited to the simulation of building fabric components such as framed walls and thermally massive structures such as basements but can also be applied to the long timescale simulation of other conduction problems. The theoretical basis of the method and its discretized form is outlined in this paper and a new numerical procedure for the calculation of the necessary weighting factor data is presented. Such data has previously been generated for three-dimensional bodies by a heuristic process of blending analytical solutions and numerical data. The numerical approach reported here has the advantage of accommodating parametric representations of multi-dimensional geometries and allows the data to be produced in an automated fashion and so more easily incorporated into simulation tools. Enhancements to the data reduction procedure and a generalised approach to representing complex boundary conditions are also presented. The numerical procedure has been validated by a series of comparisons with analytical conduction heat transfer solutions and discretization errors were found to be acceptably small. Compared to numerical methods, calculations using the DTN method were found to be up to four orders of magnitude quicker but with comparable accuracy.

Theoretical–experimental analysis of heat transfer in nonhomogeneous solids via improved lumped formulation, integral transforms and infrared thermography

Theoretical and experimental methodologies for the identification of spatially variable thermophysical properties and for simulating multidimensional heat transfer in heterogeneous materials are illustrated by using plate samples with aluminum oxide nanoparticles dispersed in a polymeric matrix. First, the heterogeneous nanocomposite plate is thermally characterized by means of a fairly simple experimental setup which can be modeled by a one-dimensional heat conduction formulation with space variable properties. Non-intrusive temperature measurements are obtained via infrared thermography, while the direct problem is handled by an error-controlled integral transform solution with an improved lumped-differential formulation, and the inverse analysis is undertaken via Bayesian inference, making use of the Markov Chain Monte Carlo method. Then, in order to illustrate the application of the methodologies here presented, an experimental multidimensional demonstration is provided consisting of a small electrical resistance attached to the plate, simulating a heat generating electronic device installed on the nanocomposite substrate, which in such situation works as a heat spreader modeled by an improved lumped-differential two-dimensional heat conduction formulation. The integral transform solution of the lumped-differential two-dimensional problem is then critically compared against the infrared thermography experimental results.

Dynamic problems of composite mechanics

A procedure for applying Green’s tensor method to the averaging of dynamic problems in the theory of elasticity of nonhomogeneous solids with both periodic and nonperiodic structures is presented.

Studies of adsorption equilibria and kinetics of o-, m-, p-nitro- and chlorophenols on microporous carbons from aqueous solutions

The studies of adsorption of o-, m-, and p-nitrophenols and chlorophenols from aqueous solutions on microporous activated carbon are presented. The adsorption isotherms were measured at different temperatures: 288, 298, 308 and 318 K. The thermal analysis was applied in order to find the differences in solute interactions with adsorbent surface. The kinetic dependences were measured and the relations between position of adsorbate functional groups and adsorption rate were discussed. The adsorption data are analyzed in terms of the theory of physical adsorption on energetically non-homogeneous solids. The relations between the adsorbate structure and adsorption effectiveness, and the influence of temperature are discussed.

Evaluation of the Stress Intensity Factors for Cracks in Continuously Nonhomogeneous Solids, Part II: Meshless Method

A meshless method based on the local Petrov-Galerkin approach is proposed for crack analysis in two-dimensional (2-D), orthotropic and linear elastic solids with continuously varying material properties. Both quasi-static thermal and transient elastodynamic problems are considered. For time-dependent problems, the Laplace-transform technique is utilized. A unit step function is used as the test functions in the local weak-form. It leads to local boundary integral equations (LBIEs) involving only a domain-integral in the case of transient dynamic problems.The analyzed domain is divided into small subdomains with a circular shape. The moving least-squares (MLS) method is adopted for approximating the physical quantities in the LBIEs. The accuracy of the present method for computing the mode-I stress intensity factors is discussed by comparison with available analytical or numerical solutions.

Evaluation of the Stress Intensity Factors for Cracks in Continuously Nonhomogeneous Solids, Part I: Interaction Integral

The interaction integral method is an accurate and robust scheme for evaluating mixed-mode stress intensity factors. The method makes use of a conservation integral in an elastic orthotropic and continuously nonhomogeneous solid. Two independent equilibrium states are considered. State 1 corresponds to the actual state for given boundary conditions, and State 2 corresponds to an auxiliary state, which can be either the mode-I or the mode-II near-tip displacement and stress fields. Path-independent contour-domain integral form for evaluation of stress intensity factors is derived for cracked bodies described by uncoupled thermoelasticity and elastodynamics. The advantage of the presented method for the computation of stress intensity factors through the interaction integral technique is the high accuracy because the contour integral is evaluated at points far away from the crack-tip. Numerical errors of the computed quantities in the crack-tip vicinity in a direct computation of stress intensity factors from the asymptotic expansion formulae are hence reduced in this approach.

