# CFD Simulations of Radiative Heat Transport in Open-Cell Foam Catalytic Reactors

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{−1}K

^{−1}; superficial velocities 0.1–0.5 m s

^{−1}; surface emissivities 0.1–1). Moreover, the temperature levels correspond to a range of exo- and endothermal reactions, such as CO

_{2}methanation, dry reforming of methane, and methane steam reforming. We found a significant influence of radiation on heat flows (deviations up to 24%) and temperature increases (deviations up to 400 K) for elevated temperature levels, low superficial velocities, low solid thermal conductivities and high surface emissivities.

## 1. Introduction

_{2}methanation), ignoring proper heat transport design of the reactor can lead to uncontrollable hot-spot formation or even thermal runaways. Moreover, catalysts can be harmed due to sintering effects or unwanted byproducts that might be formed, which then lead to catalyst poisoning [2]. The highly exothermic CO

_{2}methanation reaction is part of the power-to-gas (PtG) concept where renewable excess energy (e.g., from wind turbines) is stored (electro-)chemically by converting hydrogen and carbon dioxide to methane. This process, among others, has the potential to drastically reduce the dependence on fossil fuels and reduce carbon dioxide emissions. The supply of renewable energy is fluctuating, which leads to a demand of dynamic operable reactors [3]. Additionally, the power grids might not be able to withstand and transport all renewable energy during peak wind or sun hours, which makes small scale dynamic operated plants a current research topic [4]. Usually, catalytic fixed-bed reactors that contain pellets are used for steady conversion of hydrogen to methane. Recent studies have shown, that, for small-scale reactors and low flow rates, structured catalyst carriers (such as open-cell foams) have advantageous heat transport properties over conventional packed bed reactors [5,6,7]. Open-cell foams are characterized by an interconnected solid matrix, allowing for unhindered radial heat transport as well as high porosities and relatively high specific surface areas, yielding low pressure drop and proper catalyst inventory, respectively [8].

_{s}the solid body temperature. This relationship underlines that the contribution of radiation to the overall heat transport is low for reactions at low to medium temperature and explains why radiation is often neglected in modeling. Studies dealing with thermal radiation in open-cell foams mainly focused on deriving optical parameters and deploying analytical or pseudo-homogeneous models that are based on the so-called Rosseland approximation [20,21,22,23,24,25]. The pseudo-homogeneous approaches do not distinguish between solid and fluid phase and utilize effective transport properties (e.g., effective or two-phase thermal conductivity) which can be used to estimate general contributions of radiation to the overall heat transport. For instance, the general influence of temperature levels, window diameter, or thermal conductivity on the total heat flows can be evaluated. However, homogeneous models are only suitable for a certain parameter range (i.e., velocity, foam properties, and temperature range) and, compared to experiments, can deviate in the order magnitude of about 30–40% [26,27]. Furthermore, homogeneous models are not always suitable for prediction of temperatures when heat is produced in the solid (i.e., exothermic reaction) [28]. An overview of currently available pseudo-homogenous models for open-cell foams as well as their range is for example given in [6]. Furthermore, experimental techniques and pseudo-homogeneous models generally cannot resolve occurring heat flows between the solid foam, fluid, and reactor wall parts that might give valuable insight in the foam’s heat transport ability. In contrast, three-dimensional CFD simulations of an open-cell foam embedded in a tube can supply information about the different heat flows that are mandatory to fully understand structured reactors and, hence, improve the general design [28]. Some researchers considered radiation in their CFD models, e.g., for solar receivers [29] and for catalytic exothermic reactions (catalytic partial oxidation of methane [30]; dry reforming of methane [31]; CO oxidation [32]; methane steam reforming [33]). The explicit consideration of radiation in catalytic gas-phase reactors has a significant influence on the computed concentration and temperature fields which was shown in a honeycomb reactor for the partial oxidation of methane [34] and in a pellet reactor for the methane steam reforming reaction [35]. Hettel et al. [34] found temperature increases of up to 56 K and Wehinger and Flaischlen [35] found a maximum yield increase of up to 70% with temperature differences below 40 K. The influence of radiation modeling on simulated temperature and yields decreased for a higher Reynolds number (i.e., superficial velocity) as the relative contribution of dispersion increases [35]. In a different study, a pure conjugate heat transfer case in a fixed-bed pellet reactor also indicated the importance of radiation modeling for the design of catalytic reactors since neglecting radiation led to a 6% temperature increase for a wall temperature of 800 K [36]. To sum up, radiation in CFD simulations of catalytic reactors should generally be considered for elevated temperatures and low superficial velocities [37].

_{2}methanation reaction in a 25 × 24 mm foam, we estimated a heat source intensity of 50 W (i.e., 1.9 × 10

^{7}W m

^{−3}). Generally, this approach supplies the desired heat flows and allows to study thermal effects decoupled from chemistry. However, radiation was not considered in the study. As the explicit simulation of thermal radiation is computationally expensive, it is therefore important to quantify under which conditions radiation can be neglected.

