Slug bubble growth and dissolution by solute exchange

In many environmental and industrial applications, the mass transfer of gases in liquid solvents is a fundamental process during the generation of bubbles for specific purposes or, vice versa, the removal of entrapped bubbles. We address the growth dynamics of a trapped slug bubble in a vertical glass cylinder under a water barrier. In the studied process, the ambient air atmosphere is replaced by a ${\mathrm{CO}}_2$ atmosphere at the same or higher pressure. The asymmetric exchange of the gaseous solutes between the ${\mathrm{CO}}_2$-rich water barrier and the air-rich bubble always results in net bubble growth. We refer to this process as solute exchange. The dominant transport of ${\mathrm{CO}}_2$ across the water barrier is driven by a combination of diffusion and convective dissolution. The experimental results are compared to and explained with a simple numerical model, with which the underlying mass transport processes are quantified. Analytical solutions that accurately predict the bubble growth dynamics are subsequently derived. The effect of convective dissolution across the water layer is treated as a reduction of the effective diffusion length, in accordance with the mass transfer scaling observed in laminar or natural convection. Finally, the binary water-bubble system is extended to a ternary water-bubble-alkane system. It is found that the alkane ($n$-hexadecane) layer bestows a buffering (hindering) effect on bubble growth and dissolution. The resulting growth dynamics and underlying fluxes are characterized theoretically.

Figure 1

(a) Schematic overview of the experimental setup. (b) Sketch of the cylinders containing the water-bubble and water-bubble-alkane systems. These are placed inside the chamber, which is subsequently pressurized with ${\mathrm{CO}}_2$ gas.

Figure 2

A typical experiment involving a binary and ternary system. Experimental snapshots at (a) $t=0$, equilibrium state, (b) $t=t_f$, immediately after ${\mathrm{CO}}_2$ flushing and pressurization, (c) halfway through the experiment, and (d) at the end of the experiment. The blue line indicates the bubble-water interface detected by the contour detection algorithm. Similarly, in the right cylinder with the alkane, the red line indicates the detected bubble-alkane interface. (e) Time evolution of the pressure in the tank during the first 200 s of the experiment. Before $t_{f-\mathrm{start}}$ (green stage) the tank is filled with ambient air. Between $t_{f-\mathrm{start}}$ and $t_f$ (orange stage) the tank is flushed with ${\mathrm{CO}}_2$ and the bubbles are compressed. (f) Evolution in time of the bubble volume for the two systems.

Figure 3

Experimental and theoretical results for water-bubble systems in an upright configuration [cases (i) and (ii)] and an inverted configuration [cases (iii) and (iv)] for two distinct water layer heights; the experiment pressure level is $P_0(t>t_f)=1$ bar and bubble volumes are initially $V_f=35\mu \mathrm{L}$. (a) Bubble growth dynamics ($\mathrm{\Delta }{V}_b\equiv V_b\left(t\right)-V_f$), according to experiment (circles), numerical model (black and white markers), and analytical solutions (green lines); the latter are given by Eq. (13) for cases (iii) and (iv) and Eq. (17) for cases (i) and (ii). The onset of convective dissolution for upright cases (i, ii) is evident. (b) Time evolution of the ${\mathrm{CO}}_2$ concentration profiles in the water layer according to the model for the convective dissolution case (ii) and the pure diffusion case (iv). The uniform profile outside the diffusion boundary layer in case (ii) illustrates the model assumption of instant convective transport. (c) Time evolution of the gas composition in the bubble according to the model for upright case (ii) (white markers and lighter curves), and inverted case (iv) (black markers and darker curves).

Figure 4

(a) Sketch of the quasisteady concentration profile of ${\mathrm{CO}}_2$ across the water layer after the onset of convective dissolution. The main concentration drop occurs across the top diffusion boundary layer. Transport through the convective layer is assumed instant. The concentration drop across the bottom boundary layer is neglected. This is equivalent to (b), where the water layer is treated as a single diffusion layer of thickness $\delta \left(t\right)\sim H_w{\mathrm{Ra}}_w{\left(t\right)}^{-1/4}$, where ${\mathrm{Ra}}_w\left(t\right)\propto [P_0-P_{{\mathrm{CO}}_2}^b\left(t\right)]$, the difference in the partial pressure of ${\mathrm{CO}}_2$ in the ${\mathrm{CO}}_2$-ambient and the bubble.

