B

where v is kinematic viscosity (momentum diffusivity), is the length scale and tK is the time scale. The strongly dissipative nature of fluid motions at these smallest scales indicates 'viscous mixing'. This formation of smaller scales constitutes the essence of mixing. Complete viscous mixing, however, does not ensure complete molecular mixing because in general, momentum diffusivity and molecular diffusivity are different. Molecular mixing is usually characterized by the Batchelor length scale, A.B, which is the penetration depth of the scalar by diffusion in the Kolmogorov time scale, tK and can be written:

XK v

where Sc is the Schmidt number and D is the molecular diffusion coefficient. For most gases, the Schmidt number is of the order of unity and therefore molecular mixing is as fast as viscous mixing. However, for most liquids, the Schmidt number is of the order 1000, which implies much slower molecular mixing. Typical energy and concentration spectra (and dissipation of turbulent kinetic energy and concentration fluctuations) for isotropic turbulence are shown in Fig. 5.2. The upper two curves are for turbulent kinetic energy and the lower two curves are for concentration fluctuations. It can be seen that the spectrum for concentration extends further to the right (towards scales smaller than Kolmogorov scales) than the energy spectrum (for systems with v/D > 1). For accurate simulation of mixing, it is necessary to resolve all the scales contributing to the dissipation of concentration fluctuations. This means that for simulations of reactive mixing in liquids, an even wider (than simulations of turbulent flows) range of length scales, encompassing inertial-convective, viscous-convective and viscous-diffusive sub-ranges, need to be modeled and resolved.

In order to gauge the relative importance and possible interaction between turbulence and chemical reactions, it is necessary to evaluate the various processes involved in reactive mixing. When a fluid element of different component (tracer) is added to the turbulent flow field, molecular mixing (and reaction, if possible) proceeds through several steps/mechanisms, some of which are listed below:

Step 1: convection by mean velocity Step 2: turbulent dispersion by large eddies Step 3: reduction of segregation length scale Step 4: laminar stretching of small eddies Step 5: molecular diffusion and chemical reaction

The fluid element of a tracer is transported within the solution domain by the mean flow field. During this process, turbulent fluctuating motions reduce the characteristic scales of 'lumps' of tracer (turbulent dispersion by large eddies). Generally, chemical engineers use the scale of segregation and intensity of segregation to characterize turbulent mixing (Danckwerts, 1953). The scale of segregation is a measure of the size of the unmixed lumps. Intensity of segregation is a measure of the difference in concentration between neighboring lumps of fluid. The lower the intensity of segregation the more the extent of molecular mixing. These two parameters are demonstrated qualitatively in Fig. 5.3 (for rigorous definitions, see Brodkey, 1975). Convection and turbulent dispersion by large eddies lead to macroscale mixing and do not cause any small-scale mixing. Fluid motions in the inertial sub-range reduce the characteristic inertial inertial

wavelength in ln (k)

FIGURE 5.2 Energy and concentration spectra for isotropic turbulence (from Bakker, 1996).

wavelength in ln (k)

FIGURE 5.2 Energy and concentration spectra for isotropic turbulence (from Bakker, 1996).

scales of lumps of tracer via vortex stretching. This is step 3 mentioned above. Such a reduction in scale increases the interfacial area between segregated lumps of tracer fluid and the base fluid, which increases the rate of mixing by molecular diffusion. However, the increase in interfacial area resulting from inertial sub-range eddies may not be substantial. The mixing caused by this step is typically called 'meso-mixing'. Meso-mixing reduces the scale of mixing substantially but does not significantly affect the intensity of mixing. Engulfment and viscous stretching by Kolmogorov scale eddies lead to substantial increases in the interfacial area for molecular diffusion and therefore, contribute significantly to molecular mixing. A schematic representation of steps 3 and 4 is shown in Fig. 5.4. Inertial sub-range motions reduce the characteristic scale of mixing to the Kolmogorov length scale, XK. Viscous-convective motions (engulfment and stretching) create a large interfacial area for molecular diffusion and reduce the characteristic scales to Batchelor length scale, . The final step

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