# Investigation of Point-Contact Strategies for CFD Simulations of Pebble-Bed Reactor [PERSON]
## 2 Materials and Methods
### Numerical Solvers
The heat flux is linearized with the following form:  
\[q_{0}^{\prime\prime}=A_{0}+B_{0}T_{f0}+C_{0}T_{w0}+D_{0}T_{w0}^{4} \tag{16}\]  
where \(A\), \(B\), \(C\), and \(D\) are net wall heat flux coefficients, \(T_{f0}\) is the bulk fluid temperature, and \(T_{w0}\) is the interfacial temperature. The heat transfer coefficient is defined locally based on the local flow conditions (fluid density, specific heat capacity, wall shear stress, and local Reynolds number), local fluid temperature, and local solid temperature. And this results in a local heat transfer coefficient for each cell face along the interface [23]. In Case 2, PLOFC, thermal radiative heat transfer was also modeled using the surface radiation exchange model (S2S) [23]. The heat transport fluid was assumed to be non-participating with no internal absorption, emission, or scatter (100% transparent to thermal radiation). The pebble wetted surfaces were assigned as the participating faces for thermal radiation, and these were used to calculate the appropriate view factors. No radiative heat transfer was modeled within the solid pebbles (100% opaque). The radiation exchange between surfaces can be accounted for by tracking the amount of energy emitted from and absorbed from pairs of surface mesh faces as  
\[P_{1-2}=i_{1}^{\prime}dS_{1}cos(\beta_{1})(\frac{dS_{2}cos(\beta_{2})}{L^{2}}) \tag{17}\]  
where \(P_{1-2}\) is the power emitted from face 1 and absorbed by face 2, \(i_{1}^{\prime}\) is radiation intensity flux per unit area and per unit steradian, \(\beta\) is the angle between faces, and \(L\) is the distance between faces. Conservation of radiative energy is balanced only when using a closed set of surfaces. The total energy transfer can be calculated by summing over all faces that represent the closed set. For face \(i\), this becomes  
\[Q_{1,i,\lambda}=S_{i}I_{i,\lambda}=\sum_{j=1}^{N_{p}}\frac{F_{j-i,\lambda}}{(1 -\tau_{i,\lambda}-\rho_{s,i,\lambda})}I_{d,j,\lambda}S_{j}+\frac{F_{e-i, \lambda}}{(1-\tau_{i,\lambda}-\rho_{s,i,\lambda})}I_{e,\lambda}S_{e} \tag{18}\]  
where \(S_{i}\) and \(S_{j}\) are the indexed face surface areas, \(I_{i,\lambda}\) is the indexed radiation flux, \(N_{p}\) is the total number of faces, \(F_{j-i,\lambda}\) is the view factor from face \(j\) to \(i\), \(I_{d,j,\lambda}\) is the diffuse component of radiation leaving face \(j\), \(F_{e-i,\lambda}\) is the view factor from the environment to face \(i\), \(\tau_{i,\lambda}\) is the transmissivity, and \(\rho_{s,i,\lambda}\) is the specular component of radiation. Other radiative assumptions included  
1. Each pebble surface mesh face was used in the view factor calculation;
2. Gray radiation with no wavelength dependence;Diffuse radiation with no angular emission dependence;
* The effective radiation temperature of the environment was specified as 250 \({}^{\circ}\)C;
* A pebble surface emissivity of 0.8; and
* Satisfaction of Kirchhoff's Law, such that emissivity + reflectivity + transmissivity = 1. This forces the surface reflectivity to 0.2 because the transmissivity is 0.