Time constant, Corrsin

The characteristic time constant for mesomixing by the inertial-convective process (Corrsin, 1964) is given by... 

The general equations for chemical reaction in a turbulent medium are easy to write if not to solve (2). In addition to those for velocities (U = U + uJ and concentrations (Cj = Cj + Cj), balance equations for q = A u, the segregation ( , and the dissipations e and eg can be written (3). Whatever the shape of the reactor under consideration (usually a tube or a stirred tank), the solution of these equations poses difficult problems of closure, as u S, 5 cj, cj, and also c cj, c Cj in the reaction terms have to be evaluated. The situation is even more complicated when the temperature and the density of the reacting mixture are also fluctuating. Partial solutions to this problem have been proposed. In the case of instantaneous reactions (t << Tg) the "e-quilibrium assumption" applies the mixed reactants are immediately converted and the apparent rate of reaction is simply that of the decrease of segregation, with Corrsin s time constant xs. For instance, with a stoichiometric proportion of reactants, the extent of reaction X is given by 1 - /T ( 2), a simple result which can also be found by application of the IEM model (see (33)). 

Conclusion. There are still uncertainties in the final inter-pretation of mixing and chemical reaction at the molecular level. The IEM model seems to provide a simple basis for representing interaction between particles, even by molecular diffusion. The problem is to decide what is hidden behind the micromixing time tm Corrsin s time constant ts (32) A diffusion constant based on Kolmogorov microscale (113, 114) Further research should be developed in the following directions ... 

The last three steps control the small-scale or micromixing in the reactor. The characteristic time constant for the third step, that is for reduction in segregation scale (inertial-convective mixing), is (Corrsin, 1964 Baldyga, 1989)... 

It is generally accepted that l/o) is proportional to Corrsin s time constant T. This may also be represented by the lEM model with... 

Taylor, Corrsin time constants (Table 1) dissipation time constants (Table 1) initiator availability... 

The general equivalence of time constants estimated over a wide variety of experiments involving gases and liquids and a variety of geometries was shown in a review by Brodkey (1975). He showed that nearly a 10000 fold range in mixing times could be adequately estimated if one has some idea as to the proper value of ro (the characteristic dimension) to be used. This lends some credibility to Corrsin s theory and motivates further examination of Ls and e. 

Example 2-4a Blend Time. Now that we have ways to estimate s and the characteristic length scale Lg, we return to Corrsin s equations in Table 2-4. Probably the most important practical point is that the time constant of mixing scales with (Lg/e). All the rest of the terms in the equation for (Sc 1) are either constants or relatively minor effects of the Schmidt number. For mixing in a pipe, we take the radius of the feed pipe, ro, as the initial integral length scale, and the fluctuating velocity, u, as a measure of the turbulent energy. Thus we can write... 

The inertial mixing (first term) is controlling since its time constant is about eight times the time constant of the Batchelor scale mixing. The Corrsin scale mixing is much faster than the reaction time constant. 

Where T/D is held constant on scale-up, this result reduces to 1/(N N). There are many different ways to make the scaling arguments (see, e.g., Grenville et al., 1995 Grenville and Tilton, 1996, 1997 or Nienow, 1997). The point is that the end result agrees well with Corrsin s approach. The most important thing to recognize is that Lj/e, however it is estimated, must be constant on scale-up to maintain constant blend time. If the dissipation (e) is held constant on scale-up, the blend time will always increase. 

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