^{−1}K

^{−1}), superficial velocity (0.1–0.5 m s

^{−1}), surface emissivity (0–1) and temperature level that are relevant for heterogeneous catalysis and industrial process conditions. For this close-to-reactor setup, the temperature levels and the analyzed range correspond roughly with prominent reactions like the Fischer–Tropsch synthesis (500 K, [39]), CO

_{2}methanation (700 K, [2]), dry reforming of methane (900 K, [40]), and steam reforming of methane (1200 K, [41]). The geometry information of the open-cell foam is based on a µCT scan, and the simulation is carried out in the open source CFD framework OpenFOAM. Further, the model is verified against a commercial CFD code (STAR-CCM+). In particular, we analyze the influence of radiation modeling on occurring heat flows as well as solid temperature distributions. We expect this study to guide through the conditions under which the modeling of thermal radiation in a foam reactor is necessary.

## 2. Results and Discussion

#### 2.1. Model Verification

_{SF}denotes the heat flow transferred from the solid to the fluid and Q

_{SW}denotes the conductive heat flow from the solid to the wall. The bar graphs in Figure 2a,b, thus all individually sum up to 50 W. For a wall temperature of T

_{w}= 900 K (a) and without considering radiation, the convective part of the stacked bar (blue: solid/fluid) is less than 20% of the overall heat flow for both, the OpenFOAM as well as the STAR-CCM+ model. When radiation is accounted for, this ratio becomes larger than 20%. The consideration of radiation in the conjugate heat transfer model enables another heat transport path for the thermal energy. Hence, the solid temperature distribution becomes more homogeneous (see Figure 3). Here, the heat flow from solid to wall is purely conductive whereas computed heat flow from solid to fluid combines convective as well as radiative heat transfer. Consequently, the computed heat flow from solid to fluid increases. This is because the more homogeneous solid temperature causes a larger (i.e., distributed over a larger foam part) temperature gradient between solid and fluid at the parts of the foam that were cooler when radiation was neglected. For a wall temperature of T

_{w}= 1200 K (b), the same behavior can be observed. In general, the calculated heat flows from OpenFOAM and STAR-CCM+ are comparable for both temperature levels and with or without radiation, indicating the applicability of either software.

#### 2.2. Quantification of Heat Flows and Temperature Distributions

#### 2.2.1. Influence of the Wall Temperature and Solid Thermal Conductivity

_{w}= 900; λ

_{s}=5 W m

^{−1}K

^{−1}). Here, the fluid temperature drastically increases when radiation in the model is neglected. Furthermore, the overall increased fluid temperature also enhances the development of the temperature wake behind the foam. Consequently, the solid foam temperature increases when radiation is neglected (Figure 5). Keep in mind that a constant heat source (50 W) is set in the solid volume. A perfect wall contact between foam and wall ensures unhindered heat transport [42], which results in a relatively cool outer zone of the foam. In contrast, inside the solid, hot spots develop which strongly depend on the thermal conductivity [28]. The drastic changes in both fluid and solid temperatures, due to radiation effects, can cause dramatically different results when an actual catalytic chemical reaction is simulated. This is especially true when the overall conversion is mainly influenced by the temperature (i.e., kinetically controlled regime). The homogeneously distributed heat source approach cannot determine the actual maximum temperature of a chemical reaction, because the heat production varies locally due to temperature-dependent kinetics. Nevertheless, the maximum temperature increase, calculated by the heat source approach, can at least be expected in an actual exothermal reaction when the same amount of heat is released.

_{w}= 900 K and λ

_{s}= 5 W m

^{−1}K

^{−1}), the absolute deviation amounts to almost 7 W (14%). Furthermore, we observe that, from a threshold on (λ

_{s}= 5 W m

^{−1}K

^{−1}), the absolute deviations in radiative heat flow decrease with increasing solid thermal conductivities. This is due to the less pronounced heat transport limitations in the foam center for high conducting materials. Here, heat is more likely transported via conduction through the continuous solid strut network and thus, the two models yield more comparable results. Even for quite high solid thermal conductivities the absolute deviations in radiative heat flow can be substantial. For instance, for λ

_{s}= 50 W m

^{−1}K

^{−1}at T

_{w}= 1200 K the absolute deviation amounts to 8 W (16%) and for λ

_{s}=200 W m

^{−1}K

^{−1}still to approx. 4 W (8%), respectively. That means even for highly conducting materials (e.g., metals) radiation in open-cell foams cannot generally be neglected at low flowrates.

