Diffusion effects longitudinal

Van Deemter considered peak dispersion results from four spreading processes that take place in a column, namely, the Multi-Path Effect, Longitudinal Diffusion, Resistance to Mass Transfer in the Mobile Phase and Resistance to Mass Transfer in the Stationary Phase. Each one of these dispersion processes will now be considered separately... 

The three contributions to dispersion are also shown as separate curves in figure 1. It is seen that the major contribution to dispersion at the optimum velocity, where the value of (H) is a minimum, is the multipath effect. Only at much lower velocities does the longitudinal diffusion effect become significant. Conversely, the mobile phase velocity must be increased to about 0.2 cm/sec before the resistance to mass transfer begins to become relatively significant compared to that of the multipath effect. 

Giddings (5) also determined theoretically that for a well packed column (y) should be about 0.6 so again the longitudinal diffusion effect confirms that the column was reasonably well packed. 

However, at high velocities the effective value of the diffusivity of the solute dramatically increases as a result of induced radial flow, eventually reducing the resistance to mass transfer factor to virtually zero. This results in a corresponding dramatic reduction in the value of (H). Finally, at very high velocities, the greatly reduced longitudinal diffusion effect again dominates. At this point, the value of (H) is very small and, in fact, decreases even further as the mobile phase velocity is further increased. ... 

It should be pointed out that for a low pressure gas the radial- and axial diffusion coefficients are about the same at low Reynolds numbers (Rediffusion effects may be important at velocities where the dispersion effects are controlled by molecular diffusion. For Re = 1 to 20, however, the axial diffusivity becomes about five times larger than the radial diffusivity [31]. Therefore, the radial diffusion flux could be neglected relative to the longitudinal flux. If these phenomena were also present in a high-pressure gas, it would be true that radial diffusion could be neglected. In dense- gas extraction, packed beds are operated at Re > 10, so it will be supposed that the Peclet number for axial dispersion only is important (Peax Per). 

Liquid-solid chromatography is representative of this theory because nonlinearity effects are usually appreciable. Mass transfer is fast and longitudinal diffusion effects may be ignored in describing the system. The net result is that the bands (zones) develop self-sharpening fronts and diffuse rear boundaries. It is because of this tailing that this technique is to be regarded... 

Diffusion in liquids is four to five orders of magnitude less than that found in gases. For this reason we may neglect longitudinal diffusion effects in the liquid phase in zone broadening. However, longitudinal diffusion must be considered in equilibrium effects because it may determine the rate of mass transfer. 

Longitudinal diffusion occurs in all directions the molecules at the front of the zone will move forward into the next zone, the molecules at the end of the zone will fall back into the previous zone, and diffusion will also occur toward the column walls. As diffusion is a time-dependent process, the longitudinal diffusion effect increases at low mobile-phase flow rates. 

Three mechanisms produce dispersion of a band of solute in a chromatographic system as it passes through the separation column 1) eddy diffusion 2) longitudinal diffusion and 3) mass transfer effects. These effects are discussed, in some detail, in this article. 

The open-tubular column or capillary column is the one most commonly used in gas chromatography (GC) today. The equation that describes dispersion in open tubes was developed by Golay [1], who employed a modified form of the rate theory, and is similar in form to that for packed columns. However, as there is no packing, there can be no multipath term and, thus, the equation only describes two types of dispersion. One function describes the longitudinal diffusion effect and two others describe the combined resistance to mass-transfer terms for the mobile and stationary phases. 

A section of a fixed-bed catalytic reactor is shown in Fig. 13-4. Consider a small volume element of radius r, width Ar, and height Ar, through which reaction mixture flows isothermally. Suppose that radial and longitudinal diffusion can be expressed by Pick s law, with and Dj as effective diffusivities, based on the total (void and nonvoid) area perpendicular to the direction of diffusion. We want to write a mass balance for a reactant over the volume element. With radial and longitudinal diffusion and longitudinal convection taken into account, the input term is... 

Since only flow and diffusion effects couple, while longitudinal diffusion and stationary phase mass transfer don t, the complete equation for the dependence of the HETP on the linear velodty diould be written as... 

In a continuous tank-type reactor, the flow should not follow preferential paths. In the continuous tubular reactor, the flow can be in extreme cases laminar (not desired) or turbulent (desired), but without dead volume. The type of flow may cause radial and longitudinal diffusion effects causing radial or axial temperature and concentration gradients and consequently affecting the chemical reaction. 

In CSTR reactors, the flow has preferential paths. In continuous tubular reactors, the flow may be laminar, turbulent, and have dead volumes. The flow may cause radial and longitudinal diffusion effects and therefore to result temperature gradients and radial/axial concentration. Therefore, the flow may affect the chemical reaction. 

The picture involved in the one-dimensional dispersion model is the onedimensional process of flow in a tube. There is a flow velocity in direction z, which, in the ideal case, is constant over the reactor cross section %. Because of molecular diffusion, turbulent convection, and the parabolic velocity profile that results from boundary friction (roughness, s), there are large deviations from a uniform flow front. The effective longitudinal dispersion coefficient... 

Some authors have been concerned with influence of flow or diffusion on measurements of and Anderson et al. discussed diffusion of spins between compartments, characterized by different states of longitudinal magnetization, leading to diffusion-driven longitudinal relaxation. The effects were explored experimentally and analyzed quantitatively. Herold and co-workers described an on-line NMR rheometer, able to measure NMR relaxation data. The corrections required for the analysis of relaxation data measured under flow conditions were discussed. The opposite problem how to avoid the detrimental effects of unequal relaxation rates on the diffusion measurements in complex mixtures - was discussed by Barrere et Relaxation can also cause problems in other kinds of NMR experiments. Skinner and co-workers described the optimal control design of band-selective excitation pulses that accommodates both relaxation and inhomogeneity of rf fields. 

Rg. 2. Illustration of the three principal causes of band broadening (a) multiple-path effect (b) longitudinal diffusion effect (c) mass-transfer (non-equillbrium) effect. Reproduced from A. Braithwaite F.J. Smith, Chromatographic Methods, 5th edn, 1996, first published by Blackie Academic Professional. 

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