Porous solids effective diffusivity

R S Cunningham, C J Geankoplis, Effects of Different Structures of Porous Solids on Diffusion of Gases in the Transition Region, Ind Eng Chem. Fund. 7.535 (1968). 

Drff Effective diffusivity witbin a porous solid = pDIZ mvs ft /h... 

Diffusion within the largest cavities of a porous medium is assumed to be similar to ordinary or bulk diffusion except that it is hindered by the pore walls (see Eq. 5-236). The tortuosity T that expresses this hindrance has been estimated from geometric arguments. Unfortunately, measured values are often an order of magnitude greater than those estimates. Thus, the effective diffusivity D f (and hence t) is normally determined by comparing a diffusion model to experimental measurements. The normal range of tortuosities for sihca gel, alumina, and other porous solids is 2 < T < 6, but for activated carbon, 5 < T < 65. 

The ratio of the overall rate of reaction to that which would be achieved in the absence of a mass transfer resistance is referred to as the effectiveness factor rj. SCOTT and Dullion(29) describe an apparatus incorporating a diffusion cell in which the effective diffusivity De of a gas in a porous medium may be measured. This approach allows for the combined effects of molecular and Knudsen diffusion, and takes into account the effect of the complex structure of the porous solid, and the influence of tortuosity which affects the path length to be traversed by the molecules. 

Porous solid as pseudo-homogeneous medium 635 effective diffusivity 635 Portable mixers 306 Positive displacement pumps. 315... 

Many investigators have studied diffusion in systems composed of a stationary porous solid phase and a continuous fluid phase in which the solute diffuses. The effective transport coefficients in porous media have often been estimated using the following expression ... 

The effective diffusivity depends on the statistical distribution of the pore transport coefficients W j. The derivation shows that the semi-empirical volume-averaging method can only be regarded as an approximation to a more complex dynamic behavior which depends non-locally on the history of the system. Under certain circumstances the long-time (t —> oo) diffusivity will not depend on t (for further details, see [191]). In such a case, the usual Pick diffusion scenario applies. The derivation presented above can, with minor revisions, be applied to the problem of flow in porous media. When considering the heat conduction problem, however, some new aspects have to be taken into accoimt, as heat is transported not only inside the pore space, but also inside the solid phase. 

One must understand the physical mechanisms by which mass transfer takes place in catalyst pores to comprehend the development of mathematical models that can be used in engineering design calculations to estimate what fraction of the catalyst surface is effective in promoting reaction. There are several factors that complicate efforts to analyze mass transfer within such systems. They include the facts that (1) the pore geometry is extremely complex, and not subject to realistic modeling in terms of a small number of parameters, and that (2) different molecular phenomena are responsible for the mass transfer. Consequently, it is often useful to characterize the mass transfer process in terms of an effective diffusivity, i.e., a transport coefficient that pertains to a porous material in which the calculations are based on total area (void plus solid) normal to the direction of transport. For example, in a spherical catalyst pellet, the appropriate area to use in characterizing diffusion in the radial direction is 47ir2. 

If it is assumed that the gaseous reactant is consumed by reaction with the porous solid at a rate i A and that transport of reactant within the solid is by a diffusive mechanism characterised by an effective diffusivity Dg, then the conservation equation for a reactant A in an isothermal... 

Use of the proper diffusion coefficient Replace the molecular diffusion coefficient by the effective diffusion coefficient of fluid in the porous structure. Representative values for gases and liquids in porous solids are given by Weisz (1959). 

In this paper only phenomena occurring on level (iii) will be discussed. In general, if sorption kinetics on porous solid particles are considered, one is dealing with complex physical phenomena which are often superimposed by external, e.g. apparatus effects. In order to obtain the correct diffusion data for the intracrystalline MS void volume, one should start therefore, from... 

In real porous solids, the pores are not straight, and the pore radius can vary. Two parameters are used to describe the diffusion path through real porous solids the void fraction, e, defined as fhe ratio of pore area to fofal cross-sectional area, and the tortuosity, r, which corrects for the fact that pores are not straight. The resulting effective diffusivity is then... 

The overall reaction rate may be affected by diffusion if the mass transfer from the bulk fluid to the coal surface and/or the pore diffusion steps are relatively slow. I have made tentative estimates of both these diffusion effects. The following equation can be written (2) for the rate of oxygen uptake of a porous solid such as coal, assuming the oxygen to be consumed in an irreversible first-order reaction. 

