Elliptic random-walk equation for suspension and tracer transport in porous media

We propose a new approach to transport of the suspensions and tracers in porous media. The approach is based on a modified version of the continuous time random walk (CTRW) theory. In the framework of this theory we derive an elliptic transport equation. The new equation contains the time and the mixed dispersion terms expressing the dispersion of the particle time steps. The properties of the new equation are studied and the fundamental analytical solutions are obtained. The solution of the pulse injection problem describing a common tracer injection experiment is studied in greater detail. The new theory predicts delay of the maximum of the tracer, compared to the velocity of the flow, while its forward “tail” contains much more particles than in the solution of the classical parabolic (advection-dispersion) equation. This is in agreement with the experimental observations and predictions of the CTRW theory.     

Fluctuations of the number of neighboring pores and appearance of multiple nonergodic states of a nonwetting liquid confined in a disordered nanoporous medium

Multiple nonergodic states have been observed for nonwetting liquid in the Fluka100 C18 and Fluka100 C8 porous media with broad pore size distributions having different widths. The dispersion transition where the volume of confined liquid depends critically on the degree of filling of the porous medium and temperature is observed in the temperature range 293–343 K under study for the Fluka100 C18 porous medium and is not observed for the Fluka100 C8 porous medium. A mechanism of the appearance of multiple nonergodic states has been proposed. It has been shown that fluctuations of the number of the nearest neighbors in the disordered system are mainly responsible for the features of confinement of nonwetting liquid and nonergodic states of nonwetting liquid in the nanoporous media under investigation with wide pore size distributions.     

Polydisperse particle size characterization by ultrasonic attenuation spectroscopy in the micrometer range

The theoretical advantages of ultrasonic attenuation spectroscopy for particle size are currently not fully utilized. Especially in the region of larger particles, there is a lack of experimental confirmation of applicable models which may be used to infer particle sizes from measured attenuation spectra. With the present work, an attempt is made to supply experimental data, obtained with a commercially available ultrasonic attenuation spectrometer, and model calculations, which are based on the resonant scattering theory. It is shown that measured attenuation results for various combinations of disperse and continuous phase for both polydisperse emulsions and suspensions are reproducible by calculation. The approach is further examined for suspensions of porous particles. Here, the resonant scattering approach is combined with the Biot model for poroelasticity to obtain attenuation results with several fractions of titania aggregates, differing in particle size and pore diameter. The results indicate that the theory of resonant scattering is a valid approach if applied to particle size characterization in the large particle limit.     

An NMR protocol for determining ice crystal size distributions during freezing and pore size distributions during freeze-drying

Nuclear magnetic resonance water proton relaxometry is widely used to investigate pore size distributions and pore connectivity in brine-saturated porous rocks and construction materials. In this paper we show that, by replacing water with acetone, a similar method can be used to probe the porous structure of freeze-dried starch gels and therefore the ice crystal size distribution in frozen starch gels. The method relies on the observation that the starch surface acts as a powerful relaxation sink for acetone proton transverse magnetization so that Brownstein-Tarr theory can be used to extract the pore size distribution from the relaxation data. In addition the relaxation time distribution is found to depend on the spectrometer frequency and the Carr-Purcell-Meiboom-Gill pulse spacing, consistent with the existence of large susceptibility-induced field gradients within the pores. The potential of this approach for noninvasively measuring ice crystal size distributions during freezing and pore size distributions during freeze-drying in other food systems is discussed.     

Parallel capillary-tube-based extension of thermoacoustic theory for random porous media

Thermoacoustic theory is extended to stacks made of random bulk media. Characteristics of the porous stack such as the tortuosity and dynamic shape factors are introduced into the thermoacoustic wave equation in the low reduced frequency approximation. Basic thermoacoustic equations for a bulk porous medium are formulated analogously to the equations for a single pore. Use of different dynamic shape factors for the viscous and thermal effects is adopted and scaling using the dynamic shape factors and tortuosity is demonstrated. Comparisons of the calculated and experimentally derived thermoacoustic properties of reticulated vitreous carbon and aluminum foam show good agreement. A consistent mathematical model of sound propagation in a random porous medium with an imposed temperature is developed. This treatment leads to an expression for the coefficient of the temperature gradient in terms of scaled cylindrical thermoviscous functions.     

