Kolmogoroff-theory

The final result is very well known in experimental studies and is usually used to evaluate a32 from known and s values. Other studies in this field include the work by Shinnar and Church (S7) who used the Kolmogoroff theory of local isotropy to predict particle size in agitated dispersions, and an analysis by Levich (L3) on the breakup of bubbles. Levich derived an expression for the critical bubble radius (the radius at which breakup begins) ... 

The Kolmogoroff theory implies that the mass-transfer coefficients are the same at equal power inputs. Results to the contrary are, however, noted. In a... 

Levins, D.M. Glastonbury, J. Application of Kolmogoroffs theory to particle-liquid mass transfer in agitated vessels. Chem. Eng. Sci. 1972, 27, 537-542. 

Figure 17.6 Correlation of stirred tank data based on Kolmogoroffs theory - ...

The Kolmogoroff theory can account for the increase in mass transfer rate with increasing system turbulence and power input, but it does not take into consideration the important effects of the system physical properties. The weakness of the slip velocity theory is the fact that the relationship between terminal velocity and the actual slip velocity in a turbulent system is really unknown. Nevertheless, on balance, the slip velocity theory appears to be the more successful for solid-liquid mass transfer in agitated vessels. 

Middleman, S. (1965) Mass transfer from particles in agitated systems applications of Kolmogoroff theory. AIChEJ, 11, 750-752. 

This correlation is obtained based on the surface renewal model and KolmogorofFs theory of isotropic turbulence. In the model, the liquid-solid suspension is considered as a homogeneous phase, and consequently an estimation scheme of the physical properties of the suspension from the individual phase is required. 

The full meaning of these relationships are not yet fully understood, but it appears from these preliminary results that this form of the Kolmogoroff theory can be used to predict the amount of coagulum formed during emulsion polymerizations. Further experiments are in progress. 

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... 

Kolmogoroff A (1958) Collected works on the statistical theory of turbulence. Akademic Verlag, Berlin... 

Stresses acting on micro-organisms in (a) to (c) are derived on the premise that the flow forces originate from the turbulent motion of the carrier medium. In almost all cases, turbulence is assumed to be locally isotropic and homogeneous which greatly simplifies the analysis and allows the application of the Kolmogoroff s theory of turbulence to the problem [81]. The Kolomogoroff micro-scale of turbulence,... 

Kolmogoroff, A. N. Sammelband zur Theorie der statistischen Turbulenz Akademieverlag Berlin, 1958. 

Reynolds (Rl, R2) was one of the earlier investigators to appreciate the random nature of turbulence. The dimensionless parameter bearing his name is widely used as a measure of the physical characteristics of steady, uniform flow. Such a measure is essentially macroscopic and does not describe the local or transient behavior at a point in the stream. In recent years much effort has been devoted to understanding the basic mechanism of momentum transport by turbulence. The early work of Prandtl (P6), Taylor (Tl), Karmdn (Kl), and Howarth (K4) laid a basis for the statistical theory of turbulence which is apparently in reasonable agreement with experiment. More recently Onsager (03), Corrsin (C6), and Kolmogoroff (K10) extended the statistical theory of turbulence to describe the available experimental data in terms of kinetic-energy... 

We have in this way obtained a generalization of Einstein s theory of the interaction between matter and radiation including multiple photon processes and involving transition probabilities. But there is a basic difference. The operator definite positive. We no longer have a simple addition of transition probabilities. This corresponds exactly to the interference of probabilities discussed in Section IV. The process is not of the simple Chapman-Smoluchowski-Kolmogoroff type (Eq. (11)) the operator transition probability. As the result, the second of the two sequences discussed above may decrease the effect of the first one. It is very interesting that even in the limit of classical mechanics (which may be performed easily in the case of anharmonic oscillators) this interference of probabilities persists. This is in agreement with our conclusion in Section IV. 

Several workers (Kolbel et al. [40, 41], Deckwer et al. [17], Michael and Reicheit [42]) have investigated the heat transfer in BSCR versus solid concentration and particle diameters. Deckwer et al. [17] applied Kolmogoroff s theory of isotropic turbulence in combination with the surface renewal theory of Higbie [43] and suggested the following expression for the heat transfer coefficient in the Fischer-Tropsch synthesis in BSCR ... 

Accepting Kolmogoroff s theory of local isotropy, they find for the critical drop size the relation... 

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 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... 

Most recently, Sano et al.121 derived a relation for the liquid-solid mass-transfer coefficient (or Sherwood number) based on Kolmogoroff s theory for isotropic turbulence. The Reynolds number based on this theory is defined in terms of , the rate of energy dissipation per unit mass of liquid, dp, the specific surface diameter, and vL, the kinematic viscosity of liquid. Thus, the modified Reynolds number Re was defined as Re = Edp/vl. The Sherwood number was correlated as... 

As outlined above, steady-state theories for the liquid-solid mass transfer are largely classified into two categories i.e., those based on Kolmogoroff s theory and those based on the terminal velocity-slip velocity approach. 

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