Predicting Marangoni convection caused by transient gas diffusion in liquids

Abstract

The onset of convection driven by surface tension during gas diffusion in a liquid is investigated. Gas diffusion at the gas–liquid interface results in the variation of concentration of the solute that may cause an increase in surface tension leading to Marangoni convection. The onset of convection for unsteady-state gas desorption can be predicted from the maximum transient Ma t, which is here derived by analogy with its equivalent in thermal convection. It is a function of the transient Biot number ( Bi D) for interfacial gas diffusion, which depends strongly on the state of vapour–liquid equilibrium at the interface. The transient Marangoni numbers, critical times for stable mass diffusion and the critical sizes of convection cells have been formulated. The desorption of ethyl-ether from chloro-benzene in L.M. Blair’s [The onset of cellular convection in a fluid layer with time-dependent density gradients, PhD thesis, University of Illinois, Urbana, 1968] experiments is liquid phase-controlled, hence, the highly soluble system is characterized by Bi D = 0. Therefore, his experiments that were initiated with a step-change in pressure cannot be analyzed by a step-function boundary that is characterized by Bi D = ∞. The surface concentration may change very slowly, it has been approximated to be about 0.1% of the initial pressure change at the point of onset of convection. The average critical Marangoni number for this condition was estimated to be 53.3, which is fairly close to the theoretical value of 67 for an interface with a Biot number of 0. Therefore, the high value of 3100 calculated by I.F. Davenport and C.J. King [The initiation of natural convection caused by time-dependent profiles, Lawrence Berkeley Lab, Report NBR LBL-600, 1972] is wrong, who wrongly assumed a fixed surface-concentration boundary that is applicable only to a sparingly soluble solute. The critical sizes of convection cells predicted by theory are generally less than 1 mm for reported critical times of less than 20 s, they would be difficult to measure.