## 3 Results and Discussions
### Lifetime Estimate of the Prototype Reformer  
In this study, the Larson-Miller parameter method [33] is applied to predict the catalyst tube lifetime. Catalyst tube failure may be attributed to many factors, e.g., manufacturing, casting, welding, local deformation, creep, etc. In the design phase of a catalyst tube under high temperature operation, the creep failure is usually of particular concern to engineers and metallurgists because most of the other factors are field effects and are not easy to evaluate in the design phase. Creep is severer for components operating under high temperatures or high stresses. With given operating conditions, Larson-Miller parameter method may be applied to estimate the corresponding lifetimes of the catalyst tubes. This would be helpful to find out an optimal operating condition for the catalyst tubes in high temperature operation. The Larson-Miller parameter is a correlation based on the Arrhenius rate equation. Its value is usually defined as follows. \[LMP=T\left(C+\log_{10}t_{r}\right)\times 10^{-3} \tag{14}\]  
where  
\(T\) represents temperature (K),  
\(t_{r}\) represents stress-rupture time (hr),  
\(C\) represents a constant, usually of the order 20. After mathematical manipulation, Equation (14) can be rewritten as the following equation for the stress-rupture time. \[t_{r}=10^{(1000\times LMP/T-C)} \tag{15}\]  
For the catalyst tube material investigated in this paper, the correlation diagram of the Larson-Miller parameter is shown in Figure 3, and the constant \(C\) is 22.96, as suggested by the manufacturer [34]. The design operating conditions of the pressure and temperature for the catalyst tube investigated in this study are 24 kg/cm\({}^{2}\) and 925 \({}^{\circ}\)C, respectively. The stress inside the catalyst tube can then be preliminarily estimated by the following equation to be 11.9 Mpa. \[S=0.5\;P\times Do/t \tag{16}\]  
where  
\(S\) represents the stress inside the catalyst tube,  
\(Do\) represents the catalyst tube outside diameter,  
\(t\) represents the catalyst tube thickness. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Front Wall** (\({}^{\circ}\)**C**) & **Back Wall** (\({}^{\circ}\)**C**) \\ \hline Experiment & 923 & 934 \\ Prototype Simulation & 891 & 892 \\ Periodic Model Simulation & 712 & 714 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Average temperatures at the front wall and back wall of the reformer. The design lifetime for the catalyst tube corresponding to the design operating conditions can then be calculated from Figure 3 and Equation (15) to be around \(5\times 10^{6}\) h (or 570 year). Note that this lifetime is evaluated on the basis of design operating conditions which have taken a safety factor into account. Using the design operating conditions, the catalyst tube will have a very long lifetime. However, in practical operations, there are some operational, human and environmental factors, which will yield a shorter lifetime. On the other hand, from the simulation results, the lifetimes for the catalyst tubes of the prototype reformer can be estimated from the calculated temperatures and pressures of the catalyst tubes shown in Figure 4. The results are shown in Figure 5.  
Figure 4 shows the simulation results of the average and maximum outer surface temperatures as well as the average and maximum inner surface pressures of the catalyst tubes for the prototype reformer. The average outer surface temperatures and the average inner surface pressures of the catalyst tubes for the periodic model are also shown for comparison. It can be observed that the average and maximum inner surface pressures of the catalyst tubes for the prototype reformer nearly coincide. This is because the catalyst tube is filled with catalyst. The flow motion is very slow in the tube, and therefore the average and maximum inner surface pressures are nearly equal.