Knudsen Diffusion Coefficient Research Articles

Simulation of the transport phenomenon occurring in membrane pores during vacuum membrane distillation

Basic knowledge regarding flow at conventional scales is relatively familiar to us. However, micro-scale flow is distinctly different. To explore the flow characteristics and mass transfer process in a membrane micro pore, a Knudsen-molecular diffusion mechanism with sliding boundary condition is used to simulate the treatment of ethyl acetate solution using vacuum membrane distillation. The diffusion coefficient is expressed by the Knudsen diffusion coefficient and molecular diffusion coefficient. The simulation results are consistent with the experimental data. The flow fields in the membrane pore are obtained, and the effects of the membrane structure parameters on the membrane pore separation performance are investigated. The results show that the increase in the porosity and pore diameter can enhance the membrane flux and increase the tortuosity and pore length against permeate flux. The optimal choice of the separation factors is presented as a function of the increase in the porosity, pore diameter, and tortuosity. Pore length does not enhance the membrane separation performance. The simulation of the phenomenon occurring in membrane pores should contribute to appropriate membrane preparation and improve membrane separation performance.

Pore and Particle Size Control of Gas Sensing Films Using SnO2 Nanoparticles Synthesized by Seed-Mediated Growth: Design of Highly Sensitive Gas Sensors

Gas sensing is an important application of metal oxides. The gas sensor response of metal oxide films is greatly influenced by particle size, pore size, thickness, and surface states. To study the effects of particle and pore sizes of sensing films on sensitivity, we fabricated SnO2-based films with different particle and pore sizes and studied sensor responses to three different gases: H2, CO, and H2S with different Knudsen diffusion coefficients. The pore size radii of the gas sensing films were successfully controlled from 2.8 to 5.5 nm using SnO2 nanoparticles of different sizes (4–17 nm diameter) that were synthesized by seed-mediated growth under hydrothermal conditions. Sensor response to H2 increased with decreasing particle size because of the formation of an electron depletion layer within the nanosized crystals. In contrast, the response to CO and H2S increased with increasing particle size and the resultant pore size. Using the Knudsen diffusion-surface reaction equation, we simulated a gas co...

Multiscale Modeling of Membrane Distillation: Some Theoretical Considerations

First, it is shown that the effective thickness of the membrane is the sum of the actual thickness, k0/UL, and λC/(UL) where k0 is the thermal conductivity of the membrane matrix, λ is the latent of vaporization of water, C is a parameter (defined as flux per unit thickness of membrane per unit of temperature driving force), and UL is a coefficient combining the heat-transfer coefficients on the feed side and the permeate side film. For typical conditions, the sum of the additional terms exceeds 100 μm, which clearly shows that the flux is not inversely proportional to membrane thickness. Also, to a first approximation, the thermal efficiency is independent of membrane thickness. This work and the development of an overall mass-transfer coefficient for direct contact membrane distillation build upon the pioneering work of Giulio Sarti. Second, a reassessment of the traditional method for combining the Knudsen diffusion coefficient and the molecular diffusion coefficient suggests that the traditional sum of resistances approach engages in some double counting and thereby overestimates the resistance and consequently underestimates the flux.

Active layer of fuel cell electrode with polymer electrolyte: Modeling of support grains, calculation of overall cathode characteristics

The active layer of the cathode of a fuel cell with polymer electrolyte (Nafion) is considered. The optimum carbon support structure is constructed using computer simulation: its carbon “skeleton” possesses the maximum outer surface area and provides electronic conductivity of the grains, support cubes, along the three coordinate axes. Nafion is absent in the support grain, so that the grain is capable of participating only in the transport of oxygen molecules, it possesses no proton conductivity. An estimate of all parameters of an optimum support grain is provided; in particular, the value of the effective Knudsen diffusion coefficient of oxygen is established. After this, effective proton conductivity and effective Knudsen diffusion coefficient are calculated already on the whole active layer scale, according to the model of equally sized cube grains of three types. In conclusion, the overall current in the active layer of a cathode with a polymer electrolyte was calculated for the percolation cluster consisting only of Nafion grains and the Knudsen diffusion of oxygen created only by a combined gas percolation cluster consisting of void grains and all support grains. The overall current value for t = 80°C and pressure of p* = 101 kPa proved to be low, hundreds of mA/cm2. The current value can apparently be increased to several A/cm2 if the support grains are developed that would simultaneously possess both proton conductivity and ability to sustain oxygen diffusion.

