## 2 System Detectability Based on Bayesian Networks
The ARL of a Shewhart control chart on \(F_{j}\), in terms of detecting the mean shift in \(F_{i}\), denoted by \(\Phi_{1,\varepsilon_{j}}\), is defined as follows [29]:  
\[\Phi_{1,\varepsilon_{j}}=\sum_{r=1}^{\infty}r\cdot\left[1-\psi\big{(}z_{\alpha /2}-\varepsilon_{j}\big{)}+\psi\big{(}-z_{\alpha/2}-\varepsilon_{j}\big{)} \right]\cdot\left[\psi\big{(}z_{\alpha/2}-\varepsilon_{j}\big{)}-\psi\big{(}-z _{\alpha/2}-\varepsilon_{j}\big{)}\right]^{r-1}, \tag{5}\]  
where \(\psi(\cdot)\) is the cumulative distribution function, \(z_{\alpha/2}\) is the upper \(\alpha/2\) percentile of the standard normal distribution, and \(\alpha\) is a type-I error of the Shewhart control chart. The simplified mathematical expression can be obtained for Equation (5) through integral calculation. Mathematically, this equation can be expressed as  
\[\Phi_{1,\varepsilon_{j}}=\frac{1}{1-\psi\big{(}z_{\alpha/2}-\varepsilon_{j} \big{)}+\psi\big{(}-z_{\alpha/2}-\varepsilon_{j}\big{)}} \tag{6}\]  
Assuming that the set of sensors used to detect the single mean shift of the group of physical variables \(\mathcal{F}=\{F_{1},\,F_{2},\,\ldots,\,F_{n}\}\) is \(\mathcal{S}=\{S_{k}\},k\in\{1,\,2,\,\ldots,\,n\}\), the mean shift \(\varepsilon_{k}\) of \(F_{k}\) is monitored by sensor \(S_{k}\). If \(S_{k}\in\mathcal{S}\) is the sensor such that the causal effects of \(F_{i}\) on \(S_{k}\) are the largest among all the sensors in \(\mathcal{S}\), \(F_{k}=argmax_{S_{k}\in\mathcal{S}}\) (\(\pi_{i\to j}\)); then, the control chart on \(F_{k}\) has the minimum ARL for detecting \(\varepsilon_{k}\). Therefore, the following proposition can be given under a BN framework to ensure the inclusion of all the physical variables, \(\mathcal{F}\), whose mean shifts are detectable by \(\mathcal{S}\). **Proposition. **_Given the sensor set \(\mathcal{S}=\{S_{1},\,S_{2},\,\ldots,\,S_{m}\}\) to detect the assignable variation in the series of physical variables, \(\mathcal{F}=\{F_{1},\,F_{2},\,\ldots,\,F_{n}\}\) (\(m\leq n\)), a mean shift \(\varepsilon_{i}\) in \(F_{i}\in\mathcal{F}\) can be detected by \(\mathcal{S}\), if the following conditions are fulfilled:_  
\[\varepsilon_{j}\geq z_{\alpha/2}-\psi^{-1}\bigg{(}1-\Big{(}\Phi_{1,\varepsilon _{i}}^{U}\Big{)}^{-1}\bigg{)}, \tag{7}\]  
_where \(\Phi_{1,\varepsilon_{i}}^{U}\) is an upper bound according to specific domain standards of Shewhart control charts._  
**Proof of Proposition. ** As shown in the control chart, \(F_{k}\) has the minimum ARL in detecting \(\varepsilon_{k}\), and \(\Phi_{1,\varepsilon_{i}}^{U}\geq\Phi_{1,\varepsilon_{j}}\) can be obtained, meaning that  
\[\psi\big{(}z_{\alpha/2}-\varepsilon_{j}\big{)}-\psi\big{(}-z_{\alpha/2}- \varepsilon_{j}\big{)}\leq 1-\Big{(}\Phi_{1,\varepsilon_{i}}^{U}\Big{)}^{-1} \tag{8}\]  
For both the positive and negative mean shift \(\varepsilon_{i}\), \(\psi\big{(}-z_{\alpha/2}-\varepsilon_{j}\big{)}\approx 0\), the above equations can be summarized as \(\psi\big{(}z_{\alpha/2}-\varepsilon_{j}\big{)}\leq 1-\Big{(}\Phi_{1, \varepsilon_{i}}^{U}\Big{)}^{-1}\). Taking the inverse function of \(\psi(\cdot)\) on both sides of the above equation and reorganizing the abovementioned equations results in Equation (7). Therefore, the detectability of the set of physical parameters can be determined according to Equation (7) when the average running length and the range of a single mean shift are specified. \(\square\)  
**3. Accelerometer Allocation Based on Coupled Computational Fluid Dynamics-Discrete Element Method (CFD-DEM) Simulations**  
The wet stirred-media mill is more efficient than conventional grinding techniques, such as tumbling mills, for fine and ultra-fine grinding. There is a dearth of online internal state information for the manufacturer to effectively optimize the grinding process. Computer simulations based on the CFD-DEM coupling method provide a means to estimate hidden process variables online, such as the bead-bead impact energy, which cannot be directly measured during the operation of an actual wet stirred-media mill.