Monte-Carlo simulations of surface and gas phase diffusion in complex porous structures

A new procedure for estimating surface diffusivities and tortuosities within realistic models of complex porous structures is reported. Our approach uses Monte-Carlo tracer methods to monitor mean-square displacements for molecules restricted to wander on pore walls within model random mesoporous solids typical of those used as adsorbents, heterogeneous catalysts, and porous membranes. We consider model porous solids formed from initial packings of spheres with unimodal, Gaussian, or bimodal distributions of size; changes in pellet porosity are achieved by increasing microsphere radii and by randomly removing spheres from highly densified packings in order to simulate densification and coarsening, respectively. Geometric tortuosities for the surface phase reached large values at void fractions near 0.04 and 0.42 for densified solids; the surface tortuosity gave a minimum value of 1.9 at a void fraction of ∼0.26. These high tortuosities correspond to percolation thresholds for the void and solid phases, which in turn reflect packing densities at which each phase becomes discontinuous. Surface tortuosities for coarsened solids at low void fractions were similar to those in densified solids; however, at void fractions above ∼0.3, surface tortuosities of coarsened solids increased only gradually with void fraction, because coarsening retains significant overlap among spheres at void fractions above those giving disconnected solids in densified structures. Simulations of bulk diffusion within voids were used to compare the transport properties and connectivity of the void space with those of surfaces that define this void space. Surface and void tortuosities were similar, except for void fractions near the solid percolation threshold, because unconnected solid particles interrupt surface connectivity but not gas phase diffusion paths. Surface and void tortuosities were also similar for channels within linear chains of overlapping hollow spheres as both tortuosities increased with decreasing extent of sphere overlap. These simulations provide a basis for estimates of surface and void tortuosities, which are essential in the interpretation and extrapolation of diffusion rates in complex porous media. Surface and void diffusivity estimates differed significantly from those obtained from lattice and capillary models of complex porous structures.     

Comparison of Wicke-Kallenbach and Graham's diffusion cells for obtaining transport characteristics of porous solids

Counter-current gas diffusion measurements on a series of porous solids covering a broad range of pore sizes (mean pore radii between 78 nm and ) were performed in the Wicke-Kallenbach and Graham's diffusion cells. Mutual agreement of diffusion fluxes from both cells was found in the whole range of tested pore radii and inert gas systems. For pore materials with mean pore radii exceeding the experimentally unavoidable tiny total pressure gradient induces additional permeation flow which precludes the use of Graham's law for evaluation of transport parameters of the porous solids. Transport parameters together with 95% confidence regions were determined for porous materials with pore radii up to and the prevailing diffusion mechanism, intimately connected with the shape of confidence regions, was estimated.     

Wicke-Kallenbach and Graham's diffusion cells: Limits of application for low surface area porous solids

The influence of permeation flux induced by tiny pressure gradients on the counter-current gas diffusion in porous solids is discussed. Binary counter-current gas diffusion in porous samples with a broad range of pore radii (78 nm- was studied in arrangements of the Wicke-Kallenbach and Graham's diffusion cells. A small pressure difference between the cell compartments was observed both in the Wicke-Kallenbach cell (7 Pa) and Graham's cell (9 Pa). For samples with pore radii above significant deviations of diffusion fluxes were observed. Limits of pore sizes above which the additional permeation transport influences the diffusion results beyond acceptable level were obtained by simplified calculations. The distorted diffusion fluxes can be numerically freed from the contribution of permeation flux if permeation characteristics are available.     

Composition dependent transport diffusion coefficients of CH4/CF4 mixtures in carbon nanotubes by non-equilibrium molecular dynamics simulations

Understanding transport diffusion on a molecular level helps to develop improved adsorbents with tailored rate and equilibrium properties. Dual-control-volume grand canonical molecular dynamics (DCV-GCMD) simulations allow the direct simulation of transport diffusion on a molecular level. DCV-GCMD simulations of CH₄/CF₄ mixtures in carbon nanotubes were carried out. An approach to get composition dependent transport diffusivities directly from DCV-GCMD simulations is introduced. Composition dependent transport diffusivities and fluxes are calculated for varying driving forces in order to investigate the influence of the very large driving force in simulations which is about four orders of magnitude larger than in real experimental systems. Whereas, the flux depends on the driving force the transport diffusivity is independent of it so that DCV-GCMD simulation can be used to simulate transport under experimental conditions. Furthermore, the results of composition dependent diffusivities at four different temperatures are presented. A linear function describes the composition dependence and reproduces the simulated concentration profiles very well. The analysis of the temperature dependence indicates that the transport in the investigated system is due to liquid-like molecular diffusion and not to activated diffusion.     

