Cahn–Hilliard modeling of particles suspended in two‐phase flows

In this paper, we present a model for the dynamics of particles suspended in two‐phase flows by coupling the Cahn–Hilliard theory with the extended finite element method (XFEM). In the Cahn–Hilliard model the interface is considered to have a small but finite thickness, which circumvents explicit tracking of the interface. For the direct numerical simulation of particle‐suspended flows, we incorporate an XFEM, in which the particle domain is decoupled from the fluid domain. To cope with the movement of the particles, a temporary ALE scheme is used for the mapping of field variables at the previous time levels onto the computational mesh at the current time level. By combining the Cahn–Hilliard model with the XFEM, the particle motion at an interface can be simulated on a fixed Eulerian mesh without any need of re‐meshing. The model is general, but to demonstrate and validate the technique, here the dynamics of a single particle at a fluid–fluid interface is studied. First, we apply a small disturbance on a particle resting at an interface between two fluids, and investigate the particle movement towards its equilibrium position. In particular, we are interested in the effect of interfacial thickness, surface tension, particle size and viscosity ratio of two fluids on the particle movement towards its equilibrium position. Finally, we show the movement of a particle passing through multiple layers of fluids. Copyright © 2011 John Wiley & Sons, Ltd.     

A nonlocal species concentration theory for diffusion and phase changes in electrode particles of lithium ion batteries

A nonlocal species concentration theory for diffusion and phase changes is introduced from a nonlocal free energy density. It can be applied, say, to electrode materials of lithium ion batteries. This theory incorporates two second-order partial differential equations involving second-order spatial derivatives of species concentration and an additional variable called nonlocal species concentration. Nonlocal species concentration theory can be interpreted as an extension of the Cahn–Hilliard theory. In principle, nonlocal effects beyond an infinitesimal neighborhood are taken into account. In this theory, the nonlocal free energy density is split into the penalty energy density and the variance energy density. The thickness of the interface between two phases in phase segregated states of a material is controlled by a normalized penalty energy coefficient and a characteristic interface length scale. We implemented the theory in COMSOL Multiphysics\(^{\circledR }\) for a spherically symmetric boundary value problem of lithium insertion into a \(\hbox {Li}_x\hbox {Mn}_2\hbox {O}_4\) cathode material particle of a lithium ion battery. The two above-mentioned material parameters controlling the interface are determined for \(\hbox {Li}_x\hbox {Mn}_2\hbox {O}_4\), and the interface evolution is studied. Comparison to the Cahn–Hilliard theory shows that nonlocal species concentration theory is superior when simulating problems where the dimensions of the microstructure such as phase boundaries are of the same order of magnitude as the problem size. This is typically the case in nanosized particles of phase-separating electrode materials. For example, the nonlocality of nonlocal species concentration theory turns out to make the interface of the local concentration field thinner than in Cahn–Hilliard theory.     

Stress-mediated lithiation in nanoscale phase transformation electrodes

Development of high-performance phase transformation electrodes in lithium ion batteries requires comprehensive studies on stress-mediated lithiation involving migration of the phase interface. It brings out many counter-intuitive phenomena, especially in nanoscale electrodes, such as the slowing down migration of phase interface, the vanishing of miscibility gap under high charge rate, and the formation of surface crack during lithiation. However,it is still a challenge to simulate the evolution of stress in arbitrarily-shaped nanoscale electrodes, accompanied with phase transformation and concurrent plastic deformation. This article gives a brief review of our efforts devoted to address these issues by developing phase field model and simulation. We demonstrate that the miscibility gap of two-phase state is affected not only by stress but also by surface reaction rate and particle size. In addition, the migration of phase interface slows down due to stress. It reveals that the plastic deformation generates large radial expansion, which is responsible for the transition from surface hoop compression to surface hoop tension that may induce surface crack during lithiation.We hope our effort can make a contribution to the understanding of stress-coupled kinetics in phase transformation electrodes.     

Effects of Particle Size Non-Uniformity on Transport and Retention in Saturated Porous Media

Short-pulse injection experiments are investigated to study the effects of particle size non-uniformity on the transport and retention in saturated porous media. Monodisperse particles (3, 10, and 16 \(\upmu \hbox {m}\) latex microspheres) and polydisperse particles (containing 3, 10, and 16 latex microspheres) were explored. The obtained results suggest considering not only the particle sizes but also their polydispersivity (particle size non-uniformity) in transport and retention. Although, the density of the suspended particles is close to that of water, results reveal a slow transport of particles compared to the dissolved tracer whatever their size and flow velocity. The recovered particles in the mixture experiments show that the retention of large particles (10 and 16 \(\upmu \hbox {m}\)) enhances the retention of small ones (3 \(\upmu \hbox {m}\)). However, the straining of 10 and 16 \(\upmu \hbox {m}\) particles in “mixture experiments” is smaller than their straining in “monodisperse experiments”. A linear relationship summarizing the simultaneous effect of particle sizes and flow velocity on deposition kinetics coefficient is proposed.     

