Pavel Čapek

Gas transport in porous media under dynamic conditions

Abstract The dynamic version of the Wicke-Kallenbach diffusion cell with one compartment closed and equipped with a sensitive pressure gauge was used for determination of sets of Mean Transport Pore Model and Dusty Gas Model parameters for an industrial catalyst (ICI 52-1 in reduced form). The dynamic pressure responses due to gas composition step changes of inert gases (H2, He, N2, Ar) were measured and fitted to the system of partial differential equations which describe the transport (mass balances with Maxwell-Stefan constitutive equations). The optimum textural parameters were obtained by simultaneous matching of all experiments. The model parameters are material constants of the porous solid and, thus, do not depend on temperature, pressure and kind of the transported gases. Both diffusion models gave a good agreement between experiments and calculations. The parameter reliability is discussed.

Prediction and Evaluation of Time-Dependent Effective Self-diffusivity of Water and Other Effective Transport Properties Associated with Reconstructed Porous Solids

We reconstructed pore structures of three porous solids that differ from each other in morphology and topology of pore space. To achieve this, we used a stochastic method based on simulated annealing and X-ray computed microtomography. Simulated annealing was constrained by the following microstructural descriptors sampled along the principal and diagonal directions: the two-point probability function for the void phase and the lineal-path functions for both void and solid phases. The stochastic method also assumed the isotropic pore structures in accordance with a recent paper (Čapek et al. in Transp Porous Media 88(1): 87–106 (2011)). With the exception of the solid with the widest pores, we made tomographic volume images in high and low resolution, which enabled us to study the effect of resolution on microstructural descriptors and effective transport properties. A comparison of the two-point probability function and the lineal-path function sampled in the principal directions revealed that the pore structures derived from the tomographic volume images were slightly anisotropic, in opposition to the assumption of the stochastic method. Besides the anisotropy, other microstructural descriptors including the pore-size function and the total fraction of percolating cells indicated that the morphological and topological characteristics of the pore structures depended on the reconstruction method and its parameters. Particularly, the pore structures reproduced using the stochastic method contained wider pores than those obtained using X-ray tomography. Deviations between the pore structures derived from low- and high-resolution tomographic volume images were also observed and imputed to partial volume artefacts. Then, viscous flow of incompressible liquid, ordinary diffusion, Knudsen flow and self-diffusion of water in the reconstructed pore spaces were simulated. As counterparts, experimental data were measured by means of permeation and Wicke–Kallenbach cells and pulsed field gradient NMR. Deviations between the simulated quantities on the one hand and experimental data on the other hand were generally acceptable, which corroborated the pore-space models. As expected, the predictions based on the tomographic models of pore space were more successful than those derived from the stochastic models. The stationary effective transport properties, i.e. the effective permeability, the effective pore size and the geometric factor, were sensitive to a bias in long-range pore connectivity. Furthermore, the time-dependent effective diffusivity was found to be especially sensitive to relatively small morphological deviations between the real and reconstructed pore structures. It is concluded that the combined predictions of the effective permeability, the effective pore size, the geometric factor and time-dependent effective self-diffusivity of water are needed for the reliable evaluation of pore-space reconstruction.

Permeation of gases in industrial porous catalysts

An experimental method for determination of porous solid texture was proposed and verified. The method is based on unsteady permeation of single gases in a porous medium. Transport parameters characterizing porous materials were determined by fitting the experimental data to the theoretical transient responses by minimizing the objective function. Confidence limits of the transport parameters were evaluated on the base of the Beale criterion. Results were discussed in terms of different mechanisms of mass transport. Advantages of this technique were demonstrated for three porous catalysts of different textures.

On the Importance of Simulated Annealing Algorithms for Stochastic Reconstruction Constrained by Low-Order Microstructural Descriptors

In the past two decades, simulated annealing has played an important role in stochastic reconstruction of porous media. In this study, we compare simulated annealing algorithms constrained by low-order microstructural descriptors and controlled by four annealing schedules that use different ways of temperature reduction. Besides the plain exponential decay of temperature, three adaptive schedules deducing the optimum cooling speed from statistical measures, such as mean system energy and standard deviation of energy, are investigated. Unlike the first three schedules, which modify temperature by a stepwise manner, the fourth one takes into account the effect of move generation strategies and decreases temperature continuously maintaining quasi-equilibrium. The performance of the algorithms is exemplified by reconstructing pore structures of five porous samples. The fourth annealing schedule can significantly reduce the total time required for the reconstruction process itself and for parameter tuning because the same values of annealing parameters are used for both 2D and 3D reconstruction. None of the schedules affects the quality of the reconstructed pore structures.

Reconstructing the microstructure of polyimide–silicalite mixed‐matrix membranes and their particle connectivity using FIB‐SEM tomography

Properties of a composite material made of a continuous matrix and particles often depend on microscopic details, such as contacts between particles. Focusing on processing raw focused‐ion beam scanning electron microscope (FIB‐SEM) tomography data, we reconstructed three mixed‐matrix membrane samples made of 6FDA‐ODA polyimide and silicalite‐1 particles. In the first step of image processing, backscattered electron (BSE) and secondary electron (SE) signals were mixed in a ratio that was expected to obtain a segmented 3D image with a realistic volume fraction of silicalite‐1. Second, after spatial alignment of the stacked FIB‐SEM data, the 3D image was smoothed using adaptive median and anisotropic nonlinear diffusion filters. Third, the image was segmented using the power watershed method coupled with a seeding algorithm based on geodesic reconstruction from the markers. If the resulting volume fraction did not match the target value quantified by chemical analysis of the sample, the BSE and SE signals were mixed in another ratio and the procedure was repeated until the target volume fraction was achieved. Otherwise, the segmented 3D image (replica) was accepted and its microstructure was thoroughly characterized with special attention paid to connectivity of the silicalite phase. In terms of the phase connectivity, Monte Carlo simulations based on the pure‐phase permeability values enabled us to calculate the effective permeability tensor, the main diagonal elements of which were compared with the experimental permeability. In line with the hypothesis proposed in our recent paper (Čapek, P. et al. (2014) Comput. Mater. Sci. 89, 142–156), the results confirmed that the existence of particle clusters was a key microstructural feature determining effective permeability.

Pore structure and effective permeability of metallic filters

The pore structures (microstructures) of two metallic filters were reconstructed using the stochastic reconstruction method based on simulated annealing. The following microstructural descriptors were included in the description of the real microstructures: the two-point probability function, the lineal-path functions for the void or solid phases, i.e. simulated annealing was constrained by all low-order statistical measures that were accessible through the analysis of images of polished sections. An effect of the microstructural descriptors on the course of reconstruction was controlled by modifying two parameters of the reconstruction procedure [1]. Their values resulted from repeated reconstruction of two-dimensional microstructures in such a way that the reference (experimental) and calculated two-point cluster functions deviated negligibly. It was tacitly assumed that the parameters adjusted during two-dimensional reconstruction had the same influence on the formation of the three-dimensional microstructures. Since connectivity of phases is a critical property of the stochastically reconstructed media, clusters of pore and solid voxels were determined using the Hoshen-Kopelman algorithm. It was found that the solid phase formed one large cluster in accordance with the physical feasibility. The void phase created one large cluster and a few small clusters representing the isolated porosity. The percolation properties were further characterised using the local porosity theory [2]. Effective permeability of the replicas was estimated by solving the Stokes equation for creeping flow of an incompressible liquid in pore space. Calculated permeability values matched well their experimental counterparts.