Compressible Flow Basis of Gas Chromatography

For many packed and porous media (chromatographic columns, ultrafiltration membranes) the pressure drop Ap associated with flow is important. From Eq. 4.22 we get 

Thus for a given mean flow velocity (v), Ap increases rapidly with decreasing particle (or pore) size dp. 

As a gas stream is driven down a column from a position of high pressure to one of low pressure, it expands considerably. Since in steady flow the same mass of gas must pass through each cross section in unit time, a greater volume flux of gas will be observed at points where the gas is more expanded. This means that the flow velocity, which is proportional to volume flux, is largest where the pressure is least—at the column outlet. In view of the proportionality of the mean velocity (v) and the molar volume V, Boyle s law can be rewritten as p(v) = const, where p(v) replaces pV. 

To determine the pressure and velocity distributions, we must relate mean cross-sectional flow velocity (v) to the pressure gradient. This is done by means of Eq. 4.11 for capillary tubes and Eq. 4.22 for packed columns. To work with these equations, we must replace the overall pressure gradient Ap/L by the local pressure gradient - dpldx, where the negative sign arises because pressure decreases as one moves along the positive flow coordinate x. With this substitution, Eq. 4.11 can be rearranged to 

Since according to our modified Boyle s law the product p(v) is constant throughout the column or tube, we may write 

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