Effects of compression on porous texture of clay powder: Application to uranium diffusion

This paper presents the results of tests conducted on the compaction properties of clay. Swelling clays of Algeria are being examined as components of the buffer material. Knowledge of the porous texture in clay samples is a useful element in the microstructural characterization of such materials. The prepared materials were characterized by X-ray diffraction, Differential Thermal and Thermo Gravimetric Analysis, Laser Granulometry, Scanning Electron Microscopy, mercury porosimetry and nitrogen adsorption/desorption to obtain information about their structure and surface texture. The cationic exchange of clay minerals, in particular smectites, influences the cation retention and diffusion processes. These processes influence the migration of contaminants from waste disposal sites through clay barrier. In the present work we studied the effect of cation exchange and compaction on the textural properties of clay. Pellets of the Na-saturated and bulk clay samples were prepared by compression of the powder samples using uniaxial stress at different pressures in order to obtain different densities. The pressures of 31–127 MPa are applied for the production of cylindrical samples with diameter of 20 mm and heights of 15–30 mm. The effect of compaction of clay on the diffusion behavior of uranium was studied for the safety assessment of the radioactive waste. The diffusion process of uranium in compacted clay has been modeled by second Fick's law taking into account the effect of sorption and considering the non-steady state. The diffusion coefficients and profiles concentration values of uranium were calculated by computational method using numerical program based on Newton–Raphson algorithm.     

Analysis and extension of the theory of multicomponent fluid diffusion

Recently we have presented an extended theory for molecular mass transport (Kerkhof and Geboers, 2005). We have indicated the possibility, and the need, to define a new framework for studying and teaching the theory of multicomponent molecular transport in fluids (Kerkhof and Geboers, 2004). Here a new, structured, overview of the subject is presented. An analysis is given of complicated issues in textbooks and review papers.     

The influence of nanoscale microstructural variations on the pellet scale flow properties of hierarchical porous catalytic structures using multiscale 3D imaging

The total activity, selectivity and lifetime of heterogeneous silica–alumina catalysts depends on the flow of molecules through complex three-dimensional (3D) hierarchical pore structures that can span length scales from micro- to nano-meters. The transport of molecules to and from active sites ultimately determines the catalyst pellets performance. In this study a multiscale tomography (MT) methodology was developed by combining x-ray microtomography (XMT), dual beam focused ion beam tomography (DBFIB) and electron tomography (ET) to compare the flow properties of two hierarchically porous silica–alumina catalysts, one sintered/calcined at 580 °C and the other at 800 °C.The use of MT not only allowed visualization and quantification of the effect of sintering temperature on the pore structures at a number of length scales, but also allowed the influence of this pore structure on the flow effects to be calculated from the nanometer to millimeter scale. The lower temperature calcined catalyst exhibited an open, rounded, interconnected pore structure, while the high temperature one had a finer plate/crack-like pore structure, with ∼15-20% lower porosity. The permeability of both catalysts was calculated via direct meshing of the MT volumes and coupled across all three length scales (∼10⁴ m), giving a good agreement to experimentally measured values.It was concluded that the higher calcining temperature significantly reduces the permeability across all length scales investigated. The very fine and tortuous paths in this high temperature calcined catalyst will be more vulnerable to deactivation by coking, illustrating how MT coupled with flow simulations can give new insights into the processing of sintered catalysts and potentially other functional materials.     

Strategy for predicting effective transport properties of complex porous structures

Measurements of effective transport properties of porous media, such as effective diffusivity and permeability, are well established by several experimental techniques. Effective transport properties can be also calculated from the spatially 3D reconstructed porous media, where the morphology characteristics required for the reconstruction are obtained from electron microscopy images. Here we demonstrate the reconstruction of porous alumina catalyst carrier with bimodal pore size distribution. Multi-scale concept is employed for the computation of effective diffusivity and permeability of reconstructed porous media and calculated effective transport properties are compared with transport parameters experimentally determined in Graham diffusion and simple permeation cell. The limitations of current state-of-the-art reconstruction techniques for porous media with broad pore size distribution are discussed. We show that the contribution of nano-pores towards the total diffusion flux is significant and cannot be neglected, but it is reasonable to neglect the contribution of nano-pores towards the sample permeability.     

An approximate model for diffusion and reaction in a porous pellet

A method of derivation of linear driving force approximation for diffusion and reaction processes in porous catalysts based on Laplace-Carson transform has been presented. The approximate model for any type of kinetic expression has been derived. Accuracy of developed model is good, especially in the range of small and intermediate Thiele modulus values that is used most often in cases of practice. This approximation yields a substantial simplification of analysis and computations and does not require any iterative and trial-and-error calculations.     