Figure 5

(a) Effect of increasing the experiment pressure level $P_0(t>t_f)$ on the bubble growth dynamics. Water layer height is $H_w\approx 4.5$ mm and postpressurization volumes are $V_f\approx 35\mu \mathrm{L}$. Experimental, numerical, and analytical—Eq. (17)—results are compared. (b) Time evolution of the boundary layer thickness for marginal stability, $\delta \left(t\right)$ (black solid line), as established by its concomitant Rayleigh number (dotted line) according to the numerical model for experiment (iii) where $P_0(t>t_f)=3.1$ bars. The red solid line represents ${\delta }_{99}$, the distance over which 99% of the numerical concentration drop takes place. Equilibrium, flushing, and experiment stages are shaded accordingly (see Fig. 2).

Figure 6

(a) Time-averaged dimensionless bubble growth rate as a function of the time-averaged ${\mathrm{Ra}}_w$ for every single water-bubble experiment (colored markers) and water-bubble-alkane experiment (grey markers). The error bars indicate the range over which ${\varphi }_b\left(t\right)$ and ${\mathrm{Ra}}_w\left(t\right)$ decay during that particular experiment. The solid green line and surrounding shaded region provide the ${\mathrm{Sh}}_w\left({\mathrm{Ra}}_w\right)$ dependence in Eq. (4): ${\mathrm{Sh}}_w={(1+k{\mathrm{Ra}}_w/{\mathrm{Ra}}_c)}^{1/4}$, with $k=2.75\pm 1.25$. (b) Bubble growth rate of the water-bubble-alkane experiments corrected with $q_a$, the dimensionless ${\mathrm{CO}}_2$ leakage flux from the bubble to the alkane according to Eq. (25).

Figure 7

(a) Comparison of the bubble growth dynamics for water-bubble system [cases (i) and (ii)] and ternary water-bubble-alkane systems [cases (ii) and (iv)] for two distinct water layer heights. The alkane layer height is $H_a\approx 3.5$ mm. In all four cases, $V_f=35\mu \mathrm{L}$ and $P_0=1$ bar. Experiments (i) and (ii) are overlaid with numerical and analytical (17) solutions, experiments (iii) and (iv) with numerical and ordinary differential equation (ODE) (27) solutions. (b) Time evolution of the ${\mathrm{CO}}_2$ concentration profile in the alkane layer according to the numerical model for case (iii) (top panel) and case (iv) (bottom panel).

Figure 8

(a) Bubble growth dynamics for the water-bubble-alkane system at higher pressure levels ($V_f=27\mu$ L). Case (i), $P_0=2.0$ bars, $H_w=3.0$ mm, $H_a=3.2$ mm; case (ii), $P_0=4.2$ bars, $H_w=3.0$ mm, $H_a=3.2$ mm. Experimental, numerical, and ODE (27) results are compared. (b) Close-up of the pressurization stage and initial growth. Inset: Concomitant pressure time history.

Figure 9

Bubble growth dynamics for a binary and ternary system ($V_f=35$ $\mu \mathrm{L}$, $H_w=3$ mm, and $H_a=3$ mm) at $P_0=1$ bar. After $t=1$ h, the ${\mathrm{CO}}_2$ atmosphere inside the chamber is replaced by a ${\mathrm{N}}_2$ atmosphere at the same pressure, thereby enforcing dissolution.

Figure 10

(a) Diffusion outflux (blue line) and its time integral (red line) as functions of time. The transient period can be thought to last $\tau \sim 0.4$, after which the flux overcomes the 95% mark of the steady-state value. (b) Influx ratios: $q_{ss}$ denotes the (self-similar) diffusion influx into the unit-concentration boundary of an infinite sheet, $q$ is the influx into the unit-concentration boundary of a sheet of unit length delimited by a zero-concentration boundary, and $q_{\mathit{nf}}$ is the influx into the unit-concentration boundary of a sheet of unit length delimited by a no-flux boundary. In both cases, self-similarity breaks down after a time $\tau \sim 0.3$, where the flux ratios drop below 95%.