## 2 Modeling of Containment Thermal Hydraulics
### Modeling for Passive Autocatalytic Recombiner  
In general, PAR modeling is composed of thermodynamic and gas dynamic models. The catalytic body of a PAR has two modes such as heat generation by the surface reaction and heat transfer between gas and the catalytic body.  
The catalytic surface reaction of hydrogen and oxygen can be described by Equation (13) [28]  
\[H_{2}+\frac{1}{2}\mathrm{O}_{2}\Rightarrow H_{2}\mathrm{O}+121\ \mathrm{MJ/kg} \tag{13}\]  
Depending on the hydrogen removal rate R of a PAR, the hydrogen and oxygen consumption rates and water vapor production rate are defined as follows.  
\[\frac{d}{dt}m_{h2}=-R \tag{14}\]  
\[\frac{d}{dt}m_{o2}=-8R \tag{15}\]  
\[\frac{d}{dt}m_{h20}=9R \tag{16}\]  
As a PAR catalytic reaction rate, a correlation equation based on hydrogen removal rate data obtained from PAR performance tests is generally used.  
\[R=correlation(p,T,x_{h2},x_{o2},x_{h20}) \tag{17}\]  
In the PAR model based on the correlation equation, the hydrogen removal rate R is obtained using the correlation equation. Table 1 shows the hydrogen removal rate correlation equations for commercial PARs.  
\begin{table}
\begin{tabular}{c c c} \hline
**PAR Vendor** & **PAR Correlation** & **Unit** \\ \hline AREVA & \(R=\eta x_{min}(A\times p+B)tanh(100x_{min,lim})\) & X[-], p[bar] \\ \hline AECL & \(R=f_{size}k\big{(}a_{1}x_{h2}+a_{2}x_{h2}^{2}\big{)}\times\big{(}\frac{298}{ T}\big{)}^{c}\times p^{d}\) & X[\%], p[bar] \\ \hline NIS & \(R=a\times x_{h2}^{b}\times\frac{p}{RT}\) & X[-], p[Pa] \\ \hline KNT & \(R=0.66N\big{(}a_{1}+a_{2}x_{h2}+a_{3}x_{h2}^{2}\big{)}\times\big{(}\frac{p}{T} \big{)}\) & X[\%], p[bar] \\ \hline CERACOMB & \(R=Sk(x_{h2}-0.15)^{1.16}\times p\left(\frac{273}{T}\right)\times 10^{-3}\) & X[\%], p[bar] \\ \hline \end{tabular}
\end{table}
Table 1: Hydrogen depletion correlations of commercial PARs [29; 30; 31].  
Figure 1: One-dimensional virtual mesh for wall heat conduction model.  
The heat released from the hydrogen recombination is shared by the catalytic body and the gas flowing through the body because the surface catalyst participates in the reaction. The heat transfer model is similar to the previous study [22], and the equation is as follows.  
\[\big{(}mC_{p}\big{)}_{par}\frac{d}{dx}T_{par}=(1-\varphi_{par})\frac{121\times 1 0^{6}}{V_{par}}\times\frac{d}{dt}m_{h2}-Ah(T_{par}-T_{gas}) \tag{18}\]  
\(\phi_{par}\) in Equation (18) is used to split the thermal energy, and it may vary depending on the shape of the catalyst body.  
The hydrogen removal rate of a PAR is physically related to the mass flow rate of the hydrogen mixture gas flowing into the PAR housing. The gas flow into the catalytic body is induced continuously by the heat released from the catalytic hydrogen recombination but is limited by the frictional resistance between the catalytic body and the flowing gas.  
In this study, the unsteady Darcy-Forchheimer friction model [33], which is an extension of the Darcy-Forchheimer model, was applied to consider a transient phenomenon of a PAR. The first term of Equation (19) is the same as the virtual mass term used in the Euler two-phase flow model, and the default value of the coefficient \(C_{\wp}\) is 0.5. The second and third terms are Darcy friction and Forchheimer friction, respectively.  
\[\
abla p=C_{\wp}\rho\bigg{(}\frac{d}{dt}\mathbf{U}+\mathbf{U}\cdot\
abla\mathbf{U}\bigg{)} +\mu\mathbf{D}\cdot\mathbf{U}+\frac{1}{2}\rho|\mathbf{U}|\mathbf{F}\cdot\mathbf{U} \tag{19}\]