The effective diffusivities in porous media with and without nonlinear reactions

The question of whether effective diffusivities in porous materials under reactive and nonreactive conditions are equal is addressed. Previous studies had considered the problem with first-order reactions. We study the issue with two nonlinear reactions—a second-order reaction and one governed by the Michaelis-Menten kinetics. Pore network and continuum models of porous media are utilized to estimate the effective diffusivities under reactive and nonreactive conditions. We show that the two effective diffusivities are significantly different. The difference is due to the heterogeneities of the porous material, and the fluctuations that they cause in the spatially varying local concentrations and diffusivities, and can be as large as a few orders of magnitude. Theoretical analysis of diffusion and reactions in porous media is also presented that supports the results of the simulations. In particular, it is shown that the results of pore network simulations cannot be fitted to the classical continuum equation of diffusion and reaction, and that a more complex continuum equation should be used for this purpose.     

Study of molecular shape and non-ideality effects on mixture adsorption isotherms of small molecules in carbon nanotubes: A grand canonical Monte Carlo simulation study

The sorption isotherms for binary mixtures of methane, ethane, propane and tetrafluoromethane have been determined in carbon nanotubes using configurational bias Monte Carlo simulation techniques. At high loadings, a curious maximum for equimolar gas-phase mixtures occurs with increasing pressure in the absolute adsorption isotherm of one or both adsorbing species. It was detected that there exist two fundamentally different reasons for this maximum. First, due to a higher packing efficiency, one component is able to displace the other component at high loadings. Here, it must be stressed that the displaced component is not necessarily the larger molecule. Second, non-ideality effects of the bulk gas phase can be made responsible for this maximum. The acceptance probability of a molecule insertion in a grand canonical Monte Carlo step is proportional to the component fugacity. If, owing to non-ideality effects of the gas phase, the fugacity of one component does not increase as steeply with pressure as the other component, a maximum can occur in the absolute adsorption isotherm of this component. These findings were demonstrated for various binary mixtures of CH₄, CF₄, C₂H₆ and C₃H₈.     

On the relationship between Aris and Sherwood numbers and friction and effectiveness factors

We present a general solution of the diffusion-reaction problem for linear kinetics and an expression for the effectiveness factor (η) in terms of the shape normalized Thiele modulus (Φ) for a catalyst particle of arbitrary shape and with an arbitrary activity profile. We also show that the Sherwood number (Sh_Ω) or the dimensionless mass transfer coefficient between the interior of a particle or region (denoted by Ω) and its boundary (∂Ω) is related to the effectiveness factor and Thiele modulus by η=1/(1+Φ²/Sh_Ω). Further, the coefficients in the expansion of η in terms of Φ (the Aris numbers, Ar_i) are related to the asymptotic Sherwood number (Sh_Ω∞) obtained in the limit of slow reaction. We show that the curve Sh_Ω versus Φ is universal for most common particle or channel geometric shapes and for the case of uniform activity is described by the two asymptotes Sh_Ω=Sh_Ω∞=1/Ar₁ for Φ?1 and Sh_Ω≈Φ for Φ?1. For two-dimensional ducts we show that the friction factor times Reynolds number (fRe) is equal to 8Sh_Ω∞ and provide a physical interpretation of this result. In the second part of this work, we derive low-dimensional models for solving multicomponent nonlinear diffusion-reaction problems using the concept of an internal mass transfer coefficient. We also present low-dimensional models for catalytic reactors using external and internal mass transfer coefficients. Finally, the Aris/Sherwood numbers are presented for some commonly used catalyst particles in packed-bed reactors and washcoat shapes in catalytic monoliths.     

From pore scale to wellbore scale: Impact of geometry on wormhole growth in carbonate acidization

Acid injection in carbonate reservoir is commonly used in the oil industry to improve, or at least recover, its productivity. The aim of this stimulation technique is to create empty channels called wormholes which, if successful, would bypass the damaged area near the wellbore. During production, wormholes become pathways for the reservoir oil to reach the well. This technique increases near-wellbore permeability, and therefore improves oil production. The interaction between the transport of acid, chemical reaction, and heterogeneities encountered at different scales, controls the unstable behaviour of wormholing and, thus, the success of the treatment. Most of the experimental and numerical studies done on this subject in the past have been limited in their observations because they only considered the dissolution process at a small scale (from pore scale to core scale). The purpose of this work is to study how the geometry of the domain can constrain wormhole competition, and influence wormholing dynamics in a core submitted to acidizing.After a short review of the literature on wormholing to see how the geometry effect could have influenced previous experiments, we study specifically the question of wormhole density. We emphasize that two mechanisms are involved in wormhole competition, with one of them being effective only at small scale. Thus we conclude that wormholing is not a full-scale independent process. We describe differences in the wormhole growth dynamics between “confined” and “unconfined” domains for different dissolution regimes. We focus on optimum conditions and their transition from “confined” to “unconfined” domain to realize that the flow rate in the dominant wormhole does not depend on geometric effects. We conclude by a comparison between 2D and 3D simulations, in both linear and radial flow, and observe changes in the wormholing process. All our results serve as a discussion about definitions of optimum conditions in the literature.     

