Micromixing, local—

Thus, the reactor will be perfectly mixed if and only if = at every spatial location in the reactor. As noted earlier, unless we conduct a DNS, we will not compute the instantaneous mixture fraction in the CFD simulation. Instead, if we use a RANS model, we will compute the ensemble- or Reynolds-average mixture fraction, denoted by ( ). Thus, the first state variable needed to describe macromixing in this system is ( ). If the system is perfectly macromixed, ( ) = < at every point in the reactor. The second state variable will be used to describe the degree of local micromixing, and is the mixture-fraction variance (maximum value of the variance at any point in the reactor is ( )(1 — ( )), and varies from zero in the feed streams to a maximum of 1/4 when ( ) = 1/2. 

Note that we have used the fluid velocity U to describe convection of particles, which is valid for small Stokes number. In most practical applications, / is a highly nonlinear function of c. Thus, in a turbulent flow the average nucleation rate will depend strongly on the local micromixing conditions. In contrast, the growth rate G is often weakly nonlinear and therefore less influenced by turbulent mixing. 

For non-linear chemical reactions that are fast compared with the local micromixing time, the species concentrations in fluid elements located in the same zone cannot be assumed to be identical (Toor 1962 Toor 1969 Toor and Singh 1973 Amerja etal. 1976). The canonical example is a non-premixed acid-base reaction for which the reaction rate constant is essentially infinite. As a result of the infinitely fast reaction, a fluid element can contain either acid or base, but not both. Due to the chemical reaction, the local fluid-element concentrations will therefore be different depending on their stoichiometric excess of acid or base. Micromixing will then determine the rate at which acid and base are transferred between fluid elements, and thus will determine the mean rate of the chemical reaction. 

For fast equilibrium chemistry (Section 5.4), an equilibrium assumption allowed us to write the concentration of all chemical species in terms of the mixture-fraction vector c(x, t) = ceq( (x, 0). For a turbulent flow, it is important to note that the local micromixing rate (i.e., the instantaneous scalar dissipation rate) is a random variable. Thus, while the chemistry may be fast relative to the mean micromixing rate, at some points in a turbulent flow the instantaneous micromixing rate may be fast compared with the chemistry. This is made all the more important by the fact that fast reactions often take place in thin reaction-diffusion zones whose size may be smaller than the Kolmogorov scale. Hence, the local strain rate (micromixing rate) seen by the reaction surface may be as high as the local Kolmogorov-scale strain rate. 

A New Chemical Method for the Study of Local Micromixing Conditions in Industrial Stirred Tanks... 

At a specified point in the tank, 100 cm of hydrochloric acid 1.2N(Cg0 = 0.83 moles. m- after complete mixing in the tank in the absence of reaction) were injected by means of a cylinder obturated at its lower end by an elastic membrane which becomes inflated and bursts out when submitted to the pressure of the liquid pushed by a piston. The acid was thus injected without any preferential direction. This locally released acid B triggers reactions f2j and 3]. If the local micromixing state is perfect, the acid is totally and instantaneously neutralized, as it is in stoichiometric defect with respect to A. The first reaction being very fast as compared to the second one, the precipitate S does not appear. Conversely, if mixing of the acid is not instantaneously... 

When local micromixing is slow compared to the reaction time scale and the macromixing time scale is smaller than the process time scale, the performance of a reactive flow process is controlled only by the micromixing. In such cases, though there is no macroscopic segregation, reactants are not mixed on a molecular scale (see the right bottom case of Fig. 5.5). Several micromixing models have been developed to simulate such reactive flow processes. Some of the widely used models are ... 

Crystal-crystal impact a function of both the local micromixing environment and the overall macromixing circulation... 

Barth OLE ).P., David R., Villermaux J, A new chemical method for the study of local micromixing conditions in industrial stirred tanks, ACS Symp. Ser. 196 (1982), p. 545-554... 

For non-isothermal or non-linear chemical reactions, the RTD no longer suffices to predict the reactor outlet concentrations. From a Lagrangian perspective, local interactions between fluid elements become important, and thus fluid elements cannot be treated as individual batch reactors. However, an accurate description of fluid-element interactions is strongly dependent on the underlying fluid flow field. For certain types of reactors, one approach for overcoming the lack of a detailed model for the flow field is to input empirical flow correlations into so-called zone models. In these models, the reactor volume is decomposed into a finite collection of well mixed (i.e., CSTR) zones connected at their boundaries by molar fluxes.4 (An example of a zone model for a stirred-tank reactor is shown in Fig. 1.5.) Within each zone, all fluid elements are assumed to be identical (i.e., have the same species concentrations). Physically, this assumption corresponds to assuming that the chemical reactions are slower than the local micromixing time.5... 

The rapid local micromixing and poor macromixing of the reactants... 

In the preceding two sections, it was shown how to measure local micromixing intensity using intermaterial area distribution [H(log p)] and length scale distributions [H(logs)]. These tools are not directly applicable to 3D flows, because the resolution of the smallest length scales in 3D mixture structures is, to date, computationally prohibitive. In this section we present a predictive method for striation thickness distributions applicable to either 2D or 3D chaotic flows. 

Barthole, J., R. David, and J. Villermaux (1982). A New Chemical Method for the Study of Local Micromixing Conditions in Industrial Stirred Tanks, ISCRE, Boston. 

Radna A, Kind M (2006) Specific power input and local micromixing times in turbulent Taylor-Cotrette flow. Exp Fluids 41 513-522... 

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