Highly pressurized partially miscible liquid-liquid flow in a micro-T-junction. II. Theoretical justifications and modeling

This is the second part of a two-part study on a partially miscible liquid-liquid flow (carbon dioxide and deionized water) that is highly pressurized and confined in a microfluidic T-junction. In the first part of this study, we reported experimental observations of the development of flow regimes under various flow conditions and the quantitative characteristics of the drop flow including the drop length, after-generation drop speed, and periodic spacing development between an emerging drop and the newly produced one. Here in part II we provide theoretical justifications to our quantitative studies on the drop flow by considering (1) $\mathrm{C}{\mathrm{O}}_2$ hydration at the interface with water, (2) the diffusion-controlled dissolution of $\mathrm{C}{\mathrm{O}}_2$ molecules in water, and (3) the diffusion distance of the dissolved $\mathrm{C}{\mathrm{O}}_2$ molecules. Our analyses show that (1) the $\mathrm{C}{\mathrm{O}}_2$ hydration at the interface is overall negligible, (2) a saturation scenario of the dissolved $\mathrm{C}{\mathrm{O}}_2$ molecules in the vicinity of the interface will not be reached within the contact time between the two fluids, and (3) molecular diffusion does play a role in transferring the dissolved molecules, but the diffusion distance is very limited compared with the channel geometry. In addition, mathematical models for the drop length and the drop spacing are developed based on the observations in part I, and their predictions are compared to our experimental results.

Figure 1

(a) Indications of the Cartesian coordinates (x: perpendicular to the liquid $\mathrm{C}{\mathrm{O}}_2$ stream; y: tangential to the interface and in the channel depth direction; z: tangential to the interface and in the flow direction; origin: one point-of-interest on interface at a half channel depth), scale bar: 150 μm. (b) Schematic of the transport of the dissolved $\mathrm{C}{\mathrm{O}}_2$ molecules from the interface (solid line) into water driven by dissolution and diffusion (in the x-y plane). The region outlined by a solid line and a dash line represents a diffusive film of the $\mathrm{C}{\mathrm{O}}_2$ molecules. Note that this diffusive film is enlarged for easy viewing and is actually very thin compared to the channel depth $D$ ($X_D/D\sim {10}^{-2}$, where $X_D$ is the thickness of the diffusion film.). The schematic shows a cross-sectional view of the two phases separated by two interfaces, one (the solid line) is between $\mathrm{C}{\mathrm{O}}_2$ and the $\mathrm{C}{\mathrm{O}}_2$ aqueous solution, and the other (the dash line) is a hypothetical one between pure water and the $\mathrm{C}{\mathrm{O}}_2$ aqueous solution where $\mathrm{C}{\mathrm{O}}_2$ concentration is nonzero but approaching zero.

Figure 2

Liquid $\mathrm{C}{\mathrm{O}}_2$ drop length increases during the main three stages, namely, the stagnating and filling stage, the elongating and squeezing stage, and the truncating stage of one period of drop generation. (a) Drop length increase $\mathrm{\Delta }{L}_{\mathrm{sf}}$ from the beginning to the end of the filling; (b) the time estimate of the elongating and squeezing stage by observing the advancing distance ($Y$) of the water front from the filling end to the end of elongating and squeezing, the right frame shows that (1) the conjuncture between the clear and the shading section is located in the vicinity of the midpoint of the channel width and (2) the shading sectional line intersects the channel sideline with a characteristic angle θ; (c) the truncating time estimate by considering the pinching off of the rest $W$/2 thick $\mathrm{C}{\mathrm{O}}_2$ stream.

Figure 3

Schematics of the development of spacing between an emerging drop and the adjacent formed one within one period of drop generation: (a) spacing increases from $S_0$ (at the beginning, solid lines) to $S$ (at the end, dash lines) during the filling stage and (b) spacing increases from $S$ (at the beginning, solid lines) to $S^{\prime }$ (at the end, dot lines) during the elongating and squeezing and the truncating stage.

Figure 4

Drop spacing development within one period (8.4 ms) of drop generation under drop flow case 14. The experimental data (•) herein are averaged from those in the Fig. 9(b) in paper I, and each error bar indicates two standard deviations from the averaged spacing upon the corresponding time moment. Dashed lines are the fitting lines from the experimental data, and solid lines are the plots of the linear mathematical model in Eq. (50).