Cotransport of Suspended Colloids and Nanoparticles in Porous Media

The objective of this study is to develop a model for cotransport of colloids and nanoparticles (NPs) in porous media under two particle capture mechanisms. The particle capture rate is proportional to the capture probability, which is a function of retained concentration, called the filtration function. Laboratory bench-scale experiments of individual transport of NPs and colloidal-size kaolinite clay particles through packed columns produced breakthrough curves (BTCs) that monotonically increased with time and stabilised at some value lower than the injected concentration. We discuss the filtration function that corresponds to BTCs stabilising at the concentration lower than the injected value. This so-called binary filtration function incorporates two particle capture mechanisms. The analytical transport model with a binary filtration function was capable to fit successfully BTCs obtained from individual transport experiments using kaolinite and NPs conducted by Chrysikopoulos et al. (Transp Porous Med 119(1):181–204, 2017). Assuming that the electrostatic particle–solid matrix interaction and the fraction of the solid matrix surface area occupied by a single attached particle (kaolinite or NP) are the same for individual transport of either kaolinite particles or NPs and for simultaneous cotransport of kaolinite particles and NPs, the proposed binary filtration function was extended for the cotransport case. Although the breakthrough data from cotransport experiments with kaolinite particles and NPs have six degrees of freedom, the developed cotransport model successfully matches the BTCs by tuning two constants only. This validates the developed model for cotransport of two colloidal populations with different attachments and straining rates.

     

Numerical Modelling of Free-Surface Flow-Porous Media Interaction Using Divergence-Free Moving Particle Semi-Implicit Method

A divergence-free moving particle semi-implicit method is introduced for free-surface flow through porous media. Numerical incompressibility is conserved by solving additional pressure Poisson equation (PPE). Depending on current particle coordinates, a porosity-based factor is introduced to incorporate the effect of solid volume inside the porous domain. A hybrid formulation containing specified boundary condition and PPE is utilized on free-surface particles. The current framework is tested for four different problems. The first problem shows the effect of the proposed factor in vertical flow through a rectangular porous block and its representative volume change for different phases. Second and third problems validate the numerical model for dam break through a rectangular block of homogeneous porous media. In the fourth problem, flow through a trapezoidal porous block consisting of different porous media with variable effective porosity and permeability is simulated.     

Optical measurement uncertainties due to refractive index mismatch for flow in porous media

Application of optical techniques such as PIV, PTV, and LDA for velocity field estimation in porous media requires matching of refractive indices of the liquid phase to that of the solid matrix, including the channel walls. The methods most commonly employed to match the refractive indices have been to maximize the transmitted intensity through the bed or to rely on direct refractometer measurements of the indices of the two phases. Mismatch of refractive indices leads to error in estimation of particle position, ε _PD, due to refraction at solid–liquid interfaces. Analytical ray tracing applied to a model of solid beads placed randomly along the optical path is used to estimate ε _PD. The model, after validating against experimental results, is used to generate expression for ε _PD as a function of refractive index mismatch for a range of bead diameters, bed widths, bed porosity, and optical magnification. The estimate of ε _PD, which is found to be unbiased, is connected to errors in PIV measurement using the central limit theorem. Mismatch in refractive indices can also lead to reduction in particle density, N _s, detected light flux, J, and degrade the particle image. The model, verified through experiments, is used to predict the reduction in N _s and J, where it is found that particle defocusing caused by spherical beads in refractive index mismatched porous bed is the primary contributor to reductions of N _s and J. In addition, the magnitude of ε _PD is determined for the use of fluorescent dye emission for particle detection due to wavelength-dependent index of refraction.     

Refraction through cylindrical tubes

When viewed from the outside, objects placed inside containers with cylindrical walls (i.e. test tubes, beakers, pipes) appear distorted because of the curvature of the interfaces and the differing refractive indices of the media. We have developed a method for measuring the coordinates of a particle located inside a straight cylindrical tube by viewing it from two directions using a camera. A set of equations is derived which maps all points inside the tube to the camera image plane; ray tracing diagrams are shown for several important cases, indicating variable distortion, hidden regions, multiple images, and critical reflections. An experimental test was performed to check the calculations; excellent verification was obtained.     

