Knudsen diffusion model

The approach is similar to G/S hydrogenation, but here the pores are filled with a stagnant liquid. H2 molecules move by a pure diffusional process no Knudsen diffusion. Modelling remains basically the same as in G/S, with notorious differences ... 

The mass transfer in the boundary layers can be described by a mass transfer coefficient. In the membrane phase, the diffusion of water vapor can be described by either of the four mechanisms, namely molecular/Knudsen diffusion model, or Poiseuille flow, or by the dusty gas model (DGM). Heat transfer coefficients are used to describe the heat transfer in the boundary layer on either side of the membrane. In the membrane heat transfer occurs through the vapor and by conduction. These aspects have been explained in detail in the following sections. 

Holt et al. [16] measured water and gas flow through the pores of double-walled carbon nanotubes. These tubes had inner diameters less than 2 nm with nearly defect-free graphitic walls. Five hydrocarbon and eight non-hydrocarbon gases were tested to determine flow rates and to demonstrate molecular weight selectivity compared with helium. Water flow was pressure driven at 0.82 atm and measured by following the level of the meniscus in a feed tube. The results for both gas and liquid show dramatic enhancements over flux rates predicted with continuum flow models. Gas flow rates were between 16 and 120 times than expected according to the Knudsen diffusion model in which fluid molecule-wall collisions dominate the flow. 

As the pore size decreases, the Knudsen diffusion model may become important. This model describes diffusion when the gas molecules collide more often with the pore walls than with each other, and it is these molecule-pore wall interactions that start to dominate the rate of diffusion. Knudsen diffusion should be considered when pore diameters are typically less than 100 nm under atmospheric conditions, and is important in the catalyst layer and possibly in the GDL [21]. 

Fig. 8.3 Gas selectivity (indicated as permeability relative to He) data for sub 2 nm DWNT triangles) and MWNT circles) membranes. Open symbols denote nonhydrocarbon gases (H, He, Ne, N, O, Ar, CO, Xe) solid symbols denote hydrocarbon gases (CH, C H, C H, C,Hp C Hj). This solid line is a power-law fit of the nonhydrocarbon gas selectivity data, showing a scaling predicted by the Knudsen diffusion model (exponent of-0.49 0.01). The dashed line is power-law fit of hydrocarbon gas data, showing a deviation from the Knudsen model (exponent of -0.37 0.02). The insert shows the full mass range ofthe nonhydrocarbon gas data, again illustrating agreement with the Knudsen model scaling. (From [3])...

Fig. 8.6 Gas transport properties of CNT nanocomposite membrane. Gas transport properties of CNT/PS/PDMS membrane (triangle). CNTs/PS membranes (square), and Knudsen diffusion model (solid line), (a) Effeet of the pressure drop on the permeance of helium through CNTs/PS membrane, (b) Single-gas permeability as a funetion of the inverse square root of the molecular weight of the penetrant, (c) Single gas seleetivity with respect to He calculated from singe-gas permeability data, (d) Mixed-gas selectivity (CO /CH ) of CNTs/PS membrane. The composition of gas mixture was COjiCH =1 1. The feed pressure was 50 psi, and the pressure differential across the membrane was maintained by drawing a vaeuum on the permeate side. Operating temperature was maintained at 308 K. (From [8])...

Ruthven, D. M., Desisto, W. J. and Higgins, S. (2009) Diffusion in a mesoporous silica membrane validity of the Knudsen diffusion model. Chemical Engineering Science, 64,3201-3203. 

In general, tests have tended to concentrate attention on the ability of a flux model to interpolate through the intermediate pressure range between Knudsen diffusion control and bulk diffusion control. What is also important, but seldom known at present, is whether a model predicts a composition dependence consistent with experiment for the matrix elements in equation (10.2). In multicomponent mixtures an enormous amount of experimental work would be needed to investigate this thoroughly, but it should be possible to supplement a systematic investigation of a flux model applied to binary systems with some limited experiments on particular multicomponent mixtures, as in the work of Hesse and Koder, and Remick and Geankoplia. Interpretation of such tests would be simplest and most direct if they were to be carried out with only small differences in composition between the two sides of the porous medium. Diffusion would then occur in a system of essentially uniform composition, so that flux measurements would provide values for the matrix elements in (10.2) at well-defined compositions. 

The above estimates of pressure variations suggest that their magni-tude as a percentage of the absolute pressure may not be very large except near the limit of Knudsen diffusion. But in porous catalysts, as we have seen, the diffusion processes to be modeled often lie in the Intermediate range between Knudsen streaming and bulk diffusion control. It is therefore tempting to try to simplify the flux equations in such a way as to... 

In chapter 5 we showed that the dusty gas model flux relations could be solved (fjite easily at the limit of Knudsen diffusion control, when they reduce to the form given in equation (5.25), namely... 

Many theoretical embellishments have been made to the basic model of pore diffusion as presented here. Effectiveness factors have been derived for reaction orders other than first and for Hougen and Watson kinetics. These require a numerical solution of Equation (10.3). Shape and tortuosity factors have been introduced to treat pores that have geometries other than the idealized cylinders considered here. The Knudsen diffusivity or a combination of Knudsen and bulk diffusivities has been used for very small pores. While these studies have theoretical importance and may help explain some observations, they are not yet developed well enough for predictive use. Our knowledge of the internal structure of a porous catalyst is still rather rudimentary and imposes a basic limitation on theoretical predictions. We will give a brief account of Knudsen diffusion. 

