## 3 Results
### Power Input Correlation
#### 3.1.2 Effect Reynolds Number on Newton Number  
In the _P/V\({}_{L}\)_ correlation process, the number of variables is reduced by the theory of dimensionless analysis. We used the Buckingham \(\pi\)-theorem [57] to classify the variables as the following four dimensionless groups:  
Newton number: \(Ne=\frac{p}{\rho\cdot N^{3}\cdot D^{4}\cdot V_{L}^{1/3}}\)  
[PERSON] number: \(\text{Re}=\frac{p\cdot N\cdot D^{2}}{\eta}\)  
[PERSON] number: \(Fr=\frac{2(\pi N)^{2}\cdot D}{\delta}\)  
Volume number: \(Vol=\frac{V\cdot N}{D^{3}}\)  
Figure 2: Effects of different filling volumes on local power consumption. (**a**) Water; (**b**) 5 vol% glycerin; (**c**) 10 vol% glycerin; (**d**) 20 vol% glycerin.  
The power law is utilized as a mathematical correlation between the dimensionless number:  
\[Ne=B\cdot\text{Re}^{\alpha}\cdot F^{\beta}\cdot(\frac{V_{L}}{D^{3}})^{\gamma} \tag{9}\]  
where \(\alpha\), \(\beta\) and \(\gamma\) exponents are determined by experiments. One dimensionless number is modified in each experiment and other number remain the same.  
The exponent in the Equation (9) must be determined by experiment. Thus, each dimensionless number is individually changed to distinguish the effects of each number in Equation (9). The correlation between Newton number and Reynolds number is first, evaluated, and the \(\alpha\) exponent of Equation (9) is determined. \(P/V_{L}\) is measured in 20 L vessel applying the liquids of different viscosity. The changes of liquid viscosity result in changes of Reynolds number without altering any other dimensionless number in Equation (9). Under different conditions, the \(\alpha\) value is verified using the shake frequency from 100 rpm to 120 rpm and the fill volume from 4 L to 10 L. The relationship between Newton number and Reynolds number of different filling volumes is presented in Figure 3. As a result, the increase in Reynolds number corresponded to a decrease in Newton number [18]. The Newton number decreased with the increase in Reynolds number and Newton number of all liquids was the highest in 4 L, such as reported for an unbaffled mixing reactor [58]. The functional correlation between Newton number and Reynolds number is  
\[\text{Ne}=\text{C}_{2}\cdot\text{Re}^{\alpha}=\text{C}_{2}\cdot\text{Re}^{-2.66} \tag{10}\]  
where the ranges of Re is from \(4.6\times 10^{4}\) to \(1.75\times 10^{5}\) in the turbulent regime. This experiment results in a coefficient of determination greater than 0.9999.  
Figure 3: Relationship between Newton number and Reynolds number of different filling volumes. (a) Water; (b) 5 vol% glycerin; (c) 10 vol% glycerin; (d) 20 vol% glycerin.  
Equation (9) is confirmed by the research of [PERSON] et al. [18] at shaking frequencies of 100 rpm, 120 rpm and 160 rpm in a 20-L vessel using polypropylene liquid with different viscosities and shaking diameters of 2.5 cm and 5 cm and consistent with the research of [PERSON] et al. [17] at shaking frequencies 160 to 340 rpm in 250 and 500-mL flasks in a viscous range of shear rates of polyvinylpyrrolidone fermentation broths between 0.1 s\({}^{-1}\) and 3000 s\({}^{-1}\). A variation of the liquid viscosity results in a variation of the Reynolds number without changing any other dimensionless number of Equation (9).