A two-interface transport model with pore-size distribution for predicting the performance of direct contact membrane distillation (DCMD)

We investigated fundamental aspects of heat and mass transfer of direct contact membrane distillation. Molar flux of water vapor through a membrane pore was analytically obtained by solving Fick's law in the original differential form. Axial variation of the temperature profile was derived as exponentially decreasing, and was found to be linear due to small membrane thickness and dominant heat conduction through the solid part of the membrane. An alternative expression of water vapor pressure at a constant temperature was developed using experimental data of water latent heat for evaporation, and was used to calculate the concentration of water vapor in the membrane pore. The effective diffusion coefficient was obtained by combining Knudsen and Brownian diffusion coefficients with Bosanquet's assumption. The effective diffusivity and mean free path of water vapor slowly decrease in the axial direction, and the vapor concentration increases along the membrane pore primarily due to the linearly decreasing temperature. We found that the required heat flux monotonously increases with the vapor flux through membrane pores. Finite variance of a pore size distribution provides less vapor flux than that of mono-dispersed pores. This is because a number of smaller pores than the average pore size significantly hinders the vapor transport across the porous membrane. Theoretical prediction of permeate flux agrees very well with experimental observations reported in the literature.

An analytically predictive model for moderately rarefied gas flow

AbstractIn microducts deviation from continuum flow behaviour of a gas increases with rarefaction. When using Navier–Stokes equations to calculate a flow under slightly and moderately rarefied conditions, slip boundary conditions are used which in turn refer to the tangential momentum accommodation coefficient (TMAC). Here we demonstrate that, in the so-called slip and transition regime, the flow in microducts can be reliably described by a consistently non-empirical model without considering the TMAC. We obtain this equation by superposition of convective transport and Fickian diffusion using two-dimensional solutions of Navier–Stokes equations and a description for the Knudsen diffusion coefficient as derived from kinetic theory respectively. For a wide variety of measurement series found in the literature the calculation predicts the data accurately. Surprisingly only size of the duct, temperature, gas properties and inlet and outlet pressure are necessary to calculate the resulting mass flow by means of a single algebraic equation. From this, and taking the discrepancies of the TMAC concerning surface roughness and nature of the gases into account, we could conclude that neither the diffusive proportions nor the total mass flow rates are influenced by surface topology and chemistry at Knudsen numbers below unity. Compared to the tube geometry, the model slightly underestimates the flow rate in rectangular channels when rarefaction increases. Likewise, the dimensionless mass flow rate and the diffusive proportion of the total flow are distinctly higher in a tube. Thus the cross-sectional geometry has a significant influence on the transport mechanisms under rarefied conditions.

Contribution of the Dusty Gas Model to Permeability/Diffusion Tests on Partially Saturated Clay Rocks

