## 2 System Detectability Based on Bayesian Networks
Experimental studies and computer simulations have demonstrated that the impact energy among beads is related to the grinding rate [30,31]. Moreover, it has been recently reported that the impact energy among beads in a wet-stirred mill can be determined through the impact energy of the bead wall, which can be detected by accelerometers mounted on the mill's shell via proper signal processing [32]. Although the CFD-DEM coupling method allows a massive amount of bead-wall impact data to be collected over the entire wet-stirred mill boundary, data collection must be constrained in a systematic approach that is generally only sensitive to bead-wall impact events on the mill's shell. This action can ensure that accelerometers are placed at the positions of strong surface vibrations generated by repeated bead-wall impacts. The communication process in wet stirred-media mills is a time-varying dynamic process. Two conditions must be fulfilled to fully break product particles in a specific time. The first is that the number of bead-wall collisions per unit time must be sufficient, and the second is that the intensity of each bead-wall collision must be sufficiently high. To meet these conditions, a stress model is developed to better describe the grinding mechanism in a wet-stirred media mill [33]. Two parameters can demonstrate the basic concept underlying the stress model: the first is the stress number, \(SN_{bw}\), which denotes the number of bead-wall collisions per unit time, where every bead-wall collision is called a stress event. The second is the stress intensity, \(SI_{bw}\), which denotes the magnitude of the bead-wall impact energy or force at each stress event. Thus, a measure for \(SN_{bw}\) can be derived according to Equation (9):  
\[SN_{bw}=\alpha_{n}\cdot n_{s}\cdot N_{b}\cdot t_{g}, \tag{9}\]  
where \(\alpha_{n}\), \(n_{s}\), \(N_{b}\), and \(t_{g}\) are the scale factor, rotation speed of the stirrer, number of beads in the chamber, and grinding time, respectively. Derived from the kinetic energy of the beads, the stress intensity \(SI_{bw}\) is the other essential parameter that characterizes a comminution process, and it can be represented as follows:  
\[SI_{bw}=\beta_{n}\cdot d_{b}^{3}\cdot v_{b}^{2}\cdot\rho_{b}\cdot\left(1+\frac {Y_{p}}{Y_{b}}\right)^{-1}, \tag{10}\]  
where \(\beta_{n}\), \(d_{b}\), \(v_{b}\), and \(\rho_{b}\) are the scale factor, the bead's diameter, the bead's absolute velocity, and the bead's density, respectively. \(Y_{p}\) and \(Y_{b}\) are the Young's moduli of the product particles and the bead material, respectively. Therefore, the impact energy of the beads to the chamber wall, \(E_{bw}\), can be described by the product of the stress number, \(SN_{bw}\), and the stress intensity, \(SI_{bw}\), which can be determined using Equations (9) and (10), respectively:  
\[E_{bw}=SN_{bw}\cdot SI_{bw} \tag{11}\]  
The selected measurement parameter is the impact energy of the beads on the chamber wall, which can be detected using proper signal processing by accelerometers mounted on the chamber shell. The zones where the accelerometers are installed must be constrained systematically and are generally only sensitive to bead-wall impact events under certain operating conditions. Therefore, the following definitions can be given under the bead-wall impact events used to identify the sensitive areas on the chamber wall under different operating conditions. **Definition**. : _Supposing that the accelerometers are mounted on \(n\) zones of the chamber shell, under \(m\) disparate operating conditions, the sensing data are detected by accelerometers in the \(n\) zones of the chamber shell, and the average impact energy, \(\overline{E}_{bw,j}^{i}(1\leq i\leq m\), \(1\leq j\leq n)\), is obtainedvia proper signal processing.