Thermo-mechanical modelling of cellular hybrid refractories

Refractory materials are subjected to both quasi-static and dynamic thermal loading (thermal shock) causing damage up to mechanical failure. Typical refractories are magnesia carbon bricks consisting of periclase (MgO) and carbon inclusions. Recently, a significant improvement of the thermo-mechanical behaviour could be achieved by cellular hybrid composites made of periclase-filled carbon foams. The present contribution focuses on MgO-filled carbon foams and the investigation and optimisation of the structure-property relationship with respect to a reduction of thermally induced stresses and damage. It is a transient as well as static, fully coupled thermo-mechanical problem. According to the fact that, in general, refractories are brittle materials a linear elastic model, with a damage criterion was used. To optimise the structural morphology of the cellular refractories, the effect of micro structural changes has been determined. For the investigation of the thermal shock! behaviour, the results correlate very well with the experimentally motivated Hasselman relation. There is a significant size effect depending on the pore size. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An approach to constructing models of multiphase elastic porous media

A novel approach to describing the behaviour of multiphase elastic porous media is proposed. The average values of the physical quantities needed to describe the motions of porous media are formulated using an integral relation. The validity of this relation is taken as the fundamental hypothesis. The integral definition of the average values enables integral relations to be devised for the average values from the integral laws of conservation of mass, momentum and energy and the increase in entropy. Along with the average values, the integral relations contain new variables that can be identified with generalized thermodynamic forces, which can be used to take into account the phase interaction in a porous medium. The integral relations are used to derive differential equations for the rate of entropy change and Gibbs relations for a porous medium as a basis for obtaining the constitutive relations. Relationships between the thermomechanical parameters of the model are established from the Gibbs relations under additional assumptions. The equation for the rate of entropy change can be used to establish relations between the generalized thermodynamic forces and fluxes. A complete system of differential equations in the defining parameters, which describes the motion of multiphase elastic porous media, is finally obtained.     

Simulation of a compositional model for multiphase flow in porous media

In this article, we consider the simulation of a compositional model for three‐dimensional, three‐phase, multicomponent flow in a porous medium. This model consists of Darcy's law for volumetric flow velocities, mass conservation for hydrocarbon components, thermodynamic equilibrium for mass interchange between phases, and an equation of state for saturations. A discretization scheme based on the block‐centered finite difference method for pressures and compositions is developed. Numerical results are reported for the benchmark problem of the third comparative solution project (CSP) organized by the society of petroleum engineers (SPE). © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.     

Stochastic modeling of non-equilibrium multiphase flow in porous media

Non-equilibrium phenomena arising from the pore scale dynamics can have considerable effect on the large scale dynamics of multiphase flow in porous media. In such cases, the relative permeabilities and capillary pressure curves are not just simple functions of phase saturations, rather are dynamic functions of space and time. The present work proposes a stochastic approach in which particle mobility can be modelled based on the actual conductance of fluid volume inside a pore space. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Development of a compressible multiphase cavitation approach for diesel spray modelling

The influence of in-nozzle phenomena including cavitation on the morphology of the spray from a diesel injector with a sharp nozzle inlet is investigated numerically. A compressible, multi-phase Volume of Fluid Large Eddy Simulation is implemented in the OpenFOAM environment. The volume fraction transport equations for liquid and gas phases are reformulated to include mass transfer source terms. These source terms are modelled with two cavitation models by Schnerr and Kunz, which are extended to eliminate non-physical mass transfer rates. Validation is carried out only for the Schnerr cavitation model due to its independence of empirical parameters. The numerical method is validated by comparing the simulated mass flow rates, pressure and velocity profiles at different cavitation conditions against published experimental data obtained using a slightly converging square channel. Favourable comparison between simulations and experiments is achieved with minor discrepancies attributable to uncertainties in fuel properties, experimental artefacts and assumptions made in numerical models. Application of the method to calculation of in-nozzle phenomena and primary breakup of a diesel spray reveals that in-nozzle flow separation, wall shear and cavitation contribute greatly to the fragmentation of the jet. Comparison of the two cavitation models shows that after the onset of complete flow detachment, the Kunz implementation predicts higher air inflow at the nozzle outlet than the Schnerr model.     

