## 2 Modeling
### Mathematical Models and Correlations
#### 2.2.1 Connecting Pipe Model  
The connecting pipe is heat insulated. Consequently, the models solely consider Pdt, and its influence on x is negligible, thereby dismissing Pda.  
* Pressure drop for single-phase flow
* Frictional pressure drop (Pdfsp)  
The Darcy-Weisbach equation is commonly used for determining the frictional pressure drop (Pdf) for single-phase, incompressible fluids, as depicted in Equation (3) [28]:  
\[\mathrm{Pdfsp}=\ 10^{-3}.\mathrm{f}.\frac{\mathrm{L}:\rho}{2}.\frac{\mathrm{v}^{2 }}{\mathrm{D}} \tag{3}\]  
In this equation, L denotes the length of the connecting pipe, \(\rho\) represents the density of the single-phase refrigerant, v signifies the refrigerant's flow velocity, D is the hydraulic diameter, and f is the friction factor. This factor is theoretically calculated for a circular tube and is defined in Equation (4) [29]:  
\[\left\{\begin{array}{ll}\mathrm{f}=\frac{64}{\mathrm{Re}}&:\mathrm{Laminar \ flow}\quad(\mathrm{Re}<2000)\\ \mathrm{f}=0.316\cdot\mathrm{Re}^{-0.25}&:\mathrm{Turbulent\ flow}\quad(\mathrm{Re}\geq 2 000)\end{array}\right. \tag{4}\]  
The Reynolds number, denoted by \(\mathrm{Re}\), is significant in these computations and is obtained via Equation (5):  
\[\mathrm{Re}=\frac{\rho\cdot\mathrm{V}\cdot\mathrm{D}}{\mathrm{\mu}} \tag{5}\]  
Here, \(\mathrm{\mu}\) denotes the dynamic viscosity of the refrigerant.
* Gravitational pressure drop (Pdgsp) The gravitational pressure gradient is specified by Equation (6): \[\mathrm{Pdg}_{\mathrm{sp}}=\rho\cdot\mathrm{g}\cdot\mathrm{H}\] (6)  
In this equation, g stands for the acceleration due to gravity, and H is the vertical height of the connecting pipe.  
* Pressure drop for two-phase flow
* Frictional pressure drop (Pdfsp)The calculation of the frictional pressure drop in a two-phase flow within the connecting pipe utilizes the associative model proposed by [PERSON] and [PERSON] [30], as recommended by [PERSON] [31] and [PERSON] [32]. This model is encapsulated in Equation (7):  
\[\left\{\begin{aligned} &\mathrm{Pdf_{tp}=F\cdot(1-x)^{\frac{1}{5}}+Pdf_{g} \cdot x^{3}}\\ &\mathrm{F=Pdf_{1}\;+2\cdot(Pdf_{g}\;-Pdf_{l})\cdot x}\end{aligned}\right. \tag{7}\]  
\(\mathrm{Pdf_{l}}\) and \(\mathrm{Pdf_{g}}\) denote the pressure drops computed by Equations (3)-(5) when the refrigerant is entirely in the liquid phase and entirely in the gas phase, respectively.  
* Gravitational pressure drop  
The calculation of gravitational pressure drop in a two-phase region is depicted in Equation (8), which structurally parallels Equation (6):  
\[\mathrm{Pdg_{tp}=\rho_{tp}\cdot g\cdot H} \tag{8}\]  
Here, \(\rho_{tp}\) represents the density of the two-phase mixture, which depends on the vapor fraction \(\alpha\), computed as per Equation (9):  
\[\rho_{tp}=\alpha\cdot\rho_{g}+(1\;-\;\alpha)\cdot\rho_{1} \tag{9}\]  
The [PERSON] model [33] is utilized to determine the void fraction \(\alpha\) in two-phase flow, as outlined in Equation (10):  
\[\left\{\begin{aligned} &\mathrm{A=\left[1+F\cdot\left(\frac{1-x}{x} \right)\cdot\left(\frac{\rho_{g}}{\rho_{1}}\right)\right]^{-1}}\\ &\mathrm{F=0.4+0.6\cdot\sqrt{\frac{\left[\frac{\rho_{g}}{\rho_{g }}+0.4\cdot\left(\frac{1-x}{x}\right)\right]}{[1+0.4]}\cdot\left(\frac{1-x}{x} \right)}}\end{aligned}\right. \tag{10}\]