Chemical regime effectiveness factor

For situations where the reaction is very slow relative to diffusion, the effectiveness factor for the poisoned catalyst will be unity, and the apparent activation energy of the reaction will be the true activation energy for the intrinsic chemical reaction. As the temperature increases, however, the reaction rate increases much faster than the diffusion rate and one may enter a regime where hT( 1 — a) is larger than 2, so the apparent activation energy will drop to that given by equation 12.3.85 (approximately half the value for the intrinsic reaction). As the temperature increases further, the Thiele modulus [hT( 1 — a)] continues to increase with a concomitant decrease in the effectiveness with which the catalyst surface area is used and in the depth to which the reactants are capable of... 

In order to understand and modify the functions of a catalyst in a process, it is necessary to determine whether or not rates are determined by physical or chemical steps. Responses to process parameters and catalyst adjustments are different for the two regimes. Diffustonal resistance, in particular, causes unexpected complicrations. We have seen how low effectiveness factors decrease conversion and disguise kinetics, but selcc tivity also can be decreased. In addition, poisoning of pore mouth sites in conjunction with low diffusion results in a much more rapid activity decline than otherwise. 

After drying and reduction, the Pd-Ag/C catalysts are composed of bimetallic Eilloy nanoparticles ( 3 nm). CO chemisorption coupled to TEM and XRD analysis showed that that, for catalysts 1.5% wt. in each metal, the bulk composition of the alloy is close to 50% in each metal, whereas the surface is 90% in Ag and 10% in Pd [9]. Mass transfer limitations can be detected by testing the same catalyst with various pellet sizes [18]. Indeed, if the reactants diffusion is slow due to small pore sizes, the longer the distance between the pellet surface and the metal particle, the larger the influence of the difiusion rate on the apparent reaction rate. Pd-Ag catalysts with various pellet sizes were thus tested in hydrodechlorination of 1,2-dichloroethane. Results were compared to those obtained with a Pd-Ag/activated charcoal catalyst. Fig. 4 shows the evolution of the effectiveness factor of the catalysts, i.e. the ratio between the apparent reaction rate and the intrinsic reaction rate, as a function of the pellet size. The intrinsic reaction rate was considered equal to the reaction rate obtained with the smallest pellet size. When rf = 1, no diffusional limitations occur, and the catalyst works in chemical regime. When j < 1, the observed reaction rate is lower than the intrinsic reaction rate due to a slow diffusion of the reactants and products and the catalyst works in diffusional regime [18]. 

Hence, it is not possible to redefine the characteristic length such that the critical value of the intrapellet Damkohler number is the same for all catalyst geometries when the kinetics can be described by a zeroth-order rate law. However, if the characteristic length scale is chosen to be V cataiyst/ extemai, then the effectiveness factor is approximately A for any catalyst shape and rate law in the diffusion-limited regime (A oo). This claim is based on the fact that reactants don t penetrate very deeply into the catalytic pores at large intrapellet Damkohler numbers and the mass transfer/chemical reaction problem is well described by a boundary layer solution in a very thin region near the external surface. Curvature is not important when reactants exist only in a thin shell near T] = I, and consequently, a locally flat description of the problem is appropriate for any geometry. These comments apply equally well to other types of kinetic rate laws. 

It is rather straightforward to employ numerical methods and demonstrate that the effectiveness factor approaches unity in the reaction-rate-controlled regime, where A approaches zero. Analytical proof of this claim for first-order irreversible chemical kinetics in spherical catalysts requires algebraic manipulation of equation (20-57) and three applications of rHopital s rule to verify this universal trend for isothermal conditions in catalytic pellets of any shape. 

If 4 a( 7 = 0) = 1 as indicated in part (b), then diffusion does not hinder the ability of reactants to populate the central core of the catalyst. Furthermore, chemical reaction does not deplete reactant A because its molar density at the center of the catalyst is equivalent to that on the external surface. This situation occurs when A 0 and the catalyst operates in the reaction-rate-controlled regime. Hence, the effectiveness factor is unity under isothermal conditions. This result can be obtained mathematically from the integral expression for the effectiveness factor by setting = 0) = 1 — where s < 10 . ... 

Two expressions are given below to calculate the effectiveness factor E. The first one is exact for nth-order irreversible chemical reaction in catalytic pellets, where a is a geometric factor that accounts for shape via the surface-to-volume ratio. The second expression is an approximation at large values of the intrapellet Damkohler number A in the diffusion-limited regime. 

Thermal effects that reduce reaction times and which can be observed for the reactions under microwave irradiation are related to different temperature regime under microwave conditions in comparison to conventional conditions. However, they can result in a seemingly faster course of chemical reactions, the proper temperature measurement and its analysis for the entire sample of the material (i.e., the bulk reaction mixture) leads to the reaction rates that and comparable to reaction rates observed under conventional conditions. Three factors that can cause thermal effects are considered [3] ... 

We note that a special regime of the so-called autonomous motion is possible, where the drop drifts at a constant nonzero velocity in the absence of any exterior factors [147,148]. In this case the other possible regime (no motion) proves to be unstable. Effects similar to the ones considered in this section can be produced by the chemoconcentration-capillary mechanisms [149], as well as other factors different from surface chemical reactions, for example, by heat production within the drop [390]. 

Non-catalytic reactions involving two phases are common in the mineral industry. Reactions such as the roasting of ores or the oxidation of solids are carried out on a massive scale but the rates of these processes are often controlled by physical, not chemical, effects. Reactant or product diffusion is the main rate controlling factor in many cases. As a result, mechanisms of reaction become models of reaction with consideration of factors such as external diffusion film control or the shrinking core yielding the various models. Matters are further complicated by considerations regarding particle shape and external fluid flow regimes. 

---