FIGURE 3 Schematic diagram of (A) bubble equivalent diameter
calculation and (B) bubble velocity calculation
Significance analysis was calculated by IBM SPSS Statistics 21.0. The
rising behavior characteristics of bubbles was analyzed by kinovea
(version:0.8.22). The rising trajectory of the bubble was observed by
tracking the path from image sequences. Since Δt (10 ms) is short enough
and distance could be replaced by displacement, the velocity (v )
of bubble is calculated according to the time interval of two adjacent
images and the straight-line distance of the center of mass of the
bubble as illustrated in Figure 3B. The v was calculated by
measuring the distance between the centroids of the fitted ellipse in
the two successive images and the corresponding time difference from a
known image capturing frame rate as follows:
\begin{equation}
v=\frac{\text{Δs}}{\text{Δt}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)\nonumber \\
\end{equation}The Weber (W ) number represents the ratio of inertia force to
surface tension effect. The smaller the Weber number is, the more
important the surface tension is. The calculation equation of Wis as follows:
\begin{equation}
W=\frac{\rho v^{2}l}{\sigma}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)\nonumber \\
\end{equation}where ρ is the fluid density (kg/m3), v is the
characteristic flow velocity (m/s), l is the characteristic
length (m), here refers to the inner diameter of the ceramic membraned (m), and σ is the surface tension coefficient of the
fluid (N/ m).