# Design Point Performance and Optimization of Humid Air Turbine Power Plants
## Appendix A Humidifier Design  
The methodology, discussed by [PERSON] [27] and originally developed by [PERSON] [28], defines the design point saturator performance using the inlet pressure and temperature of the two streams (water and dry air) at the inlet and the pinch temperature difference \(\Delta T_{ps}\). The pinch point temperature difference is defined as the minimum temperature difference between the operating line of the water and the saturated air enthalpy line across the saturator [28]. For this modelling process, the air-vapor mixture is treated as an ideal mixture of ideal gases, while the air at the outlet of the saturator is assumed to be fully saturated [9]. From the conservation of the dry air and water mass as well as conservation of energy two invariants of the system, (\(M^{+}\), \(H^{+}\)) can be defined as:  
\[\frac{m_{w}}{m_{a}}-\omega=M^{+}, \tag{1}\]  
\[h_{a}\big{(}T_{g}\big{)}+\ \omega\big{[}h_{p}\big{(}T_{g}\big{)}-h_{w}\big{(}T_{ w}\big{)}\big{]}-M^{+}h_{w}\big{(}T_{w}\big{)}=\ H^{+}. \tag{2}\]  
Under the assumption of ideal mixture of ideal gas, absolute humidity is defined as:  
\[\omega=\frac{\Phi\ p_{sat}\big{(}T_{g}\big{)}}{p-\Phi\ p_{\text{sat}}\big{(}T _{g}\big{)}}\ \frac{R_{a}}{R_{v}}. \tag{3}\]  
For the calculation of the saturated gas thermodynamic properties, the above-defined system of equations needs to be solved for three sections along the saturator as follows:  
**Station \"0\"** corresponds to the pinch condition between the operating and saturated enthalpy lines of the saturator. The temperature difference between the two abovementioned lines is characteristic of the available technology level, which corresponds to the height of the saturation tower. The thermodynamic properties of the saturated gas (temperature \(T_{0}\) and absolute humidity \(\omega_{0}\)) in this section are the unknowns of the system. The relative humidity of the gas on the saturation line is \(\Phi=1\), and \(T_{w,0}\) in the operating line is determined as \(T_{w,0}=T_{0}+\Delta T_{ps}\) once the pinch point temperature difference is defined; therefore, the invariant and absolute humidity equations can be rewritten as the following in agreement with [28] as follows:  
\[\omega_{0}\ -\ \frac{p_{sat}\big{(}T_{0}\big{)}}{p_{1}-\Phi\ p_{\text{sat}} \big{(}T_{0}\big{)}}\ \frac{R_{a}}{R_{v}}\ =0, \tag{4}\]  
\[M^{+}-\frac{cp_{a}}{cp_{w}}-\ \omega_{0}\bigg{(}\frac{cp_{v}-cp_{w}}{cp_{w}} +\ \frac{p_{1}}{p_{1}-ps\big{(}T_{0}\big{)}}\ \frac{R_{v}}{cp_{w}}\bigg{(}\frac{h_{fg} \big{(}T_{0}\big{)}}{R_{v}\ T_{0}}\bigg{)}^{2}\bigg{)}=0, \tag{5}\]  
\[H^{+}-\ h_{a}\big{(}T_{0}\big{)}-\omega_{0}h_{fg}\big{(}T_{0}\big{)}+\ \omega_{0}\big{[}h_{w}\big{(}T_{0}+\ \Delta T_{ps}\big{)}-h_{w}\big{(}T_{0}\big{)}\big{]}+M^{+}h_{w}\big{(}T_{0}+ \ \Delta T_{ps}\big{)}=\ 0. \tag{6}\]  
**Station \"1\"** corresponds to the bottom section of the saturator--i.e., air inlet/water outlet, for which the unknowns are the water temperature \(T_{w,1}\) and the water mass flow \(m_{w,1}\) whereas the gas properties are known. Therefore, the invariant equations can be written as follows:  
\[M^{+}+\ \omega_{1}-\frac{m_{w,1}}{m_{a}}=0, \tag{7}\]  
\[H^{+}-\ h_{a}\big{(}T_{g,1}\big{)}-\omega_{1}h_{fg}\big{(}T_{g,1}\big{)}+\ \omega_{1}\big{[}h_{w}\big{(}T_{w,1}\big{)}-h_{w}\big{(}T_{g,1}\big{)}\big{]}+ M^{+}h_{w}\big{(}T_{w,1}\big{)}=\ 0. \tag{8}\]  
**Station \"2\"** corresponds to the top section of the saturator--i.e., water inlet/air outlet--for which the unknowns are the outlet gas conditions, namely temperature \(T_{g,2}\) and absolute humidity \(\omega_{2}\), hence the gas mass flow. Air in this section is assumed to be saturated.