Eddy diffusivity of mass

The eddies bring about a transfer of dissolved solute, as we have mentioned before. The average concentration gradient between 1 and 2 in Fig. 3.5 is Ac y//, proportional to a local gradient, - dcjdz. The flux of A due to the interchange, Ac j and the concentration gradient can be used to define an eddy diffusivity of mass length /time... 

The region close to the wall where molecular diffusivity exceeds eddy diffusivity of mass has been termed the dijfusional suble er [38], and we can estimate its extent in particular instances. For example, for flow of water at 25 C, in the pipe considered above, v = 8.964 x 10 m /s, and typically for solutes in water, D is of the order of 10 m /s. When o is set equal to Z), Eq. (3.37) yields = 1.08 so the diffusional sublayer lies well within the viscous sublayer, the more so the smaller the D and the larger the Sc. At this for the pipe considered above and Re = 150 000, the distance from the wall is 0.008 mm. 

Sherwood TK, Woertz BB. The role of eddy diffusion in mass transfer between phases. Trans Am Inst Chem Eng 1939 35 517-540. 

According to the theory of linear stability analysis, infinitesimally small perturbations are superimposed on the variables in the steady state and their transient behavior is studied. At this stage the difference between turbulent fluctuations and perturbations may be noted. Turbulence is the characteristic feature of the multiphase flow under consideration the mean and fluctuating quantities were given by Eq. (2). The fluctuating components result in eddy diffusivity of momentum, mass, and Reynolds stresses. The turbulent fluctuations do not alter the mean value. In contrast, the perturbations are superimposed on steady-state average values and another steady... 

During food engineering operations, many fluids deviate from laminar flow when subjected to high shear rates. The resulting turbulent flow gives rise to an apparent increase in viscosity as the shear rate increases in laminar flow, i.e., shear stress = viscosity x shear rate. In turbulent flow, it would appear that total shear stress = (laminar stress + turbulent stress) x shear rate. The most important part of turbulent stress is related to the eddies diffusivity of momentum. This can be recognized as the atomic-scale mechanism of energy conversion and its redistribution to the dynamics of mass transport processes, responsible for the spatial and temporal evolution of the food system. 

In contrast to the equilibrium-dispersive model, which is based upon the assumptions that constant thermod3mamic equilibrium is achieved between stationary and mobile phases and that the influence of axial dispersion and of the various contributions to band broadening of kinetic origin can be accounted for by using an apparent dispersion coefficient of appropriate magnitude, the lumped kinetic model of chromatography is based upon the use of a kinetic equation, so the diffusion coefficient in Eq. 6.22 accounts merely for axial dispersion (i.e., axial and eddy diffusions). The mass balance equation is then written... 

In CE, there are no eddy diffusion or mass transfer effects, only molecular diffusion broadening. The separation efficiency of CE is 10-100 times that of HPLC. 

As briefly mentioned earlier, multistage separation processes involve not only equilibrium considerations, but kinetic factors as well. The efficiency of a chromatographic process is subject to a number of these factors, such as longitudinal diffusion (B), eddy diffusion (A), mass transfer in the immobile (Cj), and mobile (Cj ) phases. A useful relation, proposed by van Deemter and subsequently elaborated by many others, sums all of these factors... 

Additional modifications to the original van Deemter equation have been proposed by other workers. For example, one can argue that eddy diffusion (the A-term) is part of mobile phase mass transfer (the Cm -term) or is coupled with it. Giddings [7] has thoroughly discussed mass transfer and prefers a coupled term combining eddy diffusion and mass transfer to produce a new equation. 

The three eddy diffusivities of momentum, heat, and mass can be computed from measured velocity, temperature, and concentration gradients, respectively, in that order of increasing difficulty. 

Heat and mass eddy diffusivity The evidence is that with Prandtl and Schmidt numbers close to unity, as for most gases, the eddy diffusivities of heat and mass are equal to the momentum eddy diffusivity for all regions of turbulence [15], For turbulent fluids where Prandtl and Schmidt numbers exceed unity, the ratios E jand E /E will vary with location relative to the wall and in the turbulent core will lie generally in the range 1,2 to 1.3, with E and essentially equal [44, 62], For = 0 to 45, with Pr and Sc > 1, a critical analysis of the theoretical and experimental evidence [44] led to... 

In the case of turbulent flow, the differential equations will contain time-averaged velocities and in addition the eddy diffusivities of momentum, mass, and heat transfer. The resulting equations cannot be solved for lack of information about the eddy diffusivities, but one might expect results of the form... 

Noncatalytic gas-solid reactions in mixed bed systems usually involve the movement of a reaction front in the direction of the flow and radial gradients of concentration are usually not very signfiicant. It follows that radial dispersion usually plays an insignificant role in mass transfer problems. However, radial eddy diffusion of heat (eddy thermal conductivity) may play an important role in reactors that are heated or cooled through the bounding walls. An interesting example of this type has been presented by Amundson [20]. 

A closer look at the Lewis relation requires an examination of the heat- and mass-transfer mechanisms active in the entire path from the hquid—vapor interface into the bulk of the vapor phase. Such an examination yields the conclusion that, in order for the Lewis relation to hold, eddy diffusivities for heat- and mass-transfer must be equal, as must the thermal and mass diffusivities themselves. This equahty may be expected for simple monatomic and diatomic gases and vapors. Air having small concentrations of water vapor fits these criteria closely. 

Eddy diffusion as a transport mechanism dominates turbulent flow at a planar electrode ia a duct. Close to the electrode, however, transport is by diffusion across a laminar sublayer. Because this sublayer is much thinner than the layer under laminar flow, higher mass-transfer rates under turbulent conditions result. Assuming an essentially constant reactant concentration, the limiting current under turbulent flow is expected to be iadependent of distance ia the direction of electrolyte flow. 

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. 

The energy of large and medium-size eddies can be characterized by the turbulent diffusion coefficient. A, m-/s. This parameter is similar to the parameter used by Richardson to describe turbulent diffusion of clouds in the atmosphere. Turbulent diffusion affects heat and mass transfer between different zones in the room, and thus affects temperature and contaminant distribution in the room (e.g., temperature and contaminant stratification along the room height—see Chapter 8). Also, the turbulent diffusion coefficient is used in local exhaust design (Section 7.6). 

The term mass transfer is used to denote the transference of a component in a mixture from a region where its concentration is high to a region where the concentration is lower. Mass transfer process can take place in a gas or vapour or in a liquid, and it can result from the random velocities of the molecules (molecular diffusion) or from the circulating or eddy currents present in a turbulent fluid (eddy diffusion). 

In addition to momentum, both heat and mass can be transferred either by molecular diffusion alone or by molecular diffusion combined with eddy diffusion. Because the effects of eddy diffusion are generally far greater than those of the molecular diffusion, the main resistance to transfer will lie in the regions where only molecular diffusion is occurring. Thus the main resistance to the flow of heat or mass to a surface lies within the laminar sub-layer. It is shown in Chapter 11 that the thickness of the laminar sub-layer is almost inversely proportional to the Reynolds number for fully developed turbulent flow in a pipe. Thus the heat and mass transfer coefficients are much higher at high Reynolds numbers. 

It should be stressed that in the case of linear isotherm, the peak broadening effect results from eddy diffusion and from resistance of the mass transfer only, and it does not depend on Henry s constant. In practice, such concentration profiles are observed for these analyte concentrations, which are low enough for the equilibrium isotherm to be regarded as linear. 

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