Geometrical optics approximation of light scattering by large air bubbles

For large spherical bubbles in water,geometrical optics approximation is considered a better method for calculating light scattering patterns.In this paper,the basic theory of geometrical optics approximation is clarified.The change of phase for bubbles is calculated when total reflection occurs,which is different from particles with relative refractive indices larger than 1.Verification of the method was achieved by assuming a spherical particle and comparing present results to Mie scattering and Debye calculation.Agreement with the Mie theory was excellent in all directions when the dimensionless size parameter is larger than 50.Limitations of the geometrical optics approximation are also discussed.     

Investigation on mass diffusion process in porous media based on Lattice Boltzmann method

The mass diffusion process inside a porous medium is difficult for numerical simulation due to complex and stochastic nature of its structure. Based on the lattice Boltzmann method and reconstruction technology, this article presents an approach for simulating mass diffusion process and predicting the effective mass diffusivity in porous media, which is validated by comparing theoretical and experimental data. The concentration distribution and effective mass diffusivity inside porous media can be obtained.     

Surface Permeability of Particulate Porous Media

The dispersion process in particulate porous media at low saturation levels takes place over the surface elements of constituent particles and, as we have found previously by comparison with experiments, can be accurately described by superfast nonlinear diffusion partial differential equations. To enhance the predictive power of the mathematical model in practical applications, one requires the knowledge of the effective surface permeability of the particle-in-contact ensemble, which can be directly related with the macroscopic permeability of the particulate media. We have shown previously that permeability of a single particulate element can be accurately determined through the solution of the Laplace–Beltrami Dirichlet boundary value problem. Here, we demonstrate how that methodology can be applied to study permeability of a randomly packed ensemble of interconnected particles. Using surface finite element techniques, we examine numerical solutions to the Laplace–Beltrami problem set in the multiply-connected domains of interconnected particles. We are able to directly estimate tortuosity effects of the surface flows in the particle ensemble setting.

     

Smoothed Particle Hydrodynamics Model for Diffusion through Porous Media

A smoothed particle hydrodynamics (SPH) model is presented for the study of diffusion in spatially periodic porous media. The method of SPH is formulated to solve the convection–diffusion equation for tracer diffusion under steady state and transient conditions. Solutions obtained using SPH are compared with other available solutions and the model is used to calculate diffusion coefficients of spatially periodic porous media for the steady state diffusion problem. Diffusion coefficients are then used to calculate nondimensional diffusivities of the media. The effects of media properties on the values of nondimensional diffusivity are also presented.     

Lateral migration mechanisms in capillary hydrodynamic chromatography

In all chromatographic systems that achieve separation of colloidal particles based on the particles' hydrodynamic behavior, there are partitioning mechanisms that promote lateral migration of particles across solvent streamlines. In this work, solvent inertia and particle electrostatics are incorporated in Brenner & Gaydos' general diffusive transport theory for particles and solvent flowing in a capillary, yielding the mean axial velocity of particles as a function of particle size. A comparison is made with capillary hydrodynamic chromatography results. The lateral migration of small particles is primarily due to diffusion, while large particles are focused by the inertial force at one equilibrium radial position, as observed in “tubular pinch” experiments. The transition from diffusion- to inertia-controlled lateral migration can be tuned to specific particle size ranges through variation of solvent ionic strength, flowrate and capillary radius. Poor prediction of the separation behavior of large particles is attributed to inaccuracy in the calculation of the inertial radial velocity, suggesting the need for further theoretical analysis and experimental study of inertial migration.     