The corona ignition voltage of an electrical discharge in a gas

The corona ignition voltage of an electrical discharge in air of atmospheric pressure depends on the presence of (moisture) particles, which may increase the corona losses. A relation between the corona ignition voltage and the particle size when tested shows unexpected results. With the corona ignition voltage in air as observed by Rose and Wood our calculations do not give particle sizes of 50 m (as used by Rose and Wood), but sizes of about 1 Å corresponding to the diameters of the molecules of the component gases in air. Our conclusion is that these molecules align in a conductive channel from the axial wires to the particle considered. In this way the charge transfer from the axial wire to the particles may be explained.     

Effect of the silicon-carbide micro- and nanoparticle size on the thermo-elastic and time-dependent creep response of a rotating Al–SiC composite cylinder

The history of stresses and creep strains of a rotating composite cylinder made of an aluminum matrix reinforced by silicon carbide particles is investigated. The effect of uniformly distributed SiC micro- and nanoparticles on the initial thermo-elastic and time-dependent creep deformation is studied. The material creep behavior is described by Sherby’s constitutive model where the creep parameters are functions of temperature and the particle sizes vary from 50 nm to 45.9 μm. Loading is composed of a temperature field due to outward steady-state heat conduction and an inertia body force due to cylinder rotation. Based on the equilibrium equation and also stress-strain and strain-displacement relations, a constitutive second-order differential equation for displacements with variable and time-dependent coefficients is obtained. By solving this differential equation together with the Prandtl–Reuss relation and the material creep constitutive model, the history of stresses and creep strains is obtained. It is found that the minimum effective stresses are reached in a material reinforced by uniformly distributed SiC particles with the volume fraction of 20% and particle size of 50 nm. It is also found that the effective and tangential stresses increase with time at the inner surface of the composite cylinder; however, their variation at the outer surface is insignificant.     

A micromechanical model for effective conductivity in granular electrode structures

Optimization of composition and microstructure is important to enhance performance of solid oxide fuel cells （SOFC） and lithium-ion batteries （LIB）. For this, the porous electrode structures of both SOFC and LIB are modeled as a binary mixture of electronic and ionic conducting particles to estimate effective transport properties. Particle packings of 10000 spherical, binary sized and randomly positioned particles are created numerically and densified considering the different manufacturing processes in SOFC and LIB： the sintering of SOFC electrodes is approximated geometrically, whereas the calendering process and volume change due to intercalation in LIB are modeled physically by a discrete el- ement approach. A combination of a tracking algorithm and a resistor network approach is developed to predict the con- nectivity and effective conductivity for the various densified structures. For SOFC, a systematic study of the influence of morphology on connectivity and conductivity is performed on a large number of assemblies with different compositions and particle size ratios between 1 and 10. In comparison to percolation theory, an enlarged percolation area is found, es- pecially for large size ratios. It is shown that in contrast to former studies the percolation threshold correlates to varying coordination numbers. The effective conductivity shows not only an increase with volume fraction as expected but also with size ratio. For LIB, a general increase of conductivity during the intercalation process was observed in correlation with increasing contact forces. The positive influence of cal- endering on the percolation threshold and the effective conductivity of carbon black is shown. The anisotropy caused by the calendering process does not influence the carbon black phase.     

Sediment–Water Interface Flow with the Multiparticle Collision Dynamics

This paper presents an experimental study of particle transport in porous medium using a self-developed sand layer transportation–deposition testing system, aiming at delineating the detachment characteristics of deposited particles in porous medium. Two experimental modes, increase flow velocity and change flow direction, were adopted in this study. The tests were conducted using quartz powder as the particles and quartz sand as the porous media to study the response of detachment characteristics to changes in particle diameter (\(d_{s}\), with median diameter 18 and 41 \(\upmu \)m) and grain diameter (\(d_{p}\), with median diameter 0.36 and 1.25 mm). Breakthrough curves after the second peak were well described by a double exponential model with parameters of weight coefficient and detachment coefficient. This study shows that both modes can change the detach rate of deposited particles observably, and detach rate is affected by the value of flow velocity greatly.     

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.     

