# Investigation of Point-Contact Strategies for CFD Simulations of Pebble-Bed Reactor [PERSON]
## 1 Introduction
\[Re_{p}=\frac{\rho_{f}U_{int}d_{h}}{\mu_{f}}, \tag{2}\]  
where \(\epsilon_{i}\) is the calculated porosity for each contact strategy, \(d_{p}\) is the pebble diameter, and \(\rho_{f}\) and \(\mu_{f}\) are the density and dynamic viscosity of the heat transfer fluid (helium gas), respectively. \(U_{int}\) is the average interstitial velocity between the pores related to the superficial velocity, \(U\), through \(U_{int}=U/e\). The superficial velocity or the Darcian flow is defined as \(U=\frac{Q}{\lambda}\), where \(Q\) is the volumetric flow rate, and \(A\) is the cross-sectional area of an empty bed. The heat transfer coefficient \(h_{sf}\) and the Nusselt number \(Nu_{sf}\) are defined as  
\[h_{sf}=\frac{q_{p}^{\prime\prime}}{\left(\overline{T}_{p}-\overline{T}_{f} \right)}, \tag{3}\]  
\[Nu_{sf}=\frac{h_{sf}d_{p}}{\lambda_{f}}, \tag{4}\]  
where \(q_{p}^{\prime\prime}\) is the pebble surface heat flux, and \(\overline{T}_{p}\) and \(\overline{T}_{f}\) are the volume-averaged pebble surface and fluid temperature. The fluid thermal conductivity is denoted as \(\lambda_{f}\). For each contact strategy, the calculated porosity, \(\epsilon_{i}\), was compared to an analytically computed true porosity using perfectly spherical pebbles with point singularity contact. Table 1 summarizes the total pebble volume, calculated porosity, pore-scale hydraulic diameter, and the porosity error for each contact strategy relative to the true porosity. Some of the tested contact strategies yielded a significant porosity error relative to the true porosity. Decreasing or increasing the pebble diameter by 1% yielded a porosity error of 3.2%. The bridging method with the smallest bridge diameter, equal to 10% of the pebble diameter, showed the smallest porosity error of 0.012%. This strategy (Strategy 3a (bridge 10%)) was used as the baseline numerical result against which all the other strategies were compared. Strategy 2a, increasing the pebble diameter by 0.1%, yielded a small porosity error, but the simulations were not numerically stable unless a significantly finer mesh than required for the other strategies was used. Therefore, the results of this strategy were not discussed and denoted as not available where applicable. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Strategies**} & \multicolumn{2}{c}{**Pebble Volume**} & \multirow{2}{*}{**Calculated Porosity**} & \multirow{2}{*}{**\(d_{h}\) [m]**} & \multirow{2}{*}{**Porosity Error [\%]**} \\  & [\(\times\)**10\({}^{-3}\) m\({}^{3}\)]** & & & \\ \hline
1a—Decrease 0.1\% & 4.5678 & 0.4780 & 0.0458 & 0.3284 \\
1b—Decrease 0.5\% & 4.4979 & 0.4860 & 0.0473 & 2.0051 \\
1c—Decrease 1.0\% & 4.4454 & 0.4920 & 0.0484 & 3.2647 \\ \hline
2a—Increase 0.1\% & N/A & N/A & N/A & N/A \\
2b—Increase 0.5\% & 4.6501 & 0.4686 & 0.0441 & \(-\)1.6459 \\
2c—Increase 1.0\% & 4.7184 & 0.4608 & 0.0427 & \(-\)3.2844 \\ \hline
**3a—Bridge 10\%** & **4.5820** & **0.4763** & **0.0455** & \(-\)**0.0123** \\
3b—Bridge 15\% & 4.5839 & 0.4761 & 0.0454 & \(-\)0.0578 \\
3c—Bridge 20\% & 4.5892 & 0.4755 & 0.0453 & \(-\)0.1850 \\ \hline
4a—Cap 10\% & 4.5802 & 0.4766 & 0.0455 & 0.0309 \\
4b—Cap 15\% & 4.5790 & 0.4767 & 0.0455 & 0.0597 \\
4c—Cap 20\% & 4.5737 & 0.4773 & 0.0457 & 0.1869 \\ \hline True & 4.5815 & 0.4764 & 0.0455 & \\ \hline \hline \end{tabular}  
* Strategy 2a yielded a small porosity error but a significantly finer mesh was required for numerical stability than that required for the other strategies. Results of this strategy were not discussed and denoted as not available (N/A).  
\end{table} 
Table 1: Calculated porosity, pore-scale hydraulic diameter \(d_{h}\), and porosity error relative to the true porosity for four different pebble contact strategies of various sizes. STAR-CCM+ v16.02.009, a commercial finite-volume CFD software package developed by Siemens (Munich, Germany), was selected for this study. A steady RANS-based approach to turbulence modeling was chosen for this effort given the size of the domain of interest and the available computational resources.