Knudsen Diffusion Coefficient Research Articles

Transmission probability of gas molecules through porous layers at Knudsen diffusion

Gas flow through layers of porous materials plays a crucial role in technical applications, geology, petrochemistry, and space sciences (e.g., fuel cells, catalysis, shale gas production, and outgassing of volatiles from comets). In many applications the Knudsen regime is predominant, where the pore size is small compared to the mean free path between intermolecular collisions. In this context common parameters to describe the gas percolation through layers of porous media are the probability of gas molecule transmission and the Knudsen diffusion coefficient of the medium. We show how probabilistic considerations on layer partitions lead to the analytical description of the permeability of a porous medium to gas flow as a function of layer thickness. The derivations are made on the preconditions that the molecule reflection at pore surfaces is diffuse and that the pore structure is homogenous on a scale much larger than the pore size. By applying a bi-hemispherical Maxwell distribution, relations between the layer transmission probability, the half-transmission thickness, and the Knudsen diffusion coefficient are obtained. For packings of spheres, expressions of these parameters in terms of porosity and grain size are derived and compared with former standard models. A verification of the derived equations is given by means of numerical simulations, also providing evidence that our analytical model for sphere packing is more accurate than the former classical models.

Numerical analysis of the relation between the porosity of the fuel electrode support and functional layer, and performance of solid oxide fuel cells using computational fluid dynamics

The development of the hydrogen industry brings new technological challenges. Solid oxide fuel cells (SOFCs) seem to be one of the leading devices for hydrogen conversion and its production while operating in its reverse mode (as an electrolyser). Fine-tuning of SOFC microstructure might extend cells life and boost their performance. The article presents original numerical studies of the performance of solid oxide fuel cells, based on the OpenFuelCell model, extended by fuel and oxidant concentration loss equations. Transport phenomena in the porous structure of the anode functional layer (AFL) and the anode support layer (ASL) were simulated using the modified Fick's law (MFL) and the Knudsen diffusion coefficient. The presented research focuses on the influence of fuel distribution in fuel electrode of SOFC together with reacting zones and on the influence of AFL and ASL microstructure characteristics on cell performance. The studies included porosity in the range of 30%–60% for both types of anode layers. The obtained results showed a significant impact of the porosity characteristics on the SOFC performance and indicated the possible direction for further improvement. Numerical studies have shown that an increase in anode layers porosity of 30% allows for an increase in cell efficiency of even 15%. Experimental, as well as numerical results, showed that above a porosity of 49%, further increasing the porosity has no strong effect on the results. Also, a novelty in the work and an important part of it is the comparison of the impact of microstructure manipulation of the two anodic layers. Improvement of ASL microstructure allows to better, more homogeneous, fuel distribution inside the anode. Additionally, it increases the concentration of the fuel in the functional layer by up to 8%, which leads to increase in SOFC performance.

Validation of gas flow experiments for porous media by means of computer simulations

A profound understanding of gas flow in porous media is of great interest for various technological and scientific fields. Its investigation by laboratory measurements, however, poses several challenges. In particular, the determination of macroscopic flow parameters from pressure and gas flow measurements is prone to various error influences, some of which are very difficult to analyze experimentally. Computer simulations are a solution in this context as they facilitate modifications of the underlying geometry and boundary conditions in a flexible way. Here we present a simulation framework for the analysis of a recent experiment for determining the Knudsen diffusion coefficient and viscous permeability of various porous granular materials. By combining the finite element method with analytical models and other numerical methods, we were able to identify previously neglected physical effects that increase the uncertainty of the measurements. In particular, the porosity increase due to finite sample dimensions, in a layer of about a grain diameter thickness near the container wall, creates a deviation of the measured pressure gradient. This deviation amounts to ca. 5% for a sample width of about 100 grains and a porosity of 0.5, and is indirectly proportional to the porosity. The second most prominent error source, the sample support sieve, causes a slight constriction of the flow volume. Simulations of this effect show an error around 4%–7%, dependent on the grain size. Based on these findings we recommend an overall sample dimension of 100 grains or larger. As an example of failures of the sample homogeneity, we elaborate how channels through the sample influence the flow properties. Respective suggestions for keeping all discussed effects negligible are discussed in detail. Our methodology demonstrates how the combination of finite element computations with analytical representations of the involved macroscopic parameters can assess the validity and accuracy of laboratory experiments.