_{s}, but the values for very low thermal conductivities, i.e., λ

_{s}= 1 W m

^{−1}K

^{−1}, are all lower than for intermediate values. Here, the absolute deviation caused in radiative heat flow can be almost 50% lower than at λ

_{s}= 5 W m

^{−1}K

^{−1}(which is the second data point, compare for instance purple line in Figure 6). The reason for that diverging behavior certainly lies in the drastic temperature increase of both phases (fluid and solid) at very low solid thermal conductivity. The corresponding maximum and mean deviations in temperature increases support this hypothesis (Figure 7). Between the two smallest investigated solid thermal conductivities, i.e., when λ

_{s}is increased from 1 W m

^{−1}K

^{−1}to 5 W m

^{−1}K

^{−1}, the deviations in maximum and mean temperatures decrease tremendously. As a consequence, for the given boundary conditions, the impact of radiation modeling on heat flows is negligible at very low λ

_{s}as the entire system itself (fluid and solid temperatures) forms a temperature hot spot. Here, the relative difference between the phase temperatures decrease and hence the absolute deviation of the heat flows.

_{s}= 50 W m

^{−1}K

^{−1}, the absolute deviation for both maximum and mean temperature increase are negligible regardless of the applied temperature level (Figure 7a,b). In contrast, the absolute deviations in heat flows can still be as high as 8 W (of 50 W total heat flow). Even at λ

_{s}= 200 W m

^{−1}K

^{−1}, there is a 4 W (i.e., almost 10%) deviation in radiative heat flow at 1200 K wall temperature while deviations in mean and maximum temperature increase are virtually non-existent. This indicates that the ability of the foam to transport heat via conduction (i.e., the thermal conductivity) is not limiting and can balance the missing contribution of the radiation. Concluding, even though no temperature changes can be identified between models with and without radiation, the absolute and relative deviation in computing the heat flows can still be significant. Hence, one should consider these findings before omitting radiation modeling in open-cell foams at elevated temperatures. Once again, we want to stress that, for actual chemical reactions, the absolute deviations for maximum temperatures between models with and without radiation should be even more severe (when equal amounts of heat production are compared).

#### 2.2.2. Influence of the Superficial Velocity

_{w}= 900 K and a solid thermal conductivity of λ

_{s}= 5 W m

^{−1}K

^{−1}, the corresponding absolute deviations in heat flow as well as deviations in maximum and mean temperature are plotted against the superficial velocity in Figure 8a. Both the absolute deviation for heat flows and temperatures decrease with increasing superficial velocity. In the investigated range of superficial velocities, the absolute error for heat flows increases from about 7 W (14%, for 0.5 m s

^{−1}) to about 12 W (24%, for v = 0.1 m s

^{−1}). Due to increasing convective heat transfer between the fluid and solid at higher superficial velocities, the foam is cooled more efficiently. Therefore, the contribution of radiation as heat transport mechanism obviously decreases with increasing superficial velocity, as convection becomes more dominant. This is underlined by the specific heat flow from solid to wall plotted against the superficial velocity (Figure 8b). The transition line indicates the dominant heat removal mechanism. A value greater than 0.5 means conduction is dominant whereas a value lower than 0.5 mean convection is dominant. Here, all simulations are conduction dominated, although the case with highest investigated velocity (0.5 m s

^{−1}) and considered radiation becomes close to convection dominated. Concluding, for foam reactors with conditions where convection is dominant, the consideration of radiation is definitely less important. Here, it can only be assumed that the absolute error, which can reach values up to 24% (Figure 6), would have an asymptotic behavior for even higher superficial velocities. This should be addressed in further studies. We would like to note that the results from Figure 6 and Figure 7 would have shown even larger absolute deviations at smaller superficial velocities. Hence, the magnitude of superficial velocities plays a major role in the consideration of radiation.

#### 2.2.3. Comparison with a Homogeneous Model

_{SW}S

^{−1}). This approximation should be valid as long as temperature differences remain low between the cases (i.e., high thermal solid conductivities). The heat flow ratios and ratios of the effective thermal conductivities, respectively, are plotted against the full range of thermal conductivities at different temperature levels in Figure 9. Both approaches show the same qualitative trend regardless of the consideration of radiation or the applied temperature level. The homogeneous model reflects the trends of decreasing importance of radiation modeling for increasing thermal conductivities as well as decreasing temperature levels. It can also be seen that the simulation results as well as the analytical model agree better at high thermal conductivity and low temperature level. With increasing temperature levels, the contribution of radiation in the analytical model seems to be more pronounced than in the CFD simulations. When compared to experimental data, the homogeneous model was able to match approximately 80% of the data points with less than 30% deviation [26]. Hence, the homogeneous model can give a convenient fast glance on the general contribution of radiation on the total heat transport. However, for a more thorough quantification, CFD simulations or experiments are needed. Further, the need for even more accurate (engineering) models that can be used for the estimation of heat transport contributions becomes obvious.

#### 2.2.4. Influence of the Surface Emissivity

_{w}= 900 K and T

_{w}= 700 K, heat flows and solid temperature (mean and maximum) are almost independent of ε. For the elevated wall temperature, T

_{w}= 1200 K, changes in heat flows and solid temperatures are more pronounced and heat flows and temperatures become independent of ε only for ε almost 1. Concluding, increased wall temperatures also increase the influence of ε on heat flows and temperature fields. In the surface emissivity range 0 < ε < 0.5, the changes in heat flows and temperatures are more severe. Hence, inaccurately determined emissivity values influence simulation results more significantly for materials with general lower emissivity.

