Multicomponent diffusion and chemical reaction of gases in porous catalysts—I: Ternary gasdiffusion and chemical reaction without volume change (ν = 0)

The overall conversion rate for catalytic reactions of gases in porous catalysts is often controlled by transport processes in the porous pellet. Since in general more than two gases take part in the reaction, one has to calculate the conversion rate by using the transport equations for multicomponent mixtures. These equations, however, are rather complicated; thus calculations of conversion rates are commonly done by using Fick's transport law.First of all it is shown that the equations for multicomponent diffusion can be reduced to simple linear laws by using the stoichiometric coupling of the fluxes. With these equations, the calculation of the conversion rate for a reaction of the general type: ν₁R₁ + ν₂R₂ ? ν₃R₃ is as simple as for the case of binary diffusion. Using the example of hydrogenolysis of ethane it is then demonstrated that in general Fick's law leads to higher conversion rates than the Maxwell-Stefan, diffusion equations. With this example some basic questions arising in the discussion of heterogeneous catalytic reactions in porous systems (e.g. the choice of the “key component”) are discussed in detail.     

The permeability of gases through reacting solutions: the carbon dioxide—bicarbonate membrane system

The transport of a gas across a stationary liquid film containing reactive species is investigated for the purpose of determining gas permeabilities or mass transfer coefficients in reacting solutions. Under limiting conditions when the reaction time constant far exceeds the diffusional time constant, the flux of the transported gas follows Fick's law of diffusion. Analytical series solution for the contribution of the chemical reaction to the transport process is obtained using the technique of perturbation analysis; criteria for the validity of various terms in the series solution are presented. The permeability of carbon dioxide in water and in 1N NaHCO₃?Na₂CO₃ solution is estimated. It is shown that a high degree of accuracy in the data is necessary for obtaining separate estimates of diffusivity and solubility by this technique.     

Etude de la validite de la loi de Fick pour le transfer de matiere dans un fluide viscoelastique

An experimental study has been realized in a viscoelastic fluid to know if the Fick's law can be applied. This study also permitted to give some precisions about the differencies observed between the mass transfer measurements and the theorie results from the Ostwald's law.     

Studies of composites made by impregnation of porous bodies. 1. Sulphur impregnant in portland cement systems

Porous bodies formed by autoclaving portland cement-silica mixtures and by normally curing portland cement were characterized by measuring Young's modulus, microhardness and porosity. These bodies were impregnated almost completely with molten sulphur. The bodies were characterized again. It was found that the mechanical properties of the composites could be described by a form of Reuss' mixing law. Equations relating the improvement of the mechanical properties of the composite to the properties of the porous body were derived for both Young's modulus and microhardness.     

New technique for determination of diffusivities of volatile hydrocarbons in semi-solid bitumen

The diffusivities of volatile hydrocarbons in semi-solid Athabasca bitumen were determined by a technique which involves diffusion of hydrocarbon from a bitumen-hydrocarbon solution into a flowing stream of nitrogen. The diffusion method developed is new and can be used for the determination of the diffusivity of any volatile solute in any semi-solid or viscous liquid. For the hydrocarbon solutes tested, the diffusivity was found to decrease with increased molecular weight and the presence of branching and cyclic rings. The presence of branching and ring-shaped structures was more significant in reducing the diffusivity than increase in molecular weight. Results can be explained by Eyring's ‘hole’ theory of diffusion.     

A Two‐dimensional Network Study on Oxygen Transport in Porous Gas Diffusion Layer

Oxygen transport in the porous gas diffusion layer (GDL), which is generally characterised by the oxygen effective diffusivity, is of great importance for the performance of proton exchange membrane fuel cells (PEMFCs). The determination of the oxygen effective diffusivity is challenging due to the complex structure of the porous GDL samples. In the present study, a two‐dimensional network consisting of arms and nodes is adopted to illustrate how oxygen effective diffusivity is affected by the GDL structure under the condition with/without water invasion. Water permeation in the network is simulated using the invasion percolation algorithm and oxygen transport in the arms is described by Fick's law. The simulation results reveal that oxygen effective diffusivity under dry condition decreases with increase in the network heterogeneity. With water permeation, the oxygen effective diffusivity goes to zero even though water saturation is rather less than unity. The critical water saturation, above which the oxygen effective diffusivity becomes zero, is found to decrease with increasing heterogeneity. To enhance oxygen transport, four different modified networks are introduced in the present study. It is found that the network with large arms in oxygen transport direction has the best oxygen and water transport properties.     

