Motivated by geological carbon dioxide (CO 2 ) storage, many recent studies have investigated the fluid dynamics of solutal convection in porous media. Here we study the convective dissolution of CO 2 in a closed system, where the pressure in the gas declines as convection proceeds. This introduces a negative feedback that reduces the convective dissolution rate even before the brine becomes saturated. We analyse the case of an ideal gas with a solubility given by Henry's law, in the limits of very low and very high Rayleigh numbers. The equilibrium state in this system is determined by the dimensionless dissolution capacity, Π, which gives the fraction of the gas that can be dissolved into the underlying brine. Analytic approximations of the pure diffusion problem with Π > 0, show that the diffusive base state is no longer self-similar and that diffusive mass transfer declines rapidly with time. Direct numerical simulations at high Rayleigh numbers show that no constant flux regime exists for Π > 0; nevertheless, the quantity F/C 2 s remains constant, where F is the dissolution flux and C s is the dissolved concentration at the top of the domain. Simple mathematical models are developed to predict the evolution of C s and F for high-Rayleigh-number convection in a closed system. The negative feedback that limits convection in closed systems may explain the persistence of natural CO 2 accumulations over millennial timescales.

Reverse Osmosis (RO) is a membrane-based technology for water desalination. Of paramount importance is the understanding of ion selectivity in mixtures of salts, i.e., to what extent the membrane retains one ion more than another in a multicomponent salt solution. We apply continuum transport theory to describe a large set of data for the ion selectivity of RO membranes treating brackish ground water with more than ten different mono-and divalent ions. The model is based on the Donnan steric partitioning pore model extended to include ions of multiple charge states, such as bicarbonate/carbonic acid, ammonia/ammonium, and the hydroxyl/hydronium ions, and the acid-base reactions between them and with the membrane charge. By adjusting for each ion the ratio of ion size over pore size, we can fit the model to the data. We note that the fitted ion sizes do not always follow a logical order based on the ionic or hydrated size of the ions and that rejection of divalent cations is overestimated in some cases. We discuss possible theoretical improvements to address these discrepancies. Our results highlight the potential of continuum transport theory to describe in detail multicomponent ion transport in RO membranes. The development of a detailed and validated physics-based model is an important step towards achieving improved operation and design of RO-based desalination systems.

Previous work on solute transport with sorption in Poiseuille flow has reached contradictory conclusions. Some have concluded that sorption increases mean solute transport velocity and decreases dispersion relative to a tracer, while others have concluded the opposite. Here we resolve this contradiction by deriving a series solution for the transient evolution that recovers previous results in the appropriate limits. This solution shows a transition in solute transport behavior from early to late time that is captured by the first-and zeroth-order terms. Mean solute transport velocity is increased at early times and reduced at late times, while solute dispersion is initially reduced, but shows a complex dependence on the partition coefficient k at late times. In the equilibrium sorption model, the time scale of the early regime and the duration of the transition to the late regime both increase with ln k for large k. The early regime is pronounced in strongly-sorbing systems (k ≫ 1). The kinetic sorption model shows a similar transition from the early to the late transport regime and recovers the equilibrium results when adsorption and desorption rates are large. As the reaction rates slow down, the duration of the early regime increases, but the changes in transport velocity and dispersion relative to a tracer diminish. In general, if the partition coefficient k is large, the early regime is well-developed and the behavior is well characterized by the analysis of the limiting case without desorption.

In this work, we develop an extended uniform potential (UP) model for a membrane nanopore by including two different charging mechanisms of the pore walls, namely by electronic charge and by chemical charge. These two charging mechanisms will generally occur in polymeric membranes with conducting agents, or membranes made of conducting materials like carbon nanotubes with surface ionizable groups. The electronic charge redistributes along the pore in response to the gradient of electric potential in the pore, while the chemical charge depends on the local pH via a Langmuir-type isotherm. The extended UP model shows good agreement with experimental data for membrane potential measured at zero current condition. When both types of charge are present, the ratio of the electronic charge to the chemical charge can be characterized by the dimensionless number of surface groups and the dimensionless capacitance of the dielectric Stern layer. The performance of the membrane pore in converting osmotic energy from a salt concentration difference into electrical power can be improved by tuning the electronic charge.

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