## 3. Materials and Methods

#### 3.1. General Model and Meshing

_{2}methanation reaction in a foam [6,28], which was adopted for this study. We note that, the effect of heat source can be easily extrapolated to other heat source intensities as it was shown in [28] for the range of 5 ≤ S ≤ 150 W.

#### 3.2. Governing Equations and Thermal Radiation Modeling

_{f}denoting the fluid’s density and

**U**denoting the velocity field. The conservation of momentum can be described by:

_{f}being the fluid’s thermal conductivity. The solid phase, in contrast, is described only by the conservation of energy:

_{s}being the solid’s thermal conductivity, T

_{s}the solid’s temperature, and S the specific artificial heat source. The set of equations is completed by the ideal gas law.

^{−2}), that a real solid body emits, can be expressed by the Stefan–Boltzman law (Equation (1)) extended by the surface emissivity ε:

^{−2}sr

^{−1}):

_{rel}= 50%; Appendix C). In this study, only fluid participating approaches are used as they have been successfully applied in literature for high temperature heat transfer and are also suitable for symmetry boundary conditions [34,35,47]. We also note that the fluid participation should not play a significant role in this study for the model fluid (air) as well as the model conditions (i.e., atmospheric pressure, temperatures and geometrical dimensions) [34,35]. Both models basically solve the radiative transfer equation (RTE) that describes the change of the radiant intensity at any point along a path through a participating medium (fluid) depending on scattering, emission and adsorption effects [43]. For further explanation and discussion the reader is referred to [43] and [48]. The RTE reads [34,48]:

_{b}denotes the black body intensity, σ

_{s}the scattering coefficient, I

_{incident}the incident intensity, Φ(s

_{i},s) the scattering phase function and Ω

_{i}the solid angle.

_{w}, the initial intensity from all possible directions s

_{i}reads [48]:

## 4. Conclusions

_{rel}= 50%). Secondly, an influence of radiation on the simulated heat flows and temperature increases was found for the following parameters in decreasing order: 1. overall temperature level (here, wall temperature T

_{w}and inlet temperature T

_{in}), 2. solid thermal conductivity, 3. superficial velocity, and 4. surface emissivity. At four temperature levels corresponding with a range of industrially-relevant chemical reactions in heterogeneous catalysis (500 K: Fischer–Tropsch synthesis; 700 K: CO

_{2}methanation; 900 K: dry reforming of methane; 1200 K: methane steam reforming), the influence of radiation modeling was quantified. At most, we found maximum temperature deviations of up to 400 K and a discrepancy in heat flows of about 12 W (24%).

_{2}) with potential scattering. The heat source approach further underlined its suitability to effectively study thermal effects of catalyst carriers decoupled from chemistry. A deeper understanding of heat transport processes in foams in particular and structured reactors in general can lead to more accurate foam reactor models and consequently to intensified processes. This knowledge might help to facilitate the launch of structured reactors into industrial application.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Geometrical Foam Properties

Parameter | Symbol | Value |
---|---|---|

pore count | 10 ppi | |

open porosity | ε_{0} | 0.77 |

specific surface area | S_{V} | 521.3 m^{−1} |

cell diameter | d_{c} | 5.76 ± 1.9 mm |

window diameter | d_{w} | 3.3 ± 0.9 mm |

strut diameter | d_{s} | 1.5 ± 0.5 mm |

## Appendix B. Geometry for Verification

## Appendix C. Analysis of P1 and fvDOM Radiation Models for Suitability in Open-Cell Foams

**Figure A2.**Test for suitability of P1 and fvDOM models used for radiation modeling in open-cell foams. Left: illustration of case (geometry = cube) and temperature boundary condition. Right: relative error of radiative heat flow between numerical solution and analytical solution plotted against cube dimensions.