A kinetic Monte Carlo approach to diffusion in disordered nanoporous carbons

The efficiency of different technological processes where nanoporous carbons are used depends on their storage capacity and an appropriate gas transport. Different experimental and theoretical works that relate storage to global structural parameters of the solid such as specific surface area (SSA) or total pore volume (V_t) can be found in literature. The structure–transport relationships have been less studied. The combined use of the truncated pore network model (TPNM) and the kinetic Monte Carlo (KMC) method is proposed in this work to find hydrogen and methane effective self-diffusivities (D_eff) and to delve into those relationships. It was found that for Knudsen and free molecular diffusion in the simulated materials, the D_eff/V_t vs. SSA graphic nearly follows a power law. The KMC/TPNM approach was also used to predict the self-diffusion coefficients of hydrogen in Vulcan XC-72 and of methane in a carbon aerogel. The obtained values are within the expected range. KCM/TPNM is computationally fast and it allows a study of the diffusion synchronously and globally in the network, avoiding thus its fractionation in single pores and the use of just one geometric model to describe the porous spaces.     

Transmission probabilities and particle-wall contact for Knudsen diffusion in pores of variable diameter

Knudsen dynamics simulations were performed in cylindrical pores with multiple sections of different diameters. The number and location of hits between particles and the pore walls can be related to the known distributions in the uniform diameter cases. Both sweep-gas and pass-through modes of operation were examined. The contact of the particles traveling inside the pores with the pore walls can be regulated by adjusting the aspect ratio (section length/section diameter) of the pore sections and the ratio of their cross-sectional areas. The results were found to be independent of the transition angle between sections except for very smooth transitions. Analytical expressions were developed to relate the transmission probability in pores of multiple sections to the transmission probabilities of the constituent sections.     

Effect of speciation on the apparent diffusion coefficient in nonreactive porous systems

A combined theoretical and experimental study of the effect that concentration and ionic speciation have on the apparent diffusion coefficient is performed using a nonreactive porous material in a divided cell diffusion apparatus. Varying the ionic species concentration over two orders of magnitude changes the apparent diffusion coefficient by no more than 20% for the systems studied. By contrast, at fixed ionic concentration, varying the ionic species changes the initial apparent diffusion coefficient by a factor of two. Over longer periods of time, the apparent diffusion coefficient varies in time, increasing by a factor of ten or more. For one system, the macroscopic diffusion potential across the specimen induces a transient negative apparent diffusion coefficient; iodide ions are transported from regions of low iodide concentration to regions of high iodide concentration. The theoretical analysis shows that, in nonreactive porous systems, the behavior of all the concentrations and species studied can be completely characterized by an electro-diffusion system of equations that contain two time-independent constants: the porosity and the formation factor. The relationship between these results and the prediction of concrete performance in the field is discussed.     

Water vapor/propylene sorption and diffusion behavior in PVA-P(AA-AMPS) blend membranes by GCMC and MD simulation

Detailed atomistic structures of blend membranes (poly vinyl alcohol (PVA)/(acrylic acid-co-2-acrylamido-2-methylpropylsulfonic acid) (P(AA-AMPS)) were constructed to investigate the sorption and diffusion behavior of gas molecules (water and propylene) in the membranes. Interaction and miscibility between PVA and P(AA-AMPS) were calculated, and it was found that strong intermolecular interaction resulted in good miscibility of PVA and P(AA-AMPS) in the blend. The polymer chains mobility and free volume properties of the blend membranes were characterized. The sorption quantities and sorption sites of water and propylene in the blend membranes were calculated using Grand Canonical Monte Carlo (GCMC) method. The diffusion coefficients of water in the blend membranes were calculated by molecular dynamics (MD) simulation. The simulated results of the membrane structure (chain mobility, free volume properties), the sorption quantities and diffusion coefficients of water/propylene in the blend membranes showed the identical changing trends as the experimental results. Hopefully, this study could offer qualitative insight into the mass transport phenomena within the blend membranes.     

Determination of the effects of the pore size distribution and pore connectivity distribution on the pore tortuosity and diffusive transport in model porous networks

An efficient computation method to study flow and transport process of small molecules in porous media using a dual site-bond lattice model, DBSM, is described. The microscopic properties of the porous network take into account the influence of local heterogeneities during the simulations. The numerical experiments demonstrated the combined effect of pore size distribution and connectivity distribution on the mass transport properties and the structural tortuosity. The results indicate that the pore size distribution and percolation phenomena related with pore shielding effects, influence significantly the tortuosity and the effective diffusivity of the porous network. Also, the simulations raise the important role of the connectivity distribution among the various pores in the gas diffusive properties of the poorly connected networks.