## 5 Results and Discussion
### Calibration  
Figure 3 shows the posterior distribution for the parameters and their dependencies. The optimum values and the 95% confidence interval of each parameter are as well listed in Table 6. All the parameters display a Gaussian posterior distribution, that is, the model output is sensitive to all the selected parameters. Otherwise, the posterior distribution would resemble the prior, uniform distribution. The narrower the width of the Gaussian distribution, the more sensitive the model output is on the respective parameter. The heat-transfer coefficient has a narrow distribution and lies with \(\alpha_{HTT}=337\) W/m\({}^{2}\)K as mean value at the upper bond of the selected range. The larger the heat transfer coefficient, the more heat is transported to the heat-transfer channel given the same temperature conditions. The heat transfer is, however, correlated to the heat-loss mechanism that transports the heat to the top of the reactive bed. It is described by the combination of the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Symbol** & **Unit** & **Mean** & **+** & \(-\) \\ \hline Heat-transfer coefficient & \(\alpha_{HTT}\) & W/m\({}^{2}\)K & 337 & 1.30 & 1.38 \\ Heat-loss coefficient & \(\alpha_{loss}\) & W/m\({}^{2}\)K & 13.2 & 0.40 & 0.38 \\ Temperature distribution Param1 & \(k_{1}\) & Km\({}^{-2}\) & 7.54 & 1.51 & 1.49 \\ Temperature distribution Param2 & \(k_{2}\) & Km\({}^{-1}\) & \(-\)4.71 & 1.34 & 1.35 \\ Temperature distribution Param3 & \(k_{3}\) & K & 739 & 1.39 & 1.41 \\ Reaction coefficient & \(k_{R,1}\) & s\({}^{-1}\) & 20.9 & 1.46 & 1.45 \\ Reaction exponent & \(k_{R,2}\) & - & 5.58 & 0.26 & 0.26 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Optimized set of parameters with mean 95% confidence interval. Figure 3: Histograms for the parameter distribution after inverse modeling based on the experimental data of Reference [31]. On the diagonal are the histograms for the single parameters including mean and 95% confidence interval (indicated by the dashed lines). The scatter plots show the covariance between the parameters of the respective row and column. parameters \(\alpha_{HTF}\), \(\alpha_{Loss}\) and k\({}_{1}\)-k\({}_{3}\). Both mechanisms depend on the temperature at the boundary of the reactive bed and the outer temperature of both the HTF channel and the vapor vessel. However, the temperature distribution in the HTF channel changes according to the progress of conversion. In early stages, the reaction occurs everywhere in the reactive bed until the local temperature reaches the equilibrium temperature. Afterwards, reaction occurs only where the reaction bed is cooled down. Heat is transported to the HTF channel, where cooler air is injected from the left. The flowing air takes up the heat of the reactive bed. The temperature difference between bed and channel is largest at the inlet and, thus, the reactive material is converted the fastest at this point. Eventually, a reaction front develops between the region, that has already been converted, and the region, where the conditions are still at equilibrium. Only at the front, the conditions of temperature and pressure are such that the chemical reaction can occur. If no heat was lost and all the heat was transported to the HTF channel, the reaction front would move diagonally from the bottom left to the top right corner of the reactive bed. However, heat is also lost at the top boundary dependent on the temperature in the outer vessel and, thus, the reaction is fast also close to the top boundary. In the center of the bed, the front moves slowly and a kind of parabolic shape of the reaction front establishes. The exact shape is not tracked by the experiment, as only the temperatures at location T1, T3, and T7 are measured. The temperature distribution in the vapor vessel is described by the parameters k\({}_{1}\)-k\({}_{3}\). Via the temperature profiles at the locations T1, T3 and T7, it is thus inferred how much heat is transferred to the HTF channel or lost to the vapor vessel.