Transverse diffusion and heat conduction in a granular layer

The process of non-steady-state transverse diffusion of a passive additive in a granular layer described by a cellular model is investigated. The general results obtained in [1] are applied to an analysis of concrete transport processes of matter and heat in a granular layer. The following four cell models are treated: (1) ideal mixing cells without stagnation zones; (2) ideal mixing cells with stagnation zones; (3) ideal mixing cells with diffusive stagnation zones; (4) ideal mixing cells with diffusive stagnation zones having a finite exchange rate between the free volume and the stagnation zone. The conditions of applicability for each of the above models are found. The time to establish a normal distribution in the transverse diffusion process is determined for all the models. This quantity is then connected with the physical characteristics of transport processes of matter in a layer of nonporous and porous particles, the transport of heat in a granular layer, and the transport of matter in a layer of particles which adsorb an additive.     

Analytical Solutions to Gas Flow Problems in Low Permeability Porous Media

In most of conventional porous media the flow of gas is basically controlled by the permeability and the contribution of gas flow due to gas diffusion is ignored. The diffusion effect may have significant impact on gas flow behavior, especially in low permeability porous media. In this study, a dual mechanism based on Darcy flow as well as diffusion is presented for the gas flow in homogeneous porous media. Then, a novel form of pseudo pressure function was defined. This study presents a set of novel analytical solutions developed for analyzing steady-state and transient gas flow through porous media including effective diffusion. The analytical solutions are obtained using the real gas pseudo pressure function that incorporates the effective diffusion. Furthermore, the conventional assumption was used for linearizing the gas flow equation. As application examples, the new analytical solutions have been used to design new laboratory and field testing method to determine the porous media parameters. The proposed laboratory analysis method is also used to analyze data from steady-state flow tests of three core plugs. Then, permeability (k) and effective diffusion coefficient (D _e) was determined; however, the new method allows one to analyze data from both transient and steady-state tests in various flow geometries.     

Modeling of Single Porous Char Particle Gasification in Supercritical Water

A deep understanding of the gasification behavior of porous char particle is the premise of the reactor-scale modeling, but there are few studies on the gasification characteristics in supercritical water. Thus, a numerical model for porous char particle gasification in supercritical water was developed in this work, and the effects of particle size, inflow temperature and inflow velocity were studied. Simulation results showed that gasification of the small particle of 0.1 mm lay in zone I regime where the particle radius kept unchanged due to uniform reactions inside the particle and the effectiveness factor increased rapidly after the gasification began due to easy accessibility of supercritical water into the particle. The gasification of the large particle of 1 mm showed typical characteristics in zone II regime that the particle began to shrink at a certain conversion degree, and smaller effectiveness factor was observed due to larger supercritical water concentration gradient inside the particle. As the increase of temperature and particle size, ambient fluid became difficult to flow through the unreacted core, and the Stefan flow was observed to obviously modify the hydrodynamic boundary layer at low Reynolds number. Besides, it is unreasonable to assume isothermal particle for gasification with large particle and high temperature because of the significantly overestimated particle consumption rate.

     

Backscatter phase-Doppler anemometry for transparent non-absorbing spheres

The application of phase-Doppler anemometer for scattering angles between the main rainbow and direct backscatter, was examined by calculating the spatial intensity distribution and the phase of the light scattered by a particle crossing the measuring volume. Geometrical optics was assumed and contributions to the scattered light due to reflection on the external surface of the particle and first internal reflection were considered. The response curve of the technique was calculated for different particle refractive indices, beam intersection angles, collection angles and spacings between the collection apertures. Linear response curves were obtained after integration of the intensity of the scattered light over sufficiently large rectangular collection apertures, but they became non-monotonic after a critical value of phase shift, which varied with the optical arrangement between around 220° and 360°, did not scale with common scaling parameters used for forward scatter light, and limited the possible size range of the instrument for one optical arrangement. The particle refractive index determined the collection angle and limited sizing to particles with little uncertainty in refractive index, since a 5% change in refractive index led to uncertainties in size of the order of 100%. An alternative sizing technique is suggested for the backscatter region.     

