An explicit expression for the retention factor and velocity dependency of the mobile zone mass transfer band broadening in packed spheres beds used in liquid chromatography.

The present study proposes a ready-to-use analytical expression to calculate the mobile zone mass transfer contribution (hCm) in packed bed columns. For this purpose, first high-accuracy computations of the band broadening in a perfectly ordered sphere array (fcc-arrangement, external porosity ε=0.40) were made using computational fluid dynamics (CFD), covering a broad range of zone retention factors (2≤k''≤18) and reduced velocities (0≤νi≤48). Subsequently, these data were used to determine the value of the geometrical constants in a number of possible analytical expressions for the hCm-contribution. This fitting exercise showed the traditional literature approach, using the Wilson-Geankoplis correlation to calculate the dimensionless Sherwood (Sh) number for the mass transfer, leads to fitting errors on the hCm-term as large as 150%. Instead, a new correlation for Sh is established. In addition, we also explored the difference in fitting accuracy between hCm-expressions based on either a plug-flow or a laminar flow profile assumption. Surprisingly, no significant difference in fitting accuracy between both assumptions was observed. Finally, a best-fit analytical expression is proposed that can represent the CFD-computed band broadening data with an average absolute fitting error of Δh=0.005, corresponding to a relative error of 2.5% on the hCm-term and of only 0.3% on the total plate height in a perfectly ordered sphere packing. Defining the presently investigated fcc-ordered sphere array with external porosity=40% as the reference geometry for a perfect sphere packing, the established expression can be used as a new yardstick expression against which the degree of eddy-dispersion can be measured.