## 2 Materials and Methods
### Theoretical Analysis
#### 2.4.3 Exergy Performance Analysis  
Exergy efficiency (\(\eta_{HE}\)) is one of the most important key indexes for evaluating the exergy performance of the HE, which can be determined by Equation (17):  
\[\eta_{HE}=\overset{\bullet}{E}x_{a}/(\overset{\bullet}{E}x_{fg,in}-\overset{ \bullet}{E}x_{fg,out}) \tag{17}\]  
where \(\overset{\bullet}{E}x_{fg,in}\) and \(\overset{\bullet}{E}x_{fg,out}\) are the specific exergy of the flue gas inlet and outlet of the HE, which can be determined by Equation (3), and \(\overset{\bullet}{E}x_{a}\) is the specific exergy outlet of the HE. According to the literature [27; 28], the specific exergy of the mixture of dry air and water vapor can be calculated by Equation (18); thus, the mixed air flux entering and leaving the HE \(\big{(}\overset{\bullet}{Ex}_{a}\big{)}\) in this work can be also calculated with Equation (18):  
\[\overset{\bullet}{Ex}_{a}=m_{a}^{\bullet}\left\{\begin{array}{c}\left(C_{a}+ \omega C_{v}\right)\left(T_{a}-T_{0}\right)-T_{0}\left[\left(C_{a}+\omega C_{v }\right)\ln\left(\frac{T_{a}}{T_{0}}\right)-\left(R_{a}+\omega R_{v}\right) \ln\left(\frac{p}{P_{0}}\right)\right]\\ \qquad+T_{0}\left[\left(R_{a}+\omega R_{v}\right)\ln\left(\frac{1+1.6078\omega _{v}}{1+1.6078\omega}\right)+1.6078\omega R_{a}\ln\left(\frac{\omega}{\omega_{ 0}}\right)\right]\end{array}\right\} \tag{18}\]  
where the \(C_{a}\) and \(C_{v}\) are the specific heat of the dry air and saturated vapor, and can be calculated by Equations (19) and (20), respectively [29]:  
\[C_{a}=1.04841-\left(3.83719\times 10^{-4}T\right)+\left(9.45378\times 10^{-7}T ^{2}\right)- \tag{19}\]  
\[\left(5.49031\times 10^{-10}T^{3}\right)+\left(7.9298\times 10^{-14}T^{4}\right) \tag{20}\]  
\[C_{v}=1.883-\left(1.6737\times 10^{-4}T\right)+\left(8.4386\times 10^{-7}T^{2} \right)-\left(2.6966\times 10^{-10}T^{3}\right) \tag{21}\]  
where the mass flow rate of the dry air \(\big{(}\overset{\bullet}{m}_{a}\big{)}\) can be calculated by Equation(21) [30]:  
\[\overset{\bullet}{m}_{a}=\rho_{a}\
u_{a}s_{p} \tag{22}\]  
where \(\rho_{a}\) is the density of the dry air, which can be determined using Equation (22) [31]:  
\[\rho_{a}=101.325/0.287T \tag{23}\]  
The humidity ratio of the air (\(\omega_{a}\)) in Equation (5) can be determined by using Equation (23) [32]:  
\[\omega_{a}=0.622\varphi P_{\text{res},a}/\left(P_{a}-\varphi P_{\text{res},a}\right) \tag{24}\]  
In order to investigate the exergy efficiency improvement potential of the HE, the concept of an exergetic \"improvement potential\", proposed by [PERSON] in 1997 [33], was employed to analyze the exergetic performance of the HE. The improvement potential in rate form (\(IP\)) is expressed as Equation (24):  
\[\overset{\bullet}{IP}=\left(1-\eta_{HE}\right)\left(\overset{\bullet}{Ex}_{ in}-\overset{\bullet}{Ex}_{out}\right) \tag{25}\]  
Moreover, owing to the fact that the HE is a typical thermal system, the sustainable development of the system is one of the key considerations when designing the HE. According to [34], the relationship between exergy efficiency and the sustainability index (\(SI\)) can be expressed as:  
\[SI=1/(1-\eta_{HE}) \tag{26}\]