Molecular dispersion. . 167 fluid mechanics

The vanishing effect of molecular diffusivity on turbulent dispersion Implications for turbulent mixing and the scalar flux. Journal of Fluid Mechanics 359, 299-312. 

The way we have presented the one-dimensional dispersion model so far has been as a modification of the plug-flow model. Hence, u is treated as uniform across the tubular cross section. In fact, the general form of the model can be applied in numerous instances where this is not so. In such situations the dispersion coefficient D becomes a more complicated parameter describing the net effect of a number of different phenomena. This is nicely illustrated by the early work of Taylor [G.I. Taylor, Proc. Roy. Soc. (London), A219, 186 (1953) A223, 446 (1954) A224, 473 (1954)], a classical essay in fluid mechanics, on the combined contributions of the velocity profile and molecular diffusion to the residence-time distribution for laminar flow in a tube. 

It is worth a short detour into fluid mechanics to explore some details of this approach and how it fits into the reactor conservation equations. For moderate flow velocities the dispersion of a tracer in laminar flow will occur by axial and radial diffusion from the flow front of the tracer and, in the absence of eddy motion, this will be via a molecular diffusion mechanism. However, the net contribution of diffusion in the axial direction can be taken as small in comparison to the contribution of the flow velocity profile. This leaves us with a two-dimensional problem with diffusion in the radial direction and convection in the longitudinal direction the situation is considered in illustrated in Figure 5.7. 

The axial dispersion coefficient [cf. Eq. (16-51)] lumps together all mechanisms leading to axial mixing in packed beds. Thus, the axial dispersion coefficient must account not only for molecular diffusion and convective mixing but also for nonuniformities in the fluid velocity across the packed bed. As such, the axial dispersion coefficient is best determined experimentally for each specific contactor. 

An analogy may be drawn between the phase behavior of weakly attractive monodisperse dispersions and that of conventional molecular systems provided coalescence and Ostwald ripening do not occur. The similarity arises from the common form of the pair potential, whose dominant feature in both cases is the presence of a shallow minimum. The equilibrium statistical mechanics of such systems have been extensively explored. As previously explained, the primary difficulty in predicting equilibrium phase behavior lies in the many-body interactions intrinsic to any condensed phase. Fortunately, the synthesis of several methods (integral equation approaches, perturbation theories, virial expansions, and computer simulations) now provides accurate predictions of thermodynamic properties and phase behavior of dense molecular fluids or colloidal fluids [1]. 

Somewhat similar measurements could be based on solid disruption [18], polymer degradation [7], or accelerated dissolution. These well-known mechanical effects of ultrasound also derive from cavitation. Thus one might measure the rate of particle size reduction under sonication of some standard solid dispersed in a given fluid. Alternatively one could measure the rate of dissolution of a standard solid in a solvent, or the reduction in molecular weight of polymer chains. Here again the initial particle size and surface conditions, together with pressure and temperature, should be carefully monitored. 

Of interest is the extent to which dispersion is the result of mechanical mixing, or of molecular diffusion, in this particular groundwater system. One way to tackle this problem is to compare the flux of a relatively conservative chemical constituent such as Cl ion as dependent entirely on the rate of fluid movement with the flux by diffusion, assuming no fluid movement. 

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