## List of Symbols

Latin | |

c_{p} | Isobaric heat capacity, J Kg^{−1} K^{−1} |

d_{c} | Cell diameter, m |

d_{s} | Strut diameter, m |

d_{w} | Window diameter, m |

E_{rel} | Relative error of heat flow, - |

I | Intensity, W m^{−2} sr^{−1} |

L | Cube dimensions, m |

Q | Heat flow, W |

Q_{SF} | Heat flow solid to fluid, W |

Q_{SW} | Heat flow solid to wall, W |

h | Specific enthalpy, J |

p | Pressure, Pa |

r | Position vector, - |

s | Direction vector, - |

S | Total heat source intensity, W |

S_{v} | Specific surface area, m^{−1} |

T | Temperature, K |

T_{w} | Wall temperature, K |

T_{max} | Maximum temperature, K |

T_{mean} | Mean temperature, K |

U | Velocity, m s^{−1} |

v | Superficial velocity, m s^{−1} |

Greek | |

α | Degree of absorption, - |

γ | Degree of reflection, - |

τ | Degree of transmission, - |

Ω | Solid angle, sr |

κ | Absorption coefficient, m^{−1} |

σ_{s} | Scattering coefficient, m^{−1} |

ε_{0} | Open porosity, - |

ε | Surface emissivity, - |

μ | Dynamic viscosity, Pa s |

λ | Thermal conductivity, W m^{−1} K^{−1} |

## References

- Kiewidt, L.; Thöming, J. Predicting optimal temperature profiles in single-stage fixed-bed reactors for CO
_{2}-methanation. Chem. Eng. Sci.**2015**, 132, 59–71. [Google Scholar] [CrossRef] - Rönsch, S.; Schneider, J.; Matthischke, S.; Schluter, M.; Götz, M.; Lefebvre, J.; Prabhakaran, P.; Bajohr, S. Review on methanation—From fundamentals to current projects. Fuel
**2016**, 166, 276–296. [Google Scholar] [CrossRef] - Kalz, K.F.; Kraehnert, R.; Dvoyashkin, M.; Dittmeyer, R.; Gläser, R.; Krewer, U.; Reuter, K.; Grunwaldt, J.-D. Future Challenges in Heterogeneous Catalysis: Understanding Catalysts under Dynamic Reaction Conditions. ChemCatChem
**2016**, 9, 17–29. [Google Scholar] [CrossRef] [Green Version] - Solomon, A.; Kammen, D.M.; Callaway, D. The role of large-scale energy storage design and dispatch in the power grid: A study of very high grid penetration of variable renewable resources. Appl. Energy
**2014**, 134, 75–89. [Google Scholar] [CrossRef] - Kiewidt, L.; Thöming, J. Pareto-optimal design and assessment of monolithic sponges as catalyst carriers for exothermic reactions. Chem. Eng. J.
**2019**, 359, 496–504. [Google Scholar] [CrossRef] - Kiewidt, L.; Thöming, J. Multiscale modeling of monolithic sponges as catalyst carrier for the methanation of carbon dioxide. Chem. Eng. Sci. X
**2019**, 2, 100016. [Google Scholar] [CrossRef] - Graf, I.; Rühl, A.-K.; Kraushaar-Czarnetzki, B. Experimental study of heat transport in catalytic sponge packings by monitoring spatial temperature profiles in a cooled-wall reactor. Chem. Eng. J.
**2014**, 244, 234–242. [Google Scholar] [CrossRef] - Reitzmann, A.; Patcas, F.C.; Kraushaar-Czarnetzki, B. Keramische Schwämme—Anwendungspotenzial monolithischer Netzstrukturen als katalytische Packungen. Chem. Ing. Tech.
**2006**, 78, 885–898. [Google Scholar] [CrossRef] [Green Version] - Meinicke, S.; Dubil, K.; Wetzel, T.; Dietrich, B. Characterization of heat transfer in consolidated, highly porous media using a hybrid-scale CFD approach. Int. J. Heat Mass Transf.
**2020**, 149, 119201. [Google Scholar] [CrossRef] - Bianchi, E.; Heidig, T.; Visconti, C.G.; Groppi, G.; Freund, H.; Tronconi, E. Heat transfer properties of metal foam supports for structured catalysts: Wall heat transfer coefficient. Catal. Today
**2013**, 216, 121–134. [Google Scholar] [CrossRef] - Razza, S.; Heidig, T.; Bianchi, E.; Groppi, G.; Schwieger, W.; Tronconi, E.; Freund, H. Heat transfer performance of structured catalytic reactors packed with metal foam supports: Influence of wall coupling. Catal. Today
**2016**, 273, 187–195. [Google Scholar] [CrossRef] - Della Torre, A.; Montenegro, G.; Tabor, G.R.; Wears, M. CFD characterization of flow regimes inside open cell foam substrates. Int. J. Heat Fluid Flow
**2014**, 50, 72–82. [Google Scholar] [CrossRef] [Green Version] - Regulski, W.; Szumbarski, J.; Łaniewski-Wołłk, Ł.; Gumowski, K.; Skibinski, J.; Wichrowski, M.; Wejrzanowski, T. Pressure drop in flow across ceramic foams—A numerical and experimental study. Chem. Eng. Sci.
