Local isotropic turbulence

In many types of contactors, such as stirred tanks, rotary agitated columns, and pulsed columns, mechanical energy is appHed externally in order to reduce the drop si2e far below the values estimated from equations 36 and 37 and thereby increase the rate of mass transfer. The theory of local isotropic turbulence can be appHed to the breakup of a large drop into smaller ones (66), resulting in an expression of the form... 

This case can also be approached using Kolmogoroff s (K9, H15) theory of local isotropic turbulence to predict the velocity of suspended particles relative to a homogeneous and isotropic turbulent flow. By examining this situation for spherical particles moving with a constant relative velocity, varying randomly in direction, Levich, (L3) has demonstrated that... 

In homogeneous, locally isotropic turbulence the velocity-gradient correlation terms in (2.127) simplify to (Pope 2000)... 

In locally isotropic turbulence, the fluctuating velocity gradient and scalar gradient will be uncorrelated, and sf will be null. Thus, at sufficiently high Reynolds number, the scalar-flux dissipation is negligible. 

Kolmogorov, A. N. (1941a). Dissipation of energy in locally isotropic turbulence. Dokl. Akad. Nauk SSSR 32, 19-21. 

From Kolmogoroff s theory of local isotropic turbulence [see Batchelor (Bl)] it follows that u cc e1/3rf1/3 where e is the turbulent energy dissipation per unit time per unit volume. Therefore,... 

The total number of collisions with the dead comers is proportional to the total number of drops in the surface layer opposite the dead comers and—when the theory of local isotropic turbulence holds here also—proportional to the turbulent fluctuation frequency u/d. Near the wall, however, the theory of local isotropic turbulence certainly will not hold and —more or less—stationary large scale eddies will occur. Therefore, centrifugal effects will strongly increase the collision rate when the dispersed phase... 

As conventional Re cannot be computed, a concept of local isotropic turbulence is introduced ... 

The rate parameters of importance in the multicomponent rate model are the mass transfer coefficients and surface diffusion coefficients for each solute species. For accurate description of the multicomponent rate kinetics, it is necessary that accurate values are used for these parameters. It was shown by Mathews and Weber (14), that a deviation of 20% in mass transfer coefficients can have significant effects on the predicted adsorption rate profiles. Several mass transfer correlation studies were examined for estimating the mass transfer coefficients (15, jL6,17,18,19). The correlation of Calderbank and Moo-Young (16) based on Kolmogaroff s theory of local isotropic turbulence has a standard deviation of 66%. The slip velocity method of Harriott (17) provides correlation with an average deviation of 39%. Brian and Hales (15) could not obtain super-imposable curves from heat and mass transfer studies, and the mass transfer data was not in agreement with that of Harriott for high Schmidt number values. 

The numerical value of ks is dictated by the local turbulence around the particles. Based on Kolmogoroff s theory of local isotropic turbulence, this leads to a Reynolds number based on the velocity of the critical eddies responsible for most of the energy dissipation. For solid particles much larger than the Kolmogoroff scale of these eddies, this leads to... 

KolmogorofTs theory Brian et al.,9 Elenkov et al.,26 and Middleman84 used Kolmogoroff s theory of local isotropic turbulence in an attempt to correlate the effective relative velocity with some macroscopic variables, such as stirrer speed and particle diameter. From the dimensionless analysis of agitated slurry reactors,45,84 they suggested a correlation... 

In his treatise "The local structure of turbulence in an incompressible viscous liquid at very high Reynolds numbers , Kolmogorov [289] considered the elements of free turbulence as random variables, which are in general terms accessible to probability theory. This assumes local isotropic turbulence. Thus the probability distribution law is independent of time, since a temporally steady-state condition is present. For these conditions Kolmogorov postulated two similarity hypotheses ... 

The laws of statistical distribution for locally-isotropic turbulence are clearly determined by the kinematic viscosity v and the power per unit mass e = P/p -Dimensional analysis gives the following relationship for the linear dimension 2 of a turbulence element ... 

On the other hand, the so-called macro-scale of turbulence A is given by the size of the primary eddies and is of the order of magnitude of the stirrer diameter. Thus the precondition for the existence of locally isotropic turbulence is sufficient difference between both scales and in a high Re number [364]. 

The division of the kinetic energy into the individual eddy regions takes the form of an energy spectrum E[k). Only those parts of the spectrum in the region of small eddies are of interest, in which locally-isotropic turbulence is expected. Two regions can be clearly distinguished, for which different relationships apply (see Section 1.4.2.2). 

The fluctuating motion of micro-turbulence results in two points separated by Ar exhibiting different velocities. However, it is also possible to define an average value for a turbulent process, which in the case of locally isotropic turbulence is independent of the spacial orientation of the distance Ar ... 

This way of proceeding has nothing to do with the approach [377] of local isotropic turbulence, since it is in no way applicable to the conditions which pertain in the liquid film on the heat transfer wall )... 

Kolmogorov AN (1941) Energy dissipation in locally isotropic turbulence. Dok-... 

An apropriate Re-number expression characteristic of local isotropic turbulence can be derived using the root-mean-square fluctuating velocity postulated by Batchelor (1951). 

Golizin G. S., Fluctuations of Energy Dissipation in locally isotropic Turbulent Flow, Paper Acad. Sd. USSR, 1962, Vol. 144. No. 3, p. 520-523 (in Russian). 

Drop break-up is usually associated with turbulence and is most prominent in sections with high tiubulent shear. Kolmogoroff [from Davies (7)] showed that the maximum diameter a drop can have in a local isotropic turbulent field is given by... 

Kolmogorov, A. (1941b). Dissipation of Energy in the Locally Isotropic Turbulence, Compt. Rend. (Dokl.) Acad. Sci. URSS, 32, 16-18. 

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