气孔分布特性对含水多孔介质气体扩散的影响

本文基于多孔介质的气孔分布特性,计算了多孔介质在含水状态下的扩散性能,并且比较了采用两种方式计算相对渗透率时的相对扩散性能。其结果表明,基于气孔分布的计算结果低于与气孔分布无关的计算结果。另外,疏水性含水多孔介质的扩散性能低于亲水性含水多孔介质的扩散性能,基于气孔分布计算含水多孔介质的气体扩散性能时,Wyllie公式并不适用。     

Lifetime statistics for a Bloch particle in ac and dc fields

We study the statistics of the Wigner delay time and resonance width for a Bloch particle in ac and dc fields in the regime of quantum chaos. It is shown that after appropriate rescaling the distributions of these quantities have a universal character predicted by the random matrix theory of chaotic scattering.     

Anomalous dispersion in correlated porous media: a coupled continuous time random walk approach

We study the causes of anomalous dispersion in Darcy-scale porous media characterized by spatially heterogeneous hydraulic properties. Spatial variability in hydraulic conductivity leads to spatial variability in the flow properties through Darcy’s law and thus impacts on solute and particle transport. We consider purely advective transport in heterogeneity scenarios characterized by broad distributions of heterogeneity length scales and point values. Particle transport is characterized in terms of the stochastic properties of equidistantly sampled Lagrangian velocities, which are determined by the flow and conductivity statistics. The persistence length scales of flow and transport velocities are imprinted in the spatial disorder and reflect the distribution of heterogeneity length scales. Particle transitions over the velocity length scales are kinematically coupled with the transition time through velocity. We show that the average particle motion follows a coupled continuous time random walk (CTRW), which is fully parameterized by the distribution of flow velocities and the medium geometry in terms of the heterogeneity length scales. The coupled CTRW provides a systematic framework for the investigation of the origins of anomalous dispersion in terms of heterogeneity correlation and the distribution of conductivity point values. We derive analytical expressions for the asymptotic scaling of the moments of the spatial particle distribution and first arrival time distribution (FATD), and perform numerical particle tracking simulations of the coupled CTRW to capture the full average transport behavior. Broad distributions of heterogeneity point values and lengths scales may lead to very similar dispersion behaviors in terms of the spatial variance. Their mechanisms, however are very different, which manifests in the distributions of particle positions and arrival times, which plays a central role for the prediction of the fate of dissolved substances in heterogeneous natural and engineered porous materials.     

Dielectric constant of porous ultra low-k dielectrics by fractal-Monte Carlo simulations

The effective dielectric constant of porous ultra low-k dielectrics is simulated by applying the fractal geometry and Monte Carlo technique in this work. Based on the fractal character of pore size distribution in porous media, the probability models for pore diameter and for effective dielectric constant are derived. The proposed model for the effective dielectric constant is expressed as a function of the dielectric coefficient of base medium and the volume fractions of pores and base medium, fractal dimension for pores, the pore size, as well as random number. The Monte Carlo simulations combined with the fractal geometry are performed. The predictions by the present simulations are shown in good accord with the available experimental data. The proposed technique may have the potential in analyzing other properties such as electrical conductivity and thermal conductivity in porous ultra low-k dielectrics.     

Stochastic langevin model for flow and transport in porous media

We present a new model for fluid flow and solute transport in porous media, which employs smoothed particle hydrodynamics to solve a Langevin equation for flow and dispersion in porous media. This allows for effective separation of the advective and diffusive mixing mechanisms, which is absent in the classical dispersion theory that lumps both types of mixing into dispersion coefficient. The classical dispersion theory overestimates both mixing-induced effective reaction rates and the effective fractal dimension of the mixing fronts associated with miscible fluid Rayleigh-Taylor instabilities. We demonstrate that the stochastic (Langevin equation) model overcomes these deficiencies.     

DEM simulation of small particles clogging in the packing of large beads

The percolation of small particles through a periodic random loose packing of large beads is studied numerically with the Distinct Element Method. The representativity of periodic mono-sized sphere packing of varying system size was first studied by comparing their pore size distributions and tortuosities with those of a larger system, considered as an infinite medium. The results show that a periodic packing of size as low as 4-grain diameters gives a reasonable representation of the porous medium and allows reducing considerably the number of particles that has to be used in the simulations. The flow and clogging of small particles of varying concentrations and friction coefficients flowing through the former packing are then studied numerically. Results show that a steady state is rapidly reached where the mean velocity and mean vertical velocity of small particles are both constant. These mean velocities decrease with an increase in friction coefficient and in small particle concentration. The influence of the friction coefficient μ is much less marked for values of μ larger than or equal to 0.5. The distribution of small particles throughout the crossed packing becomes rapidly heterogeneous. Small particles concentrate in some pores where their velocity vanishes and where the density can reach values larger than the density of the random loose packing. The proportion of particles blocked in these pores varies linearly with concentration. Finally, the narrow throats of the porous medium responsible for blocking are identified and characterized for different values of the friction coefficient.     