A multiphase single relaxation time lattice Boltzmann model for heterogeneous porous media

The lattice Boltzmann (LB) method has been shown to be a highly efficient numerical method for solving fluid flow in confined domains such as pipes, irregularly shaped channels or porous media. Traditionally the LB method has been applied to flow in void regions (pores) and no flow in solid regions. However, in a number of scenarios, this may not suffice. That is partial flow may occur in semi-porous regions. Recently gray-scale LB methods have been applied to model single phase flow in such semi-porous materials. Voxels are no longer completely void or completely solid but somewhere in between. We extend the single relaxation time LB method to model multiphase, immiscible flow (e.g., gas and liquid or water and oil) in a semi-porous medium. We compare the solution to test cases and find good agreement of the model as compared to analytical solutions. We then apply the model to real porous media and recover both capillary and viscous flow regimes. However, some deficiencies in the single relaxation time LB method applied to multiphase flow are uncovered and we describe methods to overcome these limitations.     

New modelling approach for damped mutilayed composites

The aim of this paper is to get a good approximation of damping properties of viscoelastic damped sandwich structures. Firstly we will compare the classical models based on the kinematics of Kirchhoff-Love, Mindlin, Reddy or Touratier. In the second stage these models are used to establish the zig-zag models which describe a piecewise continuous displacement field. These models are compared in the static and the dynamic fields. In all cases the Finite-Element-based solution is considered as reference. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a fully coupled nonlinear parabolic problem modelling miscible compressible displacement in porous media

We study a model describing a compressible and miscible displacement in a porous medium. It consists of a coupled system of nonlinear parabolic partial differential equations. Using nonclassical estimates and renormalization tools, we prove the existence of relevant weak solutions for the problem. This is the first existence result obtained for a transport model containing both the coupling due to the compressibility assumption and the coupling due to the concentration dependent viscosity.     

A porous media approach for plantar tissue during gait

Recently a new computational model, based on the thermodynamically constrained averaging theory, has been proposed to predict tumor initiation and proliferation and afterwards to study plantar tissue mechanics. The foot tissue is modeled as an elastic porous medium, in large strain regime and completely filled by a fluid phase. By considering the interstitial fluid, it is possible to mimic the viscoelastic behavior of the plantar tissue observed experimentally. Being the global response of the bi-phase system viscoelastic, it is shown that the duration of stance as well as of each of gait cycle has an influence on tissue stress field. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogenization results for a coupled system modelling immiscible compressible two-phase flow in porous media by the concept of global pressure

A model for immiscible compressible two-phase flow in heterogeneous porous media is considered. Such models appear in gas migration through engineered and geological barriers for a deep repository for radioactive waste. The main feature of this model is the introduction of a new global pressure and it is fully equivalent to the original equations. The resulting equations are written in a fractional flow formulation and lead to a coupled degenerate system which consists of a nonlinear parabolic (the global pressure) equation and a nonlinear diffusion–convection one (the saturation) equation with rapidly oscillating porosity function and absolute permeability tensor. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, we obtain a nonlinear homogenized problem with effective coefficients which are computed via a cell problem and give a rigorous mathematical derivation of the upscaled model by means of two-scale convergence.     

Existence and stability of weak solutions for a degenerate parabolic system modelling two-phase flows in porous media

We prove global existence of nonnegative weak solutions to a degenerate parabolic system which models the interaction of two thin fluid films in a porous medium. Furthermore, we show that these weak solutions converge at an exponential rate towards flat equilibria.     

Travelling wave solutions for degenerate pseudo-parabolic equations modelling two-phase flow in porous media

We discuss a pseudo-parabolic equation modelling two-phase flow in porous media, which includes dynamic effects in the capillary pressure. We extend the results obtained previously for linear higher order terms and investigate the existence of travelling wave solutions in the non-linear and degenerate case. This case leads to non-smooth travelling waves, as well as to a discontinuous capillary pressure.     

Evolution of microstructure in unstable porous media flow: A relaxational approach

We study the flow of two immiscible fluids of different density and mobility in a porous medium. If the heavier phase lies above the lighter one, the interface is observed to be unstable. The two phases start to mix on a mesoscopic scale and the mixing zone grows in time—an example of evolution of microstructure. A simple set of assumptions on the physics of this two‐phase flow in a porous medium leads to a mathematically ill‐posed problem—when used to establish a continuum free boundary problem. We propose and motivate a relaxation of this “nonconvex” constraint of a phase distribution with a sharp interface on a macroscopic scale. We prove that this approach leads to a mathematically well‐posed problem that predicts shape and evolution of the mixing profile as a function of the density difference and mobility quotient. © 1999 John Wiley & Sons, Inc.     

