A phenomenological model for the contact region of an evaporating meniscus on a superheated slab

Abstract

The evaporating meniscus of a perfectly wetting fluid exhibits an apparent contact angle Θ that is a function of superheat. Existing theory predicts Θ and the heat flow from the contact region as part of the solution of a free-boundary problem. That theory admits the possibility that much of the heat flow occurs at the nanometre scale [lscr ]Θ at which Θ is determined. Here, the heat flow at that scale is proved negligible in typical applications. A phenomenological model of the contact region then holds since the part of the wetting film thinner than [lscr ]Θ can be replaced by an apparent contact line. Self-consistency arguments are used to derive conditions under which (i) the phase interface can be taken as linear with assumed contact angle Θ; (ii) the heat flux to the liquid side of the phase interface is given by Newton's law of cooling with predicted heat transfer coefficient h; and (iii) the temperature satisfies Laplace's equation within the phases. When these conditions are met, prediction of the heat flow is decoupled from the physically non-trivial problem of predicting Θ. Next, this conduction theory is used to find the heat flow from the contact region of a meniscus on a conductive slab. The solution depends on Θ, the liquid–solid conductivity ratio k = K[lscr ]/Ks and a Biot number [Bscr ] = hd/K[lscr ] based on slab thickness d. Asymptotic and numerical analysis is used to find the temperature in the double limit [Bscr ]−1 → 0 and k → 0. The solution has an inner-and-outer structure, and properties of the inner region prove universal. Formulae given here for the heat flow and contact line temperature on a slab thus apply to more complex geometries. Further, the solution explains the main features seen in published simulations of evaporation from conductive solids. Near the contact line, the solid temperature varies rapidly on the scale d of the slab thickness, but varies slowly with respect to the liquid temperature. The solid temperature thus proves uniform at the scale on which Θ is determined. Lastly, the quantitative predictions of the simplified model are verified against both new and published numerical solutions of the existing theory. In typical applications, the new formulae give the heat flow and contact line temperature with an error of about 10%. This error is due to the approximations made to derive the simplified model, rather than to those made to solve it.