Porous media reaction

Though a porous medium may be described adequately under non-reactive conditions by a smooth field type of diffusion model, such as one of the Feng and Stewart models, it does not necessarily follow that this will still be the case when a chemical reaction is catalysed at the solid surface. In these circumstances the smooth field assumption may not lead to appropriate expressions for concentration gradients, particularly in the smaller pores. Though the reason for this is quite simple, it appears to have been largely overlooked,... 

The differential material balances contain a large number of physical parameters describing the structure of the porous medium, the physical properties of the gaseous mixture diffusing through it, the kinetics of the chemical reaction and the composition and pressure of the reactant mixture outside the pellet. In such circumstances it Is always valuable to assemble the physical parameters into a smaller number of Independent dimensionless groups, and this Is best done by writing the balance equations themselves in dimensionless form. The relevant equations are (11.20), (11.21), (11.22), (11.23), (11.16) and the expression (11.27) for the effectiveness factor. 

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. 

Figure 2.50 Reaction channels with a smooth surface (left) and a porous medium (right) as catalyst.

The simple pore structure shown in Figure 2.69 allows the use of some simplified models for mass transfer in the porous medium coupled with chemical reaction kinetics. An overview of corresponding modeling approaches is given in [194]. The reaction-diffusion dynamics inside a pore can be approximated by a one-dimensional equation... 

The objective of this work is to study the possible influence of the crude oil composition on the amount of coke deposit and on its ability to undergo in-situ combustion. Thus, the results would provide valuable information not only for numerical simulation of in-situ combustion but also to define better its field of application. With this aim, five crude oils with different compositions were used in specific laboratory tests that were carried out to characterize the evolution of the crude oil composition. During tests carried out in a porous medium representative of a reservoir rock, air injection was stopped to interrupt the reactions. A preliminary investigation has been described previously (8). 

Increasing the water-wet surface area of a petroleum reservoir is one mechanism by which alkaline floods recover incremental oil(19). Under basic pH conditions, organic acids in acidic crudes produce natural surfactants which can alter the wettability of pore surfaces. Recovery of incremental oil by alkaline flooding is dependent on the pH and salinity of the brine (20), the acidity of the crude and the wettability of the porous medium(1,19,21,22). Thus, alkaline flooding is an oil and reservoir specific recovery process which can not be used in all reservoirs. The usefulness of alkaline flooding is also limited by the large volumes of caustic required to satisfy rock reactions(23). 

The Effectiveness Factor Analysis in Terms of Effective Diffusivities First-Order Reactions on Spherical Pellets. Useful expressions for catalyst effectiveness factors may also be developed in terms of the concept of effective diffusivities. This approach permits one to write an expression for the mass transfer within the pellet in terms of a form of Fick s first law based on the superficial cross-sectional area of a porous medium. We thereby circumvent the necessity of developing a detailed mathematical model of the pore geometry and size distribution. This subsection is devoted to an analysis of simultaneous mass transfer and chemical reaction in porous catalyst pellets in terms of the effective diffusivity. In order to use the analysis with confidence, the effective diffusivity should be determined experimentally, since it is difficult to obtain accurate estimates of this parameter on an a priori basis. 

Upon reaction, the heterogenized catalyst can be easily separated from the reaction mixture by filtration and then recycled. The hydro-phobic substrate is microemulsified in water and subjected to an orga-nometallic catalyst, which is entrapped within a partially hydrophobized sol-gel matrix. The surfactant molecules, which carry the hydrophobic substrate, adsorb/desorb reversibly on the surface of the sol-gel matrix breaking the micellar structure, spilling their substrate load into the porous medium that contains the catalyst. A catalytic reaction then takes place within the ceramic material to form the desired products that are extracted by the desorbing surfactant, carrying the emulsified product back into the solution. 

Within the subsurface zone, two hquid phase regions can be defined. One region, containing water near the solid surfaces, is considered the most important surface reaction zone. This near solid phase water, which is affected by the sohd phase properties, controls the diffusion of the mobile fraction of the solute adsorbed on the solid phase. The second region constimtes the free water zone, which governs liquid and chemical flow in the porous medium. 

Speciation is a dynamic process that depends not only on the ligand-metal concentration but on the properties of the aqueous solution in chemical equilibrium with the surrounding solid phase. As a consequence, the estimation of aqueous speciation of contaminant metals should take into account the ion association, pH, redox status, formation-dissolution of the solid phase, adsorption, and ion-exchange reactions. From the environmental point of view, a complexed metal in the subsurface behaves differently than the original compound, in terms of its solubility, retention, persistence, and transport. In general, a complexed metal is more soluble in a water solution, less retained on the solid phase, and more easily transported through the porous medium. 

