Effect of longitudinal diffusion

Lapidus, L. and Amundson, N. R. J. Phys. Chem. 56 (1952) 373. Mathematics of adsorption in fixed beds — The rate determining steps in radial adsorption analysis ibid 56 (1952) 984. The effect of longitudinal diffusion in ion exchange and chromatographic columns. 

Acrivos, A., On the combined effect of longitudinal diffusion and external mass transfer resistance in fixed bed operations. Chem. Eng. Sci. 13, 1 (1960). 

In 4 of this paper the effect of longitudinal diffusion is estimated and is added to diffusion coefficient k/i2. Equation (41) of the present paper supplies a rigorous justification for this. 

This shows that the secohd term, i.e., the longitudinal diffusion term, is negligible compared to the first Iterm, i.e., the radial diffusion term, if the quantity Re Pr is large. The quantity RePr is termed the Peclet number, Pe. Generally, if the Peclet number is greater t ian 10, the effects of longitudinal diffusion are, indeed, negligible. 

To estimate the effect of longitudinal diffusion, the contours of constant conversion are drawn in the plane of and PJM in Fig. 9.25. If all the parameters are known the values of and PJM = v lkE can be calculated and the contour will give the conversion. If a certain conversion is required and all parameters except the length are given, this may be found from the value ofP . 

The stationary phase is supported on the wall of the column, the carrier gas therefore has an uninterrupted flow through the column. The absence of column packing means that the unequal pathways term, A term, in the van Deemter equation (see section 2.5.1) is zero and band broadening is due to the effects of longitudinal diffusion, B term, and mass transfer, C term. The reduced equation is referred to as the Golay equation [15]. 

We shall look at the boundary conditions for this equation in the next section. I want to mention here a very important model introduced by Wes-terterp and his collaborators4 and to do so will revert to the classical form of the Taylor problem, in which there is no reaction, the tube is circular (radius R), and the flow is laminar (average velocity U). Also, we ignore the effect of molecular diffusion in the longitudinal direction and are concerned only with the effects of the lateral diffusion across the flow profile. Thus, we have... 

In conclusion, it follows for the limiting case of longitudinal diffusion as the only peak-broadening effect, that the resolution depends on the following ... 

The effect of longitudinal dispersion in a column, as a factor which reduces the sharpness of breakthrough curves, has been considered by Barrow and co-workers (B3), Ledoux (L4), and others. Wicke (W5) has discussed the combined effect of longitudinal dispersion with internal diffusion, while Lapidus and Amundson (L3) have reviewed the combined effect with external diffusion. Van Deemter et al. (V2) have recently studied this problem with respect to gas chromatography. 

The selection of an optimum column diameter is also not a simple procedure. Generally speaking it is desirable to operate at high velocities in order to minimize both the convective mass transfer resistance and the undesirable effects of longitudinal and radial diffusion. However, high velocities result in high pressure drops which may be undesirable because of three concerns. 

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