## 3 Results and Discussion
### Fractal Characteristics  
The two-dimensional fractal dimension can only take values between 1 and 2. It is generally believed that the larger the value of \(D_{f}\), the closer the granular sludge is to a spherical shape and the higher the degree of space filling, indicating that the surface of granular sludge is relatively smooth and the structure is relatively dense. The smaller the value of \(D_{f}\), the more irregular the shape of granular sludge and the lower the degree of space filling, indicating that the surface is relatively rough and the structure is relatively loose [24; 25; 26]. The correlation coefficient \(R^{2}\) indicates the linear relationship of IgA-lgP; the closer the value of \(R^{2}\) is to 1, the better the linear relationship of the curve equation is, and the worse it is otherwise [27]. Figure 5 shows the fractal dimension of granular sludge and COD removal rate at each phase during start-up. At the 1 st phase of start-up, although the values of \(D_{f}\) for the four compartments were not high, the differences in the growth rate could still be shown. The microorganisms in the 1 st compartment could obtain relatively sufficient nutrients compared with those in the rear compartments, so the proliferation rate of sludge was the fastest and the value of \(D_{f}\) was the biggest among the four compartments. The proliferation rate of sludge was the slowest and the value of \(D_{f}\) was the smallest for the 4 th compartment because the low OLR meant that the microorganisms did not have enough nutrients compared with the 1 st compartment. The proliferation rate of sludge and the value of \(D_{f}\) showed a trend of steady increase since the 2 nd and the 3 rd compartments were located in the middle position and the nutrients were sufficient. Generally, the inoculated sludge had not fully adapted to the new environment and most of the sludge in each compartment was scattered and had poor sedimentation performance. The flocculent sludge with high biological activity could effectively be intercepted in the reactor to participate in the degradation of organic matter and gradually adapt to the wastewater quality due to the low OLR and lower rate of gas production in the system during this period. At the same time, the COD removal rate increased from 62.8% to 79.1%, which increased with the \(D_{f}\) of sludge flocs increasing. The correlation coefficients \(R^{2}\) of the four compartments ranged from 0.957 to 0.974, indicating that the linear relationship of the curve equations was good. Compared with the 1 st phase, the values of \(D_{f}\) for the four compartments at the 2 nd phase were all increased to different degrees. The microorganisms could gather together rapidly under the traction of external force as the amount of PAM added to the anaerobic  
Figure 5: Fractal dimension of the sludge and COD removal rate at each phase during start-up. sludge gradually increased, resulting in the fastest growth of \(D_{f}\) in the 3 rd compartment. Since the OLR was still low during this period, the reactor tended to have a plug flow pattern which resulted in a shortage of nutrients and the growth of \(D_{f}\) being slow in the 4 th compartment. The 1 st and the 2 nd compartments were located in the front of the reactor, and the nutrients were sufficient, so the proliferation rate and \(D_{f}\) could increase steadily. This phase was the key period for the formation of the granular sludge core, which showed that the amount of non-biological substances in the activated sludge gradually decreased, all kinds of bacteria collided with each other and bacteria adsorbed to the solid surface, and the large specific gravity particles as the core of the granular sludge were initially formed in each compartment. At this time, it was at the early stage of the granular sludge formation, so the sludge shape was irregular and the settling performance was poor.