Turbulent micro-mixing

In turbulent flow, micro-mixing takes place at the level of the smallest eddies. According to turbulence theory, the size of the smallest eddies is related with the specific energy dissipation e. 

Note that v is the kinematic viscosity of the liquid (v = (i/p the Greek letter v should not be confused with the itdic letter v). It is assumed that within the smallest eddies there is laminar flow where eventually almost all of the energy dissipation occurs. 

The Kolmogorov micro-scale is also indicative for the rate of micro-mixing. Concentration differences in the liquid are rapidly reduced by turbulent mixing, and molecular diffusion within the smallest eddies takes care of the fin equalization. In general, the time required for non-steady state diffusion to equalize concentrations to some extent (say about 90%) across a distance x can be estimated putting the Fourier number equal to about 0.1  

Here 0 s the diffusion coefficient of the component to be mixed. The time for diffusion within the smallest eddies can be estimated by substituting X for x  

The diffusion time can be calculated easily from the specific energy dissipation rate with eq. (4.7)  

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Cavitations generate several effects. On one hand, both stable and transient cavitations generate turbulence and liquid circulation - acoustic streaming - in the proximity of the microbubble. This phenomenon enhances mass and heat transfer and improves (micro)mixing as well. In membrane systems, increase of fiux through the membrane and reduction of fouling has been observed [56]. 

M 39] [P 37] Using an azo-type competitive reaction, the selectivities were compared for the P- and V-type micro mixers having straight and oblique fluid injection, respectively [41]. In this way, laminar- and turbulent-flow mixing achieved by vertical interdigital microstructured mixers can be compared. The selectivities of the turbulent V-type mixer are better to some extent as compared with the P-type device however, neither approaches the characteristics of the ideal tubular reactor. The micro devices, however, are better than a conventional jet mixer. 

Stirring vessels. Upon the examination of different stirring operations it was indeed found that the intensively formulated process parameter P/V represented the pertinent scale-up criterion only if the stirring power has to be dissipated in the volume as evenly as possible (micro-mixing, isotropic turbulence). Examples of this are the dispersion of a gas in a liquid or the dispersion of immiscible liquids s. [22]. 

In stirring, distinction is made between micro- and macro-mixing. Micro-mixing concerns the state of flow in the tiniest eddies. It is determined by the kinematic viscosity, v, of the liquid and by the dissipated power per unit of mass, = P/pV. Correspondingly, the so-called Kolmogorov s micro-scale k. of the turbulence is laid down as being k = (v3/e)1 4. (By the way, this equation is clearly derived from dimensional analysis )... 

From the viewpoint of dimensional analysis, the terms macro-mixing and micromixing used in the Theory of Turbulence are misleading, because they confuse the issue discussed above. In performing model experiments it does not matter whether the state of flow corresponds to the macro- or micro-mixing, but whether we succeeded in obtaining the working point of the same pi-space. 

Brodkey [56] stated that only with the advent of the modem turbulence theory a deep understanding of micro-mixing processes and turbulent scalar transfer processes on a microscopic level was possible and that this theory enabled the definition of measurable mixing criteria. Knowledge of the turbulence parameter made it possible to estimate the degree of mixing. The parameters could be estimated from the geometry of the flow system and from simple empirical expressions. The... 

It was also established that the presence of solid particles (20 < dp [pm] < 1300 and

negligible effect upon the micro-mixing [169]. In [19] it was found that the selectivity was only influenced at high mass fractions

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The Taylor micro scales of turbulence can be calculated from the same functions /(t, x) and g t,x) using (1.317) and (1.319), respectively. However, the resulting scales do not characterize any distinct group of turbulence vortices and are thus not very useful in describing micro mixing in stirred tanks. 

Always suspect micro-mixing to be important when reactions are fast and lab-scale studies show minimal side reactions. If possible, make the quantitative comparison using the appropriate reaction scheme and micro-mixing modulus. Laboratory reactors tend to run at much higher turbulence levels than plant reactors. 

The fundamental approaches to definition of turbulent flows macro-kinetics and macro-mixing processes are considered in [136-139]. Special attention was focused on micro-mixing models in the context of method based on equation for density of random variables probabilities distribution. Advantage of this method is that we can calculate average rate of chemical reaction if know the corresponding density of concentration and temperature possibility distribution. 

Figure 3.19. The values of characteristic times of turbulent mixing Tturb (1), micro-mixing Xmicro (3-6), and meso-mixing Xmeso (2) in dependence on reaction mixture movement rate V. The values of dynamic viscosity 0,001 (3), 1 (4), 10 (5), 50 (6) Pa-sec. d = 0,025m, p = 1000 kg/m, Vc = 4 m/sec.

Various operations in the field of chemical engineering and in combustion can be characterized by the simultaneous interaction of two processes, namely the transfer of heat and mass and chemical reaction. However, the determination of mean reaction rates in turbulent flows requires detailed knowledge about fluctuations of scalar quantities such as species concentrations and enthalpy. Due to the non-linear character of chemical reaction the calculation of mean reaction rates based on mean values of temperature and species concentrations is only possible in special circumstances, such as practically infinitely fast micro-mixing-rates or very small fluctuations of the scalar variable around its mean value. 

For gases, 5c 1, for hquids. Sc 1. This implies that in turbulent flow of liquids, the species concentration field contains smaller scale structures than the velocity field. Similar to the decay time of the turbulent eddies in the velocity field, Tu, (12.2-1), the decay time of the eddies in the species concentration field, the previously introduced micro-mixing time, xy, can be modeled in terms of the correlation of the species mass fraction fluctuations, the so-called scalar (co-) variance, (K F), and its dissipation rate, the so-called scalar dissipation rate, sy. 

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