Porous solids transition diffusion

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). 

During recent years, studies of a number of hydrocarbon transformations catalyzed by porous solid oxides containing a transition metal, notably platinum, have evolved some concrete examples and demonstrations of truly polystep catalytic reactions. Specifically, these reactions have been shown to be performed by catalysts which contain geometrically separate and different catalyst components, each of which catalyzes separate steps. The chemical intermediates exist as true compounds, although often at undetected concentrations. The term true is used in this context to characterize the intermediate as a normal chemical species, existing independently of, and desorbed from, the catalyst phase, and subject to ordinary physical laws of diffusion. 

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... 

One of the main purposes of developing structural models of porous solids is to predict the effects of confinement on the properties of adsorbed phases, e.g., adsorption isotherms, heats of adsorption, diffusion, phase transitions, and chemical reaction mechanisms. Once a structural model for a particular porous solid has been chosen or developed (see Section 5.3), it is necessary to assume an interaction potential between the solid (adsorbent) and the confined fluid (adsorbate), as well as a fluid-fluid potential, and to decide on a theory or simulation method to calculate the property of interest [58]. A great many such studies have been reported in the literature, particularly for simple pore geometry models, and we do not attempt to review them here. Instead we present a few examples of such stuches, with emphasis on those involving more realistic pore models. 

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. 

If conditions of pore diameter and pressure occur for which Knudsen flow prevails (d/ < 0,2), the flow will be described by Knudsen s law, Eqs. (4.17) to (4.20). There will be of course a range of conditions for a transition from hydrodynamic to Knudsen flow. If the gas is a mixture with different compositions and different total pressure on either side of the porous solid, the flow may be a combination of hydrodynamic, Knudsen, and diffusive. Younquist [20] reviews these problems. 

In the transition region between regimes I and JJ where the chemical reaction and diffusion present a comparable resistance to the overall progress of reaction, multiple solutions may occur and the possibility of instability arises when the reaction is exothermic [15]. The criteria for the existence of multiple steady state for chemical reactions in porous catalyst pellets have been studied extensively [17-21]. The effect of net gas generation or consumption on nonisothermal reaction in a porous solid was analyzed by Weekman [22]. 

The reaction kinetics approximation is mechanistically correct for systems where the reaction step at pore surfaces or other fluid-solid interfaces is controlling. This may occur in the case of chemisorption on porous catalysts and in affinity adsorbents that involve veiy slow binding steps. In these cases, the mass-transfer parameter k is replaced by a second-order reaction rate constant k. The driving force is written for a constant separation fac tor isotherm (column 4 in Table 16-12). When diffusion steps control the process, it is still possible to describe the system hy its apparent second-order kinetic behavior, since it usually provides a good approximation to a more complex exact form for single transition systems (see Fixed Bed Transitions ). 

Figure 3. Transition from the kinetic regime to the diffusion-controlled regime of a heterogeneous catalytic fluid-solid reaction carried out on a porous catalyst.

This method is also referred to as the miscible-displacement or continuous-flow method. In this method a thin disk of dispersed solid phase is deposited on a porous membrane and placed in a holder. A pump is used to maintain a constant flow velocity of solution through the thin disk and a fraction collector is used to collect effluent aliquots. A diagram of the basic experimental setup is shown in Fig. 2-6. A thin disk is used in an attempt to minimize diffusion resistances in the solid phase. Disk thickness, disk hydraulic conductivity, and membrane permeability determine the range of flow velocities that are achievable. Dispersion of the solid phase is necessary so that the transit time for a solute molecule is the same at all points in the disk. However, the presence of varying particle sizes and hence pore sizes may produce nonuniform solute transit times (Skopp and McCallister, 1986). This is more likely to occur with whole soils than with clay-sized particles of soil constituents. Typically, 1- or 2-g samples are used in kinetic studies on soils with the thin disk method, but disk thicknesses have not been measured. In their study of the kinetics of phosphate and silicate retention by goethite, Miller et al. (1989) estimated the thickness of the goethite disk to be 80 /xm. 

During this transitional period, and once full breakaway is established, growth occurs throughout the oxide and close to the scale/metal interfaceGrowth below the oxidant surface requires oxidant transport through the scale, but solid-state lattice diffusion is much too slow to account for the rates observed Therefore, a porous scale is required. Mechanisms for pore growth have been postulated by several authors who have... 

For a binary gas mixture it is frequently necessary to measure two or even three diffusivities. For example, if diffusion in the porous medium during the reaction is Knudsen type then it will be necessary to measure the Knudsen diffusivity of each component. If the diffusion is in the transition regime then the investigator will need to measure the effective molecular diffusivity within the solid, in addition to the Knudsen diffusivities. Only in the purely molecular diffusion regime will one diffusivity suffice. 

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