## 3 Shaft Furnace System Modeling
### Energy Analysis Model  
The reduction of iron ore with hydrogen is a strong endothermic effect, demanding substantial energy input to sustain the reaction. A thorough analysis of energy utilization in the HDRSF process is crucial for identifying recyclable energy sources and minimizingavoidable energy losses. This approach is essential for enhancing the general energy efficiency of the shaft furnace.  
The primary focus of existing energy analysis and research on HDRSFs has been on the reduction section, without much attention being paid to the cooling section and overall process flow. The cooling section of the shaft furnace plays a critical role in cooling high-temperature DRI to prevent re-oxidation after discharge and to protect the furnace shell. A comprehensive energy utilization analysis of both the reduction and cooling sections of the HDRSF is depicted in Figure 2, adhering to the principle of energy conservation.  
According to the energy budget shown in Figure 2, at the reduction section of the shaft furnace, the energy balance is shown in Equation (18).  
\[Q_{iron\ ore}+Q_{reducing\ gas}=Q_{top\ gas}+Q_{reduction}+Q_{vapor}+Q_{loss1}+Q_{ HDRI} \tag{18}\]  
The calculation methods of the items in Equation (18) are shown in Equations (19)-(24).  
\[Q_{iron\ ore}=M_{iron\ ore}\times\sum w_{i}C_{i}\times(T_{iron\ ore}-298) \tag{19}\]  
\[Q_{reducing\ gas}=\frac{V_{reducing\ gas}}{22.4}\times(x_{H_{2}}C_{H_{2}}+x_{ CO}C_{CO})\times(T_{reducing\ gas}-298) \tag{20}\]  
\[Q_{top\ gas}=\frac{V_{top\ gas}}{22.4}\times(X_{H_{2}}C_{H_{2}}+X_{H_{2}O}C_{ H_{2}O}+X_{CO}C_{CO}+X_{CO}C_{CO})\times(T_{top\ gas}-298) \tag{21}\]  
\[Q_{reduction}=\frac{M_{DRI}}{56}\times(x_{H_{2}}\Delta H_{H_{2}}+x_{CO}\Delta H _{CO}) \tag{22}\]  
where \(\Delta H_{H_{2}}\) is the reaction enthalpy per mole of DRI produced in the reduction of iron core with H\({}_{2}\); and \(\Delta H_{CO}\) is the reaction enthalpy per mole of DRI produced in the reduction of iron core with CO.  
The water carried in iron ore will be heated up in the furnace and then evaporated into vapor. The heat absorbed by this process is the sum of the heat absorbed when the water heats up to the boiling point of 373 K and the latent heat of the phase transition of evaporation. The calculation method is shown in Equation (23).  
\[Q_{vapor}=M_{iron\ ore}\times w_{H_{2}O}\times(C_{H_{2}O(I)}\times(373-298)+v _{H_{2}O}+C_{H_{2}O(g)}\times(T_{top\ gas}-373)) \tag{23}\]  
\[Q_{ HDRI}=M_{DRI}\times\sum W_{i}C_{i}\times(T_{ HDRI}-298) \tag{24}\]  
Figure 2: Analysis of the energy balance of the HDRSF.  
According to the energy budget shown in Figure 2, for the cooling section of the shaft furnace, the energy balance is shown in Equation (25).  
\[Q_{HDRI}+Q_{cooling\text{ {gas in}}}=Q_{cooling\text{ {gas out}}}+Q_{CDRI}+Q_{loss2} \tag{25}\]  
The calculation methods of the items in Equation (25) are shown in Equations (26)-(29).  
\[Q_{cooling\text{ {gas in}}}=V_{cooling\text{ {gas}}}\times C_{cooling\text{ {gas in}}}\times(T_{cooling\text{ {gas in}}}-298) \tag{26}\]  
\[Q_{cooling\text{ {gas out}}}=V_{cooling\text{ {gas}}}\times C_{cooling\text{ {gas out}}}\times(T_{cooling\text{ {gas out}}}-298) \tag{27}\]  
\[Q_{CDRI}=M_{DRI}\times\sum W_{i}C_{i}\times(T_{CDRI}-298) \tag{28}\]  
The specific heat of each substance in the above equations changes with the change in temperature. The relationship between the specific heat of each substance and the temperature change is shown in Equation (29).  
\[C_{i}(T)=a+b\times 10^{-3}T+c\times 10^{5}T^{-2} \tag{29}\]