Gas transfer experiments on claystone and numerical simulations have been conducted to enhance the knowledge of gas transport in nuclear waste repositories in the Callovo-Oxfordian clay formation in Bure, France. Laboratory Gas transfer experiments were performed with a specific device dedicated to very low permeability measurement (10−23 to 10−20 m2). Experiments were performed on both dry and close to saturation claystone. The Dusty Gas Model, based on multi-component gas transfer equations with Knudsen diffusion, was used to describe the experimental results. The parameters obtained are the effective permeability, the Knudsen diffusion (Klinkenberg effect) and molecular diffusion coefficients and the porosity accessible to gas. Numerical simulations were carried with various boundary conditions and for different gases (helium vs hydrogen) and were compared with experiments to test the reliability of the model parameters and to better understand the mechanisms involved in clays close to saturation. The numerical simulation fitted the experimental data well whereas simpler models cannot describe the complexity of the Knudsen/Klinkenberg effects. Permeabilities lie between 10−22 and 10−20 m2. Claystones close to saturation have an accessible porosity to gas transfer that is lower than 0.1–1% of the porosity. Analysis of the Klinkenberg effect suggests that this accessible pore network should be made of 50–200 nm diameter pores. It represents pore networks accessible at capillary pressure lower than 4 MPa.

Estimation of mechanical dispersion and dispersivity in a soil–gas system by column experiments and the dusty gas model

In a previous study, column experiments were carried out with Toyoura sand (permeability 2.05×10(-11)m(2)) and Toyoura sand mixed with bentonite (permeability 9.96×10(-13)m(2)) to obtain the molecular diffusion coefficient, the Knudsen diffusion coefficient, the tortuosity for the molecular diffusion coefficient, and the mechanical dispersion coefficient of soil-gas systems. In this study, we conducted column experiments with field soil (permeability 2.0×10(-13)m(2)) and showed that the above parameters can be obtained for both less-permeable and more-permeable soils by using the proposed method for obtaining the parameters and performing column experiments. We then estimated dispersivity from the mechanical dispersion coefficients obtained by the column experiments. We found that the dispersivity depended on the mole fraction of the tracer gas and could be represented by a quadratic equation.

Modeling of Overpotentials in an Anode-Supported Planar SOFC Using a Detailed Simulation Model

In this paper results from a parameter study of an anode-supported solid oxide fuel cell (SOFC) are presented. The effects on performance, current-voltage (I-V) characteristics, polarization voltages, and diffusion coefficients are modeled for different temperatures, electrolyte thickness, porosities, and pore sizes. The analysis is carried out for a planar SOFC with YSZ electrolyte, LSM cathode, and Ni-YSZ anode, with thicknesses 20, 50, and 500 μm respectively, and with co-flow geometry. The predicted performance is validated with measured data found in the literature with good agreement. Standard equations for binary and Knudsen diffusion in porous media, concentration overpotentials, and Ohm’s law are used in the modeling. Activation overpotential is predicted by use of temperature dependent linear equations at both the anode and cathode sides. It is found that both ohmic and activation overpotentials decrease considerably with increasing temperature, while concentration overpotentials increase moderately with increasing temperature. The effect on concentration overpotentials can be explained by the reduced gas density with increased temperature, despite the increasing diffusion coefficient. Furthermore, it was found that increasing the pore sizes decreases concentration overpotentials. At low pore size the Knudsen diffusion coefficient is a bottleneck for the diffusion coefficient since it is much lower than the binary diffusion coefficient. It has been demonstrated that by increasing the pore size the Knudsen diffusion coefficient is improved. The effect of porosity has much in common with the effect of pore size; increasing porosity leads to decreased concentration overpotential due to the improved diffusion coefficient. As a natural phenomenon for anode-supported cells, most of the concentration overpotentials take place at the anode side due to its thick structure despite the high diffusion coefficient of hydrogen. It must be underlined that in this study the effect of different porosities and pore sizes is modeled at the anode and cathode substrates only without taking into account the length of the three phase boundary (LTPB) near the electrolyte interface. However, since these microstructural parameters can have an impact on the LTPB, they can also have an impact on the activation overpotentials. This is not considered here and will be taken into account in future work.

