## 2 System Detectability Based on Bayesian Networks
Next, the sensitivity index of each zone on the chamber shell, \(ST_{j}\), \(1\leq j\leq n\), can be defined as follows:  
\[ST_{j}=\frac{\left\{\sum_{i=1}^{m}\left(\overline{E}_{bw,j}^{i}-\widetilde{E}_{ bw,j}\right)^{2}\right\}^{3}}{m^{3}\widetilde{E}_{bw}^{6}}, \tag{12}\]  
where \(\widetilde{E}_{bw,j}=\frac{1}{m}\sum_{i=1}^{m}\overline{E}_{bw,j}^{i}\) and \(\widetilde{E}_{bw}=argmax\)\(\frac{1\leq i\leq m}{1\leq j\leq n}\)\(\left\{\overline{E}_{bw,j}^{i}\right\}\). The sensitivity index \(ST\) describes the extent of sensitivity of each zone of the chamber shell to changes in the operating conditions. The larger the value, the more sensitive the zones are to changes in the operating conditions. Allocating sensors in this area is more economical and efficient. To assist in sensor allocation by determining desirable physical sensor locations that are generally sensitive to the bead-wall impact zone surrounding the chamber wall, the chamber shell of the wet stirred-media mill is split into 30 zones of different sizes (I-1, I-2,..., V-6), which produces 30 individual bead-wall impact time series data sets for each simulation. Therefore, the different zones of impact energy along the length of the chamber shell can be identified via proper signal processing. Figure 1 shows these zones. Coupled CFD-DEM simulations were performed to collect the bead-wall impact event data over the entire boundary of the wet stirred-media mill. The software packages used were EDEM2018 (version 4.0.0, academic adaption, released by the UK DEM Solutions) for the DEM simulations and Workbench (version 19.1, academic adaption released by the ANSYS, Inc., Canonsburg, PA, USA) for the CFD simulations, which were coupled using the EDEM2018-Fluent19.1 compile interface files. The degree of filling investigated for the beads ranged from 70% to 90%, the bead diameter ranged from 0.8 mm to 1.6 mm, and the stirrer speed ranged from 1000 rpm to 1400 rpm, determined in accordance with the recommendations of Root Industrial Co., Ltd. (Coimbatore, India). A set of seven simulations were completed under different loading and rotation speed combinations. The 30 zones were analyzed to determine the zones generally sensitive to the fluctuations in bead-wall impact energies, which are characterized by the aforementioned sensitivity index \(ST\). With reference to Figure 1, Figure 2 shows that Zones I-5, I-6, III-4, and III-6 are the most sensitive to changes in the operating conditions of the wet stirred-media mill. Under normal circumstances, Zone II is affected by the local filling rate of the beads and the dynamic balance of the internal bead cycle [32]. Therefore, the bead-wall impact energy  
Figure 1: Schematic plot of the chamber shell and monitoring zones. of this area is insensitive to different operating conditions. Zones IV and V are the areas where the diffusion wheel is located and are mainly applied for product particle dispersion, rather than crushing. This action also directly leads to the insensitivity of these areas to changes in the operating conditions. Therefore, considering the limited sensor location space, the cost constraints, and the goal of eliminating redundant sensing data as much as possible, the sensor layout is only performed in Zones I-5, I-6, III-4, and III-6.