Sedimentation of a single soluble particle at low Reynolds and high Péclet numbers

We investigate experimentally the dissolution of an almost spherical butyramide particle during its sedimentation, in the low Reynolds high Péclet regime. The particle sediments in a quiescent aqueous solution, and its shape and position are measured simultaneously by a camera attached to a translation stage. The particle is tracked in real time, and the translation stage moves accordingly to keep the particle in the field of the camera. The measurements from the particle image show that the radius shrinking rate is constant with time, and independent of the initial radius of the particle. We explain this with a simple model, based on the sedimentation law in the Stokes' regime and the mass transfer rate at low Reynolds and high Péclet numbers. The theoretical and experimental results are consistent within $20\%$. We introduce two correction factors to take into account the nonsphericity of the particle and the inclusions of air bubbles inside the particle, and reach quantitative agreement. With these corrections, the indirect measurement of the radius shrinking rate deduced from the position measurement is also in agreement with the model. We discuss other correction factors, and explain why they are negligible in the present experiment. We also compute the effective Sherwood number as a function of an effective Péclet number.

Figure 1

Schematic diagram of the experimental setup. In the tank, the upper layer I is saturated butyramide and the lower layer III is the NaCl aqueous solution. The transition layer is numbered II.

Figure 2

Dissolving process of the particle over time. The top line shows the original pictures, as captured by the camera during the experiment. The bottom line shows the corresponding processed images, where the white area $A$, obtained after convex closure of the binarized picture, represents the projected surface of the particle. On each of these bottom images, the blue-star point indicates the centroid of the white area, and the red circle, centered on that point, has the same surface as the white area, i.e., gives the equivalent radius $a=\sqrt{A/\pi }$ of the particle.

Figure 3

Time evolution of the equivalent radius $a$ (a) and the vertical displacement $z$ (b) of the particle. Blue circles: experimental data. The vertical dashed lines show the times separating the different stages: $t_0$ end of the saturated layer, $t_1$ end of the transition layer, $t_{\mathrm{up}}$ motion reversal. Those times were determined using the velocity deduced from the derivative of the displacement (b); see Fig. B in Supplemental Material [32]. Solid black line: time before which the volume of the air bubbles attached to the particle is less than 1%. Red line: linear fit in stage III of the radius decrease to deduce ${\dot{a}}_1$. Yellow line: linear fit in stage I of the particle position to deduce $U_0$. Green curve: fit of $z\left(t\right)$ in stage III with Eq. (9), from which another estimate ${\dot{a}}_2$ of the radius shrinking rate is obtained.

Figure 4

Reduction rate of the particle radius $\dot{a}$ from theory and data analyses for various initial particle size $a_0$. Red dots: direct measurement from the linear fitting of $a\left(t\right)$ [see Fig. 3]. These are what we denoted as ${\dot{a}}_1$. Green dots: indirect values obtained from the fitting of $z\left(t\right)$ [see Fig. 3]. These are what we denoted as ${\dot{a}}_2$. Light green points: fitting without any correction factors, i.e., using Eq. (9). Dark green points: fitting accounting for correction factors, i.e., using Eq. (12). Gray dash line: uncorrected theory (7). Black solid line: corrected theory (11). Circles with plus and cross symbols represent the experimental cases where the particle aspect ratio $E$, respectively, averages around 1.7 and 1 over $t_1-t_2$; see also Sec. 4c. These values correspond to the cases illustrated in Figs. H(b) and H(c) of the Supplemental Material [32]. Data dispersion shows the overall precision we can reach, but from the fitting process of a single experimental run, error bars are smaller than the symbol size.

Figure 5

Effective Sherwood number vs effective Péclet number. The inset displays the data in a log-log format. Symbols: experimental data, corrected by their factors ${\beta }_a$ and ${\beta }_b$ computed as explained in the text after Eqs. (10, 11, 12). The colors correspond to different runs. Gray dash line: uncorrected theory. Black solid line: theory accounting for averaged correction factors.