## 3 Initial Data, Quality Function, Independent Parameters and Their Limits  
Using a mathematical model of gas dynamics and heat exchange processes, calculations of gas temperature distribution along the length of the trajectory of their movement in the working chamber of the furnace were performed. The initial data for modeling are presented in Table 1. The most characteristic parameters of the work intended for heating steel ingots were taken as initial data. The length of the working chamber of the furnace (8 m) and height (43 m) are typical for the intended design and do not change depending on the aisles. As a rule, such furnaces are heated with a mixture of consumption of coke and blast-furnace gas with a calorific value of 5.5-812 MJ/m\({}^{3}\)[2]. At the same time, the higher the temperature of heating the fuel and air before their absorption, the lower the calorific value of the mixture of coke oven and blast-furnace gas that is used. In the assumed heating of air and visible gases, up to 800 \({}^{\circ}\)C and 300 \({}^{\circ}\)C, respectively, which is the most widely used heat leaving the flue gas furnace, the metal surface temperature of 1000 \({}^{\circ}\)C corresponds to the range of revolutions of its metal processing. The thermophysical properties of metal, gas, and air correspond to their temperature [10]. As a quality function, an indicator that characterizes the uniformity of the gas temperature in the working chamber \(F=t_{max}-t_{min}\) was chosen, where \(t_{max}\) and \(t_{min}\) are the maximum and minimum gas temperatures along the trajectory of their movement, respectively. According to the results of previous studies, six independent parameters that affect the quality function were determined, and the range of their possible change was also established. 1. Fuel temperature \(t_{f}=0,\ \ldots,\ 300\ ^{\circ}\)C. It is known from literature sources [13] that it is not safe to increase the heating temperature of gaseous fuel in the existing heat exchangers of furnaces above 300 \({}^{\circ}\)C, for the reasons of explosive safety. 2. The temperature of the air used for fuel combustion is \(t_{air}=0,\ \ldots,\ 1000\ ^{\circ}\)C. Due to the heating of air to high temperatures, significant fuel savings are achieved in method [14], ring [15], roller and shaft [16] furnaces, as well as in heating wells [17] of various designs. The maximum air heating temperature depends on the method of utilization of the heat of the flue gases leaving the furnace and the design of the heat exchangers, but usually this temperature does not exceed 1000 \({}^{\circ}\)C. At the same time, in outdated equipment that is still in operation at some enterprises, heating of the combustion air before burning is not provided. 3. Multiplicity of internal large-scale recirculation of furnace gases \(k_{p}=1,\ \ldots,\ 6\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \(B_{f}\), kg/s & \(Q_{f}\), J/kg & \(L_{u}\), kg/kg & \(v_{exp}\), kg/kg & \(t_{f}\), \({}^{\circ}\)C & \(t_{air}\), \({}^{\circ}\)C & \(t_{Me}\), \({}^{\circ}\)C & \(c_{f}\), J/(kg\(\cdot\)K) \\ \hline
1.721 & \(6.164\times 10^{6}\) & 1.943 & 2.943 & 300 & 800 & 1000 & 1240 \\ \hline \(L_{tr}\), m & \(\begin{array}{c}C_{Me},\\ \mathrm{W/(m^{2}\cdot K^{4})}\end{array}\) & \(F_{Me}\), m\({}^{2}\) & \(\xi_{gr}\) & \(\begin{array}{c}k_{long},\\ \mathrm{W/(m^{2}\cdot K)}\end{array}\) & \(n\) & \(c_{air}\), J/(kg\(\cdot\)K) & \(c_{exp}\), J/(kg\(\cdot\)K) \\ \hline
16 & \(1.731\times 10^{-8}\) & 122 & 1.344 & 2 & 8 & 1071 & 1238.2 \\ \hline \end{tabular}
\end{table} 
Table 1: Initial data. The choice of such limits for varying the multiplicity of recirculation of furnace gases is determined by the practical research data obtained by numerous experimenters. So, when modeling a recirculation circuit with a built-in fan, [PERSON] [18] discovered that increasing the recirculation ratio above 5 was impractical and led to an increase in fan power consumption.