Reaction rates in pores

Enhancement of a rate by temperature can counteract the effect of falling concentration. Exothermic reaction rates in pores, as a consequence, can be much greater than at the surface condition. Another peculiarity that can arise with adiabatic reactions is multiple steady states. 

Wheeler examined the mechanism of diffusion and flow in catalyst pores and their consequence on reaction rates in pores. He always was concerned and... 

Equations (16) to (19) complete the formal theory of reaction rates in pores. In succeeding sections these formal equations are applied to various special cases. We can apply our theory to either one single isolated pore, or to a single catalyst particle which is a composite of many interconnecting pores. For clarity and simplicity we first consider a chemical reaction occurring in a single pore. 

In conclusion we note that a rigorous discussion of the effect of forced Poiseuille flow on reaction rates in pores would start with a solution of the differential equation ... 

The solution to this equation, which is detailed in Section 10.4.1, gives the concentration at position I down a pore that has its mouth located at position (r, z) in the reactor. The reaction rate in Equation (10.3) remains based on the bulk gas-phase volume, not on the comparatively small volume inside the pore. 

Figure 5. Relative catalytic activities of CsxH3-xPWi2O40 (x=2.1,2.2 and 2.5) for various kinds of reactions in liquid-solid reaction system. Molecular size. Pore size. Catalytic activity was estimated fi-om the initial rate of the reaction. The activity of Cs2.5 for each reaction is taken to be unity. The figures in the parentheses are the reaction rates in the unit of mmol g-l h"l.

The numerator of the right side of this equation is equal to the chemical reaction rate that would prevail if there were no diffusional limitations on the reaction rate. In this situation, the reactant concentration is uniform throughout the pore and equal to its value at the pore mouth. The denominator may be regarded as the product of a hypothetical diffusive flux and a cross-sectional area for flow. The hypothetical flux corresponds to the case where there is a linear concentration gradient over the pore length equal to C0/L. The Thiele modulus is thus characteristic of the ratio of an intrinsic reaction rate in the absence of mass transfer limitations to the rate of diffusion into the pore under specified conditions. 

The porous structure of either a catalyst or a solid reactant may have a considerable influence on the measured reaction rate, especially if a large proportion of the available surface area is only accessible through narrow pores. The problem of chemical reaction within porous solids was first considered quantitatively by Thiele [1] who developed mathematical models describing chemical reaction and intraparticle diffusion. Wheeler [2] later extended Thiele s work and identified model parameters which could be measured experimentally and used to predict reaction rates in... 

Robert B. Anderson. Some information on mass transfer processes may be obtained from activation energies. If a reaction were moderately rapid and diffusion in the liquid phase were rate controlling, the apparent activation energy should be 1-2 kcal./mole. If reaction occurs in pores and were diffusion plus reaction controlled, the activation energy should be one-half that of the surface process, which should still be a large value. What apparent activation energy was found in your oxidation reactions ... 

For gas-solid heterogeneous reactions particle size and average pore diameter will influence the reaction rate per unit mass of solid when internal diffusion is a significant factor in determining the rate. The actual mode of transport within the porous structure will depend largely on the pore diameter and the pressure within the reactor. Before developing equations which will enable us to predict reaction rates in porous solids, a brief consideration of transport in pores is pertinent. 

The condition of transition to the diffusion region (point B in Fig. 1) is determined by the equality of the reaction rate in the pores and the rate of supply by diffusion ... 

The Michael addition mechanism, whereby sulfur nucleophiles react with organic molecules containing activated unsaturated bonds, is probably a major pathway for organosulfur formation in marine sediments. In reducing sediments, where environmental factors can result in incomplete oxidation of sulfide (e.g. intertidal sediments), bisulfide (HS ) as well as polysulfide ions (S 2 ) are probably the major sulnir nucleophiles. Kinetic studies of reactions of these nucleophiles with simple molecules containing activated unsaturated bonds (acrylic acid, acrylonitrile) indicate that polysulfide ions are more reactive than bisulfide. These results are in agreement with some previous studies (30) as well as frontier molecular orbital considerations. Studies on pH variation indicate that the speciation of reactants influences reaction rates. In seawater medium, which resembles pore water constitution, acrylic acid reacts with HS at a lower rate relative to acrylonitrile because of the reduced electrophilicity of the acrylate ion at seawater pH. 

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 many industrial reactions, the overall rate of reaction is limited by the rate of mass transfer of reactants and products between the bulk fluid and the catalytic surface. In the rate laws and cztalytic reaction steps (i.e., dilfusion, adsorption, surface reaction, desorption, and diffusion) presented in Chapter 10, we neglected the effects of mass transfer on the overall rate of reaction. In this chapter and the next we discuss the effects of diffusion (mass transfer) resistance on the overall reaction rate in processes that include both chemical reaction and mass transfer. The two types of diffusion resistance on which we focus attention are (1) external resistance diffusion of the reactants or products between the bulk fluid and the external smface of the catalyst, and (2) internal resistance diffusion of the reactants or products from the external pellet sm-face (pore mouth) to the interior of the pellet. In this chapter we focus on external resistance and in Chapter 12 we describe models for internal diffusional resistance with chemical reaction. After a brief presentation of the fundamentals of diffusion, including Pick s first law, we discuss representative correlations of mass transfer rates in terms of mass transfer coefficients for catalyst beds in which the external resistance is limiting. Qualitative observations will bd made about the effects of fluid flow rate, pellet size, and pressure drop on reactor performance. 

The effectiveness factor, rj, is defined as the ratio of the reaction rate with pore diffusion resistance to the reaction rate without pore diffusion resistance (i.e., all of the active catalytic sites are restricted to the external surface of the particle). In mathematical terms... 

The overall rate of reaction is equal to the rate of the slowest step in the mechanism. When the diffusion steps (1.2. 6. and 7 in Table 10-2) are very fast compared with the reaction steps (14. and 5), the concentrations in the immediate vicinity of the active sites are indistinguishable from those in the bulk Ouid. In this situation, the transport or diffusion steps do not affect the overall rate of the reaction. In other situations, if the reaction. steps are very fast compared with the diffusion steps, mass transport does affect the reaction rate. In systems where diffusion from the bulk gas or liquid to the catalyst surface or to the mouths of catalyst pores affects the rate, changing the flow conditions past the catalyst should change the overall reaction rate. In porous catalysts, on the other hand, diffusion within the catalyst pores may limit the rate of reaction. Under these circumstances, the overall rate will be unaffected by external flow conditions even though diffusion affects the overall reaction rate. 

In the general case where the active material is dispersed through the pellet and the catalyst is porous, internal diffusion of the species within the pores of the pellet must be included. In fact, for many cases diffusion through catalyst pores represents the main resistance to mass transfer. Therefore, the concentration and temperature profiles inside the catalyst particles are usually not flat and the reaction rates in the solid phase are not constant. As there is a continuous variation in concentration and temperature inside the pellet, differential conservation equations are required to describe the concentration and temperature profiles. These profiles are used with intrinsic rate equations to integrate through the pellet and to obtain the overall rate of reaction for the pellet. The differential equations for the catalyst pellet are two point boundary value differential equations and besides the intrinsic kinetics they require the effective diffusivity and thermal conductivity of the porous pellet. 

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