Characteristic time of turbulence

One should know turbulence characteristics to estimate the characteristic time of reagents mixing. They proposed a lot of various methods for estimation of mixing time in turbulent flows particularly the review on this theme and mixing models classification are presented in [130]. Since in [125] liquid movement is described by average over Reynolds Navie-Stocks equations with the use of K-e closing, so the characteristic time of turbulence mixing can be estimated as [1,32,125] ... 

Figure 3.19. The values of characteristic times of turbulent mixing Tturb (1), micro-mixing Xmicro (3-6), and meso-mixing Xmeso (2) in dependence on reaction mixture movement rate V. The values of dynamic viscosity 0,001 (3), 1 (4), 10 (5), 50 (6) Pa-sec. d = 0,025m, p = 1000 kg/m, Vc = 4 m/sec.

Temperature conductivity coefficient Dynamic coefficient of turbulent viscosity Kinematic coefficient of turbulent viscosity Characteristic time of turbulent mixing A pressure drop Input-output pressure drop... 

Lastly, we introduce the characteristic time of turbulence, xt, based on Wnns and It ... 

The last term is always negative. It corresponds to dissipation. Regarding the equation for k, the dissipation term is simply the definition of the rate of dissipation. Regarding the equation for e, dissipation takes place with the characteristic time of turbulence [8.12], since k = s Xt. 

In Table 11.1, we associate the Damkohler number with the chemical reaction, which is restrictive since that number also involves the characteristic time of turbulence, rt. It is also restrictive associating the Schmidt number only to micromixing, since this number is the ratio of the characteristic time of miciomixing to the characteristic time of small-scale maciomixing. Nevertheless, the Damkohler number is the only one that involves chemistry, just as the Schmidt number is the only one that takes into account the diffusion process. 

With the average elongational strain rate of the flow field between the eddies and the relaxation time of the polymer molecules, one can define a dimensionless characteristic number, the Deborah number, which represents the ratio of a characteristic time of flow and a characteristic time of the polymer molecule, and thus one can transfer considerations in porous media flow to the turbulent flow region. 

For laminar flow, the characteristic time of the fluid phase Tf can be deflned as the ratio between a characteristic velocity Uf and a characteristic dimension L. For example, in the case of channel flows confined within two parallel plates, L can be taken equal to the distance between the plates, whereas Uf can be the friction velocity. Another common choice is to base this calculation on the viscous scale, by dividing the kinematic viscosity of the fluid phase by the friction velocity squared. For turbulent flow, Tf is usually assumed to be the Kolmogorov time scale in the fluid phase. The dusty-gas model can be applied only when the particle relaxation time tends to zero (i.e. Stp 1). Under these conditions, Eq. (5.105) yields fluid flow. This typically happens when particles are very small and/or the continuous phase is highly viscous and/or the disperse-to-primary-phase density ratio is very small. The dusty-gas model assumes that there is only one particle velocity field, which is identical to that of the fluid. With this approach, preferential accumulation and segregation effects are clearly not predicted since particles are transported as scalars in the continuous phase. If the system is very dilute (one-way coupling), the properties of the continuous phase (i.e. density and viscosity) are assumed to be equal to those of the fluid. If the solid-particle concentration starts to have an influence on the fluid phase (two-way coupling), a modified density and viscosity for the continuous phase are generally introduced in Eq. (4.92). 

FIGURE 19.6 Relationship between characteristic times of vertical turbulent diffusion tj) and a number of atmospheric reactions tc) for a layer of thickness Az = 10 m and different thermal stability classes (Kramm et al., 1993). For example, the HNO3—NH3—NH4NO3 equilibrium (reaction 2) has a reaction timescale comparable to that of turbulent diffusion under unstable and neutral conditions. 

Compare the characteristic time of drop integration in a turbulent flow Tl"" with the characteristic time of gravitational sedimentation of the emulsion (the electric field is present in both cases) ... 