Dynamic Crack Analysis in Functionally Graded Piezoelectric Solids by Meshless Local Petrov-Galerkin Method

In the present paper, the meshless local Petrov-Galerkin (MLPG) method is extended to two-dimensional (2-D) continuously nonhomogeneous piezoelectric solids with cracks under dynamic loading conditions. To eliminate the time-dependence, the Laplace-transform technique is applied to the governing partial differential equations which are satisfied in the Laplace-transformed domain in a weak-form on small fictitious subdomains. A meshless approximation is used for spatial variations of the displacements and the electric potential.

Fracture Analyses in Continuously Nonhomogeneous Piezoelectric Solids by the MLPG

Fracture Analyses in Continuously Nonhomogeneous Piezoelectric Solids by the MLPG

An advanced numerical method for computing elastodynamic fracture parameters in functionally graded materials

A new meshless method for computing the dynamic stress intensity factors (SIFs) in continuously non-homogeneous solids under a transient dynamic load is presented. The method is based on the local boundary integral equation (LBIE) formulation and the moving least squares (MLS) approximation. The analyzed domain is divided into small subdomains, in which a weak solution is assumed to exist. Nodal points are randomly spread in the analyzed domain and each one is surrounded by a circle centered at the collocation point. The boundary-domain integral formulation with elastostatic fundamental solutions for homogeneous solids in Laplace-transformed domain is used to obtain the weak solution for subdomains. On the boundary of the subdomains, both the displacement and the traction vectors are unknown generally. If modified elastostatic fundamental solutions vanishing on the boundary of the subdomain are employed, the traction vector is eliminated from the local boundary integral equations for all interior nodal points. The spatial variation of the displacements is approximated by the MLS scheme.

A meshless local boundary integral equation method for heat conduction analysis in nonhomogeneous solids

A local boundary integral equation method (LBIEM) with meshless approximation for heat conduction analysis in non‐homogeneous solids is presented. A review of recent developments in advanced meshless LBIEM for 2‐d, 3‐d axisymmetric problems and microwave heating analysis is given. Both stationary and transient heat conduction problems are investigated in the paper. For transient problems both the Laplace transform technique and the time difference approach are applied to obtain the integral formulations using stationary or static fundamental solutions. Microwave heating is modelled by the forced heat equation with an exponential decay of the heat source intensity from an incident boundary. The thermal material properties are considered to be dependent on spatial coordinates. A pure boundary integral formulation is restricted to non‐homogeneous solids with a special material gradation, where fundamental solutions can be obtained. To remove this restriction, an LBIEM is proposed to analyse the temperature distribution in general non‐homogeneous solids. The moving least square (MLS) method is used for approximating the physical quantities in the local boundary integral equations (LBIEs). Numerical examples are given to demonstrate the high accuracy and efficiency of the implemented meshless LBIEM.

Evaluation of elastic properties in 2-D non-homogeneous solids

This paper is concerned with inverse evaluation of the material property in two-dimensional non-homogeneous elastic solids. It seeks to recover variable subsurface Lame's coefficients which are functions of depth. The method considers a temporal interval for which time dependent measurements are provided. It formulates an optimal estimation problem which seeks to minimize the error difference between the given data and the response from the system. The method leads to an iterative algorithm which, at every iteration, requires the solution to a two-point boundary value problem. Numerical results indicate that a close estimate of the unknown functions can be obtained based on the boundary measurements only.

Natural frequencies of vibration of a class of solids composed of layers of isotropic materials

In a recent paper, the Ritz method with simple algebraic polynomials as trial functions was used to obtain an eigenvalue equation for the free vibration of a class of homogeneous solids with cavities. The method presented is here extended to the study of a class of non-homogeneous solids, in which each solid is composed of a number of isotropic layers with different material properties. The Cartesian coordinate system is used to describe the geometry of the solid which is modelled by means of a segment bounded by the yz, zx and xy orthogonal coordinate planes and by two curved surfaces which are defined by fairly general polynomial expressions in the coordinates x, y and z. The surface representing the interface between two material layers in the solid is also described by a polynomial expression in the coordinates x, y and z. In order to demonstrate the accuracy of the approach, natural frequencies are given for both a two- and three-layered spherical shell and for a homogeneous hollow cylinder, as computed using the present approach, and are compared with those obtained using an exact solution. Results are then given for a number of two- and three-layered cylinders and, to demonstrate the versatility of the approach, natural frequencies are given for a five-layered cantilevered beam with a central circular hole as well as for a number of composite solids of more general shape.

A persistent random walk model for the frequency-dependent electrical conductivity

A persistent random walk model is first introduced as a coarse grained method to produce a Drude-like mobility for free charge carriers within homogeneous solids. The model is extended to nonhomogeneous solids where it is applicable for calculating the frequency-dependent effective conductivity up to microwave frequencies. The mean free path, transport time and free carrier concentration for each component phase region is accounted for by the model. The approach is demonstrated by obtaining the frequency-dependent conductivity of a two component periodically layered composite, where an exact enumeration over all Brownian paths is employed in the numerical calculations.