Knudsen diffusion in pore-scale water-bearing shales: modelling and case study

Gas diffusivity in shales is significantly affected by water. However studies on the model of diffusion coefficient for water-bearing shales are very limited. In this work, we proposed a model to calculate the Knudsen gas diffusion coefficient at the presence of water in shale nanopores. This model was used to study the gas diffusivity in a real shale pore structure composed of both organic pores and inorganic pores at different water saturations. The model was also utilized to analyze the sensitivity of the Knudsen diffusion coefficient on a variety of factors. Results indicate that the existence of water drastically reduces the Knudsen diffusion in shales. When water saturation increases from 10% to 50%, gas concentration in inorganic pores rises by 53.7%. This effect can happen both in organic pores and inorganic pores due to pore connectivity. Water in inorganic pores can transport into organic pores and hinders gas diffusion in organic pores. The sensitivity study demonstrates that the increase of temperature, pore radius and porosity can lead to the rise of Knudsen diffusion coefficient. Inversely, the increase of tortuosity induces the decrease of the Knudsen diffusion coefficient. The pore wall roughness also affects the Knudsen diffusion coefficient. The rough wall surface contributes to a lower Knudsen diffusion coefficient compared with the smooth wall surface.

Dehumidification Analysis of Rotary Solid Desiccant Wheel System for different Surface Materials

In recent decades, desiccant dehumidification has been given much interest in the air conditioning industry area. Compared to the conventional vapour compression air conditioning system, desiccant dehumidification has various advantages, including separate humidity and temperature control. In this paper, the analysis of the Rotary Solid Desiccant Wheel System has been performed. Here, we have studied three diffusion coefficients: Ordinary, Knudsen and Surface diffusion coefficients for Rotary Solid Desiccant Wheel System. Also, the analysis has been performed for three different solid desiccant materials: Silica Gel, Molecular Sieve and Activated Alumina to incorporate and to study the effect of pore size on moisture transport and to examine the impact of Tortuosity factor on moisture transport phenomenon. This analysis can correctly select the best desiccant material suitable for the desiccant dehumidification industry.

Ordered Mesoporous SnO Micro Sheets Based Highly Sensitive and Selective Ethanol Gas Sensor

Stannous oxide (SnO) micro sheets were successfully synthesized by the hydrothermal alkaline solution method and characterized by X-ray diffraction, scanning electron microscopy, transmission electron microscopy, energy-dispersive X-ray spectroscopy, and Brunauer-Emmett-Teller (BET) techniques. BET analysis confirmed a mesoporous structure with 33.8944 m2 g−1 specific surface area and an average pore diameter of 6.488 nm. The resistance-based study of SnO was performed for the various volatile organic compounds (VOCs) at 100 °C. The response of the sensor is maximum toward ethanol with good response (∼2 s)/recovery (∼11 s) time at 100 °C. The optimum working temperature of the sensor was found to be 100 °C for ethanol gas. The response of the sensor typically depends on the surface state, gas diffusion, pore diameter, working temperature of the sensor, and crystallite size of the material. The crystallite size (29.251 nm) and strain (8.952 × 10−4) of the SnO particles were analyzed by Williamson-Hall Plot. The response of the target gases was validated by the Knudsen diffusion coefficient (D κ ). The D κ value for the ethanol gas was 8.962 × 1011 nm s−1 at 100 °C, which is the maximum among the obtained values of other VOCs.

Non-isothermal pore change model predicting CO2 adsorption onto consolidated activated carbon

Accurate simulation and detailed description of the dynamics of the adsorption process play a significant role in forecasting the performance of new materials when used in various adsorption systems, like cooling/heating. The activated carbon (AC) consolidation allows improving the heat transfer rate inside the adsorption/desorption bed and compacting the systems. There are numerous mathematical models in literature for gas adsorption onto granular AC. But for consolidated AC, because of the absence of macropores, most assumptions that work well for granular AC may lead to significant discrepancies. Therefore, the present research proposes a new mathematical model for gas (CO2) adsorption onto consolidated adsorbent, a non-isothermal pore change model. The model takes into account the porosity and permeability changes due to the adsorption. The validation of the developed model is performed via comparison with the results obtained experimentally and numerically using an isothermal model. The effective Knudsen diffusion coefficient for the working pair is evaluated from porosity data. The rate of adsorption or mass transfer coefficient is estimated using the van't Hoff plot. The study results could be applied in the development of waste heat-driven cooling systems employing consolidated composite material as the adsorbent. The proposed mathematical model is also applicable for many other working pairs.

