RESUMO
The evolution of dissolved species in a porous medium is determined by its adsorption on the porous matrix through the classical advection-diffusion processes. The extent to which the adsorption affects the solute propagation in applications related to chromatography and contaminant transport is largely dependent upon the adsorption isotherm. Here, we examine the influence of a nonlinear Langmuir adsorbed solute on its propagation dynamics. Interfacial deformations can also be induced by classical viscous fingering (VF) instability that develops when a less viscous fluid displaces a more viscous one. We present numerical simulations of an initially step-up concentration profile of the solute that capture a rarefaction/diffusive wave solution due to the nonlinearity introduced through Langmuir adsorption and variety of pattern-forming behaviors of the solute dissolved in the displaced fluid. The fluid velocity is governed by Darcy's law, coupled with the advection-diffusion equation that determines the evolution of the solute concentration controlling the viscosity of the fluids. Numerical simulations are performed using the Fourier pseudospectral method to investigate and illustrate the role played by VF and Langmuir adsorption in the development of the patterns of the interface. We show that the solute transport proceeds by the formation of a rarefaction wave results in the enhanced spreading of the solute. Interestingly we obtained a nonmonotonic behavior in the onset of VF, which depends on the adsorption parameters and existence of an optimal value of such adsorption constant is obtained near b=1, for which the most delayed VF is observed. Hence, it can be concluded that the rarefaction wave formation stands out to be an effective tool for controlling the VF dynamics.
RESUMO
Viscous fingering is a hydrodynamic instability that sets in when a low viscous fluid displaces a high viscous fluid and creates complex patterns in porous media flows. Fundamental facets of the displacement process, such as the solute concentration distribution, spreading length, and the solute mixing, depend strongly on the type of pattern created by the unstable interface of the underlying fluids. In the present study, the frontal interface of the sample shows viscous fingering and the strong solvent causes the retention of the solute to depend on the solvent concentration. This work presents a computational investigation to explore the effect of the underlying physico-chemical phenomena, (i.e., the combined effects of solvent strength, retention, and viscous fingering) on the dynamics of the adsorbed solute. A linear adsorption isotherm has been assumed between the mobile and stationary phases of the solute. We carried out the numerical simulations by considering a rectangular Hele-Shaw cell as an analog to 2D-porous media containing a three component system (displacing fluid, sample solvent, solute) to map out the evolution of the solute concentration. We observed that viscous fingering at the frontal interface of the strong sample solvent intensifies the band broadening of the solute zone. Also notable increase in the spreading dynamics of the solute has been observed for less viscous and strong sample solvent as compared to the high viscous sample slices or in the pure dispersive case. On the contrary, the solute gets intensively mixed at early times for more viscous sample in comparison to less viscous one. The results of the simulations are in qualitative agreement with the behavior observed in the liquid chromatography column experiments.
RESUMO
In chromatographic columns, when the eluting strength of the sample solvent is larger than that of the carrier liquid, a deformation of the analyte zone occurs because its frontal part moves at a relatively high velocity due to a low retention factor in the sample solvent while the rear part of the analyte zone is more retained in the carrier liquid and hence moves at a lower velocity. The influence of this solvent strength effect on the separation of analytes is studied here theoretically using a mass balance model describing the spatio-temporal evolution of the eluent, the sample solvent and the analyte. The viscosity of the sample solvent and carrier fluid is supposed to be the same (i.e. no viscous fingering effects are taken into account). A linear isotherm adsorption with a retention factor depending upon the local concentration of the liquid phase is considered. The governing equations are numerically solved by using a Fourier spectral method and parametric studies are performed to analyze the effect of various governing parameters on the dispersion and skewness of the analyte zone. The distortion of this zone is found to depend strongly on the difference in eluting strength between the mobile phase and the sample solvent as well as on the sample volume.
