Porous solids Knudsen diffusion

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. 

In connection with multiphase diffusion another poorly understood topic should be mentioned—namely, the diffusion through porous media. This topic is of importance in connection with the drying of solids, the diffusion in catalyst pellets, and the recovery of petroleum. It is quite common to use Fick s laws to describe diffusion through porous media fJ14). However, the mass transfer is possibly taking place partly by gaseous diffusion and partially by liquid-phase diffusion along the surface of the capillary tubes if the pores are sufficiently small, Knudsen gas flow may prevail (W7, Bl). 

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. 

Many heterogeneous reactions give rise to an increase or decrease in the total number of moles present in the porous solid due to the reaction stoichiometry. In such cases there will be a pressure difference between the interior and exterior of the particle and forced flow occurs. When the mean free path of the reacting molecules is large compared with the pore diameter, forced flow is indistinguishable from Knudsen flow and is not affected by pressure differentials. When, however, the mean free path is small compared with the pore diameter and a pressure difference exists across the pore, forced flow (Poiseuille flow see Volume 1, Chapter 3) resulting from this pressure difference will be superimposed on molecular flow. The diffusion coefficient Dp for forced flow depends on the square of the pore radius and on the total pressure difference AP ... 

Except in the case of reactions at high pressure, the pressure drop which must be maintained to cause flow through a packed bed of particles is usually insufficient to produce forced flow in the capillaries of the solid, and the gas flow is diverted around the exterior periphery of the pellets. Reactants then reach the interior of the porous solid by Knudsen or molecular diffusion. 

When the catalyst is a porous solid, most of the surface area of the catalyst is the surface area of the inner surface of the pores. Therefore, most of the reaction proceeds in the pore. Gas molecules are transferred to the outer surface of the catalyst by diffusion. Generally speaking, the diffusion is faster than the diffusion inside the pores. Gas molecules collide with the inner wall of the pore before they collide with another molecule for the porous catalyst having an average pore radius rp of a few nm. Such diffusion is called Knudsen diffusion and its diffusion constant D is given by ... 

There are four well-known types of diffusion in solids [10] gaseous or molecular diffusion [75], Knudsen diffusion [76-80], liquid diffusion [10], and atomic diffusion. In Figure 5.27, the possible transport mechanisms in porous media are schematically shown [77], Gaseous flow (Figure 5.27a)... 

In equations (5) and (6), DM and DK are the molecular and Knudsen diffusivities, respectively, and e and x are the void fraction and the tortuosity of the porous solid, respectively. For pore dimensions significantly larger than the mean free path of the diffusant in the gas phase, the diffusivity is governed by molecular diffusion, but when the pore diameter becomes smaller than the mean free path, diffusion is properly described by Knudsen diffusion. When the pore diameter approaches that of the diffusing species, around 10, one enters the configurational regime. 

Hindered diffusion, the primary transport mechanism in porous solids, can be qualitatively described as a series of hops by the analyte, via gas-phase diffusion, from one surface site to the next. Thus, hindered diffusion is composed of two main components a pure diffusion-related term, often Fickian in nature, associated with movement of the analyte in the gas phase and a term describing the noninstantaneous equilibration between gas-phase analyte and the solid surface at each point where the analyte touches down (adsorbs). In extended porous solids (e.g., a chromatographic column tightly packed with porous beads), transport is often more complex, requiring the consideration of such factors as eddy diffusion and Knudsen effusion. This is important if there is a significant pressure drop along the path of the analyte [109]. Finally, the presence of any external fields (thermal, electric, etc.) must be considered as well. 

All these different mechanisms of mass transport through a porous medium can be studied experimentally and theoretically through classical models (Darcy s law, Knudsen diffusion, molecular dynamics, Stefan-Maxwell equations, dusty-gas model etc.) which can be coupled or not with the interactions or even reactions between the solid structure and the fluid elements. Another method for the analysis of the species motion inside a porous structure can be based on the observation that the motion occurs as a result of two or more elementary evolutions that are randomly connected. This is the stochastic way for the analysis of species motion inside a porous body. Some examples that will be analysed here by the stochastic method are the result of the particularisations of the cases presented with the development of stochastic models in Sections 4.4 and 4.5. 

