Fickian diffusion coefficients gases

In equation IV. 6, Dik represents the ordinary diffusion coefficient for binary interactions, and a, is the thermal diffusion ratio. The reactants are often present in small amounts (<1%) relative to the carrier gas thus, the multicomponent diffusion expression (equation IV. 6) may be replaced by a simple Fickian diffusion expression that includes thermodiffusion... 

The Fickian diffusion models with constant effective diffusivities presented earlier and the rigorous dusty gas model presented in this section are not the only alternatives for modelling diffusion and reaction in porous catalyst pellets. Fickian models with effective diffusion coefficients which are varying with the change of concentration of the gas mixture can also be used. This is certainty more accurate compared with the Fickian model with constant diffusivities although of course less accurate than the dusty gas model. The main problem with these models is the development of relations for the change of diffusivities with the concentration of the gas mixture without solving the dusty gas model equations. Two such techniques are presented in this section and their results are compared with dusty gas model results. 

Chapter 3 dealt with the problem of the reaction kinetics for different gas-solid reactions, while chapter 5 dealt with the mass and heat transfer problems for porous as well as non-porous catalyst pellets. In chapter 5 different degrees of complexities and rigor were used. In chapter 5, the analysis started with the simplest case of non-porous catalyst pellets where the only mass and heat transfer Coefficients are those at the external surface which depend mainly on the flow conditions around the catalyst pellet and the properties of the reaction mixture. It was shown clearly that j-factor correlations are adequate for the estimation of the external mass and heat transfer coefficients (k, h) associated with these resistances. For the porous catalyst pellets different models with different degrees of rigor have been used, starting from the simplest case of Fickian diffusion with constant diffusivity, to the rigorous dusty gas model based on the Stefan-Maxwell equations for multicomp>onent diffusion. 

The species-B balance equation includes advective transport, Fickian diffusion, and depletion by chemical reaction. The binary diffusion coefficient D represents downstream diffusion of reactant species-B relative to upstream diffusion of product species-C. The expression for Ys, the surface mass fraction of B (gas side), is obtained from a species balance at the surface on B which includes advective transport of pure B to the interface on the condensed phase side and both advective and diffusive transport of B away from the surface on the gas side. The downstream condition K(oo)=0 represents the assumption of complete conversion... 

The problem of gas diffusion in, and permeation through, inhomogeneous polymers is more complex, but has been considered by a number of investigators [3,5,6,10]. The diffusion coefficient is then also a function of position. When the polymer is highly plasticized (i.e., swelled) by the penetrant, the diffusion coefficient may also become a function of time and of sample history. Such non-Fickian diffusion has also been smdied [3,5,11-16]. 

In addition to molecular difiusion, Knudsen diffusion can also be a significant transport mechanism in the p)ore space of SOFC electrodes. In Knudsen diffusion, the interactions of gas molecules with the pore walls are of the same frequency as the interactions between gas molecules. Knudsen diffusion is typically formulated as Fickian diffusion [Eq. (26.2)], with the Knudsen diffusion coefficient being used in place of the binary diffusion coefiicient. The Knudsen diffusion coefficient of a species is independent of the other species in the system and is derived from the molecular motion of the gas molecules and the geometry of the pores [8, 11, 12). Owing to the small average pore radii of SOFCs ( 10 m [13-15]), diffusion in the pore space of the electrodes usually falls within a transition region where both molecular and Knudsen diffusion are important [16]. To model the transition region. Pick s law can be used with an effective diffusion coefficient to account for... 

Sousa et al [5.76, 5.77] modeled a CMR utilizing a dense catalytic polymeric membrane for an equilibrium limited elementary gas phase reaction of the type ttaA +abB acC +adD. The model considers well-stirred retentate and permeate sides, isothermal operation, Fickian transport across the membrane with constant diffusivities, and a linear sorption equilibrium between the bulk and membrane phases. The conversion enhancement over the thermodynamic equilibrium value corresponding to equimolar feed conditions is studied for three different cases An > 0, An = 0, and An < 0, where An = (ac + ad) -(aa + ab). Souza et al [5.76, 5.77] conclude that the conversion can be significantly enhanced, when the diffusion coefficients of the products are higher than those of the reactants and/or the sorption coefficients are lower, the degree of enhancement affected strongly by An and the Thiele modulus. They report that performance of a dense polymeric membrane CMR depends on both the sorption and diffusion coefficients but in a different way, so the study of such a reactor should not be based on overall component permeabilities. 

Studies on the sorption of some hydrocarbons have shown that above the transition temperature of EBBA (331 K) the isotherms obey Henry s law and the solubility coefficients S can be calculated. The sorption and desorption curves are similar in shape which indicates that these systems follow Fickian sorption. This fact indicates that steady state surface equilibrium is reached and that the diffusion coefficient for hydrocarbons is a function of concentration only. It follows that the membranes containing 60 wt.% of EBBA are homogeneous from the view point of gas permeation at the temperature above transition in EBBA. The permeability coefficients P show a distinct jump in the vicinity of transition temperature from crystal to nematic phase. This phenomenon was observed for hydrocarbon gases, noble gases like He, and for inert gases like N2. 

Numerical solutions were applied to the dual-mode sorption and transport model for gas permeation, sorption, and desorption rate curves allowing for mobility of the Langmuir component. Satisfactory agreement is obtained between integral diffusion coefficient from sorption and desorption rate curves and apparent diffusion coefficient from permeation rate curves (time-lag method). These rate curves were also compared to the curves predicted by Fickian-type diffusion equations. 

To determine the diffusion coefficient of a dissolved gas (e.g., of gas A in liquid B), the weight versus time curve during absorption is analyzed by comparison with modeled curves calculated on the basis of the second Fickian law. According to Crank (2003), the diffusion of a dissolved gas A in a solvent B for a one-dimensional diffusion in a slab with height L and constant concentration at the surface of the slab is described by ... 

In this section the analogy between heat and mass transfer is introduced and used to solve problems. The specific estimation relationships for permeants in polymers are discussed in Section 4.2 with the emphasis placed on gas-polymer systems. This section provides the necessary formulas for a first approximation of the diffusivity, solubility, and permeability, and their dependence on temperature. Non-Fickian transport, which is frequently present in high activity permeants in glassy polymers, is introduced in Section 4.3. Convective mass transfer coefficients are discussed in Section 4.4, and the analogies between mass and heat transfer are used to solve problems involving convective mass transfer. Finally, in Section 4.5 the solution to Design Problem III is presented. 

Some of these studies focused on the analysis of equilibrium-limited reactions, namely those in which the conditions of the respective conversion could be enhanced relatively to the value obtained in a conventional reactor, the so-called thermodynamic equilibrium conversion.i i The developed models considered generic equilibrium-limited reactions carried on in membrane reactors with perfectly mixed or plug-flow pattems. In all these studies, the main assumptions considered consisted in isothermal and steady-state operation, Fickian transport across a non-porous membrane with a homogeneously distributed nanosized catalyst with constant diffusion coefficients, Henry s law for describing the equilibrium condition at the interfaces membrane/gas, and equality of local concentrations at the interface polymer phase/catalyst surface. 

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