Mechanism of Turbulent Diffusion 487 The characteristic time of drop enlargement is thus equal to ... 

Compare the characteristic times of drop integration due to the inertial mechanism and the mechanism of turbulent difiusion ... 

The estimation of Rav for characteristic parameter values shows that Rav where Aq = d/Re /" is the internal scale of turbulence. In a turbulent flow, both heat and mass exchange of drops with the gas are intensified, as compared to a quiescent medium. The delivery of substance and heat to or from the drop surface occurs via the mechanisms of turbulent diffusion and heat conductivity. The estimation of characteristic times of both processes, with the use of expressions for transport factors in a turbulent flow, has shown that in our case of small liquid phase volume concentrations, the heat equilibrium is established faster then the concentration equilibrium. In this context, it is possible to neglect the difference of gas and liquid temperatures, and to consider the temperatures of the drops and the gas to be equal. Let us keep all previously made assumptions, and in addition to these, assume that initially all drops have the same radius (21.24). Then the mass-exchange process for the considered drop is described by the same equations as before, in which the molar fluxes of components at the drop surface will be given by the appropriate expressions for diffusion fluxes as applied to particles suspended in a turbulent flow (see Section 16.2). In dimensionless variables (the bottom index 0" denotes a paramenter value at the initial conditions). 

Comparison of turbulent diffusion coefficients Dt in various regions of tubular reactor showed (Table 3.2) that apparatus of divergent-convergent design provides in volume the field of Dt homogeneous enough (for comparison of characteristic times of diffusion Tmix and chemical reaction... 

Characteristic time of meso-mixing (mixing at the expense of exchange between large turbulent flows and being inside of them little flows) is calculated by ratio ... 

The increase of linear rate of reaction mixture movement provides optimal values of characteristic times of liquid flows mixing (Fig. 3.22-3.24), turbulent di sion coefficient (Fig. 3.26) and turbulence kinetic energy density dissipation (Fig. 3.27). High level of tubular turbulent apparatus application by dynamic characteristics of their work in this case is obviously pressure fall at apparatus ends in accordance with Ap-V and the low level is Dt 10" mVsec (transient condition). 

Concerning the length of mixing zone in turbulent flows calculations of characteristic times of mixing and chemical reaction allow derivation of criterion of possibility of fast processes realization in chemical technology in tubular turbulent apparatus ... 

Since quasi-plug flow regime in turbulent flows is formed at definitely limited reaction zone sizes, so it is necessary to consider the ratio of characteristic times of mixing XdSx and chemical... 

Since the quasi-plug flow mode, in turbulent flows, is formed at a strictly limited geometry of the reaction zone, it is necessary to examine the ratio of characteristic times of mixing and chemical reaction T heniJ the case of piperylene oligomerisation, in the presence of the studied catalysts in order to select the optimum reactor design. 

Close characteristic times of a chemical reaction also resulted from the experiments on producing chlorobutyl rubber by the reaction of BR with chlorine in a commercial Nefras solvent within turbulent flows. Calculations of BR chlorination reactor geometry were performed according to the relationships of turbulent mixing in a self-similar mode of a highly viscous mixture flow. At a 0.21 m /h rate of feeding of the BR solution in a commercial Nefras solvent, and a 2.1 m /h rate of feeding the... 

A link between the decoupling of the Reynolds stress and the temporal relaxation of turbulent fluctuations, controlled by a finite propagation of momentum at small spatial scales, has been developed for turbulent flow of a Maxwell fluid. A characteristic time between turbulent bursts from the wall... 

These space and time scales are not sufficient to describe the spatial variations in concentration, since they do not incorporate the process of domain fragmentation described in this chapter. It is necessary to consider the size Ik, which is the smallest scale of turbulence that is present within the flow. Such small vortices are associated with a velocity uk, which allows the introduction of the characteristic time of small-scale turbulence ... 

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