Measurement of Knudsen and Effective Ordinary Diffusion Coefficients in Solid Oxide Fuel Cell Anodes Fabricated by Atmospheric Plasma Spraying Using Powder, Suspension, and Solution Precursor Feedstocks

Ni and yttria-stabilized zirconia electrode layers were fabricated by atmospheric plasma spraying using powder, suspension, and solution precursor feedstocks. Permeability measurements were used to evaluate the Knudsen diffusion coefficients for N2. It was demonstrated that the forced flow behavior for a second gas, e.g. Ar, may be determined based on the microstructural parameters obtained using N2. Graham's diffusion cell was used to obtain effective ordinary diffusion coefficients for the H2/N2 system. The values obtained for the three plasma sprayed coatings did not differ significantly. Further work is required to satisfactorily separate the diffusion coefficients for the coatings from the measured results on the support and coating combinations.

Knudsen and Molecular Diffusion Coefficients for Gas Transport in Unconsolidated Porous Media

This study was an assessment of the Knudsen and molecular diffusion coefficients in relation to gas‐phase permeability and air‐filled porosity, respectively, for gas transport in unconsolidated porous media. Single gas flow experiments at various gas pore pressures and equimolar binary experiments were conducted to determine apparent, effective gas permeabilities as well as the Knudsen and molecular diffusion coefficients. Experimental results integrated with data from previous studies showed that for dry porous media, the Knudsen diffusion coefficient value for kaolin with total porosity ranging from 0.675 to 0.85 is on the order of 10−4 m2, while it is at least one order of magnitude greater for sea sand and one order less for silica flour. Additionally, the existing correlations of the Knudsen diffusion coefficient can give lower and upper limits to the order of this coefficient value for porous media with effective gas permeabilities in the order of 10−14 to 10−16 m2 The gas‐phase molecular diffusion coefficient for an unsaturated silica flour sample with total porosity of 0.386 and water saturation levels >50% was measured to be one order of magnitude less than those for the other samples with total porosities ranging from 0.402 to 0.417 at similar air‐filled porosities. The former also had lower gas diffusivity under rewetting than under desaturation conditions. The diffusivity factor values obtained from all of the unsaturated silica flour samples were far less than the predictions from other commonly used diffusivity relationships.

Measurement of Knudsen and Effective Fickian Diffusion Coefficients in Solid Oxide Fuel Cell Anodes Fabricated by Atmospheric Plasma Spraying Using Solution Precursor, Suspension, and Powder Feedstocks

not Available.

Ionomer content in the catalyst layer of polymer electrolyte membrane fuel cell (PEMFC): Effects on diffusion and performance

The percolating paths of the carbons and electrolytes in a cathode catalyst layer (CCL) could be successfully visualized in three-dimensions in order to investigate both the electronic and ionic connectivity by modeling a three-dimensional (3-D), meso-scale CCL of a polymer electrolyte membrane fuel cell (PEMFC). The effective Knudsen diffusion coefficients could also be obtained by computing pore tortuosity values. Electrochemical simulation studies were carried out by feeding air at 70 °C. Low platinum (Pt) loading (0.1 mg cm−2) catalysts with ionomer contents ranging from 14 to 50% were studied. The performance of a PEMFC electrode was affected by the ionomer content which is optimal at about 33%. In this case, both electronic and ionic connectivity produced the broadest active surface area of the Pt catalyst. The polarization drop tendency was in good agreement with the experiment, and this percolation study could successfully explain the existence of an optimum amount of ionomer.

Pore Modification in Porous Ceramic Membranes With Sol‐Gel Process and Determination of Gas Permeability and Selectivity

