Diffusional regime

When the reaction is fast enough to cause abrupt variations of concentration and temperature in a boundary layer near the catalyst surface, several methods can be used to derive the form 

Using the conditions at the pellet surface as reference =1), the following result is obtained for isothermal kinetics 

For negligible inhibition, the result for power-law kinetics is given by 

For non-isothermal kinetics, we adapt the result derived by Tavera [100] originally for Dirichlet conditions, and as expected b will be a function of kinetic descriptors (e.g., order of reaction w), nonisothermal parameters (y and P), and surface conditions (c j, and T, y). The dependence on the latter quantities (generally unknown) leads to an implicit calculation of the effectiveness factor. Substituting Equation 3.42 into 3.54a, 

In order to observe low concentration values at the pellet surface, a first estimate for b is 

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The criteria that are most often used to distinguish between a diffusional regime and a kinetic regime are... 

The reference snbstance method is based on the addition to the solntion, containing the species for which the transfer rate is going to be investigated, of another inert component for which the rate of extraction is known to be controlled only by diffnsion. By following the simultaneous transfer of the species of interest and of the reference component as function of the hydro-dynamic conditions in the extraction apparatns, a diffusional regime will be indicated by a similar functional dependence, whereas a kinetic regime is indicated by a sharply different one. 

This equation holds at the steady state. In a diffusional regime and in absence of rigid interfacial films, Ri is generally negligible relative to R and R . 

Clearly, this derivation of equation (5.1) is valid only in the case of the diffusional regime of dissolution of substance A in liquid B when the... 

As Eg is usually small the detrimental effect of gas phase dispersion on the performance of bubble columns can be neglected in columns less than 20 cm in diameter (61). For illustrating the influence of gas phase dispersion some computed conversions are presented in Fig. 10 (J ). The simulations refer to CO2 absorption in carbonate buffer in a column 5 m in length. Eq was calculated from eqn. (15). The liquid phase dispersion does not affect the conversion in the present case as the process takes place in the diffusional regime of mass transfer theory. As shown in Fig. 10, the decrease in conversion due to gas phase dispersion increases with increasing diameter and gas velocity. However, in the favorable bubbly flow regime and in small diameter columns the effect is less pronounced. 

The overall mass-transfer resistance at steady state is the sum of individual mass-transfer resistances at diffusional regime through the boundary films and chemical reactions resistances at the phases interface. 

If the same dependence of the heat transfer coefticient and the mass-transfer coefficient on the stirring or flowing rate of the phases is observed, the conclusion can be reached that the transport occurs in a diffusional regime. 

In the FT case, it was found that the additional friction due to the fast field has a different effect on the rotational versus momentum relaxation, such that, whereas T2 still behaves in a normal fashion (i.e., it is roughly proportional to the total friction, from both the solvent terms and the field terms), Tj is not much influenced by the friction generated by the fast field. These comments apply to the case in which the sources of friction are large, so that the system is always in a diffusional regime. 

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]. 

The slow reaction diffusional regime marks the transition from kinetic to mass transfer control. Thus the change is to be... 

Predicted and Experimental Rates at 25 0. On the basis of previous work ( 3) in which the same nitric acid concentration was employed, a sulphuric acid strength of 79 8 should be well inside the fast reaction diffusional regime. The initial rate according to surface renewal theory (8) should therefore be given... 

Detailed analysis conducted within this approximation shows that Eq. (9.110) works well in a much wider range than a strictly diffusional regime, while the range of vahdity of the averaged Landau Zener formula (9.101)... 

Nonlinear kinetics. The problem of calculating reactant conversion by a nonlinear wall reaction requires numerical solution. The most explicit approximations available in the literature were given by Lopes et al. [40] in limits regarding profile development (fully developed or developing profiles) and mass transfer control (kinetic or diffusional regimes). 

Figure 16.2.9 Schematic drawing of the different diffusional regimes at NEE (A) Total Overlap (B) Pure Radial (C) Linear active. The scan rate or the distance between the nanodisk elements increases from (A) to (C). Relevant equations for peak currents (A and C) and plateau current (B) refer to reversible redox systems. is the active area (nanodisk surface), is the total geometric area of the ensemble (nanodisks and insulator), q is the nanodisk density (disk cm" ), and all other symbols have their usual meaning. Reprinted with permission from reference (86). (for colour version see colour section at the end of the book).

In this paper we will first review some basic concepts and apply them to the design of isothermal reactors working in the diffusional regime.Then we will concentrate our attention on the problem of intraparticle convection in large pore catalysts.Several aspects of this question will be dealt with - effectiveness factors for iso -thermal and nonisothermal catalysts, measurement of effective diffu-sivities and the implication of intraparticle convection effects on the design and operation of fixed bed catalytic reactors. 

For zero order reactions one should take into account that r=k only if c> 0 otherwise r=0. In a situation where the concentration reaches a zero value inside the particle at a point x=x we should replace the boundary condition (Ic) by f=0 and df/dx=0 at x=x, The effectiveness factor is now simply the ratio of the "utilized" particle volume and the total particle volume,i.e., r]=l-x. In the kinetic controlled regime ri-1 and in the pure diffusional regime n-yT/((), Again for any shape we get ... 

In general for irreversible nth order reactions the catalyst effectiveness factor in the diffusional regime is ... 

Figure 1 shows the conversion obtained in a CSTR and plug flow reactor when the catalyst is working in the pure diffusional regime. 

Figure 1 - Conversion X as a function of a (catalyst working in the diffusional regime) [5]...

We observe that only one parameter a governs the steady state behavior of ideal reactors provided the catalyst is working in pure diffusion regime.Moreover,in a plug flow reactor,there is a critical value for a (or in practical terms a critical heirht for the reactor) at which complete conversion is obtained,i.e., u=2. This is because for the diffusional regime a zero order reaction is equivalent to a 1/2 order reaction in the kinetic regime. 

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