Numerical study of buoyancy-induced instability during CO2 absorption in alkaline solutions

Introduction

This work deals with the modeling and the numerical simulation of the carbon dioxide (CO₂) absorption, coupled with a chemical reaction, in an initially quiescent aqueous alkaline solution, inside a Hele-Shaw cell. Such a cell is a commonly used experimental tool to study the CO₂ transport phenomena in porous media, especially in the framework of CO₂ sequestration in deep saline aquifer processes. In our device, the liquid fills partially the gap between two parallel transparent Plexiglas plates. CO₂ is forced to flow above the liquid in the cell and, therefore, it is absorbed in the liquid, in which it takes part to chemical reactions. In previous works, two kind of aqueous alkaline solutions have been experimentally studied thanks to a Mach-Zehnder interferometer (enabling the visualization of liquid density variations): a solution of monoethanolamine (MEA) and a solution of a mixture of sodium carbonate (Na₂CO₃) and bicarbonate (NaHCO₃), hereafter called solution 1 and 2, respectively.

It has been observed that the CO₂ absorption, initially driven by diffusion and reaction, leads in both solutions to the apparition of a buoyancy-induced instability. Indeed, in a first stage, a horizontally homogeneous boundary layer (with different concentrations of the various chemical species) is formed in the liquid, and grows continuously over time. This boundary layer becomes eventually unstable, since the density might be locally increased (maximum density variation at the interface for solution 1 and 1 mm below the interface for solution 2). The buoyancy-induced instability occurs in the form of plumes sinking into the bulk liquid, hence driving an intense mixing and potentially enhancing the rate of the CO₂ absorption. These plumes appear after a few seconds for solution 1 and after several minutes for solution 2. Moreover, under certain conditions for the solution 2, instead of plunging deep inside the fresh liquid, plumes has appeared to slow down at some depth, soften, and almost vanish. Meanwhile, new plumes might be generated, repeating the same scenario.

Modeling

In order to explain these observations and to highlight further differences between solutions 1 and 2, notably concerning the mass absorption rate, a numerical study is realized thanks to a model of the reaction-diffusion-convection dynamics. Assuming that this instability is triggered by a Rayleigh-Taylor mechanism, a two-dimensional model, coupling diffusion, chemical reaction and free convection, is developed. It is considered that the reaction of the dissolved CO₂ dissolved in can be written as a single reaction, having the generic form CO₂ + ν_BB ⇌ ν_CC which has a reaction rate. Assuming that the diffusive transports can be modeled by Fick's law, the mass transport-reaction equations write:

∂_t[CO₂] + u · ∇[CO₂] = D_CO2∇² [CO₂] - r ,

∂_t[B] + u · ∇[B] = D_B∇² [B] - ν_Br ,

∂_t[C] + u · ∇[C] = D_C∇² [C] + ν_Cr .

The density variation induced by the reaction and the mass transport is assumed to follow a correlation of this form: Δρ = ∂_[B]ρ ([B]-[B]₀) + ∂_[C]ρ ([C]-[C]₀).

The flow modeling, aiming at describing the actual three-dimensional flow in the Hele-Shaw cell using a two-dimensional approach, requires averaging the flow along the gap between the two plates of the cell. The Navier-Stokes-Darcy approach is used, assuming a parabolic Poiseuille like profile of the velocity across the gap. It is assumed that the flow is incompressible and the Boussinesq approximation is implied, leading to the following equations:

ρ(∂_tu + (6/5) u · ∇u) = -∇p + μ (∇²u + 12u/h²) - Δρg ey ,

∇ · u = 0 .

The transient boundary-value problem is solved numerically using the commercial software COMSOL Multiphysics 4.4, for the two considered solutions.

Results and discussion

It is observed that the simulation succeeds to reproduce qualitatively the time evolution of the liquid density variation field in the cell observed experimentally during the onset of the instability and after, for both considered solutions. The instability starts by the formation of several plumes at the interface for solution 1 and below the interface for solution 2, the liquid near the interface remaining stagnant.

By further analyzing the simulation results, it clearly appears that the difference in dynamics between solutions 1 and 2 is linked to the existence in solution 2 of the stagnant layer near the interface, where the density stratification is stable. In both solutions however, right after the plumes formation, the rate of the chemical reaction is enhanced at horizontal locations where the liquid flows upward, because of the resulting convective supply of fresh reactant. One could therefore expect a density increase there in both solutions 1 and 2. While in solution 1, convection along the interface transport a fraction of this density excess laterally towards the falling plumes (hence sustaining this falling motion), the transport phenomena appear to be more complex for solution 2. The location of the excess of density resulting from the interaction between the chemically-enhanced diffusive transport in the stagnant layer and the convective transport induced by the instability, the dynamics of the developed instability may be variable. In some cases, after a certain time, this density field configuration is able to slow down and even reverse the flow, triggering a new generation of plumes.

In addition, another remarkable difference between cases 1 and 2 concerns the impact of convection on the interfacial mass transfer rate. The existence of a stably-stratified stagnant layer near the interface in solution 2 indeed modify considerably the mass transfer dynamics as compared to solution 1, in which the instability induce considerable mixing close to the interface. This is due to the fact that, in solution 2, mass transport in the stagnant interfacial region remains mainly diffusive. This result highlights that a buoyancy-induced instability development does not necessarily lead to a significant mass transport enhancement.

Conclusion

In this work, a two-dimensional model is use to simulate the buoyancy-induced instability development which has been observed in a Hele-Shaw cell when CO₂ is absorbed in an aqueous alkaline solution. It is observed that the simulations reproduce rather well the density variation pattern triggering the instability and that the simulated instability dynamics agree qualitatively with the experimental observation for the two considered solutions. Thanks to the simulation result analysis, the discrepancies of the instability dynamics are partially explained and their effect on the mass absorption is evaluated. It is shown that this rate may be not enhanced by the instability, contrarily to what is commonly expected.