Pore Diffusion and Chemical Reaction

Again, for the purpose of this chapter, this step is deflned as any process which is unaffected by the agitation conditions, e.g. the surface integration step in crystallization and the combined in-pore diffusion and chemical reaction at the surface associated with such particles as ion-exchange resins and catalysts. Figure 17.2 shows diagrammatically this idealized two-step mechanism. 

Equation 21.7 includes the mass transfer, pore diffusion, and chemical reaction. Figure 21.3 allows observing the effect of each step, represented by Equation 21.7. As measure parameters, one has the inverse of the mass on the abscissa and the inverse of the global rate on the ordinate. 

Figure 10. Penetration depth for chemical vapor deposition into cylindrical pores with radii of 1, 5, and 10 pm. Left-hand side comparison of experimental data with theoretical calculations. Right-hand side calculation of combined effects of diffusion and chemical reaction during the CVD process, using the Damkohler number (22-25).

At impeller speeds above the second or third term on the right-hand side of Equation (CS11.3) becomes controlling i.e., the overall rate is controlled by intrinsic chemical reaction or pore diffusion. Under such conditions, if experiments are performed at different particle sizes, then the effects of internal diffusion and chemical reaction can be elucidated. 

In other words, reactants exist everywhere within the pores of the catalyst when the chemical reaction rate is slow enough relative to intrapellet diffusion, and the intrapellet Damkohler number is less than, or equal to, its critical value. These conditions lead to an effectiveness factor of unity for zerofli-order kinetics. When the intrapellet Damkohler number is greater than Acnticai, the central core of the catalyst is reactant starved because criticai is between 0 and 1, and the effectiveness factor decreases below unity because only the outer shell of the pellet is used to convert reactants to products. In fact, the dimensionless correlation between the effectiveness factor and the intrapeUet Damkohler number for zeroth-order kinetics exhibits an abrupt change in slope when A = Acriticai- Critical spatial coordinates and critical intrapeUet Damkohler numbers are not required to analyze homogeneous diffusion and chemical reaction problems in catalytic pellets when the reaction order is different from zeroth-order. When the molar density appears explicitly in the rate law for nth-order chemical kinetics (i.e., n > 0), the rate of reaction antomaticaUy becomes extremely small when the reactants vanish. Furthermore, the dimensionless correlation between the effectiveness factor and the intrapeUet Damkohler nnmber does not exhibit an abrupt change in slope when the rate of reaction is different from zeroth-order. 

This second-order ODE for 4 a(9) with split boundary conditions, given by equations (19-11) and (19-12), cannot be solved numerically until one invokes stoichiometry and the mass balance with diffusion and chemical reaction to relate the molar densities of aU gas-phase species within the pores of the catalytic pellet... 

It is necessary to check for multiple steady-state solutions to equations (27-43), (27-44), and (27-45) for diffusion and chemical reaction within the catalytic pores at constant values of intrapeuer T- This is a difficult task. The thermal... 

When the stoichiometric relation given by (30-5) is evaluated at the external surface of the catalyst, it is possible to invoke continuity across the gas/porous-solid interface and introduce mass transfer coefficients to evaluate interfacial fluxes. Diffusion and chemical reaction within the catalytic pores are consistent with the following stoichiometric relation between diffusional mass fluxes ... 

The conversion rate of the solid is determined by the smallest value of Equations (5.18) to (5.16). Figure 5.1 shows that at elevated temperatoes (e.g., >1100 °C for hard coal) chemical reaction rates increase exponentially and pore and bulk surface diffusion occur at a much slower rate, thus becoming the rate-limiting steps. When integrating heterogeneous kinetics into the modeling, it is instructive to incorporate these effects into conversion models. A simple approach for spherical particles has been widely used in hterature [5] It imifies the bulk surface diffusion and chemical reaction control, as given in Equation (5.19). 

How this dissolution proceeds depends on the relative speed of diffusion and reaction. When the bulk of the solution next to the solid is rapidly stirred, the acid can diffuse to the solid s surface very quickly. It then reacts with the solid s surface. If the solid is essentially impermeable, containing a very few pores, then any ions produced by the dissolution are quickly swept back into the bulk solution. Because diffusion and chemical reaction occur sequentially, the overall dissolution rate is like that of a heterogeneous reaction, depending on the sum of the resistances of diffusion and of reaction (see Chapter 16). Such a process represents an important limit of corrosion and is that usually studied. 

