## 4 Discussion
### Effect of Operating Parameters on Mixing Time and Mixing Number  
The mixing time is inversely proportional to the shaking frequency. At high Reynolds number, the homogenization number remains the constant. [PERSON] et al. [27] found that the mixing time remained to constant at higher shaking frequency in a 30-L vessel utilizing polypropylene liquid with different filling volume 8, 13 and 18 L. Mixing time of this experiment increases with the increased filling volume. [PERSON] et al. [26] also found the mixing time for 100-, 250-, 500-mL nominal flask volumes increased gradually at 200 rpm shaking frequency. Consistently, the mixing in the viscous system, i.e., 20% high volume glycerin ratio solution at 10 L filling volume, was noticeably slower than in the non-viscous water system [26]. The mixing time increase phenomenon appears at a higher shaken frequency and with increasing fluid viscosity. For viscous fluids, the mixing time increase at high frequency range is also followed a decreased free surface vibration [24]. Under these conditions, the mixing number is independent of Reynolds number. This was also validated in traditional agitated tanks. The mixing (homogeneous) number increases according to increase the liquid filling volume  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Bioeractor Type** & **Shaking Frequency** & **Shaking Diameter** & **Correlation** & **Reynolds Number** & **Literature** \\ \hline
100- to 2000-mL flask & 80 to 380 min\({}^{-1}\) L/min & 2.5 and 5 cm & \(Ne^{-6}+1.5\,\text{Re}^{-0.2}\) & 90 to \(9\times 10^{4}\) & [PERSON] et al. [16] \\
250- and 500-mL flask & 160 to 340 rpm & 50 mm & Ne = 12.3 Re\({}^{-0.366}\) & 90 to \(9\times 10^{4}\) & [PERSON] et al. [17] \\
20-L orbitally shaken bioreactor & 100 rpm & 2.5 cm & Ne = 0.65 Re\({}^{-0.17}\) & \(2.5\times 10^{3}\) to \(3.5\times 10^{4}\) & [PERSON] et al. [18] \\  & 120 rpm & 2.5 cm & Ne = 0.75 Re\({}^{-0.17}\) & \(3\times 10^{3}\) to \(5\times 10^{4}\) & [PERSON] et al. [18] \\  & 160 rpm & 5 cm & Ne = 1.11 Re\({}^{-0.17}\) & \(1.8\times 10^{4}\) to \(3\times 10^{5}\) & [PERSON] et al. [18] \\
20-L orbitally shaken bioreactor & 80 to 120 rpm & 3.2 cm & Ne = C\({}_{1}\):Re\({}^{-2.66}\) & \(4.6\times 10^{4}\) to \(1.75\times 10^{5}\) & This study \\ \hline \hline \end{tabular}
\end{table}
Table 4: Correlation between Newton number and Reynolds number.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Bioeractor Type** & **Shaking Frequency** & **Shaking Diameter** & **Correlation** & **Liquid** & **Literature** \\ \hline
20-L orbitally shaken bioreactor & 100 rpm to 300 rpm & 50 mm & P\(-\)N\({}^{2.5}\) & Tap water & [PERSON] et al. [48] \\
100- to 2000-mL flask & 80 rpm to 380 rpm & 2.5 cm and 5 cm & P\(-\)N\({}^{2.8}\) & demineralized water & [PERSON] et al. [14] \\
20-L orbitally shaken bioreactor & 300 rpm & 2.5 cm & P\(-\)N\({}^{5.75}\) & Tap water & [PERSON] et al. [51] \\
20-L orbitally shaken bioreactor & 80 to 120 rpm & 3.2 cm & P\(-\)N\({}^{0.34}\) & \(4.6\times 10^{4}\) to \(1.75\times 10^{5}\) & This study \\ \hline \hline \end{tabular}
\end{table}
Table 3: Correlation between power consumption and shaking frequency.  
and liquid viscosity. The mixing time for the bioreactor volume gradually increased as the ratio of filling volume to bioreactor volume is increased. The exponent of volume number and the mixing time correlation (Equation (19)) in the experiment obtained different values for different liquids. A new set of correlations were proposed (Equation (20) to Equation (23)) for the mixing time in cylindrical orbitally shaken bioreactors.