Effect of nanoparticle concentration on zeta-potential measurement results and reproducibility

The effect of nanoparticle concentration on zeta-potential measurement results at dilute concentrations was evaluated.The values of the zeta-potential for four different types of nanoparticles,Ludox(silica),multi-walled carbon nanotubes(bamboo-shaped and hollow nanotubes)and gold,at various concentrations,were obtained using a laser Doppler electrophoresis instrument.The size of the nanoparticles on dilution was measured using dynamic light scattering(DLS).The results show that there is a concentration range within which the zeta-potential,and particle size,are not affected by nanoparticle concentration.The lower concentration limit for the system to produce consistent results was dependent on the nature of the sample under study and ranged between 10-2 and 10 4wt%.Below this concentration,there was an apparent shift in zeta-potential values to less negative values,which was accompanied by an increase in the particle size.The shift in zeta-potential was attributed to an increase in contribution of the signal from extraneous particulate matter.The increase in particle size was attributed to the nature of the homodyne optical configuration of the instrument.The aim of this study was to elucidate the range in nanopatticle concentration that allows for accurate and reliable measurement of the zeta-potential and DLS data.     

No‐slip consistent immersed boundary particle tracking to simulate impaction filtration in porous media

In this paper, we present a new method for simulating the motion of a disperse particle phase in a carrier gas through porous media. We assume a sufficiently dilute particle‐laden flow and compute, independently of the disperse phase, the steady laminar fluid velocity using the immersed boundary method. Given the velocity of the carrier gas, the equations of motion for the particles experiencing the Stokes drag force are solved to determine their trajectories. The ‘no‐slip consistent’ particle tracking algorithm avoids possible numerical filtration of very small particles due to the nonzero velocity field at the solid–fluid interface introduced by the immersed boundary method. This physically consistent tracking allows a reliable estimation of the filtration efficiency of porous filters due to inertial impaction. We illustrate and test our new approach for model porous media consisting of a structured array of aligned rectangular fibers, arranged in line and staggered. In the staggered geometry, the effect of the residual velocity at the solid–fluid interface is significant for particles with low inertia. Without adopting the developed no‐slip consistent numerical method, an artificial numerical filtration is observed, which becomes dominant for small enough particles. For both the in line and the staggered geometries, the filtration rate depends quite strongly and non monotonically on the particle inertia. This is expressed most clearly in the staggered arrangement in which a very strong increase in the filtration efficiency is observed at a well‐defined critical droplet size, corresponding to a qualitative change in the dominant particle paths in the porous medium. Copyright © 2013 John Wiley & Sons, Ltd.     

CFD–DEM simulation of particle deposition characteristics of pleated air filter media based on porous media model

In this study, the three-dimensional physical model of pleated air filtration media was simplified to porous media model, and the calculation parameters of porous media were obtained based on experimental data. The model of V-shaped pleated air filter media is constructed, the height of the media pleat is 50 mm and the pleat thickness is 4 mm, the pleat angle is 3.7°. The Hertz-Mindlin contact model was modified by Johnson Kendall Roberts (JKR) adhesion contact model. The deposition process of particles in media was simulated based on computational fluid dynamics (CFD) theory and discrete element method (DEM). Results show that the CFD–DEM coupling method can be effectively applied to the macro research of pleated air filter media. The particles will form dust layer and dendrite structure on the fiber surface, and the dust layer will affect the subsequent air flow organization, and the dendrite structure will eventually form a “particle wall”. The formation of the “particle wall” will prevent the particles from moving further in the fluid domain, which makes area of pleated angle become the “low efficiency” part about the particle deposition. Compared with area of pleated angle, the particles are concentrated in the opening area and the middle area of the pleated to agglomerate and deposit.     

