## 3 Engineering Design and off Design Performance Evaluation
### Off-Design Calculation Model  
Once the physical quantities in design conditions have been calculated, the compressors off-design analysis is carried out by means of Equations (16) and (19). According to [28], the normalized volumetric and isentropic efficiencies are influenced only by the normalized compressor speed and are independent from the pressure ratio. Then, the two efficiencies can be represented as a function of the normalized compressor speed through a second order polynomial function (Equations (16) and (17)):  
\[\frac{\eta_{p}}{\eta_{p,des}}=\text{d}_{1}+\text{d}_{2}\left(\frac{N}{N_{des}} \right)+\text{d}_{3}\left(\frac{N}{N_{des}}\right)^{2} \tag{16}\]\[\frac{\eta_{is,dss}}{\eta_{is}}=\mathrm{e_{1}}+\mathrm{e_{2}}\Bigg{(}\frac{N}{N_{ds} }\Bigg{)}+\mathrm{e_{3}}\Bigg{(}\frac{N}{N_{ds}}\Bigg{)}^{2} \tag{17}\]  
where d1, d2 and d3, e1, e2 and e3 are model parameters [29].  
Consequently, the volumetric flow rate in off-design conditions can be calculated from Equations (5) and (16) as it follows:  
\[\dot{V}_{suc,off-design}=\dot{V}_{suc,des}\frac{N}{N_{ds}}\frac{\eta_{\varphi}} {\eta_{\varphi,des}}=\dot{V}_{suc,des}\Bigg{[}\mathrm{d_{1}}\Bigg{(}\frac{N}{ N_{des}}\Bigg{)}+\mathrm{d_{2}}\Bigg{(}\frac{N}{N_{ds}}\Bigg{)}^{2}+ \mathrm{d_{3}}\Bigg{(}\frac{N}{N_{ds}}\Bigg{)}^{3}\Bigg{]} \tag{18}\]  
Furthermore, from Equations (8) and (17), the compressor shaft power in off-design conditions is evaluated as:  
\[\begin{array}{ll}W_{off-design}&=W_{des}\frac{\dot{V}_{suc,off-design}}{ \dot{V}_{suc,des}}\frac{\eta_{is,ds}}{\eta_{is}}\\ &=W_{des}\frac{\dot{V}_{suc,~{}off-design}}{\dot{V}_{suc,des}}\bigg{[}\mathrm{ e_{1}}+\mathrm{e_{2}}\Big{(}\frac{N}{N_{ds}}\Big{)}+\mathrm{e_{3}}\Big{(}\frac{N}{N_{ds}} \Big{)}^{2}\bigg{]}\end{array} \tag{19}\]  
As for the design section, the compressor model parameters come from the dataset reported in [30] and are listed in Appendix A.  
Finally, it should be highlighted that the calculation model of the temperature at the end of the compression section in design conditions (Equations (12)-(15)) can be directly applied also to off-design conditions without any modification, since the model relies on system energy balance taking into account the heat transfer loss to the ambient.  
On the other hand, in off-design conditions, the heat exchanger behavior cannot be described by \(\varepsilon\)-NTU method, but it is still governed by fluids energy balance and heat exchange equation, adjusted to the new operating conditions (Equations (20)-(22)). Evidently, considering the off-design of a specific heat exchanger, the exchange surface remains unchanged. As a consequence, the relationship between the thermal flows can be expressed only on the basis of the ratio between the products of global heat transfer coefficients and mean logarithmic temperature differences (Equation (20)):  
\[\frac{Q_{HE}}{Q_{HE,ds}}=\frac{U}{U_{ds}}\frac{\Delta T_{ml}}{\Delta T_{ml, des}} \tag{20}\]  
Being:  
\[\frac{U}{U_{des}}=\Bigg{(}\frac{\dot{m}}{\dot{m}_{des}}\Bigg{)}_{w}^{0.8} \tag{21}\]  
where the subscript \(w\) is referred to the fluid with the worst convective heat transfer coefficient. Note that the exponent can slightly vary based on the considered fluids, the type of flow and the system geometry.  
After the calculation of the global heat exchange coefficient, in off-design conditions (\(U\)), the \(NTU\) parameter can be determined as it follows:  
\[NTU=US/C_{min} \tag{22}\]  
Finally, the heat exchanger effectiveness at off-design conditions (\(\varepsilon^{\prime}\)) is still defined by Equation (15).