Turbulent diffusion: definition

The definition of X incorporates the fact that the Lagrangian integral time scale is of the order of Zilw,. The statistical theory of turbulent diffusion outlined in the beginning of Section VIII,B can be used to estimate the functional dependence of g as... 

The term A is related to the flow profile of the mobile phase as it traverses the stationary phase. The size of the stationary phase particles, their dimensional distribution, and the uniformity of the packing are responsible for a preferential path and add mainly to the improper exchange of solute between the two phases. This phenomenon is the result of Eddy diffusion or turbulent diffusion, considered to be non-important in liquid chromatography or absent by definition in capillary columns, and WCOT (wall coated open tubular) in gas phase chromatography (Golay s equation without term A, cf. 2.5). 

Another explanation of the lithium gap in the Hyades could be found in terms of turbulent diffusion and nuclear destruction. Turbulence is definitely needed to explain the lithium abundance decrease in G stars. If this turbulence is due to the shear flow instability induced by meridional circulation (Baglin, Morel, Schatzman 1985, Zahn 1983), turbulence should also occur in F stars, which rotate more rapidly than G stars. Fig. 2 shows a comparison between the turbulent diffusion coefficient needed for lithium nuclear destruction and the one induced by turbulence. Li should indeed be destroyed in F stars This effect gives an alternative scenario to account for the Li gap in the Hyades. The fact that Li is normal in the hottest observed F stars could be due to their slow rotation. 

Distributions like those in Figure 10.4, for example, indicate that Yp or T differs from Yp(Z) or T(Z), respectively. If mixing were complete in the sense that all probability-density functions were delta functions and fluctuations vanished, then differences like T — T Z) would be zero. That this situation is not achieved in turbulent diffusion flames has been described qualitatively by the term unmixedness [7]. Although different quantitative definitions of unmixedness have been employed by different authors, in one way or another they all are measures of quantities such as Yp — Yp(Z) or T — T(Z). The unmixedness is readily calculable from P(Z), given any specific definition (see Bilger s contribution to [27]). 

Turbulent diffusivity is by nonzero wind speed the function of time, location, wind speed and meteorological conditions. The turbulent diffusivity is implemented to the model on the basis of Sutton s definition of coefficients of dispersion. Coefficients of dispersion Oj = (Ty, oi are dependent on atmospheric stability and distance from the source. 

I. Definition of mass-transfer coefficient. Since our understanding of turbulent flow is incomplete, we attempt to write the equations for turbulent diffusion in a manner similar to that for molecular diffusion. For turbulent mass transfer for constant c, Eq. (6.1-6) is... 

Still air can be viewed in two ways. One somewhat catholic definition would include air velocity below a level detectable by the optomotor reaction, thus embracing discrete plume structmes similar to those generated in air speeds which can be monitored by the optomotor reaction. A second and more obvious definition would encompass chemical stimuli generated in still air. Dispersion in completely still air would be dominated by molecular diffusion, rather that turbulent diffusion in an airflow, but Mankin et al. (1980) has pointed out that in most still air situations turbulent diffusion will overwhelm molecular diffusion (see Chapter 3). The spatial distributions of chemical stimulus in these two cases will obviously be quite different. 

The dissolved constituents of seawater are grouped into two categories, the major or conservative and the trace components. By definition, the conservative constituents are not influenced significantly by biological processes, and the time required for concentration changes due to chemical and geochemical processes is very long. Therefore, their distribution is controlled mainly by the physical processes of advection and convection, turbulence diffusion, etc. 

The lack of hydrodynamic definition was recognized by Eucken (E7), who considered convective diffusion transverse to a parallel flow, and obtained an expression analogous to the Leveque equation of heat transfer (L5b, B4c, p. 404). Experiments with Couette flow between a rotating inner cylinder and a stationary outer cylinder did not confirm his predictions (see also Section VI,D). At very low rotation rates laminar flow is stable, and does not contribute to the diffusion process since there is no velocity component in the radial direction. At higher rotation rates, secondary flow patterns form (Taylor vortices), and finally the flow becomes turbulent. Neither of the two flow regimes satisfies the conditions of the Leveque equation. 