Determination of Nanoparticle Macrotransport Coefficients from Pore Scale Processes

Nanoparticle transport in porous media is modeled using a hierarchical set of differential equations corresponding to pore scale and macroscale. At the pore scale, movement and interaction of a single particle with a solid matrix is modeled using the advection–dispersion–sorption equation. A single nanoparticle entering the space encounters viscous, diffusion and surface forces. Surface forces (electrostatic and van der Waals forces) between nanoparticles and mineral grains appear as sorption propensity on solid matrix boundary condition. These local events are then transformed into a macroscale continuum by imposing periodic boundary conditions for contiguous unit cells representing porous media and using a scheme of moment analysis. At the macroscale, propagation and retention of particles are characterized by three position-independent coefficients: mean nanoparticle velocity vector \({\bar{\mathbf{U}}}^*\), macroscopic dispersion coefficient \({\bar{\mathbf{D}}}^*\), and mean nanoparticle retention rate constant \({\bar{K}}^*\). The modeling results are validated with a set of nanoparticle transport tests in porous microchips. We also present simulations of realistic porous media, where an actual image of sandstone samples is processed into binary tones. The representative unit cells are constructed from the resulting binary image by searching for areas within the sample with maximum similarities to the whole sample in terms of porosity and specific surface area, which are found to show strong correlations with the resulting \({\bar{\mathbf{U}}}^*\) and \({\bar{K}}^*\), respectively.     

Colloid Transport in Porous Media: A Review of Classical Mechanisms and Emerging Topics

To celebrate the tenth anniversary of InterPore, we present an interdisciplinary review of colloid transport through porous media. This review aims to explore both classical colloid transport and topics that fall outside that purview and thus offer transformative insights into the physics governing transport behavior. First, we discuss the unique colloid characteristics relative to molecules and larger particles. Then, the classical advection–dispersion–filtration models (both conceptual and mathematical) of colloid transport are introduced as well as anomalous transport behaviors. Next, the forces of interaction between colloids and porous media surfaces are discussed. Fourth, applications that are interested in maximizing the transport of colloids through porous media are considered. Then the concept of motile, active biocolloids is introduced, and finally, colloid swarming as a newly recognized mode of transport is summarized.

     

Measurement of permeability of microfluidic porous media with finite-sized colloidal tracers

We present two methods how the permeability in porous microstructures can be experimentally obtained from particle tracking velocimetry of finite-sized colloidal particles suspended in a liquid. The first method employs additional unpatterned reference channels where the liquid flow can be calculated theoretically and a relationship between the velocity of the particles and the liquid is obtained. The second method takes advantage of a time-dependent pressure drop that leads to an exponential decrease in the particle velocity inside a porous structure. From the corresponding decay time, the permeability can be calculated independently of the particle size. Both methods lead to results comparable with permeabilities derived from numerical simulations.     

Determination of the parameters of particles suspended in two-phase media by means of penetrating radiation

Determination of the concentration, size, and internal structure of microscopic particles suspended in two-phase media by means of contactless methods constitutes an important technological problem. If the particle sizes are on the order of the wavelength of light, methods based on light scattering by particles are widely used for this purpose. The most direct method consists in observing the optical signal scattered by an individual particle [1]. There are also several methods where the total signal from a large number of particles is recorded, but, in this case, multiple rescattering of light on particles must be negligible [2, 3]. At the same time, the complex relationship between the scattering amplitude and the refraction index, the shape of particles, etc., as well as the increasing background of multiply scattered light with greater thickness of the scattering layer, restrict the scope of application of such methods and make other measurement methods desirable, e.g., in the case of instrument calibration. Our aim is to point out the advisability of investigating two-phase media by means of penetrating radiation, which has been used successfully for radiation flaw detection [4] and for inspecting the composition and density of matter [5], We shall mention the most important advantages of the proposed method. First, the interaction between individual particles and nonrefracted radiation is described by simple expressions, which makes the interpretation of results much easier. Second, in using the most informative scheme whereby scattering media are investigated by transillumination, the background of multiply scattered radiation with a low information content (or, to borrow a term from radiation protection physics, the build-up factor [6]) increases with an increase in the scattering layer thickness much more slowly than it does for light. This makes it possible to use radiation methods for investigating optically dense two-phase media. We shall consider below the possibility of determining the distribution function of particle sizes by measuring the radiation attenuation as a function of the linear coefficient of attenuation inside the particles.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 11–14, May–June, 1979.