Length scales reaction

The characteristic times on which catalytic events occur vary more or less in parallel with the different length scales discussed above. The activation and breaking of a chemical bond inside a molecule occurs in the picosecond regime, completion of an entire reaction cycle from complexation between catalyst and reactants through separation from the product may take anywhere between microseconds for the fastest enzymatic reactions to minutes for complicated reactions on surfaces. On the mesoscopic level, diffusion in and outside pores, and through shaped catalyst particles may take between seconds and minutes, and the residence times of molecules inside entire reactors may be from seconds to, effectively, infinity if the reactants end up in unwanted byproducts such as coke, which stay on the catalyst. 

Generally, whenever fluids are processed in a confined space, two different types of phenomena are observed surface and volume effects. An example of a surface effect is a heterogeneously catalyzed reaction occurring at the walls of the vessel, whereas the motion of a fluid due to gravitational forces would be described as a volume effect. In brief, it can be stated that the surface effects gain in importance compared with the volume effects when the size of a reactor decreases. In particular, the reduction of length scale leads to a... 

Figure 1.14 Schematic representation of a stirred vessel (left) and a T-shaped micro reactor (right). Both devices can be used for liquid/liquid and for gas/liquid reactions. The length scales indicate typical physical dimensions.

The benefits refer to the ability to achieve defined thin, highly porous coatings in micro reactors. In combination with the small length scales of the channels, diffusion to the active sites is facilitated. The residence time can be controlled, accurately minimizing consecutive reactions which may reduce selectivity. 

In studies of reactions in nanomaterials, biochemical reactions within the cell, and other systems with small length scales, it is necessary to deal with reactive dynamics on a mesoscale level that incorporates the effects of molecular fluctuations. In such systems mean field kinetic approaches may lose their validity. In this section we show how hybrid MPC-MD schemes can be generalized to treat chemical reactions. 

In stirred chemical reactors, unlike in combustion and with other gas-phase reactions, these closure terms should take into account that for liquids the Schmidt number (Sc = v/D) is in the order 100-1,000, and, hence, the role of species diffusion at scales within the Kolmogorov eddies should explicitly be taken into account (Kresta and Brodkey, 2004). Essential is that diffusion of chemical species is governed by the Batchelor length scale rjB which obeys to... 

An alternative approach (e.g., Patterson, 1985 Ranade, 2002) is the Eulerian type of simulation that makes use of a CDR equation—see Eq. (13)—for each of the chemical species involved. While resolution of the turbulent flow down to the Kolmogorov length scale already is far beyond computational capabilities, one certainly has to revert to modeling the species transport in liquid systems in which the Batchelor length scale is smaller than the Kolmogorov length scale by at least one order of magnitude see Eq. (14). Hence, both in RANS simulations and in LES, species concentrations and temperature still fluctuate within a computational cell. Consequently, the description of chemical reactions and the transport of heat and species in a chemical reactor ask for subtle approaches as to the SGS fluctuations. 

As discussed in Section 4.3, the linear-eddy model solves a one-dimensional reaction-diffusion equation for all length scales. Inertial-range fluid-particle interactions are accounted for by a random rearrangement process. This leads to significant computational inefficiency since step (3) is not the rate-controlling step. Simplifications have thus been introduced to avoid this problem (Baldyga and Bourne 1989). 

Figure 5.17. A laminar diffusion flamelet occurs between two regions of unmixed fluid. On one side, the mixture fraction is unity, and on the other side it is null. If the reaction rate is localized near the stoichiometric value of the mixture fraction st, then the reaction will be confined to a thin reaction zone that is small compared with the Kolmogorov length scale.

Figure 5.18. The diffusion flamelet can be approximated by a one-dimensional transport equation that describes the change in the direction normal to the stoichiometric surface. The rate of change in the tangent direction is assumed to be negligible since the flamelet thickness is small compared with the Kolmogorov length scale. The flamelet approximation is valid when the reaction separates regions of unmixed fluid. Thus, the boundary conditions on each side are known, and can be uniquely expressed in terms of .

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