Daccord G, Lenormand R (1987) Fractal patterns from chemical dissolution. Nature 325 41 3 Daccord G, Lietard O, Lenormand R (1993) Chemical dissolution of a porous medium by a reactive fluid, 2, Convection vs. reaction behavior diagram. Chem Eng Sci 48 179-186 Darmody RG, Thorn CE, Harder RL, Schlyter JPL, Dixon JC (2000) Weathering implications of water chemistry in an arctic-alpine environment, north Sweden. Geomorphology 34 89-100 Dijk P, Berkowitz B (1998) Precipitation and dissolution of reactive solutes in fractures. Water Resour Res 34 457-470... 

Surface area is by no means the only physical property which determines the extent of adsorption and catalytic reaction. Equally important is the catalyst pore structure which, although contributing to the total surface area, is more conveniently regarded as a separate factor. This is because the distribution of pore sizes in a given catalyst preparation may be such that some of the internal surface area is completely inaccessible to large reactant molecules and may also restrict the rate of conversion to products by impeding the diffusion of both reactants and products throughout the porous medium. 

In the past, various resin flow models have been proposed [2,15-19], Two main approaches to predicting resin flow behavior in laminates have been suggested in the literature thus far. In the first case, Kardos et al. [2], Loos and Springer [15], Williams et al. [16], and Gutowski [17] assume that a pressure gradient develops in the laminate both in the vertical and horizontal directions. These approaches describe the resin flow in the laminate in terms of Darcy s Law for flow in porous media, which requires knowledge of the fiber network permeability and resin viscosity. Fiber network permeability is a function of fiber diameter, the porosity or void ratio of the porous medium, and the shape factor of the fibers. Viscosity of the resin is essentially a function of the extent of reaction and temperature. The second major approach is that of Lindt et al. [18] who use lubrication theory approximations to calculate the components of squeezing flow created by compaction of the plies. The first approach predicts consolidation of the plies from the top (bleeder surface) down, but the second assumes a plane of symmetry at the horizontal midplane of the laminate. Experimental evidence thus far [19] seems to support the Darcy s Law approach. 

In this section we shall study still another peculiarity of (3.2.3)—the occurrence of uniformly bounded, monotonic travelling waves. These waves, very common in reaction-diffusion (see, e.g., [25]—[27]), seem fairly unexpected in the reactionless diffusion under discussion. Their occurrence here is directly related to the singularity of diffusivity in (3.2.3) and thus can be viewed as the fast diffusional counterpart of the aforementioned peculiarities of the porous medium equation. 

It was observed in the previous section that a certain limit case of non-reactive binary ion-exchange is described by the porous medium equation with m = 2 in other words, a weak shock is to be expected at the boundary of the support. Recall that this shock results from a specific interplay of ion migration in a self-consistent electric field with diffusion. Another source of shocks (weak or even strong in the sense to be elaborated upon below) may be fast reactions of ion binding by the ion-exchanger. 

Simple dynamical systems have proved valuable as models of certain classes of physical systems in many branches of science and engineering. In mechanics and electrical engineering Duffing s and van der Pol s equations have played important roles and in physical chemistry and chemical engineering much has been learned from the study of simple, even artificially simple, systems. In calling them simple we mean to imply that their formulation is as elementary as possible their behaviour may be far from simple. Models should have the two characteristics of feasibility and actuality. By the first we mean that a favourable case can be made for the proposed reaction, perhaps by some further elaboration of mechanism but within the framework of accepted kinetic principles. Thus irreversible reactions are acceptable provided that they can be obtained as the limit of a consistent reversible set. By actuality we mean that they are set in an actual context, as taking place in a stirred tank, on a catalytic surface or in a porous medium. It is not usually necessary to assume the reaction to take place in a closed system with certain components held constant presumably by being in excess. 

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. 

In CVI, the pores of a porous medium are plugged by the reaction product of a precursor and an oxidising agent. For the preparation of silica layers, the precursor is a gaseous or volatile... 

When a solution containing a particular chemical species is displaced from a porous medium with the same solution but without the particular chemical species, this miscible displacement produces a chemical species distribution that is dependent on (1) microscope velocities, (2) chemical species diffusion rates, (3) physicochemical reactions of the chemical species with the porous medium, (e.g., soil), and (4) volume of water not readily displaced at saturation (this not-readily displaced water increases as desaturation increases (Nielsen and Biggar, 1961). 

Daccord et al. (1993a, b) investigated systematically the types of patterns that develop by the etching of the porous medium, when the flow rate and the reaction rate (i.e., rate of dissolution) are varied. When the kinetics of the surface reaction is limited, the porous medium is etched in a uniform way on... 

Model a process of diffusion and chemical reaction A -> B inside a porous catalyst. Assume that the reaction takes place on the catalytic surface within the porous medium. 

In catalytic systems morphological changes of the pore structure, brought upon by the reaction and sorption processes, typically result in a reduction of the available pore volume. In some instances the internal pore structure is eventually blocked and becomes completely inaccessible to transport and/or reaction. In the field of noncatalytic fluid-solid reactions and acid rock dissolution, on the other hand, the chemical reaction consumes the solid matrix of the porous medium leading eventually to fragmentation and... 

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. 

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