Understanding fast adsorption in single-solute breakthrough curves

Abstract

To design efficient adsorption separations and select high-performing materials, understanding the relationship between adsorption equilibria and the movement of solute waves along the fluid is critical. While approximate-analytical methods are useful to combat nonlinearities in the differential equations and efficiently investigate the myriad of materials, their utility is limited by the lack of a clear physical interpretation and mathematical relationship to well-established models. In this work, we show that the two popular methods can be understood with regular perturbation theory and boundary layer theory in small ɛ, where a dilute solute travels from fluid to solid much faster than along the column. We consider conditions associated with a popular experimental technique used to characterize solute wave movement, where the column is initially at steady state and the fluid concentrations exiting, or breaking through, are time-monitored in response to a step change in inlet concentration. Our transformations and asymptotic analyses illustrate the time scales associated with popular techniques. We generalize their applicability to equilibria possessing inflection points while considering both adsorption and desorption modes. We use numerical simulations to estimate convergence orders of asymptotic approximations of breakthrough concentrations. For the leading-order approximation of regular perturbation theory, rarefaction waves possess positive convergence orders, depending on the extent of nonlinearity, while shock waves do not. Boundary-layer theory affords a more accurate approximation of the breakthrough curve than a shock wave, without guaranteeing a positive convergence order. This lack of positive convergence order is likely due to the evolution of the boundary layer, as illustrated with numerical simulations. An improved understanding of fast adsorption sets the stage for more realistic selection of high-performing materials and design configurations from thousands of available candidates.