Drying characteristics of particles using thermogravimetric analyzer

The drying characteristics (critical moisture content, equilibrium moisture content, constant drying rate and effective diffusivity) of various particles (gypsum, millet, polyvinyl chloride and silica gel) using a thermogravimetric analyzer were measured. The experiments were performed in the gas temperature range of 40 to 100 °C under isothermal conditions. The drying rate curve was mainly dependent on the moisture content, particle size, gas temperature and the internal structure of the solid particles. The equilibrium moisture content of solid particles decreased with increasing gas temperature, while the critical moisture content and the effective dispersion coefficient increased with increasing gas temperature. The effective diffusion coefficients of both PVC and gypsum particles were not able to be determined by Fick’s law since the experimental data were not well matched with the model predicted values, which consisted of the drying time in the falling-rate drying period.     

Electrochemical modeling of intercalation processes with phase field models

In this paper, we apply phase field models to move beyond Fick’s law in describing Li diffusion in secondary battery electrodes. Phase field models are potentially more accurate and allow simpler tracking of phase boundaries than Fick’s equation. The phase field models are implemented using the highly accurate but fast Chebyshev-spectral method. Using the phase field we investigate to what extent non-Fickian behavior can affect results from experimental techniques for measuring diffusion coefficients, such as Galvanostatic Intermittent Titration Technique (GITT) and Potentiostatic Intermittent Titration Technique (PITT). We show that GITT and PITT can still accurately measure the diffusion coefficient in systems described by phase field models even when significant gradient energy terms are present.     

Knudsen diffusion in powders. Part I Critical examination of a gas diffusion relationship used in Knudsen flow permeametry

B. V. Derjaguin's treatment of gas diffusion in a porous medium under conditions of Knudsen flow is discussed. A semi-theoretical value obtained for the constant β in the general equation is tested experimentally. In opposition to previous conclusions, molecular diffusion seems best interpreted in terms of elastic collisions against the pore walls. The structure of the pore system is shown to be the controlling parameter of the gas flow, and the novelty of the present treatment consists in ways to take the pore structure into account.     

Diffusion of fine particles in the neighborhood of obstacles

The major mode of transport of fine particles is the so-called Brownian motion, which manifests itself as “diffusion” in the macroscopic sense. When such diffusion occurs in the neighborhood of other objects, the increase in the drag on the diffusing particle affects the transport significantly. We present here a fundamental analysis of this phenomenon starting from the dynamics of the motion of the Brownian particle. The resulting stochastic equation of motion is transformed into a deterministic partial differential equation for the concentration of the suspended particles. This analysis offers for the first time a generalization of Fick's laws of diffusion for the colloidal particle and in that process provides an interesting application of a modern area of mathematical research.     

Formation of metal composites by displacement under bubbling conditions

The present paper is aimed at studying the kinetics of copper powder deposition process onto iron—carbon alloy substrates by a ‘cementation’ type of reaction. The variables included copper sulphate concentration, reaction time, temperature, pH, rate of helium gas bubbling and the surface area of the test specimen. The properties of the composite powders were also investigated as well as the surface topography of the deposited copper powders. The properties embodied fluidity, bulk density, specific surface area, porosity and packability. The composite material — after compaction and sintering — was evaluated by both tensile and ‘Brinell’ hardness tests.It was concluded that copper deposition by cementation is controlled by the transport of the reacting species from the bulk of the solution to the test specimen. This was explained in terms of Fick's law for the rate of mass transfer by diffusion in the boundary liquid film surrounding the solid.     

