## The Methodology of Numerical Modeling  
### Overview of the Models and Their Interactions  
The heat transfer inside a CGL furnace is dominated by radiation and conduction. A developed CHT model was used to consider these two heat transfer mechanisms and determine their influence on the change in the tube's temperature distributions due to its surroundings. Convection contributes marginally to the total heat transfer from the radiant tubes to the strip, accounting for around 5 to 10 percent (depending on the heat transfer coefficient) compared to radiative heat transfer. Therefore, the convective component of heat transfer was neglected in the CHT model. The CHT model was integrated into the previously mentioned FSI approach, developed by [PERSON] et al., as illustrated in Figure 4. The selected FSI approach followed a partitioned one-way coupling between the FVM and FEM models. Detailed descriptions of boundary conditions of each model are available in the corresponding references cited. However, an outline of the chosen type of discretization and the models are given in Tables 1 and 2, respectively. An overview of the FVM, CHT, and FEM models and their interactions are delineated below. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **Mesh Type** & **Type of Elements** & **Number of Elements** & **Solution Domain** \\ \hline FVM & 3-D volumetric & 100 \% tetrahedral & 990 000 & - & Self-recuperative burner \\  & 3-D volumetric & 100 \% tetrahedral & 990 000 & - & Polarant tube mounted in test bench \\  & & & & - & Furance chamber of test bench \\  & & & & - & Heat sinks available in test bench \\ \hline CHT & 3-D surface & 100 \% triangular & 12 355 & - & Chosen radiant tube setup \\  & 3-D surface & 100 \% triangular & 12 355 & - & Strip passing between the tubes \\  & & & & - & Furance chamber as an enclosure \\  & & & & (walls, rolls and other RTs neglected \\ \hline FEM & 3-D surface & 99.44 \% quadratic 0.56 \% triangular & 6 269 & - & Central radiant tube \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Outline of chosen discretizations for the finite volume method (FVM), coupled heat transfer (CHT), and finite element method (FEM) models. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **Software** & **Solving Approach** & **(Sub)Models** \\ \hline FVM & ANSYS Fluent® & Semi-Implicit Method for Pressure-Linked Equations & - & Turbulence: Realizable k-\(t\) \\  & & Semi-Implicit Method for - & - & Combustion: Eddy Dissipation Concept \\  & & Pressure-Linked Equations & - & Radiation: Discrete Ordinates \\  & & & & with Weighted-Sum-of-Gray-Gases \\ \hline CHT & MATLAB® & Hotel’s zonal method & - & Radiation: Surface-to-Surface (full resolution) \\ \cline{2-4}  & & Abaqus® & Pure heat transfer analysis & - & Conduction: Shell heat conduction \\ \hline  & & & & Material model consisting of: \\  & & & - & Time-independent thermoelasto-plastic strain components \\ FEM & Abaqus® & Coupled-Temperature-Displacement & - & Time-dependent plastic strain included in the extended Graham-Walles creep model \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Outline of the FVM, CHT, and FEM models. Figure 4: Working principle of the coupled heat transfer (CHT) model as part of a fluid-structure-interaction (FSI) approach. The CHT model was created by combining two models: a full-resolution three-dimensional radiation model developed by [PERSON] et al. [36] and a conduction model available via the commercial FEM software Abaqus(r). The CHT model considered the effects of the neighboring tubes, the furnace chamber, and the strip of a CGL furnace on the temperature distribution of a reference radiant tube. The furnace chamber represented a virtual boundary, enveloping the radiant tube setup, including the strip. The initial temperature distribution for each tube was derived from the FVM model for a P-type radiant tube made of Alloy 602. The tube had an outer diameter of 275 mm and a wall thickness of 3 mm.