Dispersion parameters eddies

In Chapter 11, we indicated that deviations from plug flow behavior could be quantified in terms of a dispersion parameter that lumped together the effects of molecular diffusion and eddy dif-fusivity. A similar dispersion parameter is usefl to characterize transport in the radial direction, and these two parameters can be used to describe radial and axial transport of matter in packed bed reactors. In packed beds, the dispersion results not only from ordinary molecular diffusion and the turbulence that exists in the absence of packing, but also from lateral deflections and mixing arising from the presence of the catalyst pellets. These effects are the dominant contributors to radial transport at the Reynolds numbers normally employed in commercial reactors. 

The communication between off-line coupled meteorological and AQ models is often a problem of underestimated importance. The variety of modelling systems previously introduced give rise to different approaches and methods implemented within interface modules. Tasks covered by interfaces are minimized in coupled systems relying on surface fluxes, turbulence and dispersion parameters (i.e. eddy... 

The axial dispersion parameter T) accounts for mixing by both molecular diffusion processes and turbulent eddies and vortices. Because these two types of phenomena are to be characterized by a single parameter, and because we force the model to fit the form of Pick s law of diffusion,... 

MECC separations are conducted in open capillaries, hence eddy diffusion is not problematic. However, the columns behave in many ways like packed columns, with the micelles functioning as uniformly sized and evenly dispersed packing particles. In packed columns, resistance to mass transfer in the mobile phase is reduced (i.e., efficiency improved) when smaller particles are used because the "diffusion distance" between particles is decreased. Average inter-micellar" distance is the analogous parameter in MECC. This distance can be decreased by increasing surfactant concentration. 

Subscript 1 indicates continuous phase and 2 indicates dispersed phase. Cd is a parameter of the standard k-s model (0.09), k is turbulent kinetic energy and si is turbulent energy dissipation rate. The eddy lifetime seen by dispersed phase particles will in general be different from that for continuous phase fluid particles due to the so-called crossing-trajectory effect (Csnady, 1963). This can be expressed in the form ... 

The more usual approach to speeding up FCM methods for multi-body flow / dispersion calculations such as this is to use the continuum approach. This requires some estimates of turbulence parameters such as mixing lengths or eddy diffusivities for the canopy (on the neighbourhood scale), but these characteristics cannot be deduced from turbulence closure models, although they may be estimated from detailed calculations of flow/dispersion around a few typical obstacles - not a straightforward or accurate process (see Moulinec et al., 2003 [436]). 

Equation 10.115 has a considerable fundamental and practical importance. It combines parameters of fimdamentally different origins, the plate number at infinite dilution, N, which characterizes the intensity of axial dispersion taking place in the column and two parameters of thermod5mamic origin, the retention factor at infinite dilution, ICg, related to the initial slope of the isotherm, and the loading factor, proportional to the sample size and related to the saturation capacity of the isotherm. Accordingly, Eq. 10.115 indicates the extent to which the self-sharpening effect on the band profile due to the nonlinear thermodynamics is balanced by the dispersive effect of axial and eddy diffusion and of the mass transfer resistances. 

The bulk diffusion coefficient of lipase was estimated [55]. The dispersion coefficient is used to characterize the axial dispersion in a packed bed. This parameter accounts for the dispersion due to molecular diffusion as well as eddy diffusion due to velocity differences around the particles. A correlation used to estimate the dispersion coefficient Dm in fixed beds was developed by Chung and Wen [56]. 

Equation 16 shows that the peak variance or band broadening is comprised of individual contributions from different aspects of the separation process. The first term in equation 16 represents the contribution of the width of the feed band to the peak variance. The second term represents the contribution to band broadening from dispersion due to eddy diffusion. The third term represents the contribution of mass transfer effects external to the particles while the fourth term represents the contribution of diffusional resistances within the stationary phase. The significance of each term relative to the total variance depends upon the operating parameters, the column and packing dimensions and the size of the solute. 

Note that D, the dispersion coefficient, is not the molecular diffusivity, but a measure of combined dispersive effects inherent in packed bed operations, of which molecular diffusivity is a minor component. As pointed out by Kramers and Alberda (24), eddy diffusivity involves fluctuations of a statistical nature, and should not be applied to macroscopic effects, such as by-passing and mixing. This equation is important because it allows the modeling of chromatographic results using the dispersion coefficient as a free parameter. 

The K-model of Eq. (10.81) is a substantial simplihcation, because it is based on the assumption that the eddy coefficient and the wind speeds are constants. In addition, many of the parameters of influence on atmospheric dispersion presented above cannot be accounted for. This motivated numerous further developments. If the following relationship between the eddy coefficient an the atmospheric standard deviation is used... 

D, which has the same dimension unit as the molecular diffusion coefficient D. Usually is much larger than because it incorporates all effects that may cause deviation from plug flow, such as radial velocity differences, eddies, or vortices. The key parameter determining the width of the RTD is the ratio between the axial dispersion time and the space-time r, which corresponds to the mean residence time in the reactor t at constant fluid density. This ratio is often called Bodenstein number Bo). 

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