Tracer diffusion of crown ethers in chlorobenzene from 301 to 373 K

Tracer diffusivities are reported for s-trioxane, 12-crown-4, 15-crown-5, and 18-crown-6 in chlorobenzene from 301 to 373 K. The results are discussed in terms of Arrhenius equation for diffusion, Stokes' law, and Hildebrand's free-volume model. An extension of Dymond's hard sphere self-diffusion equation to tracer diffusion has been found to successfully correlate the data reported here.     

PARAMETER ESTIMATION OF FICK'S LAW DRYING EQUATION

ABSTRACT

Distributed parameter drying models such as the Fick's law diffusion model, unlike the lumped parameter model of van Meel whose parameters can be easily estimated by regression, suffer from the difficulty in estimating the parameters of the models quantitatively with accuracy. In the past they were estimated by visual inspection of the theoretical drying curves which fit the experimental drying curve best In this work, a quantitative parameter estimation technique originally suggested by Chavent, is developed by minimizing the integrated squares of error between theoretical and experimental curves over the drying lime (the criterion) subjected to the constraints that the theoretical curve is governed by the constant diffusivity Fick's taw diffusion equation (the constraint). Although the estimation of Fick's law constant diffusivity can be done by using the analytical solution developed by Crank, the use of the Fick's law model here is simply to demonstrate the utility of the proposed technique which can be used in more complex distributed models. The optimization problem is to solve for the adjoint equation for which the value of the Fick's law diffusivity minimizes the criterion. The Lagrangian derivative is solved by using a discrete derivative of the criterion. The theoretical curves are generated by using simple explicit (FSE) and modified Crank-Nicholson (FCR) algorithms The drying of oil palm kernels are used as a case study. Ii is found that the estimated diffusivities of moisture in oil palm kernels range from 0 5 to 5.0 × 10^-10 m²sol;s which are comparable with published data. It is also found that the estimated diffusivity is dependent on the initial moisture content.     

Polymethyl methacrylate plus water: sorption kinetics and volumetric changes

Sheets of polymethyl methacrylate were weighed periodically in water and, after surface mopping, in air. A case was made for dual mode sorption kinetics with 40–60% of the water accommodated in microvoids and the remainder taken up by swelling. The kinetics of sorption departed from Fick's laws in ways which appeared to be mutually inconsistent. For example, the value of the diffusion coefficient appeared to increase not only in the course of sorption but also in desorption. This inconsistency was interpreted as an artifact due to a retarded swelling component of dual mode kinetics.     

分形多孔介质中气体的非稳态扩散

通过引入平均修正系数修正Fick第二扩散定律得到了描述分形多孔介质气体非稳态扩散的理论方程。基于“塞状流”扩散实验法建立了分形多孔介质非稳态扩散实验系统,对3个孔隙结构不同的分形多孔介质样品进行了非稳态扩散实验,通过实验对理论方程进行验证和修正。结果表明,Fick第二扩散定律不适用于分形多孔介质中气体非稳态扩散,分形多孔介质中气体非稳态扩散存在一定规律,且多孔介质孔隙结构不同其扩散规律不同。     

外场作用下多组分质量传递Maxwell-Stefan方程及其应用：电场作用下球形颗粒中的扩散

Fick扩散定律只能用于没有外场作用下的二组分扩散过程,Maxwell-Stefan（MS）方程适用于外场作用下多组分系统中的质量传递过程。阐述了MS方程的物理意义,并由此将外场作用下普遍化MS方程的表达式改写为清晰且易于使用的形式。给出了在MS方程中常用的外场作用力的数学形式。Fick扩散定律是MS方程在没有外场作用下对二元理想流体扩散的应用;MS方程对离心分离和电解质扩散等过程的描述比传统的描述方法更全面。利用MS方程导出了欧姆定律,利用该结论可以通过测量电导率计算带电粒子的扩散系数。利用MS方程研究了电场作用下球形颗粒中的扩散过程,结果表明,该方程可以对电场强化质量传递过程给出有效描述。MS方程为外场作用下的质量传递过程提供了一种有效的理论研究方法。     

