Eddy transport

It is assumed throughout that no mixing takes place with the intervening fluid when an eddy transports fluid elements over a distance equal to the mixing length. 

When molecular and eddy transport both contribute significantly, it may be assumed as a first approximation that their effects are additive. Then ... 

For combined molecular and eddy transport (the buffer zone), 5 < v < 30 ... 

In the universal velocity profile a dimensionless velocity + is plotted against lnyf, where y+ is a dimensionless distance from the surface. For the region where eddy transport dominates (eddy kinematic viscosity kinematic viscosity), the ratio of the mixing length (Ag) to the distance (>>) from the surface may be taken as approximately constant and equal to 0.4. Obtain an expression for d +/dy+ in terms of y+. 

In the mid-latitude region depicted in Fig. 7-5, the motion is characterized by large-scale eddy transport." Here the "eddies" are recognizable as ordinary high- and low-pressure weather systems, typically about 10 km in horizontal dimension. These eddies actually mix air from the polar regions with air from nearer the equator. At times, air parcels with different water content, different chemical composition and different thermodynamic characteristics are brought into contact. When cold dry air is mixed with warm moist air, clouds and precipitation occur. A frontal system is said to exist. Two such frontal systems are depicted in Fig. 7-5 (heavy lines in the midwest and southeast). 

Most of the available data have been recorded under conditions such that only the terms for eddy transport and conduction through the solid are significant. Equation 12.7.19 requires that /c increase with particle diameter, mass velocity, and the conductivity of the solid. It is consistent with data for low conductivity solids, but some discrepancies arise for very high conductivity solids (108). At Reynolds numbers greater than 40, the contribution of the molecular conduction term is negligible. 

The foregoing procedure can be used to solve a variety of steady, fully developed laminar flow problems, such as flow in a tube or in a slit between parallel walls, for Newtonian or non-Newtonian fluids. However, if the flow is turbulent, the turbulent eddies transport momentum in three dimensions within the flow field, which contributes additional momentum flux components to the shear stress terms in the momentum equation. The resulting equations cannot be solved exactly for such flows, and methods for treating turbulent flows will be considered in Chapter 6. 

Now the eddies transport momentum and the corresponding momentum flux components are equivalent to (negative) shear stress components ... 

Because the eddy transport and the diffusional process always act in series, the coefiicient kg varies with Danckwerts (17) derived an equation even simpler in form ... 

Little detailed experimental information is available on the value of eddy transport properties under conditions of simultaneous thermal and material transport. If it is assumed that the Reynolds analogy is applicable, it follows that the eddy diffusivity and eddy conductivity are equal and independent of cross linking. Such an assumption is probably not true since it is to be expected that a substantial part of the eddy transport is associated with molecular transport particularly as the eddies become small in accordance with Kolmogoroff s (K10) principle. For this reason it is to be expected that temperature gradients in turbulent streams will influence to some extent the material transport in the same... 

If the radial diffusion or radial eddy transport mechanisms considered above are insufficient to smear out any radial concentration differences, then the simple dispersed plug-flow model becomes inadequate to describe the system. It is then necessary to develop a mathematical model for simultaneous radial and axial dispersion incorporating both radial and axial dispersion coefficients. This is especially important for fixed bed catalytic reactors and packed beds generally (see Volume 2, Chapter 4). 

Obtain a dimensionless relation for the velocity profile in the neighbourhood of a surface for the turbulent flow of a liquid, using Prandtl s concept of a Mixing Length (Universal Velocity Profile). Neglect the existence of the buffer layer and assume that, outside the laminar sub-layer, eddy transport mechanisms dominate. Assume that in the turbulent fluid the mixing length Xe is equal to 0.4 times the distance y from the surface and that the dimensionless velocity u1 is equal to 5.5 when the dimensionless distance y+ is unity. 

The Universal Velocity Profile is discussed in detail in Section 12.4, and in the region where eddy transport dominates ( v > 30) and making all the stated assumptions ... 

The term div (kp grad T) accounts for the conduction of heat through the packing. Since the conduction operates partly in series and partly in parallel with the eddy transport, the exact relationship with the mass velocity is complicated, and has not been successfully analyzed. Argo and Smith (A3) have given the most complete discussion of the problem. 

When the resistance to heat transfer is appreciable both within the bed and at the wall, all the ratios except k9 and fa are important, and some approximation must be made. Some progress may be made by observing that the performance of such a reactor is much more critically determined by heat transfer than by mass transfer, and that eddy transport, which provides mass transfer, makes the major contribution to heat transfer. The problem can be attacked, then, by giving attention to the... 

If the objections to a gradient-diffusion approach are valid, one should presumably use an extension of Eq. (30). A possible large-eddy transport model is... 

Figure 13-9 Schematic views of (A) meridional and vertical transport of nitrate in the upper Atlantic basin and (B) associated horizontal transport pathways. Nitrate-rich Circumpolar Deep Water (CDW) upwells in the Southern Hemisphere and the residual mean flow transports some of this upwelled water across the polar front into the regions of intermediate and mode water formation. Nitrate-rich Sub-Antarctic Mode Water (SAMW) and Antarctic Intermediate Water move northward in the thermocline, ultimately outcropping in the subpolar North Atlantic. Ekman transfer provides a source of nitrate to the subtropical gyre along with lateral and vertical eddy transport processes.

In summary, while most studies of atmospheric boundary layer flows have used local theories involving eddy transport coefficients, it is now recognized that turbulent transport coefficients are not strictly a local property of the mean motion but actually depend on the whole flow field and its time history. The importance of this realization in simulating mean properties of atmospheric flows depends on the particular situation. However, for mesoscale phenomena that display abrupt changes in boundary properties, as is often the case in an urban area, local models are not expected to be reliable. 

Hanna, S. R., A Method for Estimating Vertical Eddy Transport in the... 

A turbulent core from to where the contribution of turbulent eddy transport is much larger than the molecular contribution... 

The analysis of turbulent eddy transport in binary systems given above is generalized here for multicomponent systems. The constitutive relation for j y in multicomponent mixtures taking account of the molecular diffusion and turbulent eddy contributions, is given by the matrix generalization of Eq. 10.3.1... 

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