Formation of condensed oxide particles by combustion of metal droplets

The formation of condensed oxide particles in combustion of metal droplets is discussed; it is assumed that the characteristic diffusion time is much less than the characteristic time for the heterogeneous reaction at the condensate particle surfaces, and the structure of the reaction zone is discussed; the size spectrum is derived for the condensed oxide particles. It is found that condensation in the gas has little effect on the droplet combustion rate. Heat needed to evaporate the metal is produced directly at the surface of the drop and the rate-limiting step in the combustion is the diffusion of oxidant to the surface.     

Mathematical Modeling of Colloidal Particles Transport in the Medium Treated by Nanofluids: Deep Bed Filtration Approach

A deep bed filtration model has been developed to quantify the effect of nanoparticles (NPs) on mitigating fines migration in porous media. The filtration coefficients representing the total kinetics of particles capture were obtained by fitting the model to the laboratory data. Based on the optimum filtration coefficients, the model was utilized to history match the particle concentration breakthrough profiles observed in twelve core flood tests. In the flooding experiments, the effect of five types of metal oxide NPs, \(\upgamma \hbox {-Al}_{2}\hbox {O}_{3}\) , CuO, MgO, \(\hbox {SiO}_{2}\) , and ZnO, on migrating fines were investigated. In each test, a stable suspension was injected into the already NP-treated core and effluents’ fines concentration was measured based on turbidity analysis. In addition, zeta potential analysis was done to obtain the surface charge (SC) of the NP-treated medium. It was found that the presence of NPs on the medium surface results in SC modification of the bed and as a result, enhances the filter performance. Furthermore, the ionic strength of the nanofluid was recognized as an important parameter which governs the capability of NPs to modify the SC of the bed. The remedial effect of NPs on migrating fines is quantitatively explained by the matched filtration coefficients. The SC of the medium soaked by \(\upgamma \hbox {-Al}_{2}\hbox {O}_{3}\) nanofluid is critically increased; therefore, the matched filtration coefficient is of remarkably high value and as a result, the treated medium tends to adsorb more than 70 % of suspended particles. The predicted particle concentration breakthrough curves well matched with the experimental data.     

Experimental analysis of particle and fluid motion in ac electrokinetics

An ac electric field is applied to induce particle and fluid motion in a wedge-shaped microchannel. Micron-resolution particle image velocimetry (-PIV) is used to determine spatially resolved particle velocity and fluid velocity fields. Under steady-state conditions, the particles experience a balance between dielectrophoretic forces induced by the nonuniform electric field and Stokes drag forces due to viscous interactions with the fluid. The particle velocity is therefore different from the fluid velocity because of dielectrophoresis. A variant of -PIV, two-color -PIV, is developed and used to uniquely determine the fluid velocity from the observation of particles without a priori knowledge of the electrical properties. This technique is used to explore ac electrokinetically generated fluid motion. A series of voltage levels at fixed frequency are applied to the wedge-shaped electrodes. The dependency of fluid velocity on applied voltage at different regions in the flow is obtained by fitting power-law curves. This is used to determine the underlying physical phenomena associated with ac electrokinetics. We found that both electrothermal effects and ac electroosmosis are important for the current experimental conditions. However, the electrothermal effect is dominant in the bulk fluid.     

Numerical computation of central crack growth in an active particle of electrodes influenced by multiple factors

Mechanical degradation, especially fractures in active particles in an electrode, is a major reason why the capacity of lithium-ion batteries fades. This paper proposes a model that couples Li-ion diffusion, stress evolution, and damage mechanics to simulate the growth of central cracks in cathode particles \((\hbox {LiMn}_{2}\hbox {O}_{4})\) by an extended finite element method by considering the influence of multiple factors. The simulation shows that particles are likely to crack at a high discharge rate, when the particle radius is large, or when the initial central crack is longer. It also shows that the maximum principal tensile stress decreases and cracking becomes more difficult when the influence of crack surface diffusion is considered. The fracturing process occurs according to the following stages: no crack growth, stable crack growth, and unstable crack growth. Changing the charge/discharge strategy before unstable crack growth sets in is beneficial to prevent further capacity fading during electrochemical cycling.     

Description of liquid–gas phase transition in the frame of continuum mechanics

A new method of describing the liquid–gas phase transition is presented. It is assumed that the phase transition is characterized by a significant change of the particle density distribution as a result of energy supply at the boiling point that leads to structural changes but not to heating. Structural changes are described by an additional state characteristics of the system—the distribution density of the particles which is presented by an independent balance equation. The mathematical treatment is based on a special form of the internal energy and a source term in the particle balance equation. The presented method allows to model continua which have different specific heat capacities in liquid and in gas state.     