Many reactions taking place within catalyst or absorbent pellets in industrial plants are diffusion-limited. Under the typical operating conditions for many absorbents, diffusion of gases into the porous solid occurs in the Knudsen regime. In such circumstances the rate of gas pick-up of these materials is strongly dependent on the pore structure. The pore structure for absorbent pellets that will deliver the most efficient operation of an absorbent bed requires a pervasive system of macropores which provide rapid transport of the gas flux into the centre of the pellet. A network of ramified mesopores branching off the macropores then provides extensive surface area for absorption of gas molecules. Therefore, when manufacturing an absorbent it is necessary to be able to determine the spatial distribution of the macropore network in a product to ensure that the pore structure is the most appropriate for the peirticular duty for which it is intended. 

The inorganic membranes had until the late nineties received fairly little attention for applications in gas separation. This has mainly been due to their porous stmcmre, and therefore lack of ability to separate gas molecules. Within the group of inorganic membranes there are however the dense metallic membranes and the solid oxide electrolytes these are discussed separately in Section 4.3.5. With reference to Section 4.2, the possible transport mechanisms taking place in a porous membrane may be summarized as in Table 4.4 below, as well as the ability to separate gases (+) or not (—). Recent findings [29] have however documented that activated Knudsen diffusion may take place also in smaller pores than indicated in the table. 

Obviously, if one of the molecular or Knudsen diffusion coefficients is vastly greater than the other it may be ignored. Several formulae have been given for the transition between the two, but perhaps the simplest of them derives from thinking of the reciprocal of the diffusion coefficient as a resistance and the two modes of diffusion as being in parallel. Then, allowing for the area and tortuosity, the effective diffusion coefficient in the porous solid can be taken to be... 

In many problems of mass transfer in a solid porous medium with a large specific surface area (as with catalysts), with or without a chemical reaction, the solutes are considered to be carried only by diffusion (molecular, superficial or Knudsen diffusion), the molecular barycentric velocity being... 

Calculate effective molecular and Knudsen diffusion coefficients in porous solids. 

Calculate fluxes through porous solids when both molecular and Knudsen diffusion are important. 

For binary mixtures diffusing inside porous solids, the applicability of equation (1-100) depends upon the value of a dimensionless ratio called the Knudsen number, Kn, defined as... 

For Knudsen diffusion in porous solids of porosity e and tortuosity t,... 

Example 1.22 Combined Molecular and Knudsen Diffusion in a Porous Solid... 

When the gas is a mixture, the hydrodynamic flux given by equation (1 -112) or (1 -113) is the flux of the mixture that is, the mixture moves as a whole under the total pressure gradient. All species move at the same speed (i. e., no separation is achieved), and the individual flux of each component is the product of its mole fraction times the total flux. In that case, the viscosity in equations (1-112) and (1-113) is the viscosity of the mixture. If the gas is a mixture with different concentrations and different total pressure on either side of the porous solid, the flow may be a combination of hydro-dynamic, Knudsen, and diffusive. 

The rest of the book is dedicated to adsorption kinetics. We start with the detailed description of diffusion and adsorption in porous solids, and this is done in Chapter 7. Various simple devices used to measure diffusivity are presented, and the various modes of transport of molecules in porous media are described. The simplest transport is the Knudsen flow, where the transport is dictated by the collision between molecules and surfaces of the pore wall. Other transports are viscous flow, continuum diffusion and surface diffusion. The combination of these transports is possible for a given system, and this chapter will address this in some detail. 

The geometry of the plate does not bear much relevance to real porous solid. Thus, the above equation is not of much use in diffusion of adsorption systems. To this end, we will consider the Knudsen diffusion through a long cylindrical capillary in the next section. 

This simple equation for pure gas gives us a useful tool to study the structure of a porous solid. Making use of information such as the Knudsen diffusivity is proportional to square root of temperature and inversely proportional to the molecular weight, we can carry out experiments with different gases having different molecular weights and at different temperatures to determine the value for the tortuosity in the Knudsen relation and the viscous parameter Bq. 

In actual diffusion in porous solids the pores are not straight and cylindrical but are irregular. Hence, the equations for diffusion in pores must be modified somewhat for actual porous solids. The problem is further complicated by the fact that the pore diameters vary and the Knudsen diffusivity is a function of pore diameter. 

Gas diffusion in porous solid. In this type a gas phase is present on both sides of the membrane, which is a microporous solid. The rates of molecular diffusion of the various gas molecules depend on the pore sizes and the molecular weights. This type of diffusion in the molecular, transition, and Knudsen regions was discussed in detail in Section 7.6. 

Diffusion in a porous solid (a) straight-through pores (b) tortuous pores with branching (c) molecular diffusion (d) Knudsen diffusion. 

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