1

Mv'v

FIGURE 5.3 Scale and intensity of segregation (from Brodkey, 1975).

is a diffusion process through the interfacial area between layers of different fluids, accompanied by chemical reactions, if any. Molecular diffusion leads to complete mixing and dissipates concentration fluctuations.

In addition to an examination of length scales, it is useful to carry out quantitative examination of different relevant time scales of the mixing processes. Comparison of these time scales with the characteristic time scales of chemical reactions will be useful to determine the rate-controlling step in reactive flow processes.

The characteristic time scale for convection can be written:

c Ur Qr where LR is the characteristic length scale and UR the characteristic velocity scale of the reactor. The second term on the right-hand side is similar to the mean circulation time in the reactor, which is a ratio of reactor volume, VR, and circulatory flow within the reactor, QR. The characteristic time for the turbulent dispersion may be estimated as the ratio of the square of the characteristic length scale of the reactor to the effective turbulent dispersion coefficient (rD). Alternatively, it may be estimated as the ratio

FIGURE 5.4 Schematic representation of small-scale mixing processes (from Baldyga and Bourne, 1984). (a) Reduction of length scale due to deformations within the inertial sub-range. (b) Creation of large interfacial area by vorticity acting on fluid elements of initial thickness of order Ak.

FIGURE 5.4 Schematic representation of small-scale mixing processes (from Baldyga and Bourne, 1984). (a) Reduction of length scale due to deformations within the inertial sub-range. (b) Creation of large interfacial area by vorticity acting on fluid elements of initial thickness of order Ak.

of the characteristic length scale of the reactor to the square root of turbulent kinetic energy, k:

An estimate of the effective turbulent dispersion coefficient for any reactor is generally difficult because of the spatial variation in the dispersion coefficient within the reactor. A first-level approximation may be based on average turbulence kinetic energy and turbulent energy dissipation rates. These two time scales representing convection and turbulent dispersion determine large-scale or macromixing in the reactor.

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)

where Ls is the segregation length scale and e is the rate of dissipation of turbulent energy. The characteristic time for the engulfment step (tE) can be estimated as (Baldyga and Bourne, 1989)

where v is kinematic viscosity. This equation may be used for liquid systems with Schmidt number less than 4000. Alternatively, a modified form of Corrsin's equation can also be used (Pohoreki and Baldyga, 1993):

The diffusion time scale (tDS) can be estimated as (Baldyga and Bourne, 1984)

where Sc is the Schmidt number, defined as the ratio of kinematic viscosity to molecular diffusivity.

These time scales of turbulent mixing processes need to be examined with reference to other important time scales of interest such as reaction time scale, average residence time and so on. While doing such an analysis, it may be easier to regroup these five steps into two categories: (1) macromixing processes, characterized by tmacro and (2) micromixing processes, characterized by tmicro. When one of the micromixing step is rate controlling, tmicro can be equated to the characteristic time scale of that particular micromixing step. It is also possible to define effective time scale when both step 3 and step 4 influence the micromixing process. This effective time scale can be expressed in the form (Ranade, 1993):

where M is the ratio of tMS and tE. This effective time scale reduces to tE for small values of time and to tMS for small values of tE. When there is an interaction between macromixing and micromixing processes, it is not possible to formulate a simple expression for characteristic time scale. The time scales discussed above are used to classify different reactive mixing models and to examine the available modeling approaches in the following section.

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