Bischoff and Levenspiel (B14) present some calculations using existing experimental data to check the above predictions about the radial coefficients. For turbulent flow in empty tubes, the data of Lynn et al. (L20) were numerically averaged across the tube, and fair agreement found with the data of Fig. 12. The same was done for the packed-bed data of Dorweiler and Fahien (D20) using velocity profile data of Schwartz and Smith (Sll), and then comparing with Fig. 11. Unfortunately, the scatter in the data precluded an accurate check of the predictions. In order to prove the relationships conclusively, more precise experimental work would be needed. Probably the best type of system for this would be one in laminar flow, since the radial and axial coefficients for the general dispersion model are definitely known each is the molecular diffusivity. 

Other terms which can be defined quantitatively are introduced in the following sections. Some other terms, such as turbulence, viscosity, and diffusivity are used without definition. For a full explanation of these terms, we refer the reader to standard texts in fluid meehanies, heat transfer, and mass transfer. 

CA 42, 5229(1948)(Theory of propagation of flame. States conditions in an expl chem reaction necesssry for propagation of the flame at a const rate. Calcs this rate for a definite relationship between diffusion and heat conductance. Evaluates the effect of chain reactions on the propagation of the flame) 4) B. Karlovitz, JChemPhys 19, 541-46(1951) Sc CA 45, 9341 (1951)(Theory of turbulent flames) 5) G. Klein, Phil-TransRoySocLondon 249, 389—415 (1957)... 

The diffusion of small particles depends upon many factors. In addition to Brownian motion, we must consider the effect of gravity and the motion of the fluid in which the particles reside. Ordinary diffusion as understood in colloid chemistry must be modified considerably when we deal with turbulence. However, we still retain the usual definition of diffusion, namely that it is the number or mass of particles passing a unit cross section of the fluid in unit-time and unit-concentration gradient. That is, if dw particles (or mass) move through an area / in time dt and dC/dx is the concentration increase in the jc-directior then... 

Equation 8.2 shows how the net flux density of substance depends on its diffusion coefficient, Dj, and on the difference in its concentration, Ac] 1, across a distance Sbl of the air. The net flux density Jj is toward regions of lower Cj, which requires the negative sign associated with the concentration gradient and otherwise is incorporated into the definition of Acyin Equation 8.2. We will specifically consider the diffusion of water vapor and C02 toward lower concentrations in this chapter. Also, we will assume that the same boundary layer thickness (Sbl) derived for heat transfer (Eqs. 7.10-7.16) applies for mass transfer, an example of the similarity principle. Outside Sbl is a region of air turbulence, where we will assume that the concentrations of gases are the same as in the bulk atmosphere (an assumption that we will remove in Chapter 9, Section 9.IB). Equation 8.2 indicates that Jj equals Acbl multiplied by a conductance, gbl, or divided by a resistance, rbl. 

This form is appealing because the first term in F.-,/2 can be interpreted as a gradient diffusion of turbulent kinetic energy, and the second is negative-definite (suggestive of dissipation of turbulence energy). However, the rate of entropy production is proportional to... 

These redox cells can operate on a number of scales that depend on the length of the diffusion path from the point that the oxidised form becomes reduced to the point where it reduces another sediment constituent. In some pelagic cores these diffusion paths can be observed in linear portions of the pore-water profiles (e.g. Sawlan Murray, 1983). Here the sedimentation rate and the carbon burial rate are sufficiently low, relative to diffusion, to extend the processes of early diagenesis over tens of metres into the sediment. In coastal environments the sedimentation rate and the concentration and reactivity of the organic matter is often high, which results in a much more complex pattern. In this case, the distances between the cells are much shorter, since by definition the adjustment must occur more rapidly. Like laminar and turbulent flow, there may come a point where the flow of electrons downwards is better dispersed through eddies , which in this case are transitory micro-environments with small-scale three dimensional diffusion, rather than more stable... 

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