## 4 Case Studies and Their Setups  
The heating and soaking sections of a CGL furnace offered a broad variety of different radiant tube setups to become the subject of investigation. Three different radiant tube setups were chosen, in addition to the previously described isolated radiant tube, to form the basis of the case studies in the present work, as shown in Figure 5. Setup (a) represented the case of a single tube unaffected by its surroundings. Setup (b) added a radiant tube, horizontally adjacent to the first one. The strip--passing between both tubes--was also taken into account. In order to maximize the number of horizontal neighbors, a second adjacent tube was considered in setup (c). The number of strip-passes also increased in the case of the three-tube horizontal setup, making a total of four passing strips. In addition to the influence of the horizontally adjacent neighbors, the vertical ones also needed to be investigated. Therefore, three vertically-stacked tubes in a column, enclosed by two passing strips, were chosen for setup (d). It could be noted that the relative positions of the tubes and the strip in the setups were based on recommendations by radiant tube manufacturers [38]. These positions were also confirmed with real CGL furnace dimensions. All tubes in the chosen setups were fully surrounded by other neighboring tubes. The furnace walls of the CGL were neglected due to their unknown influence. This was the main reason to avoid choosing tube setups at the furnace corners or adjoining furnace rolls as part of the present study. The tubes used in each setup were P-type radiant tubes made of Alloy 602, having a diameter of D = 275 mm and a wall thickness of s = 3 mm. Figure 6 shows the tube geometry relative to a passing strip. The CHT and FEM models had a tube length of 1905 mm since the 250 mm thick insulation was not considered. This was due to the non-availability of the recuperator-part of the burner in the FVM model. A closer look was taken into two different strip widths: a narrow strip with a width of  
Figure 5: Heating section of a CGL and the chosen radiant tube setups. \(\mathrm{w}=825\) mm and a wide strip with \(\mathrm{w}=1385\) mm. The position of the strip relative to the insulation had an offset of 623 mm and 343 mm, respectively. The material chosen for the strip was a standard steel grade based on DIN EN 10025-2 [39]. For the calculation of radiation exchange, the knowledge of the materials' emissivities became obvious. The emissivity is temperature-dependent and was experimentally determined for Alloy 602 within the used temperature range as \(\varepsilon=0.95\)[40]. The steel strip's emissivity was assumed to be \(\varepsilon=0.6\)[41]. The built-in self-recuperative burner had a nominal power of 120 kW and could operate in both flame and flameless combustion modes. For the present study, only the flameless mode was considered. The heat input through the recirculating radiant tubes into a CGL furnace was normally controlled by operating the burners in on/off firing mode. This on/off firing led to a change of thermal loads on the radiant tubes and was one of the key aspects of the present investigations. The firing was represented by two temperature distributions, one distribution for the point where the burner was switched on (burner-on point), and a second distribution for the point where the burner was switched off again (burner-off point). The burner firing took place within a cycle time of \(\mathrm{t}=15\) min, representing high-frequency furnace operations like a change in strip's cross-section or its speed. Each firing cycle was simplified to attain a triangular form, with the ratio of burner-on to burner-off time being 75/25. During actual CGL operation, the burners were usually fired individually according to the zone temperature controls. In order to simplify these real operating conditions and still be able to quantify the influence of the burner firing, two different firing modes within a setup were considered.