## 2 Results and Discussion
### Floc Growth and Breakup/Rearrengement Stages
#### 2.2.1 Evaluation of Growth Stage  
In every case, the relationship between \(D_{90}\) and \(GT\) can be approximated by applying a scaling law. The general floc diameter \(D_{f}\) is a function of the \(G\) value in the growth stage [9; 15]:  
\[D_{f}\approx G^{-0.5}. \tag{1}\]  
The stepwise decrease in \(D_{f}\) with \(G\) value, as well as the change in the self-similarity structure is confirmed. Figure 3 shows the relationship between the slope \(\alpha\) of \(D_{90}\) and \(GT\) values and the ALT ratio for the first 80 s of the test. The term \(\alpha\) represents the flocculation rate and is positively related to the ALT ratio, with \(D_{90}\) increasing as a power of the ALT ratio. By comparing the same ALT ratio, the value of \(\alpha\) was found to be greater at 160 rpm than at 120 rpm, and at an ALT ratio of 0.10 there was little difference in \(G\) values. From this trend, it is possible to speculate that the ALT ratio affects the flocculation rate, and for the same ALT ratio, conditions with a higher shear force (\(G\) value) applied to the floc tend to have a greater flocculation rate. Clearly, \(\alpha\) is greatly influenced by the ALT ratio, followed by the \(G\) value. In the presence of an excess of flocculant, such as an ALT ratio of 0.10, the agitation conditions have little effect, and the ALT ratio dominantly determines the flocculation rate. At the optimum injection rate, the error bars are short and there is little variation in the data; at an ALT ratio of 0.10, the error bars are large and vary vertically. This reflects the larger size of the flocs and the variability of the flocs entering the analysis area. [PERSON] has shown a relationship in which the structural growth of flocs increases following a power-law relationship with respect to the ALT ratio in a two-stage mixing transition from rapid to slow mixing [3; 4]. The present report also agreed that the aggregation rate varied as a power of the ALT ratio. [PERSON] et al. [37] also investigated the properties of flocs formed by increasing salt concentration in the flocculation of kaolinite and aluminum salts by measuring the sedimentation rate. In [PERSON]'s report, precipitated aluminum hydroxide contributed to the formation of large flocs, which were more pronounced near the charge neutralization of kaolinite. Such findings appear to agree with the present report in that \(\alpha\) was found to increase as the flocculant concentration increased. This phenomenon can be considered as follows: the ALT ratio increases the concentration of aluminum hydroxide in the water. The flocculated flocs have a larger collision radius due to the attachment of aluminum hydroxide. Since the flocculation rate is proportional to the cube of the particle radius, \(\alpha\) increases and the flocculation rate increases. At the same time, flocs with large fractal dimensions and low strength are formed, which become more fragile as the \(G\) value increases. In all conditions, growth occurred while maintaining fractality, but \(\alpha\) showed different values for each condition. The reason why the \(\alpha\) value of the growth stage varies with the \(G\) value is that breakup occurs at the same time as growth. The phenomenon of floc growth according to the fractal scaling law was more pronounced at an ALT ratio of 0.05 and 120 rpm, where \(\alpha\) was close to 0.5. This shows that the growth rate and structure of the flocs can be controlled by adjusting the ALT ratio and \(G\) value.