I Molecular diffusion and reaction rates

Since the molecular diffusivities are used in (5.254), the interval length L(t) and the initial conditions will control the rate of molecular diffusion and, subsequently, the rate of chemical reaction. In order to simulate scalar-gradient amplification due to Kolmogorov-scale mixing (i.e., for 1 < Sc), the interval length is assumed to decrease at a constant rate ... 

This initial condition is rather idealized. In reality, one would expect to see partially premixed zones with f = fst and 7 = 0 which will move towards 7 = 1 along the stoichiometric line. The movement along lines of constant f corresponds to premixed combustion, and occurs at a rate that is controlled by the interaction between molecular diffusion and chemical reactions (i.e., the laminar flame speed). 

As stated above, shape selectivity due to molecular sieving depends on the relative rates of diffusion and reaction, hence on the respective sizes and shapes of molecules and pores and on the characteristics of active sites (e.g. concentration, nature and strength of acid sites). Obviously the diffusion rate, hence the selectivity depend also on the length of the diffusion path (i.e., on the size of the zeolite crystallites). The selectivity of a zeolite catalyst can be optimized by an adequate... 

The use of distributed pharmacokinetic models to estimate expected concentration profiles associated with different modes of drug delivery requires that various input parameters be available. The most commonly required parameters, as seen in Equation 9.1, are diffusion coefficients, reaction rate constants, and capillary permeabilities. As will be encountered later, hydraulic conductivities are also needed when pressure-driven rather than diffusion-driven flows are involved. Diffusion coefficients (i.e., the De parameter described previously) can be measured experimentally or can be estimated by extrapolation from known values for reference substances. Diffusion constants in tissue are known to be proportional to their aqueous value, which in turn is approximately proportional to a power of the molecular weight. Hence,... 

Sustained combustion requires a continuous supply of fresh reactants and a continuous removal of reaction products. This process is loosely known as mass transfer. Specifically, mass transfer is a consequence of three possible modes bulk fluid motion, molecular and turbulent diffusion, and reaction sources and sinks. Mass transfer due to bulk fluid motion is generally known as convection. It is similar to the convection heat transfer process. Mathematically, the rate of change for species / per unit volume, pYit via convection can be described as 3(pUjY ldxj, where p is fluid density, Yt is the mass fraction of species i, Uj is the / -component of the fluid velocity. 

Theory, Experiment, and Reaction Rates. A Personal View. J. D. Doll and A. F. Voter, Annu. Rev. Phys. Chem., 38, 413 (1987). Recent Develoinnents in the Theory of Surface Diffusion. B. J. Berne, in Multiple Time Scales, J. U. Brackbill and B. I. Cohen, Eds., Academic Press, Orlando, FL, 1987, pp. 419-436, Molecular Dynamics and Monte Carlo Simulation of Rare Events. P. HSnggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys., 62, 251 (1990). Reaction-Rate Theory Fifty Years After Kramers. 

To take into account the role of surface-active species the transport equations in the bulk and at surfaces for each of them (i = 1,2,...,AO are studied (Dukhin et al. 1995, Danov et al. 1999). In the bulk the change of concentration, C/, is compensated by the bulk diffusion flux, j bulk convective flux, C/V, and rate of production due to chemical reactions, (see Fig. 1). The bulk diffusion flux includes the flux driven by external forces (e.g. electro-diffusion), the molecular diffusive and thermodiffusion fluxes. The rate of production, r/,... 

In the case of high differences between two timescales r and to, either Dan S> 1 or Dan 1 is possible. In the case of Dan 3> 1 (to 3> ), the processes are referred to as diffusion limited, that is, diffusion rate determines the reaction rate. In the case of Dan -C 1 (Jr 3> to), the reaction partners can be mixed by molecular diffusion fast enough so that the reaction is not mixing influenced, but reaction limited. If both timescales are nearly the same (i.e., Dan 1). then diffusion and reaction compete with each other, that is, the reaction depends on the mixing intensity. These considerations can be transferred to micromixing in a microreactor. The mixing time (, ) is defined as... 

The mass transfer effect is relevant when the chemical reaction is far faster than the molecular diffusion, i.e. Ha > 1. The rapid formation of precipitate particles should then occur spatially distributed. The relative rate of particle formation to chemical reaction and/or diffusion can as yet be evaluated only via lengthy calculations. 

The second use of Equations (2.36) is to eliminate some of the composition variables from rate expressions. For example, 0i-A(a,b) can be converted to i A a) if Equation (2.36) can be applied to each and every point in the reactor. Reactors for which this is possible are said to preserve local stoichiometry. This does not apply to real reactors if there are internal mixing or separation processes, such as molecular diffusion, that distinguish between types of molecules. Neither does it apply to multiple reactions, although this restriction can be relaxed through use of the reaction coordinate method described in the next section. 

Equation (2.19), which concerns a situation without processes in the biofilm, can be extended to include transformation of a substrate, an electron donor (organic matter) or an electron acceptor, e.g., dissolved oxygen. If the reaction rate is limited by j ust one substrate and under steady state conditions, i.e., a fixed concentration profile, the differential equation for the combined transport and substrate utilization following Monod kinetics is shown in Equation (2.20) and is illustrated in Figure 2.8. Equation (2.20) expresses that under steady state conditions, the molecular diffusion determined by Fick s second law is equal to the bacterial uptake of the substrate. 

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