## 4 Case Studies
### Case 3: Rotary Hearth Furnace Combustion System  
In this section, an experimental analysis is conducted based on the data of an actual rotary hearth furnace combustion system. The rotary hearth furnace combustion system is a typical distributed parameter system with significant spatiotemporal coupling characteristics. The rotary hearth furnace divides the space into five relatively independent and mutually coupled areas: preheating area, reduction area 1, reduction area 2, reduction area 3, and reduction area 4. These areas undertake different metallurgical processes and have a strong coupling relationship in space. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & & **CNN** & **ST-LS-SVM** & **3D-OMSVR-SGD** \\ \hline RMSE & \(\sigma_{0}\) (Internal interference) & 23.0408 & 1.3803 & 0.9532 \\ \(\omega_{r}\) (External interference) & 19.0302 & 1.2614 & 0.8871 \\ RE (mean) & \(\sigma_{0}\) (Internal interference) & 0.0133 & 0.0010 & \(7.4453\times 10^{-4}\) \\ \(\omega_{r}\) (External interference) & 0.0113 & 0.0010 & \(7.2564\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of RMSE and RE in Case 2. Figure 21: TNAE comparison of different methods under internal disturbance in Case 2. Figure 22: RLNE comparison of different methods under internal disturbance in Case 2. The main application of the rotary hearth furnace Is to treat metallurgical solid waste dust, and the main products are zinc powder and metalized pellets. During operation, the green balls are evenly distributed in the rotary hearth furnace and pass through each reduction zone in turn. Under the action of preheating and multiple reduction zones, the green balls undergo a reduction reaction under specific temperature conditions and finally generate metal pellets with a high iron content. At the same time, the rotary hearth furnace also collects zinc-containing flue gas to obtain high-content metallic zinc powder. The preheating and reduction zones 1, 2, 3, and 4 are equipped with adjustable flow gas and air valves to control the temperature of the rotary hearth furnace. Two temperature measurement points are installed in each area. The gas flow rate, air flow rate, gas, and air manifold pressure in each area of the rotary hearth furnace and the speed of the powder-collecting fan are used as input variables, and the temperature measurement values in each area are used as output variables. \(S_{1}\) and \(S_{2}\) represent the temperature feedback values for the inner and outer rings of reduction zone 1, respectively. \(S_{3}\) and \(S_{4}\) correspond to the temperature feedback values for the inner and outer rings of reduction zone 2, respectively. Similarly, \(S_{5}\) and \(S_{6}\) are the temperature feedback values for the inner and outer rings of reduction zone 3, respectively. \(S_{7}\) and \(S_{8}\) represent the temperature feedback values for the inner and outer rings of reduction zone 4, respectively. Finally, \(S_{9}\) and \(S_{10}\) are the temperature feedback values for the inner and outer rings of the heating zone, respectively. The radius (\(r\)) of the rotary hearth furnace ranges from 10 m of the inner ring to 12.5 m of the outer ring, and all data are collected within this range. The schematic diagram of the rotary hearth furnace combustion system is shown in Figure 23.  
In this experiment, a total of 1400 samples of data were collected from the sensors with a sampling interval of 1 s. The first 800 samples were used to initialize the model, and the last 600 samples were used for online model verification. As shown in Figure 24, the online prediction output (Reduction zone 1) of the proposed 3D fuzzy modeling method model approximates the actual output well. The true value and expected value of sensors \(S_{1}\) and \(S_{2}\) are shown in Figure 25. It can be seen that the 3D-OMSVR-SGD model can track the changes in the system promptly, proving the effectiveness of the proposed modeling method in actual industrial systems.