AbstractSummary: The aim of the study was to investigate the variation in total surface area, porosity, pore size, Knudsen and surface diffusion coefficients, gas permeability and selectivity before and after the application of sol‐gel process to porous ceramic membrane in order to determine the effect of pore modification. In this study, three different sol‐gel process were applied to the ceramic support separately; one was the silica sol‐gel process which was applied to increase porosity, others were silica‐sol dip coating and silica‐sol processing methods which were applied to decrease pore size. As a result of this, total surface area, pore size and porosity of ceramic support and membranes were determined by using BET instrument. In addition to this, Knudsen and surface diffusion coefficients were also calculated. After then, ceramic support and membranes were exposed to gas permeation experiments by using the CO2 gas with different flow rates. Gas permeability and selectivity of those membranes were measured according to the data obtained. Thus, pore surface area, porosity, pore size and Knudsen diffusion coefficient of membrane treated with silica sol‐gel process increased while total surface area was decreasing. Therefore, permeability of ceramic support and membrane treated with silica sol‐gel process increased, and selectivity decreased with increasing the gas flow rate. Also, surface area, porosity, pore size, permeability, selectivity, Knudsen and surface diffusion coefficients of membranes treated with silica‐sol dip coating and silica‐sol processing methods were determined. As a result of this, porosity, pore size, Knudsen and surface diffusion coefficients decreased, total surface area increased in both methods. However, viscous flow and Knudsen flow permeability were detected as a consequence of gas permeability test and Knudsen flow was found to be a dominant transport mechanism in addition to surface diffusive flow owing to the small pore diameter in both methods. It was observed that silica‐sol processing method had lower pore diameter and higher surface diffusion coefficient than silica‐sol dip coating method.

Analysis of Reactant Gas Transport in Catalyst Layers; Effect of Pt-loadings

An analytical in-situ technique was utilized to evaluate reactant gas transport resistance (Rother) in catalyst layers (CLs). It was found that Rother was increased with the decrease of Pt-loadings of the CLs although their thickness was decreased. Effective Knudsen diffusion coefficient was calculated from the pore size distribution and the porosity. It was estimated almost steady regardless of the Pt-loadings. To understand the increase of Rother, a simple transport model was established including Knudsen diffusion resistance in secondary pores and the local transport resistance around the Pt surface. Knudsen diffusion resistance should be decreased in lower Pt-loadings because of the shorter transport length. On the other hand, it was considered that the local transport resistance was increased with decreasing of the effective Pt surface area. Hence, that the effective Pt surface area might be one of the important factors for Rother in the CLs, especially in the lower Pt-loadings.

Evaluation of local pore sizes and transport properties in porous catalysts

A methodology for the analysis of local pore size in an arbitrary porous medium, based on the maximum sphere inscription method, has been developed. The method has been validated for regular cylindrical pores and used for the evaluation of pore size distribution on several random media, both computer-generated and scanned using X-ray microtomography. The local pore radii have then been used for the calculation of effective diffusion across the porous medium, whereby local Knudsen diffusion coefficients were employed depending on the local pore radius. This continuum-based approach was then compared with the results of Knudsen diffusion through the same porous media simulated by the molecular dynamics method. A difference of maximum 40% was obtained, which is deemed acceptable given the significantly faster computing speed and higher versatility of the continuum-based models.

Experimental Study of Gas Transport Parameters in Unsaturated Silica Flour

An experimental study was conducted to assess the relative contributions of molecular and Knudsen diffusions for gas‐phase transport in unsaturated silica flour with an intrinsic permeability of 1 ∼ 2 × 10−14 m2 Single‐gas phase flow and binary diffusion experiments were performed on the same soils to determine the Knudsen diffusion and the effective gas‐phase diffusion coefficients, respectively. These results were further used to assess the diffusibility factor for Fick's law of diffusion and the obstruction factor for the Dusty Gas Model theory. These factors were used to account for impedance to gas diffusion caused by the tortuous nature of soil pores. The Knudsen diffusion coefficient was found to be one to two orders of magnitude greater than the effective diffusion coefficient for silica flour at water saturations ranging from 53 to 75% (v/v), which suggested that the Knudsen effect has negligible impact on the effective diffusion coefficient for the unsaturated silica‐flour system at water saturations lower than 75%. For unsaturated silica flour with air‐filled porosity <0.2, the diffusibility factor (or the obstruction factor) value can be more than 10 times smaller than the values estimated by other diffusibility correlations published in the literature.

