The surface-renewal theory of interphase transport: a stochastic treatment

Abstract

The stagnant-film, boundary-layer, and surface-renewal theories have been regarded as the cornerstone of the science of interphase mass or heat transfer in turbulent environments. The stagnant-film theory has been highly popular and remains so because of its simplicity; however, it is deemed too simplistic and unrealistic. The boundary-layer theory has been derived from a fairly rigorous and self-consistent fluid mechanical theory based on the notion of continuity; nevertheless, this theory is incapable of elucidating random disturbance or chaotic bursting at the interface under turbulent conditions. The surface-renewal theory has been conceived so that the deficiencies of the first two theories can be rectified through incorporation of some statistical components into the description of interphase mass or heat transport. Numerous variants of this theory, giving rise to various mathematical models, have been proposed; still, the acceptance or popularity of the surface-renewal theory appears to lag behind the other two theories. This is probably attributable to the fact that while the mathematical formulation of the theory is abundantly couched in statistical or stochastic parlance, the methodologies and procedures followed are those of continuum mechanics and deterministic mathematics, thereby rendering the model or theory less acceptable. The present work attempts to derive a self-consistent mathematical model of the surface-renewal theory of interphase mass transfer by resorting to the theories and methodologies of stochastic processes based on the Markovian assumption. Specifically, the expression for the contact-time distribution of fluid elements or solid particles participating in the interphase transport has been derived from the stochastic population balance of these elements or particles. Moreover, the expression for the dynamic rate of transfer of molecules or microscopic components across the interface has been derived as the continuous limit of the probability balance equation of the random walks of these entities around the interface. Proper coupling of the two expressions constitutes the desired model. By analogy, this model is applicable to the turbulent interphase heat transport and plausibly to the momentum transport under certain circumstances.