**2015**, 137, 320–337. [Google Scholar] [CrossRef] - Bracconi, M.; Ambrosetti, M.; Maestri, M.; Groppi, G.; Tronconi, E. A fundamental analysis of the influence of the geometrical properties on the effective thermal conductivity of open-cell foams. Chem. Eng. Process. Process. Intensif.
**2018**, 129, 181–189. [Google Scholar] [CrossRef] - Diani, A.; Bodla, K.K.; Rossetto, L.; Garimella, S.V. Numerical investigation of pressure drop and heat transfer through reconstructed metal foams and comparison against experiments. Int. J. Heat Mass Transf.
**2015**, 88, 508–515. [Google Scholar] [CrossRef] [Green Version] - Zafari, M.; Panjepour, M.; Emami, M.D.; Meratian, M. Microtomography-based numerical simulation of fluid flow and heat transfer in open cell metal foams. Appl. Therm. Eng.
**2015**, 80, 347–354. [Google Scholar] [CrossRef] - Bianchi, E.; Schwieger, W.; Freund, H. Assessment of Periodic Open Cellular Structures for Enhanced Heat Conduction in Catalytic Fixed-Bed Reactors. Adv. Eng. Mater.
**2015**, 18, 608–614. [Google Scholar] [CrossRef] - Iasiello, M.; Cunsolo, S.; Bianco, N.; Chiu, W.; Naso, V. Developing thermal flow in open-cell foams. Int. J. Therm. Sci.
**2017**, 111, 129–137. [Google Scholar] [CrossRef] - Nie, Z.; Lin, Y.; Tong, Q. Numerical investigation of pressure drop and heat transfer through open cell foams with 3D Laguerre-Voronoi model. Int. J. Heat Mass Transf.
**2017**, 113, 819–839. [Google Scholar] [CrossRef] - Mendes, M.; Skibina, V.; Talukdar, P.; Wulf, R.; Gross, U.; Trimis, D.; Ray, S. Experimental validation of simplified conduction–radiation models for evaluation of Effective Thermal Conductivity of open-cell metal foams at high temperatures. Int. J. Heat Mass Transf.
**2014**, 78, 112–120. [Google Scholar] [CrossRef] - Bianchi, E.; Heidig, T.; Visconti, C.G.; Groppi, G.; Freund, H.; Tronconi, E. An appraisal of the heat transfer properties of metallic open-cell foams for strongly exo-/endo-thermic catalytic processes in tubular reactors. Chem. Eng. J.
**2012**, 198, 512–528. [Google Scholar] [CrossRef] - Fischedick, T.; Kind, M.; Dietrich, B. High temperature two-phase thermal conductivity of ceramic sponges with stagnant fluid—Experimental results and correlation including thermal radiation. Int. J. Therm. Sci.
**2015**, 96, 1–11. [Google Scholar] [CrossRef] - Dietrich, B.; Fischedick, T.; Heissler, S.; Weidler, P.; Wöll, C.; Kind, M. Optical parameters for characterization of thermal radiation in ceramic sponges—Experimental results and correlation. Int. J. Heat Mass Transf.
**2014**, 79, 655–665. [Google Scholar] [CrossRef] - Zhao, C.; Tassou, S.A.; Lu, T. Analytical considerations of thermal radiation in cellular metal foams with open cells. Int. J. Heat Mass Transf.
**2008**, 51, 929–940. [Google Scholar] [CrossRef] - Patel, V.M.; Talukdar, P. Evaluation of radiative properties of a representative foam structure using blocked-off region approach integrated with finite volume method. Int. J. Therm. Sci.
**2016**, 108, 89–99. [Google Scholar] [CrossRef] - Fischedick, T.; Kind, M.; Dietrich, B. Radial two-phase thermal conductivity of ceramic sponges up to high temperatures—Experimental results and correlation. Int. J. Therm. Sci.
**2017**, 114, 98–113. [Google Scholar] [CrossRef] - Dietrich, B.; Schell, K.G.; Bucharsky, E.C.; Oberacker, R.; Hoffmann, M.; Schabel, W.; Kind, M.; Martin, H. Determination of the thermal properties of ceramic sponges. Int. J. Heat Mass Transf.
**2010**, 53, 198–205. [Google Scholar] [CrossRef] - Sinn, C.; Pesch, G.R.; Thöming, J.; Kiewidt, L. Coupled conjugate heat transfer and heat production in open-cell ceramic foams investigated using CFD. Int. J. Heat Mass Transf.
**2019**, 139, 600–612. [Google Scholar] [CrossRef] - Wu, Z.; Caliot, C.; Flamant, G.; Wang, Z. Numerical simulation of convective heat transfer between air flow and ceramic foams to optimise volumetric solar air receiver performances. Int. J. Heat Mass Transf.