Viscous and Knudsen gas flow through dry porous cometary analogue material

ABSTRACTAccording to current theories of the formation of stellar systems, comets belong to the oldest and most pristine class of bodies to be found around a star. When approaching the Sun, the nucleus shows increasing activity and a pressure increase inside the material causes sublimated and trapped gas molecules to stream away from their regions of origin towards the surface. The present work studies two essential mechanisms of gas transport through a porous layer, namely the Darcy and the Knudsen flow. Gas flow measurements are performed in the laboratory with several analogue materials, which are mimicking dry cometary surface properties. In this first series of measurements, the aim was to separate gas transport properties from internal sources like local sublimation or release of trapped gases. Therefore, only dry granular materials were used and maintaining a low temperature environment was unnecessary. The gas permeability and the Knudsen diffusion coefficient of the sample materials are obtained, thereby representing the relative importance of the respective flow mechanism. The experiments performed with air at a stable room temperature show that the grain size distribution and the packing density of the sample play a major role for the permeability of the sample. The larger the grains, the bigger the permeability and the Knudsen diffusion coefficient. From the latter, we estimated effective pore diameters. Finally, we explain how these parameters can be adapted to obtain the gas flow properties of the investigated analogue materials under the conditions to be expected on the comet.

Diffusion-Based Modeling of Gas Transport in Organic-Rich Ultratight Reservoirs

SummaryThe complex pore structure and storage mechanism of organic-rich ultratight reservoirs make the hydrocarbon transport within these reservoirs complicated and significantly different from conventional oil and gas reservoirs. A substantial fraction of pore volume in the ultratight matrix consists of nanopores in which the notion of viscous flow may become irrelevant. Instead, multiple transport and storage mechanisms should be considered to model fluid transport within the shale matrix, including molecular diffusion, Knudsen diffusion, surface diffusion, and sorption. This paper presents a diffusion-based semianalytical model for a single-component gas transport within an infinite-acting organic-rich ultratight matrix. The model treats free and sorbed gas as two phases coexisting in nanopores. The overall mass conservation equation for both phases is transformed into one governing equation solely on the basis of the concentration (density) of the free phase. As a result, the partial differential equation (PDE) governing the overall mass transport carries two newly defined nonlinear terms; namely, effective diffusion coefficient, De, and capacity factor, Φ. The De term accounts for the molecular, Knudsen, and surface diffusion coefficients, and the Φ term considers the mass exchange between free and sorbed phases under sorption equilibrium condition. Furthermore, the ratio of De/Φ is recognized as an apparent diffusion coefficient Da, which is a function of free phase concentration. The nonlinear PDE is solved by applying a piecewise-constant-coefficient technique that divides the domain under consideration into an arbitrary number of subdomains. Each subdomain is assigned with a constant Da. The diffusion-based model is validated against numerical simulation. The model is then used to investigate the impact of surface and Knudsen diffusion coefficients, porosity, and adsorption capacity on gas transport within the ultratight formation. Further, the model is used to study gas transport and production from the Barnett, Marcellus, and New Albany shales. The results show that surface diffusion significantly contributes to gas production in shales with large values of surface diffusion coefficient and adsorption capacity and small values of Knudsen diffusion coefficient and total porosity. Thus, neglecting surface diffusion in organic-rich shales may result in the underestimation of gas production.

