Transient Taylor-Aris dispersion in N-capillary systems: Convergence properties of the band broadening in...

We report on a generic mathematical study of the transient plate height regime in bundles of polydisperse capillaries with diffusional bridging. I.e., we have investigated how the plate height in such systems can be expected to vary with the residence time, or equivalently, with the column length, before reaching a long-time limit wherein the plate height becomes constant. This is important because, in case of systems with a large number of capillaries (N), the transient regime solution is practically much more relevant than the long-time limit solution. The problem has been investigated using a simplified, yet representative geometry that is amenable to a semi-analytical solution, only requiring a numerical integration of the velocity field across a line (2D geometries) or a square or rectangle (3D geometries). The availability of this semi-analytical approach allows to consider the required large number of different random capillary diameter drawings (order of 10,000) needed to establish a sufficiently averaged result (deviation from true mean on the order of 1%). The general solution could be simplified into a set of two-parameter expressions describing either the long-time limit (τ→∞) as well as that of an infinite number of capillaries (N→∞). The combinations of N and τ where these expressions are valid to within an accuracy of 5% could be delimited with simple analytical expressions as well. It has also been observed that the long-time limit solution for the additional plate height (Δh_τ→∞) remains dependant on the number of capillaries N, even when N tends to infinity. It is found that Δh_τ→∞ can be expected to increase in proportion with N-1 (2D-case) or with ln(N-1) (3D-case). This dependency of Δh_τ→∞ on N is a result that is not obtained when modelling the capillary bundle using a representative binary capillary system approach as has been done up-till-now in literature.