## 4 Results and Discussion
Figure 5a shows the axial distribution of the turbulent dissipation rate at \(x\) = 0.03 m with rotational speeds of 300 and 900 RPM. The turbulent dissipation rate at peak points near the impeller with a rotational speed of 900 RPM is about 40 times higher than that with a rotational speed of 300 RPM. The high turbulent dissipation rate leads to a rapid increase in the number density of bubbles due to the breakage of bubbles as shown in Figure 5b. The gas hold-up at 900 RPM is higher and more widely dispersed than that at 300 RPM in the upper region (A) as shown in Figure 5c. The high gas hold-up near the impeller can be verified by previous experimental data that depicted the axial distribution of the gas hold-up at different rotational speeds [33]. The Pearson correlation coefficient (\(r\)) was used to obtain the linear relationship between the volumetric mass transfer coefficient and the dissolved carbon monoxide concentration [34; 35]. Figure 5d shows the calculated Pearson correlation coefficient on the \(y\) = 0 m plane at various rotational speeds. The Pearson correlation coefficient takes a range  
Figure 2: (**a**) Velocity vectors in \(y\) = 0 m plane. Contour of the (**b**) gas hold-up and (**c**) dissolved carbon monoxide concentration with a rotational speed of 500 RPM and an inlet velocity of 0.1 m/s. (**d**) Volume averaged gas hold-up and (**e**) dissolved carbon monoxide concentration in three regions. A: below the bottom impeller, B: between each impeller, and C: above the upper impeller. of values from \(-1\) to \(+1\). The closer the Pearson correlation coefficient is to \(+1\) or \(-1\), the higher the positive or negative linear correlation. The Pearson correlation coefficient was determined as follows [35]:  
\[r=\frac{\sum_{i=1}^{n}[(x_{i}-\overline{x})\cdot(y_{i}-\overline{y})]}{\sqrt{ \sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}\cdot\sqrt{\sum_{i=1}^{n}(y_{i}- \overline{y})^{2}}} \tag{28}\]  
Here, the variables \(x_{i}\) and \(y_{i}\) relate to the volumetric mass transfer coefficient and the dissolved carbon monoxide concentration, respectively. \(\overline{x}\) and \(\overline{y}\) denote the means of the two variables, and \(n\) is the sample number. [PERSON] [36] stated that \(r\leq 0.39\) represents a weak correlation, \(r\) between 0.40 to 0.69 represents a moderate correlation, \(r\) between 0.70 to 1 represents a strong or high correlation, and \(r\geq 0.9\) represents a very high correlation [35; 37]. As the rotational speeds of the impellers increase, the Pearson correlation coefficient also increases. It can be seen that when the rotational speed of the impellers is 900 RPM, the Pearson correlation coefficient is approximately 0.52, indicating a moderate correlation between the volumetric mass transfer coefficient and the dissolved carbon monoxide concentration. On the other hand, in the cases of rotational speeds under approximately 700 RPM, a Pearson correlation coefficient less than 0.20 is obtained, indicating a weak correlation. Figure 3: Axial distribution of the (**a**) turbulent dissipation rate, (**b**) gas hold-up, (**c**) mass transfer coefficient (\(k_{\mathrm{L}}a\)), and (**d**) dissolved carbon monoxide concentration at \(x=0.03\) m with 500 RPM and an inlet velocity of 0.1 m/s. Figure 4: Contour of the volumetric mass transfer coefficient and axial distribution of the dissolved carbon monoxide with different RPMs: **(a,b)** = 300 RPM; **(c,d)** = 900 RPM. Figure 5: _Cont._