## 3 Shaft Furnace System Modeling
### Exergy Analysis Model  
The energy conservation equation on the basis of the first law of thermodynamics focuses on the quantitative conservation of energy in various forms during conversion without distinguishing qualitative differences. Exergy, on the other hand, is a physical quantity that measures the quality or usefulness of energy, allowing for an analysis from both quantitative and qualitative perspectives, and it has been widely used in the analysis and evaluation of energy systems [40; 41; 42]. Figure 3 illustrates the exergy analysis of HDRSFs based on the second law of thermodynamics, which considers energy quality and efficiency.  
According to the exergy budget shown in Figure 3, the equilibrium equations of exergy at the reduction and cooling sections of the shaft furnace are shown in Equations (30) and (31), respectively.  
\[Ex_{iron\text{ {ore}}}+Ex_{reducing\text{ {gas}}}=Ex_{top\text{ {gas}}}+Ex_{HDRI}+Ex_{loss1} \tag{30}\]  
\[Ex_{HDRI}+Ex_{cooling\text{ {gas in}}}=Ex_{cooling\text{ {gas out}}}+Ex_{CDRI}+Ex_{loss2} \tag{31}\]  
Figure 3: Analysis of the exergy balance of HDRSFs.  
The calculations of exergy in Equations (30) and (31) are shown in Equations (32)-(37). In the present study, exergy contains physical exergy and chemical exergy, as shown in Equation (32).  
\[Ex=Ex_{ph}+Ex_{ch} \tag{32}\]  
Physical exergy mainly consists of thermal exergy (related to the system temperature and latent heat of the phase transition) and mechanical exergy (related to the system pressure). The physical exergy can be calculated using Equation (33).  
\[Ex_{ph}=(H-H_{0})+T_{0}(S_{0}-S) \tag{33}\]  
where (\(H-H_{0}\)) and (\(S-S_{0}\)) represent enthalpy change and entropy change, respectively; \(T_{0}\) is the reference ambient temperature, which is 298 K in the present study.  
In Equation (33), (\(H-H_{0}\)) can be calculated by Equation (34).  
\[H-H_{0}=m\times C_{i}(T_{m})\times(T-T_{0}) \tag{34}\]  
where \(T_{m}\) is the average temperature, calculated by (\(T+T_{0}\))/2.  
In Equation (33), (\(S-S_{0}\)) can be calculated using Equations (35) and (36). Equation (35) is employed to compute the entropy change of substances whose properties remain unaffected by pressure variations. Equation (36) is utilized to calculate the entropy change of the substances whose properties are affected by pressure changes, with appropriate corrections being applied [43].  
\[S_{0}-S=m\int\limits_{T}^{T_{0}}\frac{C_{i}(T)}{T}dT \tag{35}\]  
\[S_{0}-S=m\left(\int\limits_{T}^{T_{0}}\frac{C_{i}(T)}{T}dT-R\ln\frac{P_{0}}{P}\right) \tag{36}\]  
where \(T\) is the material temperature; \(T_{0}\) is the reference ambient temperature, which is 298 K in the present study; \(R\) is the ideal gas constant, which is 8.314 J/(mol\(\cdot\)K); \(P_{0}\) is the reference ambient pressure, which is 0.1 MPa in the present study; and \(P\) is the gas pressure.  
The chemical exergy of the mixture is related to its components and is not a simple addition of the standard exergy of each component. The chemical exergy of the mixture can be calculated by Equation (37) [44].  
\[Ex_{ch}=\sum e_{i}Ex_{i}+RT_{0}\sum e_{i}\ln e_{i} \tag{37}\]  
where \(e_{i}\) is the molar percentage of the \(i\)-th component in the mixture.