Micromixing time

Consider an inert (non-reacting) scalar p in a poorly micromixed PFR. The IEM model for this case reduces to 

Since the inlet concentrations will have no effect on the final result, for simplicity we let pm 0) = 0 and //2,(0) = 1. Applying (1.18) to (1.20), it is easily shown that the scalar mean is constant and given by ( p(a)) = p2. 

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The failure of conventional criteria may be due to the fact that it is not only one mixing process which can be limiting, rather for example an interplay of micromixing and mesomixing can influence the kinetic rates. Thus, by scaling up with constant micromixing times on different scales, the mesomixing times cannot be kept constant but will differ, and consequently the precipitation rates (e.g. nucleation rates) will tend to deviate with scale-up. 

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. 

The micromixing time has an exact definition in terms of the rate of decay of concentration fluctuations. The mixture fraction is defined in Chapter 5. 

Note that in order to close (1.16), the micromixing time must be related to the underlying flow field. Nevertheless, because the IEM model is formulated in aLagrangian framework, the chemical source term in (1.16) appears in closed form. This is not the case for the chemical source term in (1.17). 

In Section 3.2, we show that under the same conditions the right-hand side of (1.24) is equal to the negative scalar dissipation rate ((3.45), p. 70). Thus, the micromixing time is related to the scalar dissipation rate e and the scalar variance by... 

Choosing the micromixing time in a CRE micromixing model is therefore equivalent to choosing the scalar dissipation rate in a CFD model for scalar mixing. 

In the CRE literature, turbulence-based micromixing models have been proposed that set the micromixing time proportional to the Kolmogorov time scale ... 

Perhaps the simplest Lagrangian micromixing model is the interaction by exchange with the mean (IEM) model for a CSTR. In addition to the residence time r, the IEM model introduces a second parameter tm to describe the micromixing time. Mathematically, the IEM model can be written in Lagrangian form by introducing the age a of a fluid particle, i.e., the amount of time the fluid particle has spent in the CSTR since it entered through a feed stream. For a non-premixed CSTR with two feed streams,100 the species concentrations in a fluid particle can be written as a function of its age as... 

Technically, there is no reason to limit consideration to homogeneous flows. However, since the micromixing time and location-conditioned expected values will be independent of particle position, we do so here in order to simplify the notation and to isolate the problems associated with the chemical kinetics. 

If the characteristic micromixing time scale is much smaller than Atf, then care must be taken in implementing tiie intra-cell processes. For example, several smaller time steps may be required to represent mixing and chemical reactions at each iteration. 

If tmicro > Tmeso > Tmacro O 011 the process is micromixing controlled. Micromixing is a complex phenomenon (Section 2.4), but for most liquids engulfment is the longest step. In this case, micromixing time is the inverse of engulfment rate (E) and can be estimated by... 

The status of micromixing is described by the parameter characteristic time constant for micromixing , fM, which can be simply called micromixing time. It represents the time needed to achieve completely uniform micromixing, and is correlated to the microscale A by... 

As mentioned above, to determine one value for fM needs an experimental curve of X, versus u0 for the case where the impinging velocity m() is taken as the governing variable. The plot of a set of typical data is indicated in Fig. 10.12. From the curve the critical point at which Xs turns from a varying value to a small constant is found at u -0.184 m/s. With the known reaction rate constant at 298 K, the micromixing time can be calculated as... 

In comparison, the micromixing time in a general stirred tank reactor (STR) ranges from 500 to 1000 ms. If it is noted that the impingement zone in the SCISR occupies only a small volumetric fraction (less than 20% of the total), the value of 192 ms for the micromixing time in the SCISR indicates clearly that impinging streams do promote micromixing very efficiently. 

The data determined for the micromixing time, rM, from various curves are illustrated in Fig. 10.13, which meet approximately the relationship below ... 

The reason for the decrease in the micromixing time with the increase in impinging velocity is clear a higher impinging velocity implies an increased energy dissipation rate and thus more efficient micromixing. 

Figure 10.13 Influence of impinging velocity on micromixing time at oe=] 5...

Comparison between measured and theoretically predicted results for micromixing time... 

The micromixing time calculated for the four sets of conditions yielding the data listed in Table 10.1 are given in the fourth column of Table 10.2 and the corresponding data measured experimentally are listed in the fifth column of the same table for comparison. 

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