Influence of the particle size and volume fraction on micro-damage of the composites

The bulk and shear modulus of metal matrix composites with various volume fractions of particles are modified based on the Eshelby’s equivalent inclusion method combined with self-consistent scheme. By introducing the modified modulus, a new model, which can predict the particle size effects on the stress–strain relation under interfacial debonding damage between matrix and particles, is established. The results obtained from the present investigation show a better agreement with the experimental data.     

Suspended Particles Transport and Deposition in Saturated Granular Porous Medium: Particle Size Effects

An experimental study on the transport and deposition of suspended particles (SP) in a saturated porous medium (calibrated sand) was undertaken. The influence of the size distribution of the SP under different flow rates is explored. To achieve this objective, three populations with different particles size distributions were selected. The median diameter $d_{50}$ of these populations was 3.5, 9.5, and $18.3~\upmu \hbox {m}$ . To study the effect of polydispersivity, a fourth population noted “Mixture” ( $d_{50} = 17.4\; \upmu \hbox {m}$ ) obtained by mixing in equal proportion (volume) the populations 3.5 and $18.3\;\upmu \hbox {m}$ was also used. The SP transfer was compared to the dissolved tracer (DT) one. Short pulse was the technique used to perform the SP and the DT injection in a column filled with the porous medium. The breakthrough curves were competently described with the analytical solution of a convection–dispersion equation with first-order deposition kinetics. The results showed that the transport of the SP was less rapid than the transport of the DT whatever the flow velocity and the size distribution of the injected SP. The mean diameter of the recovered particles increases with flow rate. The longitudinal dispersion increases, respectively, with the increasing of the flow rates and the SP size distribution. The SP were more dispersive in the porous medium than the DT. The results further showed that the deposition kinetics depends strongly on the size of the particle transported and their distribution.     

Modulation of homogeneous turbulence seeded with finite size bubbles or particles

The dynamics of homogeneous, isotropic turbulence seeded with finite sized particles or bubbles is investigated in a series of numerical simulations, using the force-coupling method for the particle phase and low wavenumber forcing of the flow to sustain the turbulence. Results are given on the modulation of the turbulence due to massless bubbles, neutrally buoyant particles and inertial particles of specific density 1.4 at volumetric concentrations of 6%. Buoyancy forces due to gravity are excluded to emphasize finite size and inertial effects for the bubbles or particles and their interactions with the turbulence. Besides observing the classical entrapment of bubbles and the expulsion of inertial particles by vortex structures, we analyze the Lagrangian statistics for the velocity and acceleration of the dispersed phase. The turbulent fluctuations are damped at mid-range wavenumbers by the bubbles or particles while the small-scale kinetic energy is significantly enhanced. Unexpectedly, the modulation of turbulence depends only slightly on the dispersion characteristics (bubble entrapment in vortices or inertial sweeping of the solid particles) but is closely related to the stresslet component (finite size effect) of the flow disturbances. The pivoting wavenumber characterizing the transition from damped to enhanced energy content is shown to vary with the size of the bubbles or particles. The spectrum for the energy transfer by the particle phase is examined and the possibility of representing this, at large scales, through an additional effective viscosity is discussed.     

Coupled Effects of Ionic Strength,Particle Size,and Flow Velocity on Transport and Deposition of Suspended Particles in Saturated Porous Media

In this study, the coupled effect of ionic strength, particle size, and flow velocity on transport and deposition of suspended particles (SP) in saturated sand was undertaken. Three polydispersive SP populations (silt particles with the median of 3.5, 9.5 and 18.3 \(\upmu \)m) were investigated using a pulse injection technique. High ionic strengths were used and vary from 0 to 600 mM (NaCl). Two high velocities were tested: 0.15 and 0.30 cm/s. Suspended particles recovery and deposition kinetics were strongly dependent on the solution chemistry, the hydrodynamics, and the suspended particles size, with greater deposition occurring for increasing ionic strength, lower flow velocity, and larger ratios of the median diameter of the SP to the median sand grain diameter. A shift between the extended Derjaguin–Landau–Verwey–Overbeek theory prediction (the particles and sand grain surfaces are considered chemically and topographically homogeneous) and the experimental results for certain ionic strength was observed. So, as reported in recent literature, effects of surface heterogeneities should be considered. The residence time of the non-captured particles is dependent on ionic strength and hydrodynamic. A relationship between the deposition kinetics, particle and grain sizes, flow velocity, and ionic strength is proposed.