Quantitative relationship between interfacial curvature due to a liquid’s surface tension and the binary gas diffusivity estimates obtained in an isothermal Stefan column

Abstract

Josef Stefan developed the theoretical framework needed to analyze the diffusion column in the second half of the 19^th century. For the next six decades after his death, his design allowed the estimation of binary gas diffusivities assuming a flat liquid–gas interface and isothermal operation at atmospheric pressure. In the 1950s, inaccuracies in the diffusivity estimates were detected. These were related to column end effects at the top, where turbulence and eddy formation at the mixing point of the sweep and gas phases occurred, and at the liquid–gas interface, where curvature due to surface tension changed the mass transfer area and the diffusion path length for gas A to the top. The present work examines quantitatively for the first time the relationship between interfacial curvature resulting from surface tension and the binary gas diffusivity estimates in the Stefan column. The hypothesis is that such a relationship exists. The dimensionless parameter N₁, which gives the ratio of surface tension to gravitational forces acting on the interface, determines the radial distribution profile, in turn affecting the curved-to-flat-interface binary gas diffusivity ratio. The generalized surface tension model was validated numerically with interfacial descent versus time data from two Stefan column runs reported recently (acetone-air and n-hexane-air). The experimental curved-to-flat-interface diffusivity ratio in both runs was 1.7–1.8, indicating an important contribution from interfacial curvature in the diffusivity calculations. The researcher now has a quantitative tool relating interfacial curvature due to surface tension and the binary gas diffusivity estimates obtained in the Stefan column.

Keywords:

• Binary gas diffusivity estimation
• diffusion and convection of gas A
• flat and curved liquid–gas interfaces
• gas-phase end effects in the Stefan column
• inaccuracies in experimental diffusivities
• stagnant species B
• Stefan column with a descending interface
• surface tension and gravitational forces acting on an interface
• theoretical transport modeling of gas A

Acknowledgements

The author derived a substantial amount of academic inspiration from University of Wisconsin-Madison Professors R. B. Bird, W. E. Stewart, and E. N. Lightfoot. This world-renowned triumvirate is responsible for the timeless text Transport Phenomena, which researchers young and old should read at some point in their lives. Thanks are also extended to the undergraduate chemical engineering students I. Moreno and M. Moreno, for carrying out and analyzing the Stefan diffusion column experiments used to validate numerically the generalized surface tension model developed in this work. A. L. Braña and J. B. Ramírez are recognized for procuring the archival references. Finally, the time and effort spent by the anonymous reviewers scrutinizing the manuscript have led to an improved final version. Their comments and suggestions are appreciated.