## 5 Results  
The stress-strain relation of a ceramic packed bed has been determined via experimental and simulation results as presented in [12] and are used as a basis for the present work. The parameters are fitted to approximate the experimental and simulation data following [PERSON]'s approach. The effective Young's Modulus, which is calculated from the stress/strain ratio of uniaxial compression tests, is plotted as a function of stress in Figure 4. Here, the experimental results are shown as well as the results from the DEM simulations. Again, details on the procedure of those tests are described in [12]. For all four different cases shown in Figure 4a-d, the effective Young's Modulus increases at a higher rate for low stresses compared to higher stress levels. This means that the particles still have the possibility to move and to rearrange themselves at low stress states due to the low compaction of the packed bed. The increase rate slows down since particle movement is increasingly restrained with continuing compression. Comparing the compression (Figure 4a,b) and decompression (Figure 4c,d) separately, the effective Young's Modulus at 600 \({}^{\circ}\)C is slightly higher than at 20 \({}^{\circ}\)C at any given stress state, which is the result of the increasing Young's Modulus of the ceramic bulk material at higher temperatures. The differences between the gradients of compression and decompression can be explained by energy dissipation effects. The maximum effective Young's Modulus is significantly higher for the 600 \({}^{\circ}\)C packed bed is about 15% to 20% higher compared to the 20 \({}^{\circ}\)C packed bed. Again, this is a result of the temperature dependency of the ceramic bulk material's Young's Modulus. Compared to the bulk material, the effective Young's Modulus for the packed bed at a given stress range is over two orders of magnitude smaller, which matches with the results of [23]. As displayed, the effective Young's Modulus of the packed bed depends on temperature and stress state and thus can be expressed as a continuous function. In this paper, [PERSON]'s approach  
Figure 4: Effective Young’s Modulus as a function of stress at 20 \({}^{\circ}\)C and 600 \({}^{\circ}\)C: (**a,b**) during compression, (**c,d**) during decompression. (see Equation (1)) is applied. The parameters \(\text{C}_{\text{i}}\) were fitted to match the simulation data. Since the compression cycle yield higher stress and stiffness compared to the decompression cycle, the following results for the Reimann parameters are only given for the compression cycle since this represents the more critical case. The resulting parameters are listed for a range of different packing factors (investigated in [12]) of the packed bed in Table 1.  
The parameters above are implemented in the simplified continuum model described in the previous paragraph. The results for the contact pressure between packed bed and container wall are displayed in Figure 5. Additionally, results of the Janssen approach [13] for the \"cold\" state of the packed bed is plotted for comparison. The relevant material properties for calculating the Janssen approach are listed in Table 2.  
For both cases (cold and hot state), DEM results from [4] for the large-scale application are also plotted. These results were obtained using a 2D-DEM model to reduce the number of simulated particles and thus computation time. While this decreases the overall accuracy of the simulation results compared to a 3D analysis, the results are still expected to be of satisfying quality. Figure 5a shows the results for the packed bed in a cold state, whereas Figure 5b displays the results for the thermally charged packed bed.