Exact analytical expressions for the band broadening in polydisperse 2-D multi-capillary columns with diffusional bridging

Using a recent generalization of the Aris dispersion method, we have derived exact analytical expressions for the long-time limit dispersion in 2-D multi-capillary packings with diffusional bridging (binary channel system). Both the plug flow and the parabolic flow case are considered. The expressions are mathematically exact over the entire range of possible values of the degree of polydispersity (σ), the retention equilibrium constant K and the diffusion coefficients in the mobile and stationary zone. They are validated by comparing them to the dispersion data obtained by numerically solving the partial differential equation describing the general advection-diffusion mass balance. A correlation coefficient of R² > 0.99995 is obtained. The form of the obtained analytical expression shows the dispersion arising from the polydispersity effect in multi-capillary systems can be split in two contributions, one related to the actual velocity difference between the adjacent channels and one related to the fact that the average of the intra-capillary C-term band broadening is larger in any σ≠0-case than in the σ=0-case. When leaving the small σ-assumption, a new factor ((1- σ²)/(1+3σ²)²) emerges, common to both the solution for the plug flow and the parabolic case. The analysis shows the parabolic flow not only leads to a larger intra-capillary dispersion (σ=0) than the plug flow case as known from literature but is also more sensitive to the polydispersity effect (up to some 10 to 20%).