## 4 Reinforcement and Stress Analysis of the Support Frame
Case 7 and 8 were designed by adding vertical or horizontal-plus-vertical braces to the horizontal beams on the top of convection section of Case 4 and were expected to be able to enhance the strength of the horizontal beams. Further, Case 8 was expected to be the most effective one among the eight cases investigated. From the results of finite element analysis, the stresses at all positions for the eight cases shown in Figure 8 were lower than the fatigue limit. As expected, Case 8 was the most effective one among the eight cases investigated. However, positions 10, 18, 31, 34, 35, and 38 were still in potential threat because their stresses were close to the fatigue limit, as shown in Figure 9. When the wind speed increases, the stresses at the above-mentioned positions will exceed the fatigue limit. To further enhance the security of the heater, Case 9 was proposed by modifying Case 8, as illustrated in Figure 10. The four braces at the top of convection section for Case 8 were removed because one end of the brace was welded to the skin of the heater and this may be harmful to the skin under the action of a strong wind, as marked in Figure 10a. Further, four braces at the bottom of the convection section were added in Case 9 to share the forces acting on the bottom of the convection section, as indicated by the green marks in Figure 10b,c. The inclined angles of the four braces at the top of convection section for Case 9 were changed from 45\({}^{\circ}\) in Case 8 to 60\({}^{\circ}\) in Case 9 because this was more effective in reducing stresses after finite element analysis, as indicated by the yellow marks in Figure 10b,c. Figure 4.0 Eight different kinds of reinforcement methods for the support frame. (**a**) Case 1; (**b**) Case 2; (**c**) Case 3; (**d**) Case 4; (**e**) Case 5; (**f**) Case 6; (**g**) Case 7; (**h**) Case 8. From the results of the finite element analysis for Case 9, as shown in Figure 11, the stresses at all positions were lower than the fatigue limit, and the stresses at positions 10, 18, 31, 34, 35, and 38 were further reduced, as compared to Case 8. The wind speed that the heater can withstand was also increased, as will be analyzed below. Because the stack bore most of the wind load, the wind flowing through a heater could be reasonably simplified as flow past a cylinder. Therefore, the relation between the wind speed and the stress could be derived from [PERSON]'s diagram of drag coefficient vs. Reynolds number [26], as shown in Figure 4.1  
Figure 4.2 Results of finite element analysis for Case 8. Figure 4.3 Results of finite element analysis for Case 9. Figure 4.4 Numerical model for Case 9. (**a**) Case 8 (The braces with green marks are removed. ); (**b**) Case 9 (The braces with green marks are added and the brace angles with yellow marks are changed, similar for (**c**)), (**c**) Case 9. The drag force for flow past a cylinder is written as:  
Figure 4.5  
where \(\mathrm{C_{D}}\) is the drag coefficient, \(\rho\) is the air density, \(\mathrm{V}\) is the wind speed, and \(\mathrm{A}\) is the frontal area of the stack. From Equation (1), the drag force \(\mathrm{F_{D}}\) is proportional to \(\mathrm{C_{D}}\times\mathrm{V^{2}}\). Because the stress (\(\mathrm{\tau}\)) is proportional to the drag force, it is also proportional to \(\mathrm{C_{D}}\times\mathrm{V^{2}}\). The drag coefficient \(\mathrm{C_{D}}\) can be found from Figure 4.6 The Reynolds number (Re) is computed from \(\mathrm{Re}=\rho\mathrm{VD}/\mathrm{\mu}\), where \(\mathrm{D}\) is the stack diameter and \(\mathrm{\mu}\) is the air viscosity. The critical wind speed, \(\mathrm{V_{c}}\), when the stress of the support frame reaches the fatigue limit, \(\mathrm{\tau_{f}}\), can then be solved from the following equation:  
Figure 4.7  
where \(\mathrm{C_{D,c}}\) and \(\mathrm{C_{D,ref}}\) are the drag coefficients corresponding to the critical wind speed, \(\mathrm{V_{c}}\), and reference wind speed, \(\mathrm{V_{ref}}\), respectively.