## 3 Results and Discussion
### Fractal Characteristics
The COD removal rate kept increasing after a short decline with the gradual adaptation of microorganisms to the changed OLR increasing from the lowest value of 73.8% to 81.8%. The correlation coefficients \(R^{2}\) of the four compartments ranged from 0.967 to 0.982, indicating that the linear relationship of the curve equations was good. At the 3 rd phase of start-up, the values of \(D_{f}\) in the 2 nd and the 3 rd compartments still maintained a high growth trend, indicating that the granular sludge in the two compartments had a smoother surface, denser structure and better sedimentation performance than the other compartments. Although the formation rate of the granular sludge in the 1 st compartment was fast and the particle size increased significantly, the values of \(D_{f}\) increased slowly due to the long-term adsorption of suspended solids and organic matter in the influent. As the increase in the OLR enabled the microorganisms in the 4 th compartment to obtain appropriate nutrients, the formation rate of granular sludge in this compartment was accelerated and the values of \(D_{f}\) gradually increased. At the early stage of this phase, flocculent sludge flocs gradually attached to the surface of the core to form micro particles. The micro particles relied on the intertwining of a large number of filamentous bacteria growing under a low OLR at the 1 st and the 2 nd phases to form large-sized granular microbial aggregates at the later stage. The acidogenic bacteria and methanogens formed the symbiotic and alternate system in their respective population, which was conducive to the degradation of the organic matter. However, the granular sludge structure during this period was relatively loose, and the ability to resist the hydraulic shear force and bubble cutting force was also weak. The values of \(D_{f}\) in each compartment continued to increase and the COD removal rate kept increasing after a brief decline with the gradual adaptation of microorganisms to the changed OLR, with it increasing from the lowest values of 75.3% to 81.7%. The correlation coefficients \(R^{2}\) of the four compartments ranged from 0.978 to 0.987, indicating that the linear relationship of the curve equations was good. At the 4 th phase of start-up, with the increase in granular sludge size, the ability of particles to resist the hydraulic shear force and bubble cutting force increased. The bacteria continuously adhered and gathered on the surface of the particles by mass transfer inside and gradually formed a stable micro-ecosystem granular sludge from the core to the outside. Each bacterium in the particle became a member of this micro-ecosystem, and the living conditions were relatively stable. Therefore, the size of the granular sludge increased and its structure became more compact and stronger. The values of \(D_{f}\) in the 3 nd and the 2 rd compartments still ranked as the first and the second during this phase. In the early stage of this phase, the values of \(D_{f}\) in each compartment maintained a high growth trend, but the value of \(D_{f}\) in the 4 th compartment began to exceed that in the 1 st compartment at the later stage, which illustrated that although the particle size of the granular sludge in the 4 th compartment was relatively small, the increase in the OLR made the structure of the granular sludge in the 1 st compartment loose due to the increase in gas production. As a result, the appearance structure and sedimentation performance of the granular sludge in the 4 th compartment were better than those in the 1 st compartment. At the later stage of this phase, the values of \(D_{f}\) in each compartment increased slowly, indicating that the granular sludge had been mature at this time, and the values of \(D_{f}\) would not increase much if the granular sludge continued to be cultivated and acclimatedin the start-up phase.