## 3 Engineering Design and off Design Performance Evaluation
### Engineering Design Mathematical Model  
In this study, a validated semi-empirical method for volumetric reciprocating compressors coming from the literature [28; 29] is employed to calculate mass flow rate, power input and temperature at the end of each compression section. This method provides an integration between physical-based models for compressors at design speed and a polynomial approach for compressor efficiency at off-design speed. Figure 5: Flow chart of the developed calculation procedure consisting in (i) thermodynamic design and optimization, (ii) engineering design and (iii) off-design calculation. First of all, as a starting point for the off-design calculation, the applied compressor model defines the involved physical quantities at the design point (Equations (3)-(11)), which can be calculated on the basis of the optimal results obtained by the thermodynamic analysis presented in Section 2 and of the equations presented in [28]. In more detail, based on the physical conditions (i.e., temperature and pressure) and on the mass flow rate at the inlet of each compressor (\(\dot{m}\), [kg/s]), the suction volumetric flow rate (\(\dot{V}_{suc,des}\), [m\({}^{3}\)/s]) can be calculated as:  
\[\dot{V}_{suc,des}=\frac{\dot{m}}{\rho} \tag{3}\]  
where \(\rho=f(p,T)\) is the density of the fluid at the inlet of the compressor, expressed in [kg/m\({}^{3}\)]. Furthermore, based on [28], the volumetric efficiency (\(\eta_{V,des}\)) can be calculated by the following equation:  
\[\dot{V}_{suc,des}=\frac{\dot{m}}{\rho}\eta_{V,des}=\mathrm{b}_{1}+\mathrm{b}_ {2}\left[\left(\frac{p_{dis}}{p_{suc}(1-\mathrm{dp})}\right)^{1/k}\right] \tag{4}\]  
where \(p_{dis}\) [kPa] is the discharge pressure (evaluated from the imposition of the compressor pressure ratio), \(p_{suc}\) [kPa] is the suction pressure, \(k\) is the isentropic coefficient and \(\mathrm{b}_{1}\), \(\mathrm{b}_{2}\) and \(\mathrm{dp}\) are model parameters [28]. As a consequence, for known compressor speed (\(N_{des}\), [Hz]) and starting from the definition of the volumetric efficiency (Equation (5)), the compressor displacement (\(V_{disp}\), [m\({}^{3}\)]) can be calculated by means of the Equation (6). \[\eta_{V,des}=\frac{\dot{V}_{suc,des}}{N_{des}V_{disp}} \tag{5}\]  
\[V_{disp}=\frac{\dot{V}_{suc,des}}{N_{des}\eta_{V,des}} \tag{6}\]  
Thus, the compressor shaft power (\(W_{des}\), [kW]) can be calculated as:  
\[W_{des}=\frac{1}{\eta_{o}}p_{suc}\dot{V}_{suc,des}a_{1}\left[\left(\frac{p_{ dis}}{p_{suc}}\right)^{a_{2}+\frac{b-1}{k}}+\frac{a_{3}}{p_{dis}}\right] \tag{7}\]  
where \(a_{1}\), \(a_{2}\) and \(a_{3}\) [kPa] are three model parameters [29]. Furthermore, in Equation (7), a power loss is accounted (by means of the coefficient \(\eta_{o}\)): as evaluated in the semi-empirical model presented in [28], indeed, a power loss equal to the 2% is registered. Once evaluated, the compressor shaft power, the isentropic efficiency (\(\eta_{is,des}\)) can be calculated by means of the following equation:  
\[\eta_{is,des}=\frac{\dot{V}_{suc,des}(h_{is}-h_{suc})}{v_{suc}W_{des}} \tag{8}\]  
where \(h_{is}\) is the discharge enthalpy in an isentropic process [kJ/kg], \(h_{suc}\) is the suction enthalpy [kJ/kg], \(v_{suc}=\frac{1}{\rho}\) is the suction specific volume [m\({}^{3}\)/kg]. All these quantities are known from the previously presented thermodynamic analysis. The evaluation of the discharge temperature (i.e., the temperature at the end of the compression section), finally, starts from the shell temperature (\(T_{shell}\), [K]), defined as shown in Equation (9):  
\[T_{shell}=\mathrm{c}_{1}+\mathrm{c}_{2}\frac{p_{dis}}{p_{suc}}+\mathrm{c}_{3} /p_{dis} \tag{9}\]  
where \(\mathrm{c}_{1}\) [K], \(\mathrm{c}_{2}\) [K] and \(\mathrm{c}_{3}\) [K\(\cdot\)kPa] are three model parameters [28].