## 2 Numerical Analysis Method  
### Check-Valve Shape and Analysis Conditions  
In this study, a solenoid-type check valve, which blocks the hydrogen flow when the pressure and temperature limits of the hydrogen refueling process are exceeded, was applied to the numerical analysis. The check valve was selected to be in the normal open state (Open ratio = 100%), which is a condition where hydrogen flows in without changing the open ratio during the normal charging process. In order to compare the results in the area where hydrogen flows in and out of the core part, which is the area blocking the hydrogen inflow of the check valve shown in Figure 1a, a total of 150 mm of pipe geometry was applied before and after the valve to simulate the fully developed flow. The detailed geometry and specifications of the check valve used in this study are shown in Figure 1a and Table 1.  
Figure 1: Schematics of check valve for hydrogen charging system. (**a**) Internal flow zone of check valve; (**b**) Measurement positions. For the numerical analysis, the charging conditions of the check valve were selected based on the hydrogen charging protocol (SAE J2601). The inlet pressure of the check valve (\(\mathrm{P_{in}}\)) was selected to be 70 MPa, and the outlet pressure (\(\mathrm{P_{out}}\)) was equally spaced at 2 MPa [8]. To select the initial inlet temperature (\(\mathrm{T_{in}}\)) condition of the hydrogen, we selected 233 K for the cryogenic temperature, 298 K for the normal temperature, and 363 K for the high temperature, referring to the charging limit temperature range specified in the hydrogen charging protocol. In addition, during the numerical analysis, the wall was subjected to the no-slip condition, the solution methods were subjected to the coupled scheme condition, and the spatial discretization was subjected to the second-order upwind scheme. The details of the check-valve differential pressure (\(\mathrm{\Delta P}\)) and inlet temperature (\(\mathrm{T_{in}}\)) conditions are shown in Table 2.  
To determine the effect of energy loss in the check valve on performance, the exergy destruction results were analyzed. Exergy represents the useful potential work of a given amount of energy in a given state and is the thermodynamic losses that occur due to irreversibility within a system. Therefore, the turbulence characteristic changes and energy losses were calculated as shown in Equations (1)-(3) by comparing the exergy destruction within the system boundary due to the entropy that flows into and out of the system and is destroyed by the irreversible effect. In addition, the flow-separation phenomenon and energy change were analyzed by comparing the pressure coefficient (\(\mathrm{C_{P}}\))--a dimensionless number that indicates how much the pressure at a given point in the flow has changed compared to the dynamic pressure of the free flow, which is calculated as shown in Equation (4). \[\psi=(h-h_{0})-T_{0}(S-S_{0})+\frac{V^{2}}{2}+gz \tag{1}\]  
\[\Delta\psi=(h_{2}-h_{1})-T_{0}(S_{2}-S_{1})+\frac{V_{2}^{2}-V_{1}^{2}}{2}+g(Z _{2}-Z_{1}) \tag{2}\]  
\[\sum\bigg{(}1-\frac{T_{0}}{T_{K}}\bigg{)}q_{K}-W+(\psi_{1}-\psi_{2})-x_{ destroyed}=0 \tag{3}\]  
Equation (1) is the exergy of the flow or stream, and Equation (2) is the exergy change of the flowing fluid from state 1 to state 2. h is enthalpy, \(S\) is entropy, \(\frac{V^{2}}{2}\) is kinetic energy, gz is potential energy, and T0 is the temperature state of the matter at rest. Equation (3) represents a single-flow exergy equilibrium with one inlet and one outlet in a steady flow process, where \(\psi_{1}\) and \(\psi_{2}\) are the flow exergy at the inlet and outlet, and \(x_{destroy}\) is the exergy loss. \begin{table}
\begin{tabular}{c c} \hline \hline
**Specification of Check Valve** & \\ \hline Valve length (mm) & 150 \\ Valve height (mm) & 29 \\ Inlet and outlet diameter (mm) & 15.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Specifications of the check valve.