A discrete droplet transport model for predicting spray coating patterns of an electrostatic rotary atomizer

Electrostatic spray (E-spray) coating is widely used for coating conductive substrates. The combination of a high-velocity shaping air, an imposed electric field and charged droplets, leads to higher transfer efficiency than conventional spray coating. In this paper, a mathematical model of droplet transport in E-spray is presented which enables simulating the coating deposition rate profile. A dilute spray assumption (no particle–particle interactions) allows modeling single-droplet trajectories resulting from a balance of electrostatic force, drag and inertia. Atomization of liquid droplets is not modeled explicitly—rather an empirical correlation is used for the mean droplet size while individual droplet sizes and starting locations are determined using random distributions. Strong coupling requires the electrostatic field and droplet trajectories be determined iteratively by successive substitution with relaxation. The influences of bell-cup voltage and atomization constant on the coating deposition rate profile, mass transfer efficiency and droplet trajectories are also shown. Using individually predicted droplet trajectories and impact locations, a static coating deposition rate profiles is determined. For the parametric values considered in this paper, the predicted spray is a cone hollow with no deposition in the center, a heavy ring near the center, and a tapering of thickness toward the outer edge.     

Characterization challenges for a cellulose nanocrystal reference material: dispersion and particle size distributions

Cellulose nanocrystals (CNCs) have high aspect ratios, polydisperse size distributions, and a strong propensity for aggregation, all of which make them a challenging material for detailed size and morphology characterization. A CNC reference material produced by sulfuric acid hydrolysis of softwood pulp was characterized using a combination of dynamic light scattering (DLS), atomic force microscopy (AFM), transmission electron microscopy, and X-ray diffraction. As a starting point, a dispersion protocol using ultrasonication was developed to provide CNC suspensions with reproducible size distributions as assessed by DLS. Tests of various methods for AFM sample preparation demonstrated that spin coating on a positively charged substrate maximizes the number of individual particles for size analysis, while minimizing the presence of agglomerates. The effects of sample-to-sample variability, analyst bias, and sonication on size distributions were assessed by AFM. The latter experiment indicated that dispersion of agglomerates by sonication did not significantly change the size distribution of individual CNCs in suspension. Comparison with TEM data demonstrated that the two microscopy methods provide similar results for CNC length (mean ~?80 nm); however, the particle width as measured by TEM is approximately twice that of the CNC height (mean 3.5 nm) measured by AFM. The individual crystallite size measured by X-ray diffraction is intermediate between the two values, although closer to the AFM height, possibly indicating that laterally agglomerated CNCs contribute to the TEM width. Overall, this study provides detailed information that can be used to assess the factors that must be considered in measuring CNC size distributions, information that will be useful for benchmarking the performance of different industrially sourced materials.     

Ultrasonic Spectrometry for Particle Size Analysis in Dense Submicron Suspensions

Ultrasonic spectrometry was applied to the particle size analysis of disperse systems. The investigations were made for acoustic conditions called the long-wavelength regime (LWR). In the LWR the acoustic behaviour is governed by dissipative effects rather than by scattering. Two principal theoretical approaches to ultrasonic spectrometry — scattering theory and coupled phase models — are introduced. A model based on a newly developed coupled phase model and the scattering theory (ECAH theory) is implemented in the ultrasonic spectrometer Acousto Phor. Experiments were carried out for several suspensions with a high density contrast. It could be demonstrated that the model successfully describes acoustic attenuation and that the inversion algorithm finds particle size distributions comparable to those given by other measurement techniques. With regard to the particle size, a lower and an upper limit for the applicability were determined, which include three decades. As a further result, the model was validated at concentrations up to 10 vol.%. The model is considered to be open to development to cover even higher concentrations.     

