## 2 Thermodynamic Design and Optimization
### Thermodynamic Modeling  
The thermodynamic optimization was realized through a physical-mathematical model, developed in Visual Basic for Application (VBA) environment. In more detail, through an iterative procedure, the model allows to determine the optimal plant layout and the physical states (pressure, temperature and mass flow rate) of each sections in order to achieve the optimal values of the performance indicators usually considered for these power plants (as will be explained in the Section 2.3). Furthermore, by means of the iterative resolution of the developed thermodynamic model, a complete thermodynamic evaluation of the power plant is carried out by varying the key parameters affecting the performance of  
Figure 1: Schematic of the proposed small-scale liquefaction process.  
the plant, namely (i) the layout, (ii) the pressure at the exit of the primary compressor line (p\({}_{6}\)) and (iii) the storage tank pressure (p\({}_{10}\)). In particular, these last two parameters are varied between a minimum and a maximum value, and all their possible combinations are considered.  
During the calculation, the developed model also defines the configuration of the secondary compressor line on the basis of the value of natural gas grid pressure. The second compressor line options are (with reference to Figure 1):  
1. compressor C2-1 and inter-cooler C2 + compressor C2-2 and after-cooler C2;
2. compressor C2-1 and inter-cooler C2;
3. no secondary compressor line.  
Furthermore, depending on the considered value of the gas network supply pressure and for given values of the pressures at the exit of the primary compressor line and at the storage tank, the total compression ratio for each compression train is completely defined, if the pressure losses through the various heat exchanger are assumed. The developed model opportunely splits the total compression ratio between the two compressor stages by applying the _minimum compressor specific work criterion_. This criterion has been modified by the Authors [24] in order to take into account also the pressure losses through the compression train.  
As it regards the cooling section, the components HE3 and HE1 are simply calculated by the method of logarithmic mean temperature difference, while HE2 performance is estimated by assuming a value of the temperature at the chiller outlet (T\({}_{8}\)) as input.  
Finally, it is possible to choose between the use of Joule-Thomson (JT) valve or a turbo-expander (TE) in the expansion section. In this last case, the expander performance is estimated by considering a constant value of isentropic efficiency.  
For further details about the mathematical models adopted in this thermodynamic optimization tool, please refer to [24].  
Four scenarios have been taken into account relating to the value of natural gas supply pressure:  
- high pressure supply network, 70 bar (HPN),  
- medium-high pressure supply network, 30 bar (MHPN),  
- medium-low pressure supply network, 15 bar (MLPN),  
- low pressure supply network, 3 bar (LPN)  
According to the most common pressure values in the various branches of Italian national gas network [25].  
Furthermore, the pressure at the exit of primary compressor line (i.e., the maximum pressure of the cycle) has been assumed varying between 150 bar and 300 bar, while the storage pressure ranges between 3 bar and 15 bar. These values have been chosen considering the current technology regarding the gas compressors and the available technology for LNG storage tanks on vehicles.  
The main flow chart of the calculation model, which accomplishes the thermodynamic design and optimization phase (level 1) is presented in Section 3, as part of the complete procedure for the design and performance evaluation. In addition, further details on the thermodynamic optimization procedure can be found in [17].