The catalytic reaction can be subdivided into pore diffusion and chemisorption of reactants, chemical surface reaction, and desorption and pore diffusion of products, the number of steps depending upon the nature of the catalyst and the catalytic reaction. 

Models of chemical reactions of trace pollutants in groundwater must be based on experimental analysis of the kinetics of possible pollutant interactions with earth materials, much the same as smog chamber studies considered atmospheric photochemistry. Fundamental research could determine the surface chemistry of soil components and processes such as adsorption and desorption, pore diffusion, and biodegradation of contaminants. Hydrodynamic pollutant transport models should be upgraded to take into account chemical reactions at surfaces. 

The reactant solid B is porous and the reaction occurs in a diffuse zone. If the rate of the chemical reaction is much slower compared to the rate of diffusion in the pores, the concentration of the fluid reactant would be uniform throughout the pellet and the reaction would occur at a uniform rate. On the other hand, if the chemical reaction rate is much faster than the pore diffusion rate, the reaction occurs in a thin layer between the unreacted and the completely reacted regions. The thickness of the completely reacted layer would increase with the progress of the reaction and this layer would grow towards the interior of the pellet). 

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. 

Abstract We formulate the balance principles for an immiscible mixture of continua with micro structure in the broadest sense for include, e.g., phenomena of diffusion, adsorption and chemical reactions. After we consider the flow of a fluid/adsorbate mixture through big pores of an elastic solid skeleton and propose suitable constitutive equations to study the coupling of adsorption and diffusion under isothermal conditions. 

Observed transport limitations in the studies given in Table I depend upon the magnitude of the intrinsic reaction rate. Petroleum hydrodesulfurization (19-21), certain types of petroleum hydrogenations (22), or chemical decomposition reactions (11) are liquid-limiting and proceed slowly enough that only internal particle diffusion or combined pore diffusion and liquid-to-solid resistances are controlling. Chemical... 

In zeolites Na X and ZSM-5, the self-diffusion coefficients were found to decrease with increasing concentration while for zeolite NaCa A they are essentially constant The highest diffusivities were observed in zeolite Na X. This is in agreement with the fact that due to the internal pore structure the steric restrictions of molecular propagation in zeolite Na X are smaller Aan those in Na Ca A and ZSM-5 (94). Mass transfer and chemical reaction in zeolite channels in which the individual molecules cannot pass each other (single-file... 

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. 

Residue hydroprocessing requires the diffusion of multiringed aromatic molecules into the pore structure of the catalyst prior to initiation of the sequential conversion mechanism. The observed diffusion rate may be influenced by adsorption interactions with the surface and a contribution from surface diffusion. Since chemical reactions between species inherently imply transfer of mass and energy, the amounts of mass and energy required by a reaction will become increasingly difficult to transfer physically as the reaction rate increases. Usually it is necessary to obtain the intrinsic kinetic information by running experiments under conditions where transport limitations are eliminated or can be neglected... 

A similar model that specifically considers the poison deposition in a catalyst pellet was presented by Olson [5] and Carberry and Gorring [6], Here the poison is assumed to deposit in the catalyst as a moving boundary of a poisoned shell surrounding an unpoisoned core, as in an adsorption situation. These types of models are also often used for noncatalytic heterogeneous reactions, which was discussed in detail in Chapter 4. The pseudo-steady-state assumption is made that the boundary moves rather slowly compared to the poison diffusion or reaction rates. Then, steady-state diffusion results can be used for the shell, and the total mass transfer resistance consists of the usual external interfacial, pore diffusion, and boundary chemical reaction steps in series. 

The initial step is the transfer of reactant (i.e., oxygen) through the layer of gas adjacent to the surface of the particle. The reactant is then adsorbed and reacts with the solid after which the gaseous products diffuse away from the surface. If the solid is porous, much of the available surface can only be reached by passage of the oxidant along the relatively narrow pores and this may be a rate-controlling step. Rate control may also be exercised by (a) adsorption and chemical reaction, which are considered as chemical reaction control and (b) pore diffusion, by which the products diffuse away from the surface. This latter phenomenon is seldom a rate-controlling step. 

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