Determination of an effective pore dimension for microporous media
Introduction
The determination of characteristics of porous media permeability like the micro and nanoporous membranes or ultra-tight shale-gas reservoirs is still a challenge up to now. The low porous media find a broad application in medicine [1], biotechnology for separation and filtration [2]. The recent development of porous ceramic media with high thermal, chemical and structural stability and the ability to have catalytic properties has opened up new horizons for membranes applications, for example, in high-temperature gas separation and catalytic reactions [3]. Unconventional resources, such as ultra-tight shale-gas reservoirs of very small pores (in nanoscale) play a significant role in securing hydrocarbon energy because of their potential to offset declines in conventional gas production [4]. The morphology of the porous structure dominates the fluid flow through a porous medium. Therefore, it is important to characterize the geometrical properties of a porous medium quantitatively. Different methods exist for the measurements of the average pore size and pore size distribution. The choice of the most appropriate method depends on the application of the porous solid, its chemical and physical nature and the range of pore size. The most commonly used methods are [5]: mercury porosimetry, where the pores are filled with mercury under pressure. This method is suitable for many materials with pores in the appropriate diameter range from μm to 300 μm. From mesopore to micropore size analysis, BET method [6], can be done by gas adsorption, usually nitrogen, at liquid nitrogen temperature. This method can be used for pores in the approximate diameter range from 1 nm to 0.1 μm. The pore size diameter can also be determined via direct observation methods: scanning electron microscopy (SEM), field-emission scanning electron microscopy (FESEM), environmental scanning electron microscopy (ESEM), and atomic force microscopy (AFM), [7], [8]. The tomography analysis of a porous structure can allow the determination of the internal structure of a sample limited by the characteristics of their spatial resolution [9]. All these methods require either preliminary sample preparation or lead to the complete sample destruction, furthermore, they only use a small part of the sample for analysis.
We propose here a simple approach for the non-destructive porous sample characterization by measuring the pressure variation in the inlet and outlet tanks (or just the pressure difference between them). The experimental methodology, based on the constant volume technique, was initially developed for the isothermal and non-isothermal measurements of the mass flow rate through the microchannels [10] and has been recently adapted for the analysis of porous samples [11]. The gas permeability of the porous sample can be easily obtained directly from the pressure evolution in time without calculation of the mass flow rate.
The measurements are analyzed by assuming the porous media have similar behavior as the classical bundle of capillaries model, first suggested by Kozeny [12] and then extended by Carman [13] to allow for torturous capillaries. In our analysis we assume that the capillary tubes have the same radius. This allows us to find a unique parameter (capillary’s radius) to characterize the porous structure. This unique parameter helps also to determine the gas flow regime, by introducing the Knudsen number as the ratio of the molecular mean free path and the capillary radius, and then by referring on this Knudsen number to distinguish the flow regimes. Recently, the models of a bundle of capillary tubes of variable shape and size cross-section were developed, [14], [15], but all the models were used either for the liquid or for two phase flows, which physics is different from the single phase flows.
The model of a bundle of capillaries with gas flow inside was considerably improved by Klinkenberg [16] taking into account the slip flow regime through the capillaries. In the present article, from the measured mass flow rate the effective pore size is estimated by using the fitting procedure via slip flow expression. The obtained effective pore sizes are then compared to mercury porosimetry and micro-computed tomography (CT) results. The proposed technique of the effective pore size measurement can be used as a non-destructive method for quality verification. Furthermore, this method is independent of the exterior sample geometry. When the effective pore size is known and by using the information about porosity the permeability, apparent permeability, and tortuosity coefficients as well as the surface-to-volume ratio can be easily obtained.
Section snippets
Experimental methodology
The experimental methodology, applied in this article, is described in details in Ref. [11]. We present here only the summary of this technique, essential to understand the data treatment. From measurements of pressure variation over time we calculate the important characteristics of porous media such as mass flow rate and permeability, and then effective pore size dimension.
Modeling of the porous structure
Different type of modeling can be used to characterize the flow through microporous media. One of the simplest and, in the same time, efficient models of a microporous medium is its representation as a bundle of several numbers of capillaries with the circular cross-section of the same or different diameters [12], [20]. All the capillaries (pores) can be parallel and have a length equal to the length L (thickness) of the porous medium, see Fig. 1 (left). However, in the real samples, this
Flow regimes
The microporous medium is modeled here as a bundle of capillaries, so it is worth to define first different possible flow regimes in a capillary and to present then the expressions of the mass flow rate through a capillary for these flow regimes. Usually the flow regimes could be identified through the Knudsen number, which is calculated as the ratio between the equivalent molecular mean free path and the characteristic dimension a of the capillary (its radius):
The equivalent molecular
Determination of porous medium characteristics from pressure measurements
In the previous Section, we introduced the complete description of the flow through a single capillary. In this Section, the model of the porous media as a bundle of capillaries is presented, and its parameters as the capillary radius, capillary number, tortuosity, and specific surface area are extracted from the measurements. The proposed geometrical model corresponds to a homogeneous porous medium with a signature of a single pore size.
Other characteristic parameters of porous sample
As it discussed in previous Section, from the mass flow rate fitting expression we can extract: the characteristic dimension of porous medium, a, and also the number N of the capillaries as
However, in Eq. (36) the capillary length, , is still unknown, so we can make two assumptions to obtain this value. One of possibilities is to assume that the capillary length is equal to the porous disc thickness, , so the tortuosity factor, , Eq. (10), is equal to 1. However, with this
Permeability
In Section 2.3 we provided the definition of the permeability as it was proposed by Darcy, i.e. for the incompressible fluid, and then its expression through the mass flow rate, Eq. (8), more adapted for the gas flows, so the permeability is calculated as
By using the same model of the porous media as a bundle of N capillaries with length and replacing the mass flow rate in the previous expression by its representation provided in Section 5.2, Eqs. (30), (31), we expresse the
Tomography analysis
To have additional information about the samples, a typical sample from the same batch was scanned with MicroXCT-400 tomograph at CEREGE,1 which uses the linear attenuation method. The focal spot size of X-ray beam was 5–7 m. The geometrical voxel size is determined by the size and number of detector elements and the source-object-detector distances
Results and comparison
In this Section we present the results obtained with the proposed methodology on the effective pore size, tortuosity, surface-to-volume ratio and the permeability. We compare these porous sample characteristics to the data obtained from the tomographic and porosimetry analyses, when they are available.
Conclusion
The classical model of the porous media presentation as a bundle of capillaries was revised. The original methodology was suggested to determine the characteristic flow dimension. The experimental procedure is developed to determine the effective pore size (characteristic flow dimension) and the number of capillaries, related to the model a bundle of capillaries. The experimentally obtained effective pore dimension is in very good agreement with the results of the mercury porosimetry and
Declaration of Competing Interest
The authors declared that there is no conflict of interest.
Acknowledgement
The project leading to this publication has received funding from Excellence Initiative of Aix-Marseille University - A*MIDEX, a French “Investissements d’Avenir” programme. It has been carried out in the framework of the Labex MEC. The authors (M.V. Johansson, P. Perrier, and I. Graur) would like to acknowledge the financial support provided by the European Union network program H2020, MIGRATE project under Grant Agreement No. 643095.
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