A methodology to predict the gas permeability parameters of tight reservoirs from nitrogen sorption isotherms and mercury porosimetry curves

A methodology is suggested for the explicit computation of the absolute permeability and Knudsen diffusion coefficient of tight rocks (shales) from pore structure properties. The pore space is regarded as a pore-and-throat network quantified by the statistical moments of bimodal pore and throat size distributions, pore shape factors, and pore accessibility function. With the aid of percolation theory, analytic equations are developed to express the nitrogen (N2) adsorption/desorption isotherms and mercury (Hg) intrusion curve as functions of all pertinent pore structure parameters. A multistep procedure is adopted for the successive estimation of each set of parameters by the inverse modeling of N2 adsorption–desorption isotherms, and Hg intrusion curve. With the aid of critical path analysis of percolation theory, the absolute permeability and Knudsen diffusion coefficient are computed as functions of estimated pore network properties. Application of the methodology to the datasets of several shale samples enables us to evaluate the predictability of the approach.

Optimization of Transports in a Proton-Exchange Membrane Fuel Cell Cathode Catalyst Layer at High Current Densities: A Coupled Modeling/Imaging Approach

For PEMFC to attain sustainable commercialization, the amount of platinum (catalyst used in conventional PEMFCs) must be significantly reduced, while the performance must be improved. At high current densities, it is found that the nature of the losses is mainly mass transport related [1]. Since the catalyst layer is a highly heterogeneous composite material, understanding its structure is major in order to proceed with the optimization [2]. Therefore, in this work, an approach coupling both numerical modeling and electron microscopy characterization is adopted. An agglomerate/pore-scale model is developed and coupled with a MEA-scale model [3]. The model includes a description of the ionic and gas transport, plus the oxygen reduction reaction under a classical Butler-Volmer formulation. Multiple agglomerate structures are reproduced on the basis of a multiscale characterization setup. From 2-nm isotropic resolution FIB-SEM images, the carbon phase arrangement is built, while from HAADF-STEM a representative platinum particle size histogram is extracted and the particles are distributed on the surface of the carbon particles accordingly. On the HRTEM images, it is possible to observe the Nafion thin film covering the Pt/C agglomerates, and therefore, an average layer thickness can be extracted from these. The local pore size distribution is then obtained using the method of maximum sphere inscription [4], which allows to then determine the local gas diffusivities through the computation of local Knudsen diffusion coefficients. The agglomerate scale structures are upscaled to the catalyst layer scale through a mirroring procedure along the catalyst layer thickness, allowing therefore to analyze the cathode catalyst layer performance for different structure arrangements. In this work, the influence of the operating conditions and of certain structural aspects is analyzed: namely the Pt particle size, carbon aggregate size and Nafion layer thickness. It is found that decreasing the Pt particle size leads to a performance improvement since it is expected that more surface is available for the electrochemical reactions to take place. Still, the improvement margin is dependent on the Pt particle spatial distribution since particles located in zones having large Nafion agglomerates and low Nafion/pore contact simultaneously are found to severely underperform when comparing to particles located in other regions. Increasing the carbon aggregate size is found to be highly detrimental for the overall performance, even though an improved electrical contact is ensured. Such performance decay is linked to the large oxygen concentration drop that arises mainly from the porosity reduction, and to a certain extent, from the mean pore size reduction and the pore tortuosity increase. The Nafion layer thickness parametrization requires a more complex analysis since there is a delicate trade-off that must be explored in the sense that it is highly structure-dependent. On the one hand, an improvement in the ionic conductivity is expected upon increasing the Nafion layer thickness, though on the other hand, an increased resistance to oxygen diffusion is also expected. This trade-off is explored and discussed along with the introduction of the Nafion swelling effect, which is surprisingly not explored in the literature, neither on agglomerate/pore-scale models, nor macroscale models. It is found that increasing the Nafion layer thickness is detrimental for performance only when the porosity is not high enough. To complement the analysis, 'close-to-ideal' structures are built while holding the composition of the reference structures (i.e. electron microscopy based structures). It is found that these idealized structures lead to large performance improvements, showing that there is still much ground for further improvements on the PEMFC cathode catalyst layer.

