Diffusion, Dispersion, and Mixing

The microreactors require mixing and dispersion of various reactants. In a lab-on-a-chip system, it is often required to mix reactants. One example application is fragmentation of proteins by mixing them with enzymes. This procedure is used for the identification of proteins by mass spectrometry. The role played by hydrodynamic instabilities and turbulence on mixing between two streams is limited or nonexistent due to the low Reynolds number in microsystem. The detailed understanding of diffusion and dispersion is necessary to investigate the mixing issues in microsystems. These issues have been discussed in this chapter. 

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Diffusion, dispersion, and mass transfer are three ways to describe molecular mixing. Diffusion, the result of molecular motions, is the most fundemental, and leads to predictions of concentration as a function of position and time. Dispersion can follow the same mathematics used for diffusion, but it is due not to molecular motion but to flow. Mass transfer, the description of greatest value to the chemical industry, commonly involves solutes moving across interfaces, most commonly, fluid-fluid interfaces. Together, these three methods of analysis are important tools for chemical engineering. 

The second part deals with mixing phenomena associated with tnrbulence. In that part, notions relating to turbulence are first presented. The problems associated with dispersion and mixing in connection with chemical reactions are then considered. The key notion, from a fundamental standpoint, regards the interrelation between the phenomena of turbulence and that of molecular diffusion, the latter being the actnal cause for mixing that allows a chemical reaction to occur. 

The parameter kg-out must account for diffusion/dispersion and advection losses at the lower boundary of a soil compartment. Advection with water that infiltrates through the soil is typically a unidirectional process, which removes chemicals with the effective velocity obtained in Equation 8.11. However, dispersion and diffusion processes such as molecular diffusion and bioturbation move chemicals both up and down within the soil, making it difficult to define a net loss factor applicable to the bulk soil. However, with a single well mixed compartment receiving chemical input at is surface, we can assume that the net diffusion is in the downward direction and proportional to the concentration gradient in the penetration depth z. In this case the parameter kg-out is obtained from a simple model for mass loss at the lower boundary of the soil compartment ... 

Glaser and Litt (G4) have proposed, in an extension of the above study, a model for gas-liquid flow through a b d of porous particles. The bed is assumed to consist of two basic structures which influence the fluid flow patterns (1) Void channels external to the packing, with which are associated dead-ended pockets that can hold stagnant pools of liquid and (2) pore channels and pockets, i.e., continuous and dead-ended pockets in the interior of the particles. On this basis, a theoretical model of liquid-phase dispersion in mixed-phase flow is developed. The model uses three bed parameters for the description of axial dispersion (1) Dispersion due to the mixing of streams from various channels of different residence times (2) dispersion from axial diffusion in the void channels and (3) dispersion from diffusion into the pores. The model is not applicable to turbulent flow nor to such low flow rates that molecular diffusion is comparable to Taylor diffusion. The latter region is unlikely to be of practical interest. The model predicts that the reciprocal Peclet number should be directly proportional to nominal liquid velocity, a prediction that has been confirmed by a few determinations of residence-time distribution for a wax desulfurization pilot reactor of 1-in. diameter packed with 10-14 mesh particles. 

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. 

Neglecting flow nonuniformities, the contributions of molecular diffusion ana turbulent mixing arising from stream splitting and recombination around the sorbent particles can be considered additive [Langer et ah, Int. J. Heat and Mass Transfer, 21, 751 (1978)] thus, the axial dispersion coefficient DL is given by ... 

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