Inﬂuence of Pressure, Velocity and Fluid Material on Heat Transport in Structured Open-Cell Foam Reactors Investigated Using CFD Simulations

: Structured open-cell foam reactors are promising for managing highly exothermic reactions such as CO 2 methanation due to their excellent heat transport properties. Especially at low ﬂow rates and under dynamic operation, foam-based reactors can be advantageous over classic ﬁxed-bed reactors. To e ﬃ ciently design the catalyst carriers, a thorough understanding of heat transport mechanisms is needed. So far, studies on heat transport in foams have mostly focused on the solid phase and used air at atmospheric pressure as ﬂuid phase. With the aid of pore-scale 3d CFD simulations, we analyze the e ﬀ ect of the ﬂuid properties on heat transport under conditions close to the CO 2 methanation reaction for two di ﬀ erent foam structures. The exothermicity is mimicked via volumetric uniformly distributed heat sources. We found for foams that are designed to be used as catalyst carriers that the working pressure range and the superﬁcial velocity inﬂuence the dominant heat removal mechanism signiﬁcantly. In contrast, the inﬂuence of ﬂuid type and gravity on heat removal is small in the range relevant for heterogeneous catalysis. The ﬁndings might help to facilitate the design-process of open-cell foam reactors and to better understand heat transport mechanisms in foams.


Introduction
Open-cell foams are promising monolithic catalyst support structures for exo-and endothermal heterogeneous gas-phase reactions such as the CO 2 methanation, due favorable characteristics such as good heat transport, low pressure drop and high porosities [1][2][3]. The CO 2 methanation currently attracts much attention in the chemical engineering community as it can be a cornerstone the future transition to a sustainable energy supply within the Power-to-X (PtX) concept [4]. Here, excess energy from wind turbines or solar panels is converted via electrolysis into hydrogen, and usually reacted catalytically with carbon dioxide to methane. As the power grids might not be resilient enough to withstand the high energy load during wind or sun peak hours, dynamically operated small-scale plants for storage of excess energy are needed in the future [5]. Small-scale plants are operated at low flow rates making heat removal traditionally difficult. The dynamic operation can cause tremendous jumps of pressure, velocity and even change the fluid composition resulting in undesired or uncontrollable temperature increases. Thus, the heat transport (and heat removal) inside these reactors needs to be addressed to protect the catalyst's activity and ensure safe as well as robust operation [6]. Generally,

Materials and Methods
The CFD model investigated in this study consists of a foam embedded in a tube (i.e., a tubular structured reactor) and is similar to our previous study [19]. The problem is the steady-state conjugate heat transfer between the flowing fluid and the foam. The simulations were carried out with the commercial CFD software STAR-CCM+ from Siemens PLM (Plano, TX, USA) [33]. Heat production through exothermal reactions is represented by volumetric homogeneously distributed heat sources in the solid (S = 12.5 W in all cases) [19]. The heat source intensity of 12.5 W was chosen as it corresponds to the heat released during the CO 2 methanation for a foam in this order of magnitude (50 W for a full sponge equals 12.5 W for one quarter of the sponge) [19].
Two foams (radius 7.5 mm; length 20 mm) are investigated in this study, modeled through Kelvin cell lattices (KC1 and KC2) that differ strongly in their strut diameter (see Figure 1 and Table 1). Only a quarter of the foams as well as the pipe are simulated due to the regularity of the structure and to further reduce computational cost. Polyhedral meshes were created using the integrated STAR-CCM+ meshing utilities (see Figure A1 in Appendix A). The utilized meshes were tested and results were found to be grid independent (see Figure A2 in Appendix B).

Materials and Methods
The CFD model investigated in this study consists of a foam embedded in a tube (i.e., a tubular structured reactor) and is similar to our previous study [19]. The problem is the steady-state conjugate heat transfer between the flowing fluid and the foam. The simulations were carried out with the commercial CFD software STAR-CCM+ from Siemens PLM (Plano, TX, USA) [33]. Heat production through exothermal reactions is represented by volumetric homogeneously distributed heat sources in the solid (S = 12.5 W in all cases) [19]. The heat source intensity of 12.5 W was chosen as it corresponds to the heat released during the CO2 methanation for a foam in this order of magnitude (50 W for a full sponge equals 12.5 W for one quarter of the sponge) [19].
Two foams (radius 7.5 mm; length 20 mm) are investigated in this study, modeled through Kelvin cell lattices (KC1 and KC2) that differ strongly in their strut diameter (see Figure 1 and Table  1). Only a quarter of the foams as well as the pipe are simulated due to the regularity of the structure and to further reduce computational cost. Polyhedral meshes were created using the integrated STAR-CCM+ meshing utilities (see Figure A1 in Appendix). The utilized meshes were tested and results were found to be grid independent (see Figure A2 in Appendix).    The basic equations that are solved for a Newtonian fluid domain are the conservation of mass, momentum and energy: with ρ f denoting the fluid's density, U the velocity field, h the enthalpy, µ the fluid's dynamic viscosity, T f the fluid's temperature and λ f the fluid's thermal conductivity. The solid phase is solely described by the conservation of energy with λ s donating the solid thermal conductivity, T s the solid temperature and S the specific artificial heat source. Furthermore, the fluid density is expressed via the ideal gas law. The solid properties are kept constant, whereas the fluid properties such as dynamic viscosity, specific heat and thermal conductivity depend on the temperature (see Table 2). The model properties and assumptions are also listed in Table 2 and the basic boundary conditions can be found in Figure 1. In this study, fluid and wall temperatures are kept constant as a fixed temperature boundary at 500 K. This way, the only energy that enters the system is due to the volumetric heat source. Thermal radiation is not explicitly considered in the simulation as the temperatures are moderate and this study wants to investigate heat transport of fluid properties independently. Moreover, the influence of radiation has already been quantified in a comparable setup [15]. Furthermore, turbulence effects were taken into account through a realizable k-ε RANS turbulence model with All y + wall-treatment [34].  Realizable k-ε RANS (All y + wall-treatment) neglected [15] For studying the influence of gravitational acceleration in different directions, gravity is considered in y direction (in flow direction, representing a vertical reactor) as well as in z direction (perpendicular to flow, representing a horizontal reactor). Additionally, the varied and investigated properties are highlighted in orange in Figure 1.
In our previous studies, similar models were validated against pressure drop correlations, heat transfer correlations and verified against CFD data [15,19]. Here, the principial changed model property is the CAD-created geometry. We therefore omit the validation as the models are virtually identical.

