Dissipation range

Materials that have the abiUty to dissipate a charged formed, by any means including tribocharging and induction, on the material are referred to as static dissipative. There is a correlation between static dissipation and surface resistance. EIA-541 currently defines the static dissipative range as 10 to 10 ... 

As is confirmed by the results in Section 6, the experimental findings correspond more closely to this stress formula, even in the dissipation range, than to Newton s stress formula (1), which is often used because of the assumption of laminar-flow eddies in this region (see e.g. [51,52,77]). 

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 ... 

The relationship dp dp found for the floccular system and the oil/ water emulsion, with b = -l/3, confirms the theoretically derived Eq. (16b) where particle disintegration is determined by the turbulent eddies in the dissipation range. 

It could be shown (see Sect. 6) that in stirred vessels with baffles and under the condition of fully developed turbulence, particle stress can be described by Eqs. (2) and (4) alone. The turbulent eddys in the dissipation range are decisive for the model particle systems used here and many biological particle systems (see Fig. 2), so that the following equation applies to effective stress ... 

It confirms again the theoretically derived Eq. (16b) and means that also particles which are much smaller as the smallest turbulent eddies ( 3 l> see Fig. 2) are disrupted by the turbulent eddies of the dissipation range. For the calculation of stress has to be used the Reynold s stress Eq. (2) and not Newton s law (1). 

The turbulent fluctuation frequency can be estimated by means of turbulent measurements. Mockel 124] found that the wave number k = 2Tlft/u in the interesting dissipation range is k>ko with the limiting value ko= (0.1. ..0.2)qL-From this becomes the frequency to ft> (0.016...0.032)u/qL. An important measure should be the related number of turbulent fluctuation z/zp which occur during the residence time of particles ti=Vi/qp inside the fictive impeller volume Vj at one circulation. It follows to ... 

Furthermore, the universal equilibrium range is composed of the inertial range and the dissipation range. As its name indicates, at high Reynolds numbers the universal equilibrium range should have approximately the same form in all turbulent flows. 

By definition, the dissipation range is dominated by viscous dissipation of Kolmogorov-scale vortices. The characteristic time scale rst in (2.74) can thus be taken as proportional to the Kolmogorov time scale rn, and taken out of the integral. This leads to the final form for (2.70),... 

For example, the vortex-stretching term is a triple-correlation term that corresponds to the rate at which dissipation is created by spectral energy passing from the inertial range to the dissipative range of the energy spectrum (see (2.75), p. 43). Letting /cdi 0.1 denote... 

However, in general, we can write the gradient-dissipation term as the product of the dissipation rate and a characteristic gradient dissipation rate. The latter can be formed by dividing the fraction of the dissipation rate falling in the dissipation range, i.e.,20... 

By definition of the dissipation range, s even when the turbulence is not in spectral equilibrium. 

However, DNS data for Schmidt numbers near unity suggest that (3.70) provides the best model for the scalar-dissipation range (Yeung et al. 2002). 

For a passive scalar, the turbulent flow will be unaffected by the presence of the scalar. This implies that for wavenumbers above the scalar dissipation range, the characteristic time scale for scalar spectral transport should be equal to that for velocity spectral transport tst defined by (2.67), p. 42. Thus, by equating the scalar and velocity spectral transport time scales, we have23 t)... 

The scalar-dissipation wavenumber /cd is defined in terms of /cdi by /cd = Sc1/2kdi-Like the fraction of the turbulent kinetic energy in the dissipation range kn ((2.139), p. 54), for a fully developed scalar spectrum the fraction of scalar variance in the scalar dissipation range scales with Reynolds number as... 

Thus, like the turbulence dissipation rate, the scalar dissipation rate of an inert scalar is primarily determined by the rate at which spectral energy enters the scalar dissipation range. Most engineering models for the scalar dissipation rate attempt to describe (kd, t) in terms of one-point turbulence statistics. We look at some of these models in Chapter 4. 

In principle, the same equation could be used for a reacting scalar. However, one would need to know the spectral distribution of the covariance chemical source term Sai, (3.141), in order to add the corresponding covariance-dissipation-range chemical source term to (3.165). 

At high Reynolds number and for Schmidt numbers near unity or larger, we are justified in assuming that Tt is nearly independent of Schmidt number. We will also need a closure for in (3.175). In general, the dissipation-range variance scales as Re, 1 = Rc, 1/2 (Fox and Yeung 1999 Vedula 2001). We will thus model the covariances by... 

Figure 4.9. Sketch of CSTR representation of the SR model for 1 < Sc. Each wavenumber band is assumed to be well mixed in the sense that it can be represented by a single variable

The condition that a must be positive limits the applicability of the model to 1 < CMRei or 12 < R>,. This corresponds to k = ku = 0.1 k, so that scalar energy is transferred directly from the lowest-wavenumber band to the dissipative range. However, at such low Reynolds numbers, the spectral transfer rates used in the model cannot be expected to be accurate. In particular, the value of Rq would need to account for low-Reynolds-number effects. 

Cs = Cb - Co, Cb = 1, and Cd = 3 (Fox 1995).36 Note that at spectral equilibrium, Vp = p, % = To = p( I - i/i)), and (with Sc = 1) R = Rq. The right-hand side of (4.117) then yields (4.114). Also, it is important to recall that unlike (4.94), which models the flux of scalar energy into the dissipation range, (4.117) is a true small-scale model for p. For this reason, integral-scale terms involving the mean scalar gradients and the mean shear rate do not appear in (4.117). Instead, these effects must be accounted for in the model for the spectral transfer rates. 

Figure 4.13. Predictions of the SR model for ReA = 90 and Sc = 1 for homogeneous scalar mixing with = 0. For these initial conditions, all scalar energy is in the dissipative range. Curves 1-3 and D correspond to the fraction of scalar energy in each wavenumber band

These variables are governed by exactly the same model equations (e.g., (4.103)) as the scalar variances (inter-scale transfer at scales larger than the dissipation scale thus conserves scalar correlation), except for the dissipation range (e.g., (4.106)), where... 

Because the parameters 0(30, phu- and Pm appearing in TD depend on the Schmidt number, the dissipation-range spectral transfer rates will be different for each covariance component. Vap is the covariance-production term defined by (3.137) on p. 90. 

Figure 4.14. Predictions of the multi-variate SR model for Re, = 90 and Sc = (1, 1/8) with collinear mean scalar gradients and no backscatter (cb = 0). For these initial conditions, the scalars are uncorrelated pap(0) = gap(0) = 0. The correlation coefficient for the dissipation range, pD, is included for comparison with pap. 

Note that, unlike pap, pD is not bounded by unity. This is because the dissipation-range cut-off wavenumber, (4.101), used to define ( a p) d depends on Scap, which is different for each scalar pair (i.e., Scap / Scaa Scpp). Indeed, for the mean-scalar-gradient case,... 

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