**2011**, 54, 1527–1537. [Google Scholar] [CrossRef] - Wehinger, G.D.; Heitmann, H.; Kraume, M. An artificial structure modeler for 3D CFD simulations of catalytic foams. Chem. Eng. J.
**2016**, 284, 543–556. [Google Scholar] [CrossRef] - Wehinger, G.D.; Kraume, M.; Berg, V.; Korup, O.; Mette, K.; Schlögl, R.; Behrens, M.; Horn, R. Investigating dry reforming of methane with spatial reactor profiles and particle-resolved CFD simulations. AIChE J.
**2016**, 62, 4436–4452. [Google Scholar] [CrossRef] - Dong, Y.; Korup, O.; Gerdts, J.; Cuenya, B.R.; Horn, R. Microtomography-based CFD modeling of a fixed-bed reactor with an open-cell foam monolith and experimental verification by reactor profile measurements. Chem. Eng. J.
**2018**, 353, 176–188. [Google Scholar] [CrossRef] - Behnam, M.; Dixon, A.G.; Wright, P.M.; Nijemeisland, M.; Stitt, E.H. Comparison of CFD simulations to experiment under methane steam reforming reacting conditions. Chem. Eng. J.
**2012**, 207, 690–700. [Google Scholar] [CrossRef] - Hettel, M.; Daymo, E.; Deutschmann, O. 3D modeling of a CPOX-reformer including detailed chemistry and radiation effects with DUO. Comput. Chem. Eng.
**2018**, 109, 166–178. [Google Scholar] [CrossRef] - Wehinger, G.D.; Flaischlen, S. Computational Fluid Dynamics Modeling of Radiation in a Steam Methane Reforming Fixed-Bed Reactor. Ind. Eng. Chem. Res.
**2019**, 58, 14410–14423. [Google Scholar] [CrossRef] - Wehinger, G.D. Radiation Matters in Fixed-Bed CFD Simulations. Chem. Ing. Tech.
**2019**, 91, 583–591. [Google Scholar] [CrossRef] - Dixon, A.G.; Partopour, B. Computational Fluid Dynamics for Fixed Bed Reactor Design. Annu. Rev. Chem. Biomol. Eng.
**2020**, 11, 109–130. [Google Scholar] [CrossRef] [Green Version] - Della Torre, A.; Lucci, F.; Montenegro, G.; Onorati, A.; Eggenschwiler, P.D.; Tronconi, E.; Groppi, G. CFD modeling of catalytic reactions in open-cell foam substrates. Comput. Chem. Eng.
**2016**, 92, 55–63. [Google Scholar] [CrossRef] - Güttel, R.; Turek, T. Improvement of Fischer-Tropsch Synthesis through Structuring on Different Scales. Energy Technol.
**2015**, 4, 44–54. [Google Scholar] [CrossRef] - Jang, W.-J.; Shim, J.-O.; Kim, H.-M.; Yoo, S.-Y.; Roh, H.-S. A review on dry reforming of methane in aspect of catalytic properties. Catal. Today
**2019**, 324, 15–26. [Google Scholar] [CrossRef] - Meloni, E.; Martino, M.; Palma, V. A Short Review on Ni Based Catalysts and Related Engineering Issues for Methane Steam Reforming. Catalysts
**2020**, 10, 352. [Google Scholar] [CrossRef] [Green Version] - Bianchi, E.; Groppi, G.; Schwieger, W.; Tronconi, E.; Freund, H. Numerical simulation of heat transfer in the near-wall region of tubular reactors packed with metal open-cell foams. Chem. Eng. J.
**2015**, 264, 268–279. [Google Scholar] [CrossRef] - Modest, M.F. Radiative Heat Transfer, 3rd ed.; Academic Press: New York, NY, USA, 2013; Available online: https://doi.org/10.1016/C2010-0-65874-3 (accessed on 6 April 2020).
- Weller, H.G.; Tabor, G.; Jasak, H.; Fureby, C. A tensorial approach to computational continuum mechanics using object-oriented techniques. Comput. Phys.
**1998**, 12, 620. [Google Scholar] [CrossRef] - STAR-CCM+ (n.d.). Available online: https://www.plm.automation.siemens.com/global/en/products/simcenter/STAR-CCM.html (accessed on 6 April 2020).
- Gräf, I.; Ladenburger, G.; Kraushaar-Czarnetzki, B. Heat transport in catalytic sponge packings in the presence of an exothermal reaction: Characterization by 2D modeling of experiments. Chem. Eng. J.
**2016**, 287, 425–435. [Google Scholar] [CrossRef] - Habibi, A.; Merci, B.; Heynderickx, G. Impact of radiation models in CFD simulations of steam cracking furnaces. Comput. Chem. Eng.
**2007**, 31, 1389–1406. [Google Scholar] [CrossRef] - Versteeg, H.K.; Malasekera, W. An Introduction to Computational Fluid Dynamics: The Finite Volume Method, 2nd ed.; Pearson Education Ltd.: Harlow, UK, 2007. [Google Scholar]
- Sazhin, S.; Sazhina, E.; Faltsi-Saravelou, O.; Wild, P. The P-1 model for thermal radiation transfer: Advantages and limitations. Fuel
**1996**, 75, 289–294. [Google Scholar] [CrossRef]

