## 1 Introduction
### Introduction to Single- and Multi-Component PT-Diagram  
To understand condensation and how it occurs, it is necessary to have a conceptual understanding of thermodynamic diagrams. When the working fluid is a pure single component, the line that separates the gas phase from the liquid phase is known as the coexistence line or saturation curve in the PT-diagram. This is indicated by the purple line in the Figure 1. The gas phase is below the saturation curve and the liquid phase is above it. The extreme point \((P,T)\) of this curve is the critical point of the fluid under study, often noted as \((P_{c},T_{c})\). The critical point of pure CO\({}_{2}\) is \(P_{c}\) = 7.377 MPa and \(T_{c}\) = 304.128 K, which is represented by a purple dot in the Figure 1. Above this critical point lies the supercritical fluid zone, where the fluid behaves in a completely different way. Indeed, recent studies have shown that a further classification between liquid-like and gas-like behavior continues in the supercritical zone, as explained in references [15; 16].  
In the case of a multi-component fluid, the thermodynamic representation and, in particular, the PT diagram of the mixture, are completely different from that of a single-component fluid. Indeed, even with a small amount of impurity added to pure CO\({}_{2}\), the coexistence line separates into two independent lines, the dew line at the vapor side and the bubble line at the liquid side, which constitute the so-called phase envelope. The zone between these lines is the two-phase region. The phase envelope of CO\({}_{2}\) as a main species, with N\({}_{2}\) as impurities of 5%, 4% and 3%, is shown in the dotted lines in Figure 1. The green dotted line shows the phase envelope with 5% N\({}_{2}\), light blue for 4% N\({}_{2}\), and red dotted line for the phase envelope with 3% N\({}_{2}\). In addition, we observe a shift in  
Figure 1: Pressure -Temperature diagrams of three cases with various % N\({}_{2}\) as impurity. This figure also shows the (P,T)-path during the gas expansion (represented by the yellow curve) of 95% CO\({}_{2}\). This curve crossing over on the dew line (on green dashed phase envelope) represents the appearance of condensation in the nozzle. Similar behaviour is seen for 96% and 97% CO\({}_{2}\). The saturation line of pure CO\({}_{2}\) is also represented for the comparison with the phase envelope location in the (P,T)-plane. Critical points are represented by different symbols depending on the quantity of impurities.  
the critical point (\(P_{c},T_{c}\)) with the phase envelope, showing that impurities also delay the subcritical/supercritical transition, which occurs at a higher pressure as the amount of impurities increases. The corresponding shift in the critical points is marked with a green square (for 5% N\({}_{2}\)), blue triangle (for 4% N\({}_{2}\)) and red star (for 3% N\({}_{2}\)), while the critical point of pure CO\({}_{2}\) is shown as a purple dot. The solid lines show the numerical results of case 2 (this is a reference to Table 1, which is discussed later in this paper) with N\({}_{2}\) as impurity, where the blue line is 5% N\({}_{2}\), the red line is 4% N\({}_{2}\) and the yellow line is for the 3% N\({}_{2}\) impurity. As the amount of impurity increases, the phase envelope becomes wider as the distance between the dew line and the bubble line increases. In the context of CO\({}_{2}\) with N\({}_{2}\) as an impurity, as the amount of N\({}_{2}\) increases, the deviation of the dew curve from the saturation line of pure CO\({}_{2}\) is not significant, whereas it becomes more significant in relation to the bubble curve. This means that the amount of impurities plays a dominant role in shifting thermodynamic properties on both sides of the phase envelope, but more so on the evaporation side than on the condensation side. Further discussion on the effects of impurities on the (CO\({}_{2}\)-impurities) phase envelope can be found in references [7; 17].