Diffusion, eddy molecular

Wind speed has velocity components in all directions so that there are vertical motions as well as horizontal ones. These random motions of widely different scales and periods are essentially responsible for the movement and diffusion of pollutants about the mean downwind path. These motions can be considered atmospheric turbulence. If the scale of a turbulent motion (i.e., the size of an eddy) is larger than the size of the pollutant plume in its vicinity, the eddy will move that portion of the plume. If an eddy is smaller than the plume, its effect will be to difhise or spread out the plume. This diffusion caused by eddy motion is widely variable in the atmosphere, blit even when the effect of this diffusion is least, it is in the vicinity of three orders of magnitude greater than diffusion by molecular action alone. 

An important mixing operation involves bringing different molecular species together to obtain a chemical reaction. The components may be miscible liquids, immiscible liquids, solid particles and a liquid, a gas and a liquid, a gas and solid particles, or two gases. In some cases, temperature differences exist between an equipment surface and the bulk fluid, or between the suspended particles and the continuous phase fluid. The same mechanisms that enhance mass transfer by reducing the film thickness are used to promote heat transfer by increasing the temperature gradient in the film. These mechanisms are bulk flow, eddy diffusion, and molecular diffusion. The performance of equipment in which heat transfer occurs is expressed in terms of forced convective heat transfer coefficients. 

If two vessels each containing completely mixed gas, one at temperature T, and the other at a temperature T2, are connected by a lagged non-conducting pipe in which there are no turbulent eddies (such as a capillary tube), then under steady state conditions, the rate of transfer of A by thermal diffusion and molecular diffusion must be equal and opposite, or. 

Since the mixing process involves a shuffling or redistribution of material either by slippage or eddies, and since this is repeated many, many times during the flow of fluid through the vessel we can consider these disturbances to be statistical in nature, somewhat as in molecular diffusion. For molecular diffusion in the x-direction the governing differential equation is given by Fick s law ... 

When representing rates of transfer of heat, mass, and momentum by eddy activity, the concepts of eddy thermal conductivity, eddy diffusivity, and eddy viscosity are sometimes useful. Extending the concepts of heat conduction, molecular diffusion, and molecular viscosity to include the transfer mechanisms by eddy activity, one can use Equations 2.13-2.15, which correspond to Equations 2.2,2.3, and 2.5, respectively. 

Corrections for instrumental broadening (also called axial dispersion) are also sometimes applied [15]. This phenomenon arises because of eddy diffusion and molecular diffusion at the leading and trailing edges of the pulse of polymer solution [16]. Hie result is a symmetrical, Gaussian spreading of the GPC... 

FIGURE 24-2 Comparison of curves for plate height against velocity for the individual terms (upper) and for the overall value Oower) of Equation (24-14). Left, the values for a liquid mobile phase right, values for a gaseous mobile phase (liquid stationary phases in both cases). The numbered curves represent eddy diffusion (1), molecular diffusion in the mobile phase (2), and resistance to mass transfer in the stationary (4) and mobile (5) phases. The contribution of the term for molecular diffusion in the stationary phase is negligible at velocities near the optimum for both liquid and gas systems. 

The axial dispersion coefficient is determined from the concentration profile of a non-penetrating tracer (Tl). A reasonable approximation for its velocity dependence goes back to van Deemter et al. (1956). The axial dispersion coefficient is the sum of the contributions of eddy diffusion and molecular diffusion (Chapter 2.3.4) ... 

The sources of band broadening of kinetic origin include molecular diffusion, eddy diffusion, mass transfer resistances, and the finite rate of the kinetics of ad-sorption/desorption. In turn, the mass transfer resistances can be sorted out into several different contributions. First, the film mass transfer resistance takes place at the interface separating the stream of mobile phase percolating through the column bed and the mobile phase stagnant inside the pores of the particles. Second, the internal mass transfer resistance controls the rate of mass transfer between this interface and the adsorbent surface. It is composed of two contributions, the pore diffusion, which is molecular diffusion taking place in the tortuous, constricted network of pores, and surface diffusion, which takes place in the electric field at the liquid-solid interface [60]. All these mass transfer resistances, except the kinetics of adsorption-desorption, depend on the molecular diffusivity. Thus, it is important to study diffusion in bulk liquids and in porous media. 

