Small Scales in Turbulence

In Chapter 8, we characterized the large scales of turbulence by introducing the tnrbnlent velocity Mnns and the integral scale It. These two quantities are used to quantify two key processes of turbulence, the turbulent diffusion coefficient Kt Urtcdt (equation [8.29] of Chapter 8) and the kinetic energy dissipation rate in a turbulent flow  

1 A more comprehensive presentation of Kolmogorov s theory can be found in the book by M. Lesieur, Turbulence in Fluids (Springer-Verlag, 2007). 

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The first group of models ( eddy models) assumes that the liquid renewal is due to small-scale eddies of the turbulent field. These models are based on idealized eddy structures of turbulence in the bubble vicinity. Lamont and Scott [1] have assumed that the small scales of turbulent motion, which extend from the smallest viscous motions to the iner-... 

Consider now motion of small particJes in turbulent flow of liquid. Assume that the volume concentration of particJes is small enough, so it is possible to neglect their influence on the flow of hquid. The large-scale pulsations transfer a particJe together with layers of hquid adjoining to it. Small-scale pulsations with A R, where R is particle radius, cannot involve the particJe in their motions -the particle behaves in this respect as a stationary body. Pulsations of intermediate scales do not completely involve the particle in their motion. Consider the case most interesting for apphcations, when respective densities of particle p and external liquid are only slightly different from one another, and radius of the particle is much less than inner scale of turbulence, that is R A . Thus, for water-oil emulsion pjp 1.1-1.5. Let Uq be the velocity of hquid at the particle s location, and Ui the velocity of particJe relative to hquid. At full entrainment of particle by the hquid, the same force would ad on the particle as on... 

These criteria can best be met on a small scale in a horizontal cylinder of high length to diameter ratio. The feed should enter the cylinder at a low velocity to avoid creating turbulence which could break up the settling pattern and close to the line of the interface between the two phases. 

As fast liquid-phase chemical reactions usually occur in the diffusion area, a macrokinetic approach should be used to reveal their aspects and specificity, as it describes reactions with consideration of the heat and mass transfer [47, 48]. This approach to the modelling of fast polymerisation processes, such as the cationic polymerisation of isobutylene [27-30,38,39,49], has revealed aspects of its behaviour. Many problems of fast chemical reactions were solved via the creation of sufficiently high turbulence in a reaction zone within small-scale tubular turbulent devices. 

Because of the low chemical reaction time (T hem the process of liqnid-phase BR chlorination with molecular chlorine in solution should be related to fast chemical reactions which have to be carried out according to fundamentally new technology using high efficiency, small-scale tubular turbulent jet reactors. 

Thus, using small-scale tubular turbulent divergent-convergent-type reactors at the stage of uniform gas-liquid mixture formation, prior to feeding this mixture into a stirred tank polymerisation reactor, results in a notable (virtually hy one order of magnitude) increase in the phase contact surface. A developed phase interface facilitates the uniform saturation of liquid products with monomers and hydrogen. In this case, it allows improved performance characteristics of the EPR in contrast to stirred tank reactors. 

In order to verify this effect, we require the ability to experimentally control the value k of the smaller scales of tuibulence. We have limited ourselves, in this chapter, to explaining the role played by the small scales of turbulence in the mixing process assoeiated with a chemical reaction. In Chapter 11, we provide some necessary eomplements from the theory of turbulence, which will enable us to quantify k ... 

Peclet number independent of Reynolds number also means that turbulent diffusion or dispersion is directly proportional to the fluid velocity. In general, reactors that are simple in construction, (tubular reactors and adiabatic reactors) approach their ideal condition much better in commercial size then on laboratory scale. On small scale and corresponding low flows, they are handicapped by significant temperature and concentration gradients that are not even well defined. In contrast, recycle reactors and CSTRs come much closer to their ideal state in laboratory sizes than in large equipment. The energy requirement for recycle reaci ors grows with the square of the volume. This limits increases in size or applicable recycle ratios. 

For a chemical reaction such as combustion to proceed, mixing of the reactants on a molecular scale is necessary. However, molecular diffusion is a very slow process. Dilution of a 10-m diameter sphere of pure hydrocarbons, for instance, down to a flammable composition in its center by molecular diffusion alone takes more than a year. On the other hand, only a few seconds are required for a similar dilution by molecular diffusion of a 1-cm sphere. Thus, dilution by molecular diffusion is most effective on small-scale fluctuations in the composition. These fluctuations are continuously generated by turbulent convective motion. 

Turbulent eddies larger than the cloud size, as such, tend to move the cloud as a whole and do not influence the internal concentration distribution. The mean concentration distribution is largely determined by turbulent motion of a scale comparable to the cloud size. These eddies tend to break up the cloud into smaller and smaller parts, so as to render turbulent motion on smaller and smaller scales effective in generating fluctuations of ever smaller scales, and so on. On the small-scale side of the spectrum, concentration fluctuations are homogenized by molecular diffusion. 

Experiments on a small scale with stoichiometric methane-air mixtures were carried out by Chan et al. (1980). Comparisons of results of these experiments with those performed by Moen et al. (1982) revealed that simple scaling is not possible for the results of explosions with very high flame speeds, in other words, flame speeds resulting from very intense turbulence. 

Mixing is accomplished by the rotating action of an impeller in the continuous fluid. This action shears the fluid, setting up eddies w hich move through the body of the system. In general the fluid motion involves (a) the mass of the fluid over large distances and (b) the small scale eddy motion or turbulence which moves the fluid over short distances [21, 15]. 

The thin film reactor for the continuous sulfonation of fatty acid esters was introduced by the Witco Technical Center in Oakland, New Jersey [46]. Hurl-bert et al. designed this type of reactor for small-scale sulfonation with S03 [47,48]. The reaction partners could be filled into the reactor through three inlets. One was for the carrier gas (air or nitrogen), one for the liquefied ester that is picked up from the carrier gas, and the last one was for the vaporized S03. The ester and the S03 reacted in a turbulent liquid film. Details of this reactor are given by Kapur et al. [46]. 

The same results are obtained from Equations (3.38) and (3.39), which apply to the turbulent flow of ideal gases. Thus, tube radius and length scale in the same way for turbulent liquids and gases when the pressure drop is constant. For the gas case, it is further supposed that the large and small reactors have the same discharge pressure. 

Although vortices of small scale, such as Kolmogorov scale or Taylor microscale, are significant in modeling turbulent combustion [4,6-9], vortices of large scale, in fhe order of millimeters, have been used in various experiments to determine the flame speed along a vorfex axis. 

Daneshyar, H. D. and Hill, F. G., The structure of small-scale turbulence and its effect on combustion in spark ignition engines. Progress in Energy and Combustion Science, 13,47-73,1987. 

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