Diffusion, Dispersion, and Flow

There are three basic approaches to measuring transport processes, which include diffusion, dispersion, and bulk flow phenomena. These are the following  

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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 description of transport is closely connected to the terms convection, diffusion dispersion, and retardation as well as decomposition. First, it is assumed that there are no interactions between the species dissolved in water and the solid phase, through which the water is flowing. Moreover, it is assumed that water is the only fluid phase. The multiphase flow water-air, water-organic phase (e g. oil or DNAPL) or water-gas is not considered here. 

Organic coatings are generally utilized to improve dispersability and flow characteristics of titania pigments. A fascinating array of substances are either used commercially or have been claimed as useful in the patent literature [6], Organics held in place simply by electrostatic forces [7] as well as those cemented to the surface by use of covalent bonds are known [8], The latter are inert to diffusion through the polymer matrix. Said diffusion can lead to end-use problems that include such phenomena as loss of low-temperature heat seal and poor printability. 

In order to improve the original model, Danckwerts boundary conditions were assumed in the inlet and outlet flows for mass and energy balances. Moczydlower (2002) has adopted as inlet boundary conditions the feed concentration of the components and the feed temperature. Since the classical mathematical modeling were employed to describe the diffusion, dispersion and convection phenomena, Danckwerts boundary conditions are required in order to satisfy the mass and energy balances. Besides providing a more... 

Here, the A term is due to eddy dispersion and flow contribution to plate height and is independent of ly it is a function of the particle size and the packing efficiency. The next term includes B, which depends on the molecular diffusion coefficient in the longitudinal direction, i.e. the mobile-phase diffusivity. The third term includes C, which results from mass transfer between the mobile and stationary phases and has contributions from (1) diffiision in the film around the particle in the column, (2) diffiision in the Uquid phase that is stagnant in the pores and (3) diffusion in the liquid-phase coating on particles. 

These three phenomena of diffusion, dispersion and convection are assumed to be additive and independent. The presence of the cross-flow does not bias the probability that the molecule will take a diffusive step to the right or the left it just adds something to that step. Therefore, the total flux of dissolved solute ... 

More disagreement exists with respect to axial dispersion—for example, regarding the applicability of the diffusion model, and regarding the influence of gas and liquid flow rates. More work on these aspects and on the influence of fluid distribution and method of packing is required. Some of the available results are compared in Fig. 3. 

The form of the effective mobility tensor remains unchanged as in Eq. (125), which imphes that the fluid flow does not affect the mobility terms. This is reasonable for an uncharged medium, where there is no interaction between the electric field and the convective flow field. However, the hydrodynamic term, Eq. (128), is affected by the electric field, since electroconvective flux at the boundary between the two phases causes solute to transport from one phase to the other, which can change the mean effective velocity through the system. One can also note that even if no electric field is applied, the mean velocity is affected by the diffusive transport into the stationary phase. Paine et al. [285] developed expressions to show that reversible adsorption and heterogeneous reaction affected the effective dispersion terms for flow in a capillary tube the present problem shows how partitioning, driven both by electrophoresis and diffusion, into the second phase will affect the overall dispersion and mean velocity terms. 

The gas flow direction was from the top to bottom of the figure. No divergence is observed in the dispersion curve of the capillary, indicating that under the given conditions the dispersion of flow is small, and that this scheme is thus adequate to study the dispersion within a device of interest. This may appear unexpected, as microfluidic devices are usually assumed to exhibit laminar flow, however it can be explained by the fast lateral diffusion of individual gas molecules, which uniformly sample the whole cross section of the tube in a very short time compared with the travel time. Below each image, its projection is shown together with an independ-... 

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