Development of an Oxygen Mass Transport Coefficient Measurement and Separation Method for Proton Exchange Membrane Fuel Cells

In this work, we use a method to separate the total oxygen mass transport coefficient into molecular, Knudsen, and ionomer contributions. Therefore, limiting current density measurements are carried out as a function of the diluent gas (He, N2, CO2), temperature (30, 50, 80°C), relative humidity (50, 75, 100%), and oxygen concentration (1, 3, 5, 7%) using state of the art membrane electrode assemblies with three platinum loadings (0.05, 0.1, 0.15 mg/cm2). As expected, the molecular diffusion coefficient is independent of the platinum loading, but increases with temperature to a varying degree depending on the humidity level. On the other hand, the Knudsen diffusion coefficient increases with increasing electrochemical active surface area and temperature, and with decreasing relative humidity. The separation procedure includes a novel feature to isolate the ionomer mass transport resistance. Its interpretation as well as the method’s reliability are critically questioned using operating condition dependencies.

Transient Flow Theory of Multiple-Fractured Horizontal Wells with Complex Mechanisms in Shale Gas Reservoirs

Shale reservoirs have the characterizations of low porosity, low permeability, and hydrocarbon organic matter self-generation and self-storage, resulting in its complex flow mechanisms. Compared with fractured vertical wells, multiple-fractured horizontal wells are widely used due to their advantages of effectively increasing the well control range and further expanding the drainage area. To further study the multiscale flow mechanisms of shale gas, a flow model was established that considered viscous flow in microfractures and inorganic pores, the diffusion of Knudsen in nanoscale porosity, the coexistence of slippage, adsorption-desorption effects under infinity, and closed outer boundary conditions; based on the continuous point source solution, a multiple-fractured horizontal well flow model was established and solved by MATLAB programming. Then, the effects of various factors were investigated. The results show that the Knudsen diffusion and slippage coefficients mainly affect the apparent permeability of the matrix pores. The more the Knudsen diffusion and slippage coefficients are, the earlier the turbulent flow occurs and the higher the gas production is.

Diffusion mechanism and release profile of a multivitamin from TiO2 hollow particles

The present study aimed to control the release rate profile of a multivitamin that has hydrophilic and hydrophobic properties and to investigate its diffusion rate from titanium dioxide (TiO2) hollow particles. The TiO2 hollow particles encapsulating the vitamins C and E were prepared via a sol-gel process using an inkjet nozzle. The obtained particles were evaluated by scanning electron microscopy and energy dispersive X-ray spectroscopy. The sustained release rate was estimated by the dialysis bag method. Additionally, a diffusion mechanism for the multivitamin from the TiO2 hollow particles was proposed.TiO2 particles prepared by this method exhibited spherical porous structures. The TiO2 hollow particles were successful in encapsulating both the hydrophilic vitamin C and the hydrophobic vitamin E, resulting in a sustained release profile for these two components from the TiO2 hollow particles. The release behavior of the multivitamin was quite different from that of each single vitamin. This is related to the adsorption state of each vitamin on the TiO2 particle surface. The effective diffusion coefficient of VC derivative (8.403 × 10−13 cm2/s) was faster than that of VE derivative (1.671 × 10−13 cm2/s). This tendency may be attributed to the difference between hydrophilic and hydrophobic properties. Additionally, the value of effective diffusion coefficient in multi-component was always larger than that in single component. The value of Knudsen diffusion coefficient in VC and VE derivative was 2.684 × 10−3 and 2.264 × 10−3 cm2/s, respectively. In all samples, Knudsen diffusion coefficient was larger than molecular diffusion coefficient. The diffusion mechanism of the active ingredients likely depends on the pore size of TiO2 and the molecular size of the active ingredients.

A simple permeability model for shale gas and key insights on relative importance of various transport mechanisms

Gas transport mechanism in a shale nanopore is investigated by considering convective flow, gas diffusion, and surface diffusion. A common practice in modeling shale gas permeability is to use Knudsen diffusion coefficient when calculating diffusive flux, but the use of Knudsen diffusion coefficient would be incorrect if the shale gas flow regime is lying either in the transition diffusion or Fick’s diffusion, in which case the diffusion coefficient must correspond to that regime. This study proposes an apparent permeability model of shale based on a unified diffusion coefficient that transforms its value per the flow regime, including the effect of molecular diffusion, viscous flow, and surface diffusion of adsorbed gas through linear superposition. The proposed model is verified by comparing against other models for shale gas permeability. Results of sensitivity analysis indicate that permeability of gas due to diffusive transport is independent of pressure and pore size when pressure is larger than 6.895 MPa, but is dominant at lower pressures and increases with pore size. For pore diameters larger than 100 nm, the permeability due to surface diffusion is independent of pressure or pore size, indicating negligible gas transport due to surface diffusion in relatively larger pores (>100 nm) at any pressure, but it is dominant in small pore sizes when pressure is below 10 MPa. Permeability of gas (with or without surface diffusion) increases with the decrease of pressure when the pore diameters are smaller than 100 nm, whereas for pore diameters larger than 300 nm it is not affected by pressure, but increases with the increase of pore size, indicating that Darcy’s law is applicable in pores with the diameters larger than 300 nm.