One of the most obvious manifestations of air pollution is the visible plume formed downwind from a stationary source, A relatively. simple model for such systems Ls the continuous point source in a turbulent fluid with a mean velocity, tl(j , z). The coordinate. T is measured downwind from the source, parallel to the ground, and z is the coordinate perpendicular to the surface (Fig. 11.6). The velocity components in the y and z directions vanish, and diffusion in the a direction can be neglected compared with convection. Brownian diffusion is also neglected compared with eddy diffusion. These are the usual simplifying assumptions made in the theory of diffusion of molecular species in turbulent stack plumes, and with them (11.38) becomes... 

The van Deemter rate theory identified three major factors that cause band or zone broadening during the chromatographic process the eddy diffusion or the multi-path effect (A-term), longitudinal diffusion or molecular diffusion of the analyte molecules (B-term), and resistance to mass transfer in the stationary phase (C-term). The broadening of a zone was expressed in terms of the plate height, H, and was described as a function of the average linear velocity of the mobile phase, u. 

A, Eddy diffusion B, molecular diffusion C, resistance to mass transfer. 

One natural approach to describing mixing is to solve the equations of motion of the fluid. In fluid systems, the type of fluid flow is obviously important, and we should consider both laminar and turbulent flow, and various mechanisms of diffusion (molecular diffusion, eddy diffusion). Using fluid mechanics to describe all cases of interest is a difficult problem, both from the modeling and computational perspectives. Rapid developments in computational fluid dynamics (CFD), however, make this approach increasingly attractive 1]. 

It turns out that turbulent diffusion can be described with Fick s laws of diffusion that were introduced in the previous section, except that the molecular diffusion coefficient is to be replaced by an eddy or turbulent diffusivity E. In contrast to molecular diffusivities, eddy dififusivities are dependent only on the phase motion and are thus identical for the transport of different chemicals and even for the transport of heat. What part of the movement of a turbulent fluid is considered to contribute to mean advective motion and what is random fluctuation (and therefore interpreted as turbulent diffusion) depends on the spatial and temporal scale of the system under investigation. This implies that eddy diffusion coefficients are scale dependent, increasing with system size. Eddy diffusivities in the ocean and atmosphere are typically anisotropic, having much large values in the horizontal than in the vertical dimension. One reason is that the horizontal extension of these spheres is much larger than their vertical extension, which is limited to approximately 10 km. The density stratification of large water bodies further limits turbulence in the vertical dimension, as does a temperature inversion in the atmosphere. Eddy diffusivities in water bodies and the atmosphere can be empirically determined with the help of tracer compounds. These are naturally occurring or deliberately released compounds with well-estabhshed sources and sinks. Their concentrations are easily measured so that their dispersion can be observed readily. 

To estimate turbulent dispersion and the connected concentrations of air pollutants, no completely satisfactory techniques are available at present. For vertical mixing, first-order closure [.ST-theory see Eq. (16)] gives reasonably good results in the case of groimd sources. The problem here is the proper description of the eddy diffusivity AT. If AT is assumed to be independent of downward distance from the source x, the plume width varies as /x. This differs from observed plume widths. The difficulty is that a constant K describes a situation in which all diffusing eddies are small compared with the scale of the plume. This is true for molecular dispersionbut not for turbulent dispersion If the eddy diffusivity K is allowed to... 

A recent model (1988) was published by J. Guttierrez Gonzalez et al. According to the authors, although the liquid fiow is laminar, due to the high Schmidt number in the liquid phase, eddy mass transfer can be significant and eddy diffusion cannot be disregarded with respect to molecular diffusion. Eddy thermal diffusion in the liquid phase is much smaller than thermal diffusion, so that it is not introduced in the microscopic heat balance. The Spanish authors needed to validate their results on practical data. The pressure drop over the reactor, heat- and mass transfer were fitted. 

If a fluid is placed between two concentric cylinders, and the inner cylinder rotated, a complex fluid dynamical motion known as Taylor-Couette flow is established. Mass transport is then by exchange between eddy vortices which can, under some conditions, be imagmed as a substantially enlranced diflfiisivity (typically with effective diflfiision coefficients several orders of magnitude above molecular difhision coefficients) that can be altered by varying the rotation rate, and with all species having the same diffusivity. Studies of the BZ and CIMA/CDIMA systems in such a Couette reactor [45] have revealed bifiircation tlirough a complex sequence of front patterns, see figure A3.14.16. 

Although molecular diffusion itself is very slow, its effect is nearly always enhanced by turbulent eddies and convection currents. These provide almost perfect mixing in the bulk of each Hquid phase, but the effect is damped out in the vicinity of the interface. Thus the concentration profiles at each... 

---