Diffusion in Porous Particles

Most of the adsorbents commercially used are porous particles. For large adsorption capacity, large surface area is preferable, as a result large numbers of fine pores, as fine as possible, are needed. Adsorbate molecules come from outside adsorbent particles and diffuse into the particle to fully utilize the adsorption sites. Depending on the structure of the adsorbent, several different types of diffusion mechanisms become dominant and sometimes two or three of them compete or cooperate. The dominant mechanism also depends on a combination of adsorbate and adsorbent and adsorption conditions such as temperature and concentration range. 

In adsorbent particles with bi-dispersed pore structures, such as activated carbon, macropores usually act as a path for the adsorbate molecules to reach the interior of the particle. In this case molecular diffusion or Knudsen diffusion takes place in the macropore this is called pore diffusion. 

When adsorbed molecules are mobile on the surface of the adsorbent, e.g. volatile hydrocarbon on activated carbon, diffusion due to migration of the adsorbed molecules may contribute more than pore diffusion to intraparticle diffusion. This type of diffusion is called surface diffusion. 

When the size of an adsorbate molecule is close to the size of the micropore, diffusion of the molecule becomes restricted and the rate of transport in the micropore may have a significant effect in the overall adsorption rate. This type of diffusion in the micropore is an activated process which depends heavily on adsorbate properties. 

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A. dispersion caused by mokcular diffusion Dispersion due to molecular diffusion in the interparticle void spaces is described by the void fraction of the bed, e, and the tortuosity of the diffusion path in the void space. Unlike the diffusion in porous particles reviewed in Chapter 4, the latter is considered close to unity for diffusion in packed beds. Then,... 

This book comprises 12 chapters covering an introduction, porous adsorbents, adsorption equilibrium, diffusion in porous particles, kinetics of adsorption in a vessel, kinetics of adsorption in a column (chromatographic analysis and breakthrough curves), heat effects, regeneration of spent adsorbent, chromatographic separation, pressure swing adsorption and adsorption for energy transport. The text comprises a blend of mathematical analysis and descriptions of plant and processes. Each chapter is fully referenced. 

Combined Pore and Solid Diffusion In porous adsorbents and ion-exchange resins, intraparticle transport can occur with pore and solid diffusion in parallel. The dominant transport process is the faster one, and this depends on the relative diffusivities and concentrations in the pore fluid and in the adsorbed phase. Often, equilibrium between the pore fluid and the solid phase can be assumed to exist locally at each point within a particle. In this case, the mass-transfer flux is expressed by ... 

Reaction, diffusion and catalyst deactivation in porous particles is considered. A general model for mass transfer and reaction in a porous particle with an arbitrary geometry can be written as follows ... 

Summarizing the considerations on particle side mass transport, slow protein diffusion in porous adsorbents seems to have the same dominant influence on the efficiency of a fluidized bed adsorption as it frequently is the case in a packed bed. Contrary to conventional protein chromatography increasing... 

For diffusion between soil and sediment particles, n represents the interstitial porosity. For diffusion within soil and sediment particles, n represents the in-traparticle porosity. Values of m close to 1 are common for diffusion in porous media [29-31]. However, in low porosity materials this value can increase [32]. A more thorough treatment of this topic can be found in Grathwohl [33]. 

The most widely used unsteady state method for determining diffusivities in porous solids involves measuring the rate of adsorption or desorption when the sample is subjected to a well defined change in the concentration or pressure of sorbate. The experimental methods differ mainly in the choice of the initial and boundary conditions and the means by which progress towards the new position of equilibrium is followed. The diffusivities are found by matching the experimental transient sorption curve to the solution of Fick s second law. Detailed presentations of the relevant formulae may be found in the literature [1, 2, 12, 15-17]. For spherical particles of radius R, for example, the fractional uptake after a pressure step obeys the relation... 

TABLE 1 Characteristic Timescales for Diffusion in Porous Catalyst Particles. 

Meyers, J.J., Liapis, A.I. Network modeling of the intraparticle convection and diffusion of molecules in porous particles packed in a chromatographic column, J. Chromatogr. A, 1998, 827, 197-213. 

Lattice models consist of spanning the system of interest by meshes or nodes. Molecules that are moving or reacting are lumped into coarser particles, and the nodes or the meshes keep a track of the extent of mobility or reaction. Examples include curing of polymers, diffusion in porous structures, and stress calculations in materials. 

A reduced reaction rate may result from external diffusional restrictions on the surface of carrier materials. In stirred tanks external diffusion plays a minor role as long as the reaction liquid is stirred sufficiently. Further, partition effects can lead to different solubilities inside and outside the carriers. Partition has to be taken into account when ionic or adsorptive forces of low concentrated solutes interact with carrier materials [81 - 83]. The most crucial effects are observed in porous particles due to internal or porous diffusion as outlined below. 

The bulk diffusion processes within the pores of catalyst particles are usually described by the Wilke model formulation. The extended Wilke equation for diffusion in porous media reads ... 

Fluid-phase pore diffusion, in porous bodies whose pores are freely accessible to the bulk fluid outside. Counter-diffusion of A through the pores of the particle to the point where exchange occurs and of B from the point of exchange in the pore surface back to the outer surface of the particle. 

Network modeling of the convective flow and diffusion of molecules adsorbing in monoliths and in porous particles packed in a chromatographic column. 

A model, frequently referred to as dusty-gas model [1-3], can be used to describe multi-component diffusion in porous media. This model is based on the Stefan-Maxwell approach for diluted gases which is an approximation of Boltzmann s equation. The pore walls are considered as consisting of giant molecules ( dust ) distributed in space. These dust molecules are treated as the n+l-th pseudo-species in a n-component gaseous mixture. The dust particles are kept fixed in space, and are treated like a gas component in the Stefan-Maxwell equations. This model analyzes the transport problem by distinguishing three separate components 1) diffusion, 2) viscous flow and 3) structure of the porous medium. 

For delivery systems formulated using MPS, the pore size and pore structure as well as the particle diameter will affect the diffusion coefficient and the surface area (Van Speybroeck et al. 2010 Horcajada et al. 2004 Shen et al. 2011). The surface area can be calculated from geometric considerations, and the diffusion coefficient can be estimated by using traditional approaches for mass transfer in porous particles. Mortera et al. (2010) detail a good illustration of this approach where D was determined using the Stokes-Einstein equation to derive the drug diffusivity in the dissolution media (Ddm) and the Renkin equation to correct the value for the steric hindrance (5) and the constrictivity (a>r) in the pores ... 

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