Optimizing Nucleic Acid Biomarker Detection in a Solid-State Nanopore Through Probabilistic Modeling

We have established a solid-state (SS-) nanopore assay with high specificity for nucleic acid targets, enabling applications like epigenetic screening, microRNA analysis, and pathogen detection . Despite its proven specificity and sensitivity, translational biomarker detection at clinically relevant concentrations is hindered by very low event rates and long acquisition times. It has been noted that most small-molecule translocations are not detectable due to low signal to noise ratio or fast timescales. Here, we describe the probability that a given translocation will produce a detectable event by considering tunable system parameters, and use it to show that these key experimental variables can influence the measured event rate in a predictable fashion. This model elucidates the dependencies of nanopore/analyte interactions and suggests methods by which measurement sensitivity can be optimized. Event probability can be determined experimentally by comparing expected (Rexp) to observed (Robs) event rates for dsDNA constructs of various lengths bound to monovalent streptavidin. Rexp can be derived analytically from the Nernst-Planck equation and Knudsen diffusion coefficient, and Robs measured as a function of fundamental experimental parameters, including the driving potential, target concentration, nanopore radius, and data acquisition rate. Using these data to generate probability distributions, we show that Robs can be predicted for arbitrary experimental conditions from first principles. With this approach, we are able to better understand and even influence the sensitivity of our SS-nanopore assay. The explicit use of the probability distribution provides insight into the mechanisms that govern event generation, and our approach may be expanded to take into account any fundamental system parameter. Tuning system parameters according to the underlying probability distribution will enable optimized detection of low-abundance or challenging nucleic acid biomarker targets.

Impact of Pore Geometry and Water Saturation on Gas Effective Diffusion Coefficient in Soil

Gas flow in soil plays a crucial role in terrestrial ecosystems, and numerical simulation of their movement needs to know their effective diffusion coefficients. How pore structure influences the effective diffusion coefficient has been studied intensively for dry porous media, but much remains unknown for unsaturated soils. Here, we employed the X-ray tomography technique at the pore scale to directly obtain the soil structures, the geometry of their pores and the water distribution under different water saturation levels were calculated using a morphological model. The results show that pore structures including porosity, interface area of gas–solid–water and pore diameter are closely related to water saturation. The increase of mean pore diameter with gas saturation can be fitted into a power law. We also investigated the impact of pore geometry and water saturation on the effective diffusion coefficients, which is independent of the molecular mass of gas after normalization. As the normalized effective Knudsen diffusion coefficient increases with average pore diameter following a power law, with the scaling factor related to pore geometry and the exponent is a constant, we explained and proved that the Knudsen diffusion coefficient increases with gas saturation, also following a power law.

Computational prediction of nanoscale transport characteristics and catalyst utilization in fuel cell catalyst layers by the lattice Boltzmann method

