Turbulence Theories

Probably Fl is a function of particle Reynolds number and concentration, but Fig. 6-33 gives Durand s empirical correlation for Fl as a function of particle diameter and the input, feed volume fraction solids, Cs = QsKQs + Ql)- The form of Eq. (6-145) may be derived from turbulence theory, as shown by Davies (Chem. Eng. Sci., 42, 1667-1670 [1987]). 

Hanratty (pipe flow) Hinze (turbulence theory) -1.2 0.6 -0.6 ... 

Baldyga, J. and Bourne, J.R., 1984b. A fluid mechanical approach to turbulent mixing and chemical reaction. Part II Micromixing in the light of turbulence theory. Chemical Engineering Communications, 28, 243-258. 

It seems probable that a fruitful approach to a simplified, general description of gas-liquid-particle operation can be based upon the film (or boundary-resistance) theory of transport processes in combination with theories of backmixing or axial diffusion. Most previously described models of gas-liquid-particle operation are of this type, and practically all experimental data reported in the literature are correlated in terms of such conventional chemical engineering concepts. In view of the so far rather limited success of more advanced concepts (such as those based on turbulence theory) for even the description of single-phase and two-phase chemical engineering systems, it appears unlikely that they should, in the near future, become of great practical importance in the description of the considerably more complex three-phase systems that are the subject of the present review. 

As a result, the turbulent-flow field in a stirred vessel may be far from isotropic and homogeneous. Some of the cornerstones of turbulence theory, however, start from the assumption that production and dissipation of turbulent kinetic energy balance locally. In many chemical engineering flows, this... 

Notwithstanding these difficulties, it is sometimes possible to manipulate a non-linear expression using the usual rules of calculus into a form that can be further simplified using high-Reynolds-number turbulence theory. For example, using... 

Micromixing in the light of turbulence theory. Chemical Engineering... 

This equation is predicted by the mixing cell model, and turbulence theories put forward by Aris and Amundson130 and by Prausnttz(31). 

General discussions of several aspects concerning the treatment of chemical reactions with diffusion are given by Damkohler (D2), Horn and Kiichler (H12), Prager (P7), Schoenemann and Hofmann (SlO), and Trambouze (Til). Corrsin (C21) has discussed the effects of turbulence on chemical reactions from the fundamental point of view of turbulence theory. We will first discuss the application of each type of model to chemical reactors. Then a short comparison will be made between the different approaches. 

The recent advances in turbulence theory have been so rapid and the experimental investigations sufficiently extensive that it is beyond the scope of the present discussion to consider these matters in detail. However, before the influence of turbulence upon thermal and material transport is considered, it is worth while to describe briefly some of the characteristics of turbulent flow from an experimental point of view. The description will be limited to the characteristics of steady uniform flow... 

The following considerations are based on an overview of horizontal diffusion theories by Peeters et al. (1996). Turbulence theory suggests a power law with unknown exponent m to describe the growth of cloud size a2 with elapsed time / ... 

Data from some authors (see [10]) suggest that the fraction Sh/Sc,/3 vs. Reynolds number, as defined above, correlate well (see Fig. 5.3-6). The eddy size from the isotropic turbulence theory is calculated from the following expression ... 

In Moffat s review [1] an excellent account is given of the relation between the elementary approach and modern statistical methods in turbulence theory. In particular, by analogy with kinetic theory we easily derive for the diffusion coefficient of a scalar admixture the expression... 

Droplet Breakup—High Turbulence This is the dominant breakup mechanism for many process applications. Breakup results from local variations in turbulent pressure that distort the droplet shape. Hinze [Am. Inst. Chem. Eng.., 1, 289-295 (1953)] applied turbulence theory to obtain the form of Eq. (14-190) and took liquid-liquid data to define the coefficient ... 

In turbulence theory, this is also known as the concentration macroscale, which plays an important role in the interpretation of micromixing phenomena. 

Turbulence theory provides a classical approach to mixing phenomena. This is a natural way for mechanical engineers and specialists of combustion, who are very familiar with the methods of fluid mechanics. However, when complex chemical reactions are involved, the use of the formalism of turbulence alone seems to lead to a deadlock, as has been pointed out by several authors. An excellent presentation of the state of the art can be found in the recent literature especially by Brodkey (16, 2, 27) and Patterson (3). These reviews reveal no major breakthrough, and only slow progress on a difficult road. 

The smallest size for turbulent eddies is given by the Kolmogorov microscale A. Energy loss below this size only occurs via viscous dissipation. Here also, several scales have been introduced in the framework of turbulence theory, depending whether velocity or concentration fluctuations are considered, namely Ak> Ag, and Ac (see Table I). In liquids, Ak is typically between 10 and 100 ym. The Kolmogorov microscale Ak is frequently used in the in-... 

Brodkey, R.S. "Application of Turbulence Theory to Mixing Operations" Acad. Press, New-York, 1975... 

Batch suspension reactors are, theoretically, the kinetic equivalent of water-cooled mass reactors. The major new problems are stabilization of the viscous polymer drops, prediction of particle size distribution, etc. Particle size distribution was found to be determined early in the polymerization by Hopff et al. (28, 29,40). Church and Shinnar (12) applied turbulence theory to explain the stabilization of suspension polymers by the combined action of protective colloids and turbulent flow forces. Suspension polymerization in a CSTR without coalescence is a prime example of the segregated CSTR treated by Tadmor and Biesenberger (51) and is discussed below. In a series of papers, Goldsmith and Amundson (23) and Luss and Amundson (39) studied the unique control and stability problems which arise from the existence of the two-phase reaction system. 

Fig. 12 shows that the description of the measured data is fairly good. This is one more example which demonstrates the usefulness of Kolmogoroff s turbulence theory to correlate hydrodynamic parameters in BCR. 

Using the isotropic turbulence theory of Kofmogoroff, Baird and Rice ( 1) deduced the following expression with a single arbitrary constant to describe mixing in large diameter bubble columns... 

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