Fig. 3.1.5 Temperature dependence of the coefficient of long-range self-diffusion of ethane measured by PFG NMR in a bed of crystallites of zeolite NaX (points) and comparison with the theoretical estimate (line). The theoretical estimate is based on the sketched models of the prevailing Knudsen diffusion...

Knozinger electrostatic model, 34 197 Knudsen diffusion, 34 73 KnuzTs law, 34 228... 

Reid, Sherwood and Prausnitz [11] provide a wide variety of models for calculation of molecular diffusion. Dr is the Knudsen diffusion coefficient. It has been given in several articles as 9700r(T/MW). Once we have both diffusion coefficients we can obtain an expression for the macro-pore diffusion coefficient 1/D = 1/Dk -i-1/Dm- We next obtain the pore diffusivity by inclusion of the tortuosity Dp = D/t, and finally the local molar flux J in the macro-pores is described by the famiUar relationship J = —e D dcjdz. Thus flux in the macro-pores of the adsorbent product is related to the term CpD/r. This last quantity may be thought of as the effective macro-pore diffusivity. The resistance to mass transfer that develops due to macropore diffusion has a length dependence of R]. 

As the pore size decreases, molecules collide more often with the pore walls than with each other. This movement, intermediated by these molecule—pore-wall interactions, is known as Knudsen diffusion. Some models have begun to take this form of diffusion into account. In this type of diffusion, the diffusion coefficient is a direct function of the pore radius. In the models, Knudsen diffusion and Stefan—Maxwell diffusion are treated as mass-transport resistances in seriesand are combined to yield... 

Molecular versus Knudsen diffusion and the Dusty-gas Model... 

In Equation (9.6), x is the direction of flux, nt [mol m-3 s 1 ] is the total molar density, X [1] is the mole fraction, Nd [mol m-2 s 1] is the mole flux due to molecular diffusion, D k [m2 s 1] is the effective Knudsen diffusion coefficient, D [m2 s 1] is the effective bimolecular diffusion coefficient (D = Aye/r), e is the porosity of the electrode, r is the tortuosity of the electrode, and J is the total number of gas species. Here, a subscript denotes the index value to a specific specie. The first term on the right of Equation (9.6) accounts for Knudsen diffusion, and the following term accounts for multicomponent bulk molecular diffusion. Further, to account for the porous media, along with induced convection, the Dusty Gas Model is required (Mason and Malinauskas, 1983 Warren, 1969). This model modifies Equation (9.6) as ... 

Mass transport inside the catalyst has been usually described by applying the Fick equation, by means of an effective diffusivity Deff a Based on properties of the interface and neglecting the composition effect, composite diffusivity of the multi-component gas mixture is calculated through the simplified Wilke model [13], The effect of pore-radius distribution on Knudsen diffusivity is taken into account. The effective diffusivity DeffA is given by... 

In order to predict correctly the fluxes of multicomponent mixtures in porous membranes, simplified models based solely on Fields law should be used with care [28]. Often, combinations of several mechanisms control the fluxes, and more sophisticated models are required. A well-known example is the Dusty Gas Model which takes into account contributions of molecular diffusion, Knudsen diffusion, and permeation [29]. This model describes the coupled fluxes of N gaseous components, Ji, as a function of the pressure and total pressure gradients with the following equation ... 

The free parameters of this model are the ratio of porosity to tortuosity, e/r, the binary molecular diffusivities, D-j, the Knudsen coefficient, Ko (which determines specific effective Knudsen diffusivities, D j) and the permeability constant, Bo. 

Studies with many types of porous media have shown that for the transport of a pure gas the Knudsen diffusion and viscous flow are additive (Present and DeBethune [52] and references therein). When more than one type of molecules is present at intermediate pressures there will also be momentum transfer from the light (fast) molecules to the heavy (slow) ones, which gives rise to non-selective mass transport. For the description of these combined mechanisms, sophisticated models have to be used for a proper description of mass transport, such as the model presented by Present and DeBethune or the Dusty Gas Model (DGM) [53], In the DGM the membrane is visualised as a collection of huge dust particles, held motionless in space. 

The constitutive equations of transport in porous media comprise both physical properties of components and pairs of components and simplifying assumptions about the geometrical characteristics of the porous medium. Two advanced effective-scale (i.e., space-averaged) models are commonly applied for description of combined bulk diffusion, Knudsen diffusion and permeation transport of multicomponent gas mixtures—Mean Transport-Pore Model (MTPM)—and Dusty Gas Model (DGM) cf. Mason and Malinauskas (1983), Schneider and Gelbin (1984), and Krishna and Wesseling (1997). The molar flux intensity of the z th component A) is the sum of the diffusion Nc- and permeation N contributions,... 

Coppens and Froment (1995a, b) employed a fractal pore model of supported catalyst and derived expressions for the pore tortuosity and accessible pore surface area. In the domain of mass transport limitation, the fractal catalyst is more active than a catalyst of smooth uniform pores having similar average properties. Because the Knudsen diffusivity increases with molecular size and decreases with molecular mass, the gas diffusivities of individual species in... 

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