Porous solids molecular 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. 

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). 

A variety of spin probe methods have also been used to study the morphological features of the nano-channels present within MCM 41, as well as dynamical aspects connected to molecular diffusion in the inner pores,186-188 EPR has been used to investigate the adsorption and interactions of nitroxide-labelled de-ndrimers within porous silica.181 This method allows one to investigate the effective porosity of a solid surface (as a host) which is determined by the accessibility of the host surface to an adsorbed guest molecule. Information on the adsorption and interaction of dendrimers with the porous surface arises from computer-aided analysis of the EPR spectra based on of the well-established procedure proposed by Schneider and Freed.189... 

The thermal movement of molecules often serves as a prototype of random motion. In fact, molecular diffusion is the result of the random walk of atoms and molecules through gaseous, liquid, solid, or mixed media. This section deals with molecular diffusion of organic substances in gases (particularly air) and in aqueous solutions. Diffusion in porous media (i.e., mixes of gases or liquids with solids) and in other media will be discussed in the following section. 

Compared to rivers and lakes, transport in porous media is generally slow, three-dimensional, and spatially variable due to heterogeneities in the medium. The velocity of transport differs by orders of magnitude among the phases of air, water, colloids, and solids. Due to the small size of the pores, transport is seldom turbulent. Molecular diffusion and dispersion along the flow are the main producers of randomness in the mass flux of chemical compounds. 

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. 

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. 

A good agreement is generally obtained between the models based on transport equations and the SDE for mass and heat molecular transport. However, as explained above, the SDE can only be applied when convective flow does not take place. This restrictive condition limits the application of SDE to the transport in a porous solid medium where there is no convective flow by a concentration gradient. The starting point for the transformation of a molecular transport equation into a SDE system is Eq. (4.108). Indeed, we can consider the absence of convective flow in a non-steady state one-directional transport, together with a diffusion coefficient depending on the concentration of the transported property... 

For pores smaller than 10 m, a molecular sieving effect can be present and the movement of one or more species inside the porous solid occurs due to the molecular interactions between the species and the network of the porous body here, for the description of species displacement, the theory of molecular dynamics is frequently used. The affinity between the network and the species is the force that controls the molecular motion at the same time, the affinity particularities, which appear when two or more species are in motion inside the porous structure, explain the separation capacity of those solids. We can use a diffusive characterisation of species motion inside a porous solid by using the notion of conformational diffusion. 

Molecular transport and tracer diffusion in porous solids are conveniently correlated in terms of diffusivities defined in accordance with Pick s first equation ... 

J. KSrger and D.M. Ruthven, Diffusion and Adsorption in Porous Solids, Ch. 5 in Molecular Sieves Science and Technology, H.G. Karge and J. Weitkamp eds., Wiley-VCH (2002). 

It has been demonstrated that the combined application of various NMR techniques for observing molecular rotations and migrations on different time scales can contribute to a deeper understanding of the elementary steps of molecular diffusion in zeolite catalysts. The NMR results (self-diffusion coefficients, anisotropic diffiisivities, jump lengths, and residence times) can be correlated with corresponding neutron scattering data and sorption kinetics as well as molecular dynamics calculations, thus giving a comprehensive picture of molecular motions in porous solids. 

In a somewhat similar paper, diffusion through a 2D porous solid modeled by a regular array of hard disks was evaluated [65] using non-equilibrium molecular dynamics. It was found that Pick s law is not obeyed in this system unless one takes different diffusion constants for different regions in the flow system. Other non-equilibrium molecular dynamics simulations of diffusion for gases within a membrane have been presented [66]. The membrane was modeled as a randomly... 

Of all the porous solids, diffusion in zeolites has certainly been studied most extensively, in part because there seemed to be an enormous difference between macroscopic and microscopic diffusion constants (from MD and from NMR). It is not practical to discuss all this work here, but references to other such molecular dynamics simulations are given in the papers of [69]. 

W. Dong and H. Luo, Fluid Diffusion through a Porous Solid Nonequilibrium Molecular Dynamics Simulation, Phys. Rev. E 52 (1995) 801-804. 

If substance B is a porous solid the molecular diffusion coefficient D has to be replaced by the effective diffusion coefficient DeB. This is smaller than the molecular diffusion coefficient because the movement of the molecules is impeded by the pores. It is common practice to define a diffusion resistance factor... 

The sources of band broadening of kinetic origin include molecular diffusion, eddy diffusion, mass transfer resistances, and the finite rate of the kinetics of ad-sorption/desorption. In turn, the mass transfer resistances can be sorted out into several different contributions. First, the film mass transfer resistance takes place at the interface separating the stream of mobile phase percolating through the column bed and the mobile phase stagnant inside the pores of the particles. Second, the internal mass transfer resistance controls the rate of mass transfer between this interface and the adsorbent surface. It is composed of two contributions, the pore diffusion, which is molecular diffusion taking place in the tortuous, constricted network of pores, and surface diffusion, which takes place in the electric field at the liquid-solid interface [60]. All these mass transfer resistances, except the kinetics of adsorption-desorption, depend on the molecular diffusivity. Thus, it is important to study diffusion in bulk liquids and in porous media. 

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... 

The acdvitities of silica-supported phosphonium are support pore size dependent with ca. 100 A typically giving the most active catalysts. This is very similar to more simple physisorbed silica-based supported reagents and seems to support the view that for liquid-phase reactions catalysed by porous solids, a reasonably large pore is required to give a good molecular diffusion rate. 

The effective diifusivity measurement of gases by tracer-pulse chromatography in porous solids has been extended to include zeolites [faujasites, mordenites, 3A and 5A molecular sieves (35)]. The measured diffusions in this case were a strong function of molecular size. 

Calculate effective molecular and Knudsen diffusion coefficients in porous solids. 

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

Steady-State Binary Molecular Diffusion in Porous Solids... 

When treating diffusion of solutes in porous materials where diffusion is considered to occur only in the fluid inside the pores, it is common to refer to an effective diffusivity, DABeg, which is based on (1) the total cross-sectional area of the porous solid rather than the cross-sectional area of the pore and (2) on a straight path, rather than the actual pore path, which is usually quite tortuous. In a binary system, if pore diffusion occurs only by ordinary molecular diffusion, Fick s law can be used with an effective diffusivity that can be expressed in terms of the ordinary diffusion coefficient, DAB, as... 

Example 1.20 Steady-State Molecular Diffusion in Porous Solid... 

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

Figure 4.18 Tortuosity as a function of porosity for randomly oriented porous media. For diffusion in porous materials, the length of the diffusional path is increased. If the pores are randomly oriented, and large enough to permit random molecular trajectories, the tortuosity is a function of total porosity. Equation 4-46 (dashed line). The tortuosity predicted for diffusion around a lattice of sparsely populated spheres is obtained from Equation 4-43 assuming completely impermeable spheres (heavy dashed line). Tortuosities for ensembles of cuboidal cells are also included (triangles). The solid lines without symbols indicate tortuosity for a Bethe lattice of coordination number 4 or 7.

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