Heterogeneous reactions. Rates of diffusion

Reactions do occur between participants that are primarily in different phases. Gases may react with liquids or solids, primarily immiscible liquids with each other, and liquids with solids. In all such cases a rate of chemical reaction exists in series with a rate of mass transfer. At steady state these rates are equal. 

The simplest mechanism postulated for transfer between phases is that a concentration gradient exists only across a stagnant film between phases. Then the rate of transfer is proportional to the interfacial area and a difference of concentrations across the film. For a power law chemical rate, at steady state, 

Elimination of the interfacial concentration from this equation results in 

Given data of (C, t) or of (r, C) and a correlation for kda, the other constants can be found as for single reactions. All three constants also can be found by nonlinear regression or by the solution of three simultaneous equations. 

Although the rate r may not be known explicitly, from Eq 2.55, that does not prevent its utility in any numerical applications. 

---

Mass Transfer. The reaction rate of heterogeneous reactions may be controlled by the rates of diffusion of the reacting species, rather than the chemical kinetics. 

For catalytic reactions carried out in the presence of a heterogeneous catalyst, the observed reaction rate could be determined by the rate of mass transfer from the bulk of the reaction mixture and the outer surface of the catalyst particles or the rate of diffusion of reactants within the catalyst pores. Consider a simple first order reaction its rate must be related to the concentration of species S at the outer surface of the catalyst as follows ... 

Heterogeneous catalysts can thus have a major influence on selectivity. Changing the catalyst can change the relative influence on the primary and by product reactions. This might result directly from the reaction mechanisms at the active sites or the relative rates of diffusion in the support material or a combination of both. 

In the case of heterogeneous surface burning of a particle, consideration must be given to the question of whether diffusion rates or surface kinetic reaction rates are controlling the overall burning rate of the material. In many cases, it cannot be assumed that the surface oxidation kinetic rate is fast compared to the rate of diffusion of oxygen to the surface. The surface temperature determines the rate of oxidation and this temperature is not always known a priori. Thus, in surface combustion the assumption that chemical kinetic rates are much faster than diffusion rates cannot be made. 

At low temperatures (T<1320 °C) and small particles, combustion regime (I) prevails [11,74,75]. Regime (I) is controlled by chemical kinetics intraparticle (reaction control), see Figure 55. The oxygen content is constant at any radius inside the particle since the rate of diffusion is fast compared to the rate of heterogeneous reaction. The particle then burns with reducing density and a constant diameter, see Figure 55. 

Basically, whenever isotopic exchanges occur between different phases (i.e., heterogeneous equilibria), isotopic fractionations are more appropriately described in terms of differential reaction rates. Simple diffusion laws are nevertheless appropriate in discussions of compositional gradients within a single phase— induced, for instance, by vacancy migration mechanisms, such as those treated in section 4.10—or whenever the isotopic exchange process does not affect the extrinsic stability of the phase. 

This study employed conventional diffusion-reaction theory, showing that with diffusion-limited reactions the internal effectiveness factor of a heterogeneous catalyst is inversely related to the Thiele modulus. Using a standard definition of the Thiele modulus [100], the observed reaction rate of an immobilized-enzyme reaction will vary with the square root of the immobilized-enzyme concentration in a diffusion-limited system. In this case, a plot of the reaction rate versus the enzyme loading in the catalyst formulation will be nonlinear. 

Reactions influenced by mass transport If the rate of a reaction is influenced by mass transport, the effect of the pressure both on the rate of the chemical reaction and on the rate of mass transport must be taken into account. As an example, a heterogeneous catalytic reaction governed by the rate of diffusion within the pores of the catalyst is considered. 

Attempts to develop a model for the digital simulation of the cyclic voltammetric behaviour of PVF films on platinum62 electrodes required inclusion of the following features (a) environmentally distinct oxidized and reduced sites within the film (b) interconversion of the above sites and interaction between them (c) rate of electrochemical reactions to depend on the rate of interconversion of redox sites, the rate of heterogeneous electron transfer between film and substrate, intrafilm electron transfer and the rate of diffusion of counter ions and (d) dependence on the nature of the supporting electrolyte and the spacing of electroactive groups within the film. 

In the experiments using a 1 1 mixture of nitric and sulphuric acids (table 4.1, column (/)) reaction occurred under heterogeneous conditions, about 50 cm3 of mixed aromatic compounds and 25 cm3 of mixed acids being used. The results are therefore complicated by differences in solubilities and rates of diffusion to the acid layer. 

