Eddy coefficients, flow dependent

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. 

These mixing processes are generally characterized by diffusion or dispersion coefficients, all with units such as m /h. Molecular diffusion coefficients in water are isotropic and typically 0.4m /h or 10 cm /s. They are usually negligible in comparison to eddy diffusion or dispersion. Eddy or turbulent diffusion coefficients are controlled by water flow, wind, biotic, and buoyancy effects and may be anisotropic. Dispersion coefficients are controlled by velocity gradients in the water and are anisotropic with vertical, lateral, and longitudinal components. These individual coefficients can be added to give a net coefficient that depends on flow conditions and the system geometry. 

It is interesting to compare equations (6.32) and (6.33) with those for a fully developed laminar flow, equations (6.29) and (6.30). In Example 5.1, we showed that eddy diffusion coefficient in a turbulent boundary layer was linearly dependent on distance from the wall and on the wall shear velocity. If we replace the diffusion coefficient in equation (6.30) with an eddy diffusion coefficient that is proportional to hu, we get... 

The A term corresponds to the eddy diffusion which describes the irregular flow through the packed particles in a column causing different pathways and different exit times for the solute molecules. The B term is the longitudinal molecular diffusion or random diffusion along the column. The last term C, corresponds to the mass transfer in the stationary phase. This mass transfer occurs between the mobile and stationary phase of the chromatographic system and is dependant on several factors such as particle size, column diameter and diffusion coefficient. 

Although Ey and are analogous to fj. and v, respectively, in that all these quantities are coefficients relating shear stress and velocity gradient, there is a basic difference between the two kinds of quantities. The viscosities n and v are true properties of the fluid and are the macroscopic result of averaging motions and momenta of myriads of molecules. The eddy viscosity and the eddy diffusivity are not just properties of the fluid but depend on the fluid velocity and the geometry of the system. They are functions of all factors that influence the detailed patterns of turbulence and the deviating velocities, and they are especially sensitive to location in the turbulent field and the local values of the scale and intensity of the turbulence. Viscosities can be measured on isolated samples of fluid and presented in tables or charts of physical properties, as in Appendixes 8 and 9. Eddy viscosities and diffusivities are determined (with difficulty, and only by means of special instruments) by experiments on the flow itself. 

Wall-to-Bed Heat Transfer. The wall-to-bed heat transfer coefficient increases with an increase in liquid flow rate, or equivalently, bed voidage. This behavior is due to the reduction in the limiting boundary layer thickness that controls the heat transport as the liquid velocity increases. Patel and Simpson [94] studied the dependence of heat transfer coefficient on particle size and bed voidage for particulate and aggregative fluidized beds. They found that the heat transfer increased with increasing particle size, confirming that particle convection was relatively unimportant and eddy convection was the principal mechanism of heat transfer. They observed characteristic maxima in heat transfer coefficients at voidages near 0.7 for both the systems. 

Here, the A term is due to eddy dispersion and flow contribution to plate height and is independent of ly it is a function of the particle size and the packing efficiency. The next term includes B, which depends on the molecular diffusion coefficient in the longitudinal direction, i.e. the mobile-phase diffusivity. The third term includes C, which results from mass transfer between the mobile and stationary phases and has contributions from (1) diffiision in the film around the particle in the column, (2) diffiision in the Uquid phase that is stagnant in the pores and (3) diffusion in the liquid-phase coating on particles. 

The next level of description, mesoscopic, involves averaging at higher levels and thus incorporates less detailed information about the internal features ofthe system of interest. This level is of particular interest for processes involving turbulent flow or flow in geometrically complex systems on a fine scale, such as porous media. The values of the dependent variables are averaged in time (turbulence) or space (porous media). Processes at this level are described by effective transport coefficients such as eddy viscosity (turbulence) or permeability (porous media). 

Air-water transfer rate of chemicals is dependent upon the rate coefficient and the equilibrium that the concentrations in each phase are moving towards. In environmental air-water mass transfer, the flow is generally turbulent in both phases. However, there is no turbulence across the interface in the diffusive sublayer, and the problem becomes one of rate of diffusion. Temporal mean turbulence quantities, such as eddy diffusion coefficient, provide a semiquantitative description of the flux across the air-water interface, however the unsteady character of turbulence near the diffusive sublayer is cmcial to understanding and characterizing interfacial transport processes. 

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