Importance of sample intraparticle diffusivity in investigations of the mass transfer mechanism in liquid chromatography

The effective diffusivity of a nonretained (thiourea) and of a strongly retained (phenol) compounds were measured with the peak parking method in two different columns (both 150 × 4.6 mm) packed with two types of porous particles having different mesopore sizes [5 μm Jupiter‐C₁₈, 320 Å and Luna(2)‐C₁₈, 100 Å]. The eluent was a methanol–water mixture (10/90 v/v) and the temperature 294 K. The effective diffusivity data acquired were used to determine the intraparticle diffusivity, D_p, based on two different diffusion models. The first one assumes that the diffusion fluxes across the particles and in the interparticle volume are additive (parallel diffusion model). The second model was rigorously derived on the basis of the effective medium theory of diffusion (diffusion model) in a binary composite medium (particles + interparticle volume). In both models, it was assumed that the rate of equilibrium between the liquid and the solid phases was infinitely faster than the rate of axial diffusion along the column at zero flow rate. Both models provide physically meaningful intraparticle diffusivity coefficients that take into account the average mesopore size of the particles, their specific surface area, and the retention factor of the analyte. Although the actual effective intraparticle diffusivity remains unknown, these result confirm that the mass transfer resistance due to diffusion through the porous particles has almost negligible effects in reversed phase liquid chromatography due to the importance of surface diffusion. Combining the results of the peak parking method with the h data measured at high linear velocities allows the unambiguous measurement of the film mass transfer and the surface diffusion coefficients. © 2010 American Institute of Chemical Engineers AIChE J, 2011     

PARAMETER ESTIMATION OF FICK'S LAW DRYING EQUATION

Distributed parameter drying models such as the Fick's law diffusion model, unlike the lumped parameter model of van Meel whose parameters can be easily estimated by regression, suffer from the difficulty in estimating the parameters of the models quantitatively with accuracy. In the past they were estimated by visual inspection of the theoretical drying curves which fit the experimental drying curve best In this work, a quantitative parameter estimation technique originally suggested by Chavent, is developed by minimizing the integrated squares of error between theoretical and experimental curves over the drying lime (the criterion) subjected to the constraints that the theoretical curve is governed by the constant diffusivity Fick's taw diffusion equation (the constraint). Although the estimation of Fick's law constant diffusivity can be done by using the analytical solution developed by Crank, the use of the Fick's law model here is simply to demonstrate the utility of the proposed technique which can be used in more complex distributed models. The optimization problem is to solve for the adjoint equation for which the value of the Fick's law diffusivity minimizes the criterion. The Lagrangian derivative is solved by using a discrete derivative of the criterion. The theoretical curves are generated by using simple explicit (FSE) and modified Crank-Nicholson (FCR) algorithms The drying of oil palm kernels are used as a case study. Ii is found that the estimated diffusivities of moisture in oil palm kernels range from 0 5 to 5.0 × 10^-10 m²sol;s which are comparable with published data. It is also found that the estimated diffusivity is dependent on the initial moisture content.     

General solution of hydraulic diffusivity from sorptivity test

A general approach to solving hydraulic diffusivity from sorptivity test is established and verified in this paper. The diffusion equation governing capillary water absorption is first converted into normalized ordinary differential and integral forms via Boltzmann transformation, which are then directly solved by the method of weighted residuals. By this method, the approximate solution of Boltzmann variable is determined for any distribution law of diffusivity. The relationship between sorptivity and diffusivity is further analytically established. It's found that initial diffusivity is proportional to square of the ratio of sorptivity to the water content difference between saturated and initial states. Ignoring the water vapor diffusion leads to the underestimation of derived water content profile and diffusivity. The Boltzmann variable and diffusivity calculated by the proposed method are verified by experimental data. Finally, the relationship between coefficients of general solution for diffusion equation and shape parameter for exponential diffusivity is also derived.