And turbulent diffusivity

Equihbrium concentrations which tend to develop at solid-liquid, gas-liquid, or hquid-liquid interfaces are displaced or changed by molecular and turbulent diffusion between biilk fluid and fluid adjacent to the interface. Bulk motion (Taylor diffusion) aids in this mass-transfer mechanism also. 

In gridpoint models, transport processes such as speed and direction of wind and ocean currents, and turbulent diffusivities (see Section 4.8.1) normally have to be prescribed. Information on these physical quantities may come from observations or from other (dynamic) models, which calculate the flow patterns from basic hydrodynamic equations. Tracer transport models, in which the transport processes are prescribed in this way, are often referred to as off-line models. An on-line model, on the other hand, is one where the tracers have been incorporated directly into a d3mamic model such that the tracer concentrations and the motions are calculated simultaneously. A major advantage of an on-line model is that feedbacks of the tracer on the energy balance can be described... 

With injection mixers (Figures 10.52b,c), in which the one fluid is introduced into the flowing stream of the other through a concentric pipe or an annular array of jets, mixing will take place by entrainment and turbulent diffusion. Such devices should be used where one flow is much lower than the other, and will give a satisfactory blend in about 80 pipe diameters. The inclusion of baffles or other flow restrictions will reduce the mixing length required. 

There are only scant data on nasal deposition. The available studies reported utilized micron sized particles and the dominant mode of deposition is impaction. This is not the case for the particles considered here and diffusion and turbulent diffusion are the mechanisms of interest. George and... 

In a CFD code, the transport rate will depend on the mean velocity and turbulent diffusivity for each zone. 

Figure 4.5. Sketch of how LEM can be applied to an inhomogeneous flow. At fixed time intervals, sub-domains from neighboring grid cells are exchanged to mimic advection and turbulent diffusion.

Inter-cell spatial transport by the mean velocity and turbulent diffusivity change the memberships of the sets 0 /, but not the particle composition vectors 4> n). 

The mean velocity and turbulent diffusivity should approach zero at solid walls. In theory, this should be enough to keep particles from crossing wall boundaries. In practice, due to the finite time step, some particles will eventually cross wall boundaries and must be accounted for. 

We will generally not be concerned here with chemical reactions, so the Ri term can be omitted, in which case the subscript i denoting species is no longer required. In addition, for the turbulent flows of interest the molecular diffusion term in Eq. (2.1) may be neglected. [Although for the spatial scales of interest to us the molecular diffusion term may be neglected, molecular and turbulent diffusion are not independent, linearly additive, physical processes (Saffman, I960).] As a result of the above two simplifications, Eq. (2.1) becomes... 

Tennekes, H. (1979). The exponential Lagrangian correlation function and turbulent diffusion in the inertial subrange. Atmos. Environ. 13, 1565-1567. 

Fluctuations due to different flow -y velocities and due to molecular / and turbulent diffusion /... 

Dispersion models, as just stated, are useful mainly to represent flow in empty tubes and packed beds, which is much closer to the ideal case of plug flow than to the opposite extreme of backmix flow. In empty tubes, the mixing is caused by molecular diffusion and turbulent diffusion, superposed on the velocity-profile effect. In packed beds, mixing is caused both by splitting of the fluid streams as they flow around the particles and by the variations in velocity across the bed. 

Turbulent diffusion is not reaUy diffusion but the mixing of chemicals through turbulent eddies created by convection. Turbulent diffusion is thus a form of convection. Although it has the appearance of diffusion in the end (i.e., random mixing similar to diffusion), the causes of diffusion and turbulent diffusion are very different. Since the end products are similar, diffusion coefficients and turbulent diffusion coefficients are often simply added together. This process will be discussed in this chapter. 

Turbulent diffusion occurs because turbulent eddies are transporting mass, momentum, and energy over the eddy scale at the rotational velocity. This transport rate is generally orders of magnitude greater than the transport rate due to molecular motion. Thus, when a flow is turbulent, diffusion is normally ignored because e Z). The exception is very near the flow boundaries, where the eddy size (and turbulent diffusion coefficient) decreases to zero. 

Prandtl s mixing length hypothesis (Prandtl, 1925) was developed for momentum transport, instead of mass transport. The end result was a turbulent viscosity, instead of a turbulent diffusivity. However, because both turbulent viscosity and turbulent diffusion coefficient are properties of the flow field, they are related. Turbulent viscosity describes the transport of momentum by turbulence, and turbulent diffusivity describes the transport of mass by the same turbulence. Thus, turbulent viscosity is often related to turbulent diffusivity as... 

Other factors also come into play in laboratory systems. For example, McMurry and Rader (1985) have shown that particle deposition at the walls of Teflon smog chambers is controlled by Brownian and turbulent diffusion for particles with Dp 0.05 yxm and by gravitational settling for particles with Dp > 1.0 yxm. However, in the 0.05- to 1.0-yxm range, the deposition is controlled by electrostatic effects Teflon tends to... 

Table 18.4 Typical Molecular and Turbulent Diffusivities in the Environment...

Exact Solution of the Diffusion Problem in a Periodic Velocity Field and Turbulent Diffusion... 

The third term on the right-hand side of the equation represents the transport of k due to molecular and turbulent diffusion. Hence, with an analogy to the laminar transport, this term may be expressed as... 

A discussion on the scale of turbulence and turbulent diffusion that can be derived from the newly defined ESD is given as follows. 

At first sight, this simple model appears to have the capability of accounting only for axial mixing effects. It will be shown, however, that this approach can compensate not only for problems caused by axial mixing, but also for those caused by radial mixing- and other nonflat velocity profiles These fluctuations in concentration can result from different flow velocities and pathways and from moleeular and turbulent diffusion. 

Note that none of the equations in Table 2-5 include wind speed, because it is assumed that the turbulence generated within the stream due to velocity shear and turbulent diffusion primarily controls gas exchange. Unfortunately, the predictions of these equations are often not in good agreement with one another, and it is difficult to know which one is best in a given situation (see Fig. 2-13). 

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