Hermite-DG methods for pdf equations modelling particle transport and deposition in turbulent boundary layers

A novel methodology is presented for the numerical treatment of multi-dimensional pdf (probability density function) models used to study particle transport in turbulent boundary layers. A system of coupled Fokker–Planck type equations is constructed to describe the transport of phase-space conditioned moments of particle and fluid velocities, both streamwise and wall-normal. This system, unlike conventional moment-based transport equations, allows for an exact treatment of particle deposition at the flow boundary and provides an efficient way to handle the 5-dimensional phase-space domain. Moreover, the equations in the system are linear and can be solved in a sequential fashion; there is no closure problem to address.A hybrid Hermite-Discontinuous Galerkin scheme is developed to treat the system. The choice of Hermite basis functions in combination with an iterative scaling approach permits the efficient computation of solutions to high accuracy. Results demonstrate the effectiveness of the methodology in resolving the extreme gradients characteristic of distributions near an absorbing boundary.     

Nuclear magnetic resonance cryoporometry

Nuclear Magnetic Resonance (NMR) cryoporometry is a technique for non-destructively determining pore size distributions in porous media through the observation of the depressed melting point of a confined liquid. It is suitable for measuring pore diameters in the range 2 nm–1 μm, depending on the absorbate. Whilst NMR cryoporometry is a perturbative measurement, the results are independent of spin interactions at the pore surface and so can offer direct measurements of pore volume as a function of pore diameter. Pore size distributions obtained with NMR cryoporometry have been shown to compare favourably with those from other methods such as gas adsorption, DSC thermoporosimetry, and SANS. The applications of NMR cryoporometry include studies of silica gels, bones, cements, rocks and many other porous materials. It is also possible to adapt the basic experiment to provide structural resolution in spatially-dependent pore size distributions, or behavioural information about the confined liquid.     

Modelling of chemical vapour deposition for optical fibre manufacture

A study of numerical modelling has been carried out for chemical vapour deposition processes with applications to manufacture of optical fibres. Temperature distributions and thermophoretic particle deposition have been calculated for the modified chemical vapour deposition (MCVD) and the outside vapour deposition (OVD) processes. A two torch formulation and a heat flux boundary condition are used for MCVD and the present model is shown to be capable of predicting tube wall temperatures and deposition profiles correctly. The present results are in agreement with experimental data. For OVD modelling, nonorthogonal body-fitted coordinates have been utilized to solve a conjugate problem including the jet flow and heat conduction through a two-layered cylinder that consists of an original target and the deposited porous layers. Surface temperatures and efficiencies of particle deposition have been obtained.     

Stochastic Kadomtsev-Petviashvili equation

The example of Kadomtsev-Petviashvili equations with a random time-dependent force (stochastic Kadomtsev-Petviashvili equations) is used to show that the theory of Brownian particle motion can be applied to the theory of the stochastic behavior of solitons of model hydrodynamic equations which are completely integrable in the absence of forces and interrelated by the generalized Galilean transformation. The Brownian motion of two-dimensional algebraic solitons of the Kadomtsev-Petviashvili equations with positive dispersion leads to their diffusion broadening similar to the broadening of one-dimensional solitons of other fully integrable hydrodynamic equations. However, for longer times the rate of decay of algebraic solitons is higher because of the degeneracy of the momentum integral for these solitons. The behavior of a periodic chain of algebraic solitons is established under the action of a random force. Tilted plane solitons of the Kadomtsev-Petviashvili equations with negative dispersion vary under the action of a random force similar to the solitons of the Korteweg-de Vries equation. Several of these solitons interact via “virtual solitons” and generate new solitons provided that resonance conditions are satisfied whose dimensions increase as a result of the influence of the random force.     

Rayleigh wave propagation along the boundary of a non-Newtonian fluid-saturated porous medium

The Frenkel-Biot theory is used to study the reflection of elastic waves from the boundary of a non-Newtonian (Maxwell) fluid-saturated porous medium. The velocity and attenuation of a Rayleigh surface wave propagating along the boundary of the medium are determined. Two models of a fluid-saturated porous medium are used for calculation: with pore channels of a fixed diameter and with a lognormal distribution of pore channels in size. The results of calculations show that, when the fluid in the porous medium is characterized by a small Deborah number (i.e., exhibits non-Newtonian properties), the velocity of Rayleigh waves exhibits a considerable frequency dispersion. The results also suggest that, in principle, it is possible to estimate the Deborah number from the measured frequency dispersion of the Rayleigh wave velocity.