**Figure 1.**Grid independence study for the OpenFOAM mesh. The mesh with approx. 4 million cells was found to be sufficient for this study. Conditions: T

_{w}= 600 K; v = 0.5 m s

^{−1}; S = 50 W; ε = 0.9; λ

_{s}= 5 W m

^{−1}K

^{−1}.

**Figure 2.**Verification of OpenFOAM results w/and w/o radiation against commercial software STAR-CCM+. Conditions: λ

_{s}= 5 W m

^{−1}K

^{−1}; v = 0.5 m s

^{−1}; ε = 0.9. (

**a**) Heat flows at T

_{W}= 900 K; (

**b**) heat flows at T

_{W}= 1200 K; (

**c**) temperatures at T

_{W}= 900 K; (

**d**) temperatures at T

_{W}= 1200 K.

**Figure 3.**Depiction of temperature fields and heat flows with and without radiation for the verification case simulated in OpenFOAM. The applied heat source S causes a conductive heat flow to the wall (Q

_{SW}) and a convective heat flow to the fluid (Q

_{SF}). Conditions: T

_{w}= 900; λ

_{s}= 5 W m

^{−1}K

^{−1}; v = 0.5 m s

^{−1}; ε = 0.9.

**Figure 4.**Histogram of temperature increases for the verification case. Conditions: T

_{w}= 900; λ

_{s}= 5 W m

^{−1}K

^{−1}; v = 0.5 m s

^{−1}; ε = 0.9. (

**a**) No radiation considered; (

**b**) radiation considered.

**Figure 5.**Depiction of temperature fields w/and w/o radiation. Conditions: T

_{w}= 900; λ

_{s}= 5 W m

^{−1}K

^{−1}; v = 0.5 m s

^{−1}; ε = 0.9.

**Figure 6.**Influence of thermal conductivity on absolute deviations in heat flows for different temperature levels. Conditions: v = 0.5 m s

^{−1}; ε = 0.9.

**Figure 7.**Influence of thermal conductivity on absolute deviations in solid temperatures for different temperature levels. Conditions: v = 0.5 m s

^{−1}; ε = 0.9. (

**a**) Maximum temperature; (

**b**) mean temperature.

**Figure 8.**Influence of the superficial velocity on transferred heat flows and maximum and mean solid temperatures. (

**a**) Absolute deviations between models w/and w/o considered radiation. (

**b**) Specific heat flow solid to wall. The transition line indicates the dominant heat removal mechanism (value > 0.5 = conduction; value < 0.5 = convection). Conditions: T

_{w}= 900 W; ε = 0.9; λ

_{s}= 5 W m

^{−1}K

^{−1}.

**Figure 9.**Comparison between heat flow ratios of CFD simulations and homogeneous model [26] for different solid thermal conductivities and temperature levels. Conditions: v = 0.5 m s

^{−1}; ε = 0.9. (

**a**) T

_{w}= 500; (

**b**) T

_{w}= 700; (

**c**) T

_{w}= 900; (

**d**) T

_{w}= 1200.

**Figure 10.**Influence of surface emissivity on transferred heat flows and maximum as well as mean solid temperatures. (

**a**) Transferred heat flow. (

**b**) Solid temperatures. Conditions: v = 0.5 m s

^{−1}; λ

_{s}= 5 W m

^{−1}K

^{−1}.

**Figure 11.**Geometrical model and main boundary conditions investigated in this study. The 10 ppi foam is similar to the one from [28].

Property | Assumption | |
---|---|---|

Fluid dynamic viscosity | µ | Sutherland equation |

Fluid heat capacity | c_{p,f} | Janaf model (OpenFOAM); |

polynomial (STAR-CCM+) | ||

Fluid thermal conductivity | λ_{f} | Eucken approximation (OpenFOAM); |

polynomial (STAR-CCM+) | ||

Fluid density | δ_{f} | ideal gas law |

Superficial velocity | v | const. (0.1–0.5 m s^{−1}) |

Pore Reynolds number | $R{e}_{p}=\frac{v\cdot {d}_{\mathrm{s}}\cdot \rho}{\mu}$ | const. (1–20) |

Fluid absorption coefficient | κ | const. (10^{−9}) |

Solid heat capacity | c_{p,s} | const. (1000 J kg^{−1} K^{−1}) |

Solid thermal conductivity | λ_{s} | const. (1–200 W m^{−1} K^{−1} [46]) |

Solid density | δ_{s} | const. (3950 kg m^{−3}) |

Solid heat source | S | const. (total: 50 W; |

specific: 1.9 × 10^{7} W m^{−3}) | ||

Solid surface emissivity | ε | const. (0.1–1) |

Wall surface emissivity | ε_{w} | const. (0.65) |

Gravitational acceleration | - | neglected |

Radiation | - | fvDOM model (OpenFOAM); |

DOM model (STAR-CCM+) | ||

Turbulence | - | neglected |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sinn, C.; Kranz, F.; Wentrup, J.; Thöming, J.; Wehinger, G.D.; Pesch, G.R.
CFD Simulations of Radiative Heat Transport in Open-Cell Foam Catalytic Reactors. *Catalysts* **2020**, *10*, 716.
https://doi.org/10.3390/catal10060716

**AMA Style**

Sinn C, Kranz F, Wentrup J, Thöming J, Wehinger GD, Pesch GR.
CFD Simulations of Radiative Heat Transport in Open-Cell Foam Catalytic Reactors. *Catalysts*. 2020; 10(6):716.
https://doi.org/10.3390/catal10060716

**Chicago/Turabian Style**

Sinn, Christoph, Felix Kranz, Jonas Wentrup, Jorg Thöming, Gregor D. Wehinger, and Georg R. Pesch.
2020. "CFD Simulations of Radiative Heat Transport in Open-Cell Foam Catalytic Reactors" *Catalysts* 10, no. 6: 716.
https://doi.org/10.3390/catal10060716