On the effective thermal conductivity of multiphase composites

While several micromechanical models have been developed today in the literature for two-phase media, the extent of their applicability to multiphase materials need yet to be investigated. The present paper studies the effective thermal conductivity of multicomponent composites, and concentrates on two methods: (a) the Mori-Tanaka model, (b) The generalized self-consistent scheme. The Mori-Tanaka method of back-stress previously developed in the context of elasticity of composites is applied here to the conduction problem. The generalized self-consistent scheme, based on a particle-matrix embedding in the effective medium, is studied in this paper in the context of multicomponent media and two variations of this method distinctly different in their imbedding procedure are proposed.Numerical results are given for three-phase composites illustrating and comparing the proposed methods.

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Resumé Plusieurs modéles de micromécanique existent aujoud'hui dans la litérature des matériaux á deux phases. L'applicabilité de ces modéles aux matériaux á multi-phases nécessite d'autre part d'être examinée. Ce travail étudie la conductivité thermale des matériaux á multi-phases et se concentre sur deux méthodes: a) le modéle de Mori-Tanaka b) la méthode auto-consistante généralisée. La méthode de Mori-Tanaka, développée auparavant dans le cadre des propriétés mécaniques des composites est appliquée ici au probléme de conduction. La méthode autoconsistante généralisée est étudiée dans le cadre des matériaux á multi-phases et deux alternatives de cette méthode distinctement différentes dans leur formalisme sont proposées. Pour des matériaux á trois phases on donne des resultats numériques qui illustrent et comparent les méthodes proposées.

     

Microwave heating of ceramic composites

The microwave heating of a ceramic composite is modelled andanalysed. The composite consists of many small ceramic particlesembedded in a ceramic cement. The composite is assumed to bewell insulated, and each particle is assumed to be in imperfectthermal contact with the surrounding cement. Based on thesetwo assumptions an asymptotic theory exploiting the small Biotnumber and small non-dimensional contact conductance is developed.Our asymptotic theory yields a set of nonlinear partial differentialequations which govern the temperature in the composite. Theseare reduced to a set of coupled nonlinear ordinary differentialequations in which the surface area of each particle entersas a parameter. Recent experiments with such composites haveshown that the steady-state temperature of the composite isstrongly dependent upon the radii of the embedded particles.Our model captures this effect. In fact, our analysis showsthat the assumption of imperfect thermal contact between theparticles and the ceramic cement is essential for this trendto be established.     

On a random scaled porous media equation

It is shown that a random scaled porous media equation arising from a stochastic porous media equation with linear multiplicative noise through a random transformation is well-posed in L^∞. In the fast diffusion case we show existence in Lp.     

A model of multiphase flow and transport in porous media applied to gas migration in underground nuclear waste repository

We prove existence of solutions for a new model of two compressible and partially miscible phase flow in porous media, applied to gas migration in an underground nuclear waste repository. This model, modeling fully and partially water saturated situations, consist of a coupled system of quasilinear parabolic partial differential equations. We seek a new set of variables in order to obtain a system which belongs to the class of equations considered by Alt and Luckhaus such that it would be possible to use their existence theorem. A simulation of a numerical test case is performed in order to numerically demonstrate the ability of this model to take in account the appearance of one phase. To cite this article: F. Smaï, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Localisation in granular media: Particle approach,homogenisation and continuum modelling

The individual motion of grains in granular material has a strong influence on the macroscopic material behaviour, which is in particular the case for the phenomena of strain localisation in shear zones and justifies the need for techniques that incorporate a micro-macro transition. In this contribution, granular media are investigated in three steps. Firstly, a microscopic particle-based modelling is set up, where individual grains are considered as rigid uncrushable particles while their motion is obtained through Newton's equations of state. The inter-particle contact forces are thereby determined via constitutive contact-force formulations, which have to account for the envisaged material behaviour. The second step is the homogenisation of the obtained particle's displacements and contact forces through a particle-centre-based strategy towards continuum quantities. Therefore, Representative Elementary Volumes (REV) are introduced on the mesoscale and the specific construction of the REV boundary leads to the understanding of granular media as a micropolar continuum. Finally, in order to verify the homogenisation strategy, a continuum based micropolar model is applied to model localisation phenomena and a comparative study of the results is carried out in a qualitative way. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)