# One-Dimensional Computation Method of Supercritical CO\({}_{2}\) Labyrinth Seal
## 3 Internal Flow Property of SCO\({}_{2}\) Labyrinth Seal
### Two-Phase Critical Flow in SCO\({}_{2}\) Labyrinth Seal  
Two-phase and critical flow is one of the typical characteristics of SCO\({}_{2}\) labyrinth seal and it is paramount for the 1-D method to the capture the position of the two-phase critical flow. Applying the isentropic model to calculate the two-phase critical flow. The curve of sealing performance of round hole is shown in Figure 11 and Table 6, and the round hole structure of [PERSON]'s experiment in Table 7 is used and the upstream condition is 10 MPa and 323 K [17]. As the outlet pressure decreases, the mass flow rate attains a peak at 7 MPa where the _Ma_ number is just 0.73. This means that the _Ma_ number at the maximum isentropic leakage point has not reached 1. Figure 11: The curve of sealing performance of round hole calculated by 1-D method. According to the definition of the _Ma_ number in Section 2.2.3 and the deduction process of the two-phase speed of sound \(a_{\text{Tivo-phase}}\) shown Equation (19), the vital point for the computation of two-phase flow _Ma_ number is the accuracy of \(a_{v}\) and \(a_{l}\) which is defined by the isentropic change of \(\frac{\partial P}{\partial\rho}\). \[ \begin{split} a_{\text{Tivo-phase}}&=\left(\frac{ \partial P}{\partial\rho}\right)_{s}=\left(\frac{\partial\rho}{\partial\rho} \right)_{s}^{-1}=\left(\frac{\partial\rho}{\partial P}\cdot\left(-\frac{1}{v^ {2}}\right)\right)_{s}^{-1}\\ &=\left(\frac{\partial\left(x_{g}c_{g}+\left(1-x_{g}\right)v_{l} \right)}{\partial P}\cdot\left(-\frac{1}{v^{2}}\right)\right)_{s}^{-1}\\ &=\left(\left(x_{g}\left(\frac{\partial v_{g}}{\partial P}\right) _{s_{g}}+\left(1-x_{g}\right)\left(\frac{\partial v_{l}}{\partial P}\right)_{s _{l}}\right)\cdot\left(-\frac{1}{v^{2}}\right)\right)^{-1}\\ &=\left(\rho^{2}\left(\frac{x_{g}}{\left(a_{g}c_{g}\right)^{2}}+ \frac{1-x_{g}}{\left(a_{l}\rho_{l}\right)^{2}}\right)\right)^{-1},s=x_{g}s_{g }+\left(1-x_{g}\right)s_{l}\\ a_{g}&=\left(\frac{\partial P}{\partial\rho_{g}} \right)_{s_{g}},a_{l}=\left(\frac{\partial P}{\partial\rho_{l}}\right)_{s_{l} }\end{split}  \tag{19}\]  
where the subscripts \(s\) represents the isentropic process of two-phase flow. The subscripts \(s_{g}\) represents the isentropic process of saturated gas. The subscripts \(s_{l}\) represents the isentropic process of saturated liquid; \(v\) is the specific volume of two-phase flow; \(\rho_{g}\) and \(\rho_{l}\) are the density of saturated gas and saturated liquid respectively; \(a_{g}\) and \(a_{l}\) are the speed of sound of saturated gas and saturated liquid respectively; \(x_{g}\) is the dryness of two-phase flow, which represents the mass fraction of saturated gas. The isentropic model is used to calculate the curve of density with a change of outlet pressure and inlet temperature (290 K-340 K) in which the inlet pressure is unchanged at 10 MPa. It can be found in Figure 12 that when the outlet condition enters the two-phase region, the change rate of density decreases suddenly. According to the definition of two-phase speed of sound \(a_{\text{Tivo-phase}}\) in Equation (19),  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(P_{out}\) & **Mass Flow Rate** & _Ma_ & **Speed of Sound** \\ \hline MPa & kg/h & & m/s \\
10.00 & 0.00 & 0.00 & 218.09 \\
9.80 & 34.86 & 0.15 & 215.74 \\
9.60 & 48.88 & 0.21 & 213.33 \\
9.40 & 59.33 & 0.27 & 210.84 \\
9.20 & 67.87 & 0.31 & 208.29 \\
9.00 & 75.1  
the sudden change of speed of sound will occur when the outlet flow enters the two-phase condition. Also, as the inlet temperature decreases, the sudden change becomes much more obvious. The original 1-D method uses the \(a_{v}\) and \(a_{l}\) provided by NIST physical property model, and it may lead to the problem of two-phase critical flow. Using the Figure 12 to further discuss the speed of sound of \(a_{v}\) and \(a_{l}\), the dotted line represents the saturation line.