# Optimizing Retention Bunkers in Copper Mines with Numerical Methods and Gradient Descent
## 3 Materials and Methods  
### Experiment Design  
Due to the substantial number of operations involved in the entire optimization process, the individual steps of the experiment have been illustrated using a flowchart. This chart is presented in Figure 2.  
Figure 1: Construction of the ore receiving bin. Figure 2: Flowchart illustrating the phases of the experiment in the optimization process. Firstly, the optimization process began with the construction of a three-dimensional model of the receiving bin. For the purposes of modeling, one of the bins used in an underground copper ore mine was selected, and its detailed geometry was transferred to a computer environment. The modeled retention bunker (Figure 3a) has a maximum capacity of 2000 Mg and is integrated with a conveyor delivering ore (Figure 3b) and a conveyor responsible for receiving ore from the bin drawer (Figure 3c). The analyzed receiving bin is presented in Figure 3d. Secondly, in the optimization process, the characteristics of the bin drawer load under real working conditions were determined. This was performed by installing a measurement system that monitored the load on the support wheels during the normal cycle of filling the retention bunker and the flow of ore. These results enabled the validation of the digital model of ore flow, which was developed in the next step of the optimization process. For modeling the ore flow in the retention bunker, material and quality parameters of the copper ore and construction materials of the bin, determined under laboratory conditions, were used. The operating parameters of the devices were set in accordance with their real values. The conveyor belt speed was set at 2.5 m/s, and the frequency of the bin drawer movement was 1.38 Hz. The extension of the bin drawer, resulting from its geometry, reached 0.105 m. The modeled length of the bin drawer was 3.98 m, the area of the outlet opening was 0.72 m\({}^{2}\), and the angle of inclination of the drawer was 4.86\({}^{\circ}\). Based on the results of the ore flow simulation in the current bin design, independent factors were identified, which were subject to modification during the experiment. The numerical experiment was divided into two stages. In both stages of the simulation, the primary aim of optimization was set to minimize the force \(F\) pressing on the drawer of the bin and the average velocity \(v\) of particles on the drawer, while maintaining the current capacity \(Q\). It was assumed that \(Q\) should not fall below 90% of its current value \(Q_{c}\). These parameters were the dependent factors in the conducted experiment. Experiments for both stages of simulation and optimization were carried out using a factorial method, analyzing all possible combinations of independent factors. Table 1 presents the range of  
Figure 3: The retention bunker subjected to analysis includes: (**a**) the storage space of the retention bunker, (**b**) the feeding conveyor, (**c**) the receiving bin chute to the receiving conveyor, and (**d**) the structure of the receiving bin. parameter modifications for the two stages of modification. Figure 4 shows the location of independent factors in relation to the receiving bin's design. In the first stage, to eliminate design flaws affecting uneven load distribution and insufficient use of the drawer's workspace, two independent factors were defined for evaluation. The first was the length of the drawer \(L\), corresponding to the total length of the bin, while the second was the offset of the retention bunker axis \(d\), aimed at ensuring an even distribution of load on the drawer's surface. These constraints included maintaining the existing geometry of the bunker and the required distance between the bins. The optimal variant was selected using a simple method of filtering responses (dependent factors) from various scenarios, and the identified modifications served as a starting point for the second stage of simulation and optimization.