4.3.2. *λ*eff,r(*r*)-Model

The obvious drawback of the *λ*eff,r-*α*<sup>w</sup> model, that the additional near wall thermal resistance is only captured with an artificial temperature drop directly at the reactor wall, can be circumvented by using a radially varying effective radial thermal conductivity. In this work, the correlation of Winterberg [61] (see Equation (23)) was used as the basis to determine the effective radial thermal conductivity. The three necessary parameters of the Winterberg correlation were determined by conducting a parameter optimization study. The basis of this study was the transport equation described by Equation (21). A summary of the optimized model parameters, including the mean squared error *MSE*, is given in Table 2.

The comparison of the radial temperature profiles for different axial positions can be found in Supplementary Material, Section S5. The spatially resolved deviations between the simplified model and the CFD results are given in Figure 10. It is obvious that in comparison to the results of the *λ*eff,r-*α*<sup>w</sup> model, the accuracy was significantly improved. The temperature close to the reactor wall was predicted with a high degree of accuracy by the model. Only sporadically, the temperatures were overestimated by more than 10 K in the vicinity of the wall, whereby the location was mostly limited to the entry zone (*z*/*d*<sup>p</sup> ≤ 20). A direct comparison of the *λ*eff,r-*α*<sup>w</sup> model and the *λ*eff,r(*r*) model in relation to the CFD results is given in Figure 11 for the loose packing of Raschig rings at *Re*<sup>p</sup> = 1000, showing the superior accuracy of the *λ*eff,r(*r*) model, especially close to the wall. Deep in the bed, deviations outside of the 10 K threshold were mostly found for packings that were characterized by a higher degree of morphological heterogeneity, like the packings of cylindrical particles, the dense bed of spheres, and the loose packing of spheres in the reactor with a random wall structure. While the former configurations were characterized by strong variations in the radial void fraction distribution, the latter showed big fluctuations in the axial void fraction profile. Bigger deviations were mostly limited to the entry zone, indicating that thermal entrance effects, which were not resolved by an axially invariant *λ*eff,r(*r*), might be the reason for this. In contrast to the *λ*eff,r-*α*<sup>w</sup> model, the deviations did not seem to increase if *Re*p was raised. Since the Winterberg correlation did not explicitly consider local variations in the void fraction or axial velocity, it was, similar to the *λ*eff,r-*α*<sup>w</sup> model, not able to capture the step-like effects of the temperature profiles.

**Figure 9.** Deviation of the *λ*eff,r-*α*<sup>w</sup> model results in comparison to particle-resolved CFD data, for loose (top) and dense (bottom) packings of (**A**) spheres, (**B**) cylinders, (**C**) rings, (**D**) four-hole cylinders, and (**E**) spheres with the wall structure.

**Figure 10.** Deviation of the *λ*eff,r(*r*) model results (correlation of Winterberg) in comparison to particle-resolved CFD data, for loose (top) and dense (bottom) packings of (**A**) spheres, (**B**) cylinders, (**C**) rings, (**D**) four-hole cylinders, and (**E**) spheres with the wall structure.

**Figure 11.** Comparison of the radial temperature profiles derived from a particle-resolved CFD simulation (solid line) and the pseudo-homogeneous model (dashed line) at different radial positions for a loose packing of Raschig rings at *Re*<sup>p</sup> = 1000 (left: *λ*eff,r-*α*<sup>w</sup> model; right: *λ*eff,r(*r*) model).

#### **5. Conclusions**

In this work, it was shown in which way the particle-resolved simulation of fixed-bed reactors can play a central role in the process the intensification of this reactor type. After a brief validation study, showing that also under harsh industrial conditions, particleresolved CFD was able to predict the temperature field accurately, the heat transfer characteristics of different particle designs were investigated. The studied designs differed in the used particle shape and the bed density. The results showed that heterogeneities in radial void fraction distribution and axial velocity increased, if packings were compacted. As a result, the overall heat transfer coefficient *U* decreased for most particle shapes. Although the wall channeling effect was most pronounced for the fixed-beds of cylindrical particles, it was found that this particle shape was among the most efficient, with respect to *U*. Furthermore, a novel reactor tube design that used random macroscopic wall structures was investigated. For packings of spherical particles, it was found that macroscopic random wall structures can significantly decrease morphological heterogeneities, leading to a significantly better heat transfer characteristic. Taking into account various process-related boundary conditions, cylindrical particles, Raschig rings, and wall-structured reactors were identified as the most promising concepts to intensify the radial heat transport.

Methods were presented to determine effective thermal transport parameters, which are needed for simplified pseudo-homogeneous models, from the particle-resolved CFD results. Depending on the degree of morphologically induced heterogeneities, an excellent to fair agreement was found for the *λ*eff,r-*α*<sup>w</sup> model in comparison to the CFD results, whereas deviations became bigger if the morphology was more heterogeneous. The known problem of underestimated temperatures close to the reactor tube, as one of the biggest drawbacks of this model, was confirmed. To circumvent this problem, parameter optimization studies, based on the Winterberg correlation, were performed to predict the radially varying effective radial thermal conductivity, which was needed for the *λ*eff,r(*r*) model. A very good agreement regarding the radial temperature profiles was found between the *λ*eff,r(*r*) model and the particle-resolved CFD results. In comparison to the *λ*eff,r-*α*<sup>w</sup> model, the *λ*eff,r(*r*) model showed its superior accuracy close to the reactor wall. Nevertheless, it was found that both pseudo-homogeneous models became less accurate if step-like temperature profiles, which were a either a result of morphological heterogeneities or a high particle solid thermal conductivity, were present.

In terms of process intensification, this work showed that particle-resolved CFD can either directly be used to study improvements on the meso-scale through:


• identifying local phenomena as hot/cold spot formation or catalyst poisoning;

or as a reliable source for parameters, and correlations of those, that are needed for process simulation. This allows a more reliable analysis of process intensification by studying:


**Supplementary Materials:** The following are available online at https://www.mdpi.com/article/ 10.3390/en14102913/s1.

**Author Contributions:** Conceptualization, N.J. and M.K.; methodology, N.J.; validation, N.J., U.S. and A.A.M.; formal analysis, N.J.; investigation, N.J., U.S. and A.A.M.; writing—original draft preparation, N.J.; writing—review and editing, N.J., U.S., A.A.M. and M.K.; visualization, N.J.; supervision, M.K. All authors read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Clariant.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author. The data are not publicly available due to large file sizes, and, partly, restrictions that might apply.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**

The following abbreviations are used in this manuscript:


Nomenclature—Greek


#### **Dimensionless Numbers**


#### **References**