In the present study, a three-dimensional lattice Boltzmann model based on the quasi-random nanostructural model is proposed to evaluate the mass transport properties and catalyst utilization of fuel cell catalyst layers in pursuance of catalyst performance improvement. A series of catalyst layers is randomly generated with statistical significance at the 95% confidence level to reflect the heterogeneity of the catalyst layer nanostructures. The nanoscale gas transport phenomena inside the catalyst layers are simulated by the D3Q19 (i.e., three-dimensional, 19 velocities) lattice Boltzmann method, and the corresponding mass transport characteristics are mathematically modeled in terms of structural properties. Considering the nanoscale reactant transport phenomena, a transport-based effective catalyst utilization factor is defined and statistically analyzed to determine the structure-transport influence on catalyst utilization. The tortuosity estimation results clearly show that the classic Bruggeman equation underestimates the tortuosity of the catalyst layers and should be modified for PEFC applications. Subsequently, the effective mass diffusion coefficient is calculated by applying the tortuosity factors to the Knudsen diffusion coefficient in the catalyst layers, and it shows good agreement with published experimental data. These results indicate that Knudsen diffusion is the dominant mass transfer mechanism for fuel cell catalyst layers and that the pre-estimated tortuosity accurately reflects the mass transfer phenomena in the catalyst layers. Furthermore, catalyst utilization can be affected by excessive Pt/C catalyst loading due to the lack of pore interconnections, and it is significantly limited by the substantive reactant mass transport path inside the fuel cell catalyst layers.

Performance analysis for a model of a multi-wing hydraulically fractured vertical well in a coalbed methane gas reservoir

This paper proposes a composite model to investigate the pressure behavior and production performance of a multi-wing hydraulically fractured multiple fractured vertical well (MFVW) in a coalbed gas reservoir with a stimulated reservoir volume (SRV). The model simultaneously considers adsorption–desorption, diffusion, and viscosity flow. The MFVW model combines line-source method with Laplace transformation and discretization method. Then, the Stehfest numerical inversion and the Gauss elimination are utilized to obtain the transient pressure responses and production dynamics.The results show that the main flow regimes for an MFVW in a coal seam gas reservoir are linear flow between hydraulic fractures, pseudo radial flow in the SRV, transition flow from the inner to the outer region, and radial flow in the outer region. Further, a slight “hump” is observed during the linear flow period, which could not be observed in the curves of MFVWs with two-wing hydraulic fractures. The effects on MFVW performance of permeability and size of the SRV, properties of hydraulic fractures (fracture angle asymmetry, different fracture angle, and fracture number), Langmuir volume, and Knudsen diffusion coefficient are discussed. The conclusions obtained in this paper will be helpful to understand the transient performance of MFVWs in coalbed gas reservoirs.

Estimation of Knudsen diffusion coefficients from tracer experiments conducted with a binary gas system and a porous medium

A previous study has reported that Knudsen diffusion coefficients obtained by tracer experiments conducted with a binary gas system and a porous medium are consistently smaller than those obtained by permeability experiments conducted with a single-gas system and a porous medium. To date, however, that study is the only one in which tracer experiments have been conducted with a binary gas system. Therefore, to confirm this difference in Knudsen diffusion coefficients, we used a method we had developed previously to conduct tracer experiments with a binary carbon dioxide–nitrogen gas system and five porous media with permeability coefficients ranging from 10−13 to 10−11 m2. The results showed that the Knudsen diffusion coefficient of N2 (DN2) (cm2/s) was related to the effective permeability coefficient ke (m2) as DN2 = 7.39 × 107ke0.767. Thus, the Knudsen diffusion coefficients of N2 obtained by our tracer experiments were consistently 1/27 of those obtained by permeability experiments conducted with many porous media and air by other researchers. By using an inversion simulation to fit the advection–diffusion equation to the distribution of concentrations at observation points calculated by mathematically solving the equation, we confirmed that the method used to obtain the Knudsen diffusion coefficient in this study yielded accurate values. Moreover, because the Knudsen diffusion coefficient did not differ when columns with two different lengths, 900 and 1500 mm, were used, this column property did not influence the flow of gas in the column. The equation of the dusty gas model already includes obstruction factors for Knudsen diffusion and molecular diffusion, which relate to medium heterogeneity and tortuosity and depend only on the structure of the porous medium. Furthermore, there is no need to take account of any additional correction factor for molecular diffusion except the obstruction factor because molecular diffusion is only treated in a multicomponent gas system. Thus, molecular diffusion considers only the obstruction factor related to tortuosity. Therefore, we introduced a correction factor for a multicomponent gas system into the DGM equation, multiplying the Knudsen diffusion coefficient, which includes the obstruction factor related to tortuosity, by this correction factor. From the present experimental results, the value of this correction factor was 1/27, and it depended only on the structure of the gas system in the porous medium.