O-nitration of cellulose with mixtures of acids is a widely applied type of esterification reaction in a heterogeneous system. One of the factors influencing the reaction rate here is the rate of diffusion of the acid into the fibre. 

In the presence of catalysts, heterogeneous catalytic cracking occms on the surface interface of the melted polymer and solid catalysts. The main steps of reactions are as follows diffusion on the surface of catalyst, adsorption on the catalyst, chemical reaction, desorption from the catalyst, diffusion to the liquid phase. The reaction rate of catalytic reactions is always determined by the slowest elementary reaction. The dominant rate controller elementary reactions are the linking of the polymer to the active site of catalyst. But the selectivity of catalysts on raw materials and products might be important. The selectivity is affected by molecular size and shape of raw materials, intermediates and products [36]. 

While the above criteria are useful for diagnosing the effects of transport limitations on reaction rates of heterogeneous catalytic reactions, they require knowledge of many physical characteristics of the reacting system. Experimental properties like effective diffusivity in catalyst pores, heat and mass transfer coefficients at the fluid-particle interface, and the thermal conductivity of the catalyst are needed to utilize Equations (6.5.1) through (6.5.5). However, it is difficult to obtain accurate values of those critical parameters. For example, the diffusional characteristics of a catalyst may vary throughout a pellet because of the compression procedures used to form the final catalyst pellets. The accuracy of the heat transfer coefficient obtained from known correlations is also questionable because of the low flow rates and small particle sizes typically used in laboratory packed bed reactors. 

A novel application of a symmetric porous membrane as a catalyst carrier but not as a permselective barrier is to use the membrane itself as the reaction zone for precise control of the stoichiometric ratio [Sloot et al., 1990]. In this case, the reactants are fed to the different sides of the membrane which is impregnated with a catalyst for a heterogeneous reaction. The products diffuse out of the membrane to its both sides. If the reaction rate is faster than the diffusion rate of the reactant in the membrane, a small reaction zone or theoretically a reaction plane will exist in the membrane. An interesting and important consequence of this type of membrane reactor is that within the reaction zone the molar fluxes of the reactants arc always in stoichiometric ratio and the presence of one reactant in the opposing side of the membrane is avoided. The reaction zone can be maintained inside the membrane as long as the membrane is symmeuic and not ultrathin. Therefore, membrane reactors of this fashion are particularly suited for those processes which require strict stoichiometric feed rates of premixed reactants. A symmetric porous a-alumina membrane of 4.5 mm thick was successfully tested to demonstrate the concept [Sloot et al., 1990]. 

While it is often possible to demonstrate that a surface process is rate limiting, identification of the step concerned is not always so readily achieved (as in heterogeneous catalysis which involve comparable mechanistic steps). Reaction rates are determined by reactant areas and are slow compared with the rate of diffusive transport of material to the appropriate boundaries. Surface limited reactions are also sensitive to the ease of removal of volatile products, which may be hampered by the presence of an inert gas. Readsorption may influence the effective concentrations of participating surface intermediates. As in catalytic heterogeneous reactions, the sequence of changes which precede product evolution may involve several interlinked steps, and the parameters which determine the overall progress of reaction are not always readily identified. 

The chemical reaction between a solid and a reactive fluid is of interest in many areas of chemical engineering. The kinetics of the phenomenon is dependent on two factors, namely, the diffusion rate of the reactants toward the solid/fluid interface and the heterogenous reaction rate at the interface. Reactions can also take place within particles, which have accessible porosity. The behavior will depend on the relative importance of the reaction outside and inside the particle. Fractal analysis has been applied to several cases of dissolution and etching in such natural occurring caves, petroleum reservoirs, corrosion, and fractures. In these cases fractal theory has found usefulness for quantifying the shape (line or surface) with only a few parameters the fractal dimension and the cutoffs. There have been some attempts to use a fractal dimension for reactivity as a global parameter. Finally, fractal concepts have been used to aid in the interpretation of experimental results, if patterns quantitatively similar to DLA are obtained. 

Often the global reaction rate of heterogeneous catalytic reactions is affected by the diffusion in the pore and the external mass-transfer rate of the reactants and the products. When the diffusion in the pores is not fast, a reactant concentration profile develops in the interior of the particle, resulting in a different reaction rate at different radial locations inside the catalytic pelet. To relate the global reaction rate to various concentration profiles that may develop, a kinetic effectiveness factor is defined [1, 3,4,7, 8] by... 

---