The effects of physical transport processes on the overall adsorption on porous solids are discussed. Quantitative models are presented by which these effects can be taken into account in designing adsorption equipment or in interpreting observed data. Intraparticle processes are often of major importance in adsorption kinetics, particularly for liquid systems. The diffusivities which describe intraparticle transfer are complex, even for gaseous adsorbates. More than a single rate coefficient is commonly necessary to represent correctly the mass transfer in the interior of the adsorbent. 

Finally, intraparticle diffusion appears to be an important factor in adsorption kinetics for many types of systems. In the past it has been customary to define such mass transfer quantitatively in terms of an effective diffusivity. However, even in gas-solid systems more than one process can be involved for porous particles. Thus, two-dimensional migration on the pore surface, surface diffusion, is a potential contribution. Liquid systems appear to be more complex, and, with electrolytes, it has been shown that the electric potential induced by counter-diffusing ions should be taken into account. A realistic description of intraparticle mass transfer in such cases requires more than a single rate coefficient for a binary system. 

One should take into account the specific features of gas diffusion in porous solids when measuring effective diffusion coefficients in the pores of catalysts. The measurements are usually carried out with a flat membrane of the porous material. The membrane is washed on one side by one gas and on the other side by another gas, the pressure on both sides being kept... 

Both Knudsen and molecular diffusion can be described adequately for homogeneous media. However, a porous mass of solid usually contains pores of non-uniform cross-section which pursue a very tortuous path through the particle and which may intersect with many other pores. Thus the flux predicted by an equation for normal bulk diffusion (or for Knudsen diffusion) should be multiplied by a geometric factor which takes into account the tortuosity and the fact that the flow will be impeded by that fraction of the total pellet volume which is solid. It is therefore expedient to define an effective diffusivity De in such a way that the flux of material may be thought of as flowing through an equivalent homogeneous medium. We may then write ... 

In the region of Knudsen flow the effective diffusivity DeK for the porous solid may be computed in a similar way to the effective diffusivity in the region of molecular flow, i.e. Dk is simply multiplied by the geometric factor. 

The shrinking core reaction mode is not necessarily limited to non-porous unreacted solids. With a fast reaction in a porous solid, diffusion into the core and chemical reaction occur in parallel. The mechanism of the process is very similar to the mechanism of the catalysed gas-solid reactions where the Thiele modulus is large, the effectiveness factor is small and the reaction is confined to a thin zone (Section 3.3.1). This combination of reaction with core diffusion gives rise to a reaction zone which, although not infinitely thin but diffuse, nevertheless advances into the core at a steady rate. 

An important problem in catalysis is to predict diffusion and reaction rates in porous catalysts when the reaction rate can depend on concentration in a non-linear way.6 The heterogeneous system is modeled as a solid material with pores through which the reactants and products diffuse. We assume for diffusion that all the microscopic details of the porous medium are lumped together into the effective diffusion coefficient De for reactant. 

Unsteady state diffusion in monodisperse porous solids using a Wicke-Kallenbach cell have shown that non-equimolal diffusion fluxes can induce total pressure gradients which require a non-isobaric model to interpret the data. The values obtained from this analysis are then suitable for use in predicting effectiveness factors. There is evidence that adsorption of the non-tracer component can have a considerable influence on the diffusional flux of the tracer and hence on the estimation of the effective diffusion coefficient. For the simple porous structures used in these tests, it is shown that a consistent definition of the effective diffusion coefficient can be obtained which applies to both the steady and unsteady state and so can be used as a basis of examining the more complex bimodal pore size distributions found in many catalysts. 

Hore importantly, the response curves are noticeably affected where one or both of the components is adsorbable, even at low tracer concentrations. The interpretation of data is then much more complex and requires analysis using the non-isobaric model. Figs 7 and 8 show how adsorption of influences the fluxes observed for He (the tracer), despite the fact that it is the non-adsorbable component. The role played by the induced pressure gradient, in association with the concentration profiles, can be clearly seen. It is notable that the greatest sensitivity is exhibited for small values of the adsorption coefficient, which is often the case with many common porous solids used as catalyst supports. This suggests that routine determination of effective diffusion coefficients will require considerable checks for consistency and emphasizes the need for using the Wicke-Kallenbach cell in conjunction with permeability measurements. 

In eq 1 Dic is the effective diffusivity of species i in the reaction mixture which can be determined on the basis of various models of the diffusion process in porous solids. This aspect is discussed more fully in Section A.6.3. Difi is affected by the temperature and the pore structure of the catalyst, but it may also depend on the concentration of the reacting species (Stefan-Maxwell diffusion [9]). As Die is normally introduced on the basis of more or less empirical models, it may not be considered as a physical property, but rather as a model-dependent parameter. 

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