## 5 Discussion  
Based on the hypothesis formulated in previous section, the results can be written as follows:  
\(f_{1}^{*}(x_{1},x_{2})=F^{*}\big{(}x_{1},x_{2},x_{3}^{0},x_{4}^{0},\ \ldots,\ x_{n}^{0}\big{)}=F(M_{0})/\big{(}g_{1} \big{(}x_{1}^{0}\big{)}\ g_{2}\big{(}x_{2}^{0}\big{)}\big{)}\)  
\(f_{2}^{*}(x_{1},x_{2},\ x_{3})=F^{*}\big{(}x_{1},x_{2},x_{3},x_{4}^{0},\ \ldots,\ x_{n}^{0}\big{)}=F(M_{0})/\big{(}g_{1} \big{(}x_{1}^{0}\big{)}\ g_{2}\big{(}x_{2}^{0}\big{)}g_{3}(x_{3}^{0})\big{)}\)  
\(f_{3}^{*}(x_{1},x_{2},\ x_{3},x_{4})=F^{*}\big{(}x_{1},x_{2},x_{3},x_{4},x_{5}^ {0},\ \ldots,\ x_{n}^{0}\big{)}=F(M_{0})/\big{(}g_{1}\big{(}x_{1}^{0}\big{)}\ g_{2} \big{(}x_{2}^{0}\big{)}g_{3}(x_{3}^{0})g_{4}\big{(}x_{4}^{0}\big{)}\big{)}\),  
\(f_{4}^{*}(x_{1},x_{2},\ x_{3},x_{4},\ \ldots,\ x_{n})=F^{*}(x_{1},x_{2},x_{3},x_{4},x_{5}, \ \ldots,\ x_{n})=F(M_{0})/\big{(}g_{1}\big{(}x_{1}^{0}\big{)}\ g_{2} \big{(}x_{2}^{0}\big{)}g_{3}(x_{3}^{0})\cdots g_{n}\big{(}x_{n}^{0}\big{)} \big{)}\). That means that moving sequentially, starting from the first step, one can obtain the entire sequence of representations of the function from these parameters. It all works when all other points are at the point \(M_{0}\big{(}x_{1}^{0},x_{2}^{0},x_{3}^{0},\ldots x_{n}^{0}\big{)}\). The simulation results given above fully reflect qualitative ideas about furnace processes, which are known from their operation. Thus, among the selected independent parameters, the most significant influence on the uniformity of the temperature field in furnaces with a U-shaped trajectory of gas movement is characterized by the multiplicity of large-scale recirculation \(k_{r}\). Changing this indicator value from \(k_{r}=1\) to \(k_{r}=6\) results in a tenfold decrease of the quality function value \(F=t_{max}-t_{min}\) meaning the uniformity  
Figure 14: Relative error on the diagonal of the definition area of the parameters \(t_{f}\), \(t_{air}\), \(k_{r}\), \(b\), \(K_{g}\), \(L_{f}\). Figure 13: Relative error on the diagonal of the definition area. of the gas temperature in the working chamber of the furnace. The simulation results confirmed the thesis of a sharp drop in the intensity of temperature equalization with an increase in the recirculation rate. The processing of the approximation curve of the quality function \(F\) shows that the uniformity of the gas temperature increases by almost 40 % when the recirculation ratio increases from \(k_{r}=1\) to \(k_{r}=2\) and by 12.5% only when the same parameter increases from \(k_{r}=5\) to \(k_{r}=6\). Small-scale recirculation has a similar quality, but smaller influence on function \(F\). Such an effect on temperature uniformity in the working chamber of the furnace is quite natural. After all, both small-scale and large-scale recirculation characterize gas-dynamic and mass exchange processes, which are described by the same equations and have the same effect on the heat exchange. Analysis of the data on the impact of small-scale recirculation shows that with an increase in its share, the unevenness of the gas temperature distribution in the furnace can increase by an additional 15%. The reduction of small-scale local recirculation is associated with the elimination of obstacles and local resistances in the path of movement of furnace gases, and a sharp change in the direction of their movement. The gas heating temperature and the parameters of the fuel combustion model have the least influence on the uniformity of the gas temperature out of the six selected parameters. The change in fuel temperature from the minimum to the maximum level (i.e., from 0 \({}^{\circ}\)C to 300 \({}^{\circ}\)C) leads to an insignificant improvement of the uniformity of gas temperature distribution in the working chamber of the furnace, as it ranges from 225 \({}^{\circ}\)C to 205 \({}^{\circ}\)C. The parameter of the model of gas combustion in the torch has almost the same effect. The length of the flare affects the quality function \(F\) in direct proportion.