An improved one-dimensional membrane-electrode assembly model to predict the performance of solid oxide fuel cell including the limiting current density

The limiting current density is an important characteristic quantity in solid oxide fuel cells (SOFCs). High concentration overpotential is often used to explain the limiting current density assuming a high tortuosity or limited surface diffusion in the vicinity of the three-phase boundary. Most membrane-electrode assembly models of SOFC fail to predict the limiting current density, even for hydrogen, when using physically reasonable values of tortuosity and considering the short residence time of the adsorbed species near the three-phase boundary. In this paper, a one-dimensional model for the transport–chemistry interactions in SOFCs is described. The model is based on a comprehensive approach that includes the dusty-gas model for gas transport in the porous electrodes, detailed heterogeneous elementary reaction kinetics for the thermo-chemistry in the anode, and detailed electrode kinetics for the electrochemistry. Correct values for the Knudsen diffusion coefficients are used. We apply the unsteady form of the conservation equations, allowing for the analysis of the response of the cell to external dynamics. Results of our model are compared with experimental data, showing good agreement over a wide range of the current density, but fail to predict the limiting current density accurately when the hydroxyl oxidation charge-transfer reaction is assumed to be the rate limiting reaction. To obtain accurate predictions of the limiting current density, we analyze the possibility that different steps can be rate-limiting reactions in the electrochemistry model of hydrogen oxidation. We use recent measurements on the three-phase boundary area and take into account the surface diffusion and competitive adsorption to determine possible rate limiting reactions at high current density. We show that a rate-limiting switchover model, in which the reaction limiting the overall kinetic rate becomes the hydrogen adsorption at the anode, may be required to explain the experimentally measured limiting current density over a range of operating conditions.

One-Dimensional Model of a Tubular Solid Oxide Fuel Cell

In this paper, a detailed model of a solid oxide fuel cell (SOFC) tube is presented. The SOFC tube is discretized along its longitudinal axis. Detailed models of the kinetics of the shift and reforming reactions are introduced in order to evaluate their rates along the SOFC axis. Energy, moles, and mass balances are performed for each slice of the components under investigation, allowing the calculation of temperature profiles. Friction factors and heat-exchange coefficients are calculated by means of experimental correlations. As for the SOFC overvoltages, the activation overvoltage is calculated using the Butler–Volmer equation and semiempirical correlations for the exchange current density, Ohmic losses are evaluated introducing an appropriate electrical scheme and material resistivities, and concentration overvoltage is calculated by means of both binary and Knudsen diffusion coefficients. On the basis of this model, a case study is presented and discussed, in which temperatures, pressures, chemical compositions, and electrical parameters are evaluated for each slice of the SOFC tube under investigation. Finally, a sensitivity analysis is performed, in order to investigate the influence of the design parameters on the performance of the system.

Effect of transition from slip to free molecular flow on gas transport in porous media

Traditional models, such as the advection-diffusion and the dusty gas models, overlook the contribution of the transition flow regime between the slip and the free molecular flow, on the gas transport in porous media. In this work we demonstrate that, due to the existence of this intermediate regime, the Klinkenberg [Drill. & Prod. Prac. 1941, 200 (1941)] parameter b depends on the pressure. Reported experiments were used to corroborate such an effect and a formulation that extends the Klinkenberg equation—to include the effect of a region at pore scale where both molecule-molecule and molecule-wall interactions are important—was developed. The mathematical form of the extended Klinkenberg equation remains the same, but the slippage Klinkenberg’s parameter b is now a generalized parameter that is a function of Knudsen’s number. It was demonstrated that the widely accepted relation between the parameter b and the Knudsen diffusion coefficient is a good approximation just for Knudsen numbers corresponding to the free molecular flow regime. The model proposed in this paper reproduces the experimental data and predicts practical situations where important errors on total flow rate can be expected if the transition flow regime is neglected in the formalism.