Kolmogorov micro scale

According to equation (1.6) the size of the material balls for a relatively intensive mixing operation of P/pV = 1 W/kg in an aqueous liquid (v = 10 m /s) is 32 pm and in a liquid with the viscosity of pure glycerine at room temperature (v = 10 m /s) is already 5.6 mm (see Table 1.1). This shows that viscous systems will always remain to a certain extent segregated, since the Kolmogorov micro-scale a can be comparatively little influenced by the mixing power 2 cc... 

Instead, the Kolmogorov micro-scales (1.328) to (1.330) characterize the dissipation scales of turbulence and might indicate the effectiveness of micromixing in the flow provided that the turbulent energy dissipation rate and the kinematic viscosity of the fluid is known. If the Kolmogorov micro length scale is much larger than the molecular scales, the molecules are not efficiently mixed by turbulent diffusion. 

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 ... 

In turbulent flow the same mechanism takes place inside the smallest eddies, on the Kolmogorov micro scale. It can be shown that the average shear rate within these eddies is proportional to the root of the specific energy dissipation ... 

The validity of Eqs. (3-5) are bond on the condition of fully developed turbulent flow which only exists if the macro turbulence is not influenced by the viscosity. This is the case if the macro turbulence is clearly separated from the dissipation range by the inertial range. This is given if the macro scale A is large in comparison to Kolmogorov s micro scale qp Liepe [1] and Mockel [24] found out by measurement of turbulence spectra s the following condition ... 

For multiphase flow that is normally encountered in fluidized bed reactors, there are two kinds of definitions of the micro-scale first, it is the scale with respect to the smaller one between Kolmogorov eddies and particles second, it is the scale with respect to the smallest space required for two-phase continuum. If the first definition is adopted, the... 

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 )... 

X Kolmogorov s micro-scale of turbulence (Section 10.1) dynamic viscosity... 

The micro-scale 2 of turbulence is according to Kolmogorov described by 2 = (v /fi) [289]. The shear condition on a small-scale is similar to that on a large-scale when the physical properties, here the kinematic viscosity v, and the geometry... 

Kolmogorov micro velocity scale (m/s) flow of gas through a bubble in bubbling bed... 

Another useful and interesting concept is the size of the eddies at which the power of an impeller is evenmally dissipated. This concept utilizes the principles of isotropic turbulence developed by Kolmogorov [1]. The calculations assume some reasonable approach to the degree of isotropic turbulence, and the estimate do give some idea as to how far down in the micro-scale size the power... 

Computational fluid dynamics enables us to investigate the time-dependent behavior of what happens inside a reactor with spatial resolution from the micro to the reactor scale. That is to say, CFD in itself allows a multi-scale description of chemical reactors. To this end, for single-phase flow, the space resolution of the CFD model should go down to the scales of the smallest dissipative eddies (Kolmogorov scales) (Pope, 2000), which is inversely proportional to Re-3/4 and of the orders of magnitude of microns to millimeters for typical reactors. On such scales, the Navier-Stokes (NS) equations can be expected to apply directly to predict the hydrodynamics of well-defined system, resolving all the meso-scale structures. That is the merit of the so-called DNS. 

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