Results and Discussion
All simulated cases have a total heat source of S = 12.5 W, which would correspond to a 50 W total heat source in a full foam setup (i.e., four quarters). As already described, the only thermal energy entering the system is due to the volumetric heat source (i.e., heat production in the solid foam). The global energy balance thus reads with Q SF being the transferred heat flow from fluid to solid and Q SW being the conducted heat flow from solid to the wall. Which heat removal mechanism dominates (i.e., convection or conduction), can be expressed through the specific heat flow from solid to wall Q SW S −1 through non-dimensionalization of Equation (4). For values above 0.5, thermal conduction is the dominant mechanism and for values below 0.5 convection is the dominant one. We note that due to the normalization by the heat source intensity, the actual implemented heat source value becomes less crucial. In our previous study it was shown that in the heat source intensity range between 5 W and 150 W temperature increases (in solid and fluid phases) and heat flows show the almost identical values. The impact of the several investigated fluid properties on the heat flows is shown in Figure 2. Panel a shows the specific heat flow for a superficial velocity of the two investigated foams up to 4 m s −1 . Previous studies only investigated heat flows with coupled heat production in the solid has only up to 0.51 m s −1 [15,19]. Obviously, both foam structures switch from being dominated by conduction to being dominated by convection at a certain velocity. KC2 switches at a superficial velocity of approximately v = 1.2 m s −1 whereas KC1 switches not until a superficial velocity of approximately v = 3 m s −1 . This is most certainly based on the different strut diameters of the foams (KC1: d s = 0.591 mm compared to KC1: d s = 0.35 mm), which ensures a better radial heat removal for KC1 through thermal conduction. This is line with the findings from the studies of Bracconi et al. [11] and Bianchi et al. [21], that investigated the role of the strut diameter in conductive heat transport.
ChemEngineering 2020, 4, x FOR PEER REVIEW 5 of 11 with QSF being the transferred heat flow from fluid to solid and QSW being the conducted heat flow from solid to the wall. Which heat removal mechanism dominates (i.e., convection or conduction), can be expressed through the specific heat flow from solid to wall QSW S −1 through nondimensionalization of Equation (4). For values above 0.5, thermal conduction is the dominant mechanism and for values below 0.5 convection is the dominant one. We note that due to the normalization by the heat source intensity, the actual implemented heat source value becomes less crucial. In our previous study it was shown that in the heat source intensity range between 5 W and 150 W temperature increases (in solid and fluid phases) and heat flows show the almost identical values.
The impact of the several investigated fluid properties on the heat flows is shown in Figure 2. Panel a shows the specific heat flow for a superficial velocity of the two investigated foams up to 4 m s −1 . Previous studies only investigated heat flows with coupled heat production in the solid has only up to 0.51 m s −1 [15,19]. Obviously, both foam structures switch from being dominated by conduction to being dominated by convection at a certain velocity. KC2 switches at a superficial velocity of approximately v = 1.2 m s −1 whereas KC1 switches not until a superficial velocity of approximately v = 3 m s −1 . This is most certainly based on the different strut diameters of the foams (KC1: ds = 0.591 mm compared to KC1: ds = 0.35 mm), which ensures a better radial heat removal for KC1 through thermal conduction. This is line with the findings from the studies of Bracconi et al. [11] and Bianchi et al. [21], that investigated the role of the strut diameter in conductive heat transport.    Figure 2a shows results using air as fluid as usually done in literature. Panel b shows the impact of the gas type, which is related to the CO 2 methanation, on heat removal. For both KC's, the difference in computed heat flows does not change distinctively, although a small difference between air and hydrogen on the one side and methane and carbon dioxide on the other side becomes apparent. The small differences between the gas types can be explained through the interplay of gas viscosity, density, heat capacity and thermal conductivity which change over temperature for each gas individually. The order of thermal conductivities for the gases at 1 bar from highest to lowest is air, carbon dioxide, methane and hydrogen which is not the same order as in Panel b.
The influence of the pressure on the simulated heat flows at v = 0.5 m s −1 is shown in Figure 2c. Again, most of the literature investigated heat transport phenomena in foams only at 1 bar absolute pressure [14,35]. With increasing pressure, the heat transport switches from conduction dominated to convection dominated for both foams. At a pressure of 6 bar, both foams dominantly remove heat over convection and hence lose their advantage over standard pellets. The consideration of the actual working pressure thus influences the performance of the catalyst carriers tremendously. Structures that are designed to work efficiently, i.e., conduction dominated, at 1 bar, might already lose their efficiency at 2.5 bar (compare KC2 in panel c). The general effect is obvious, as both gas density and thus the ability of the fluid to cool down the foam via convection increase with pressure.
Often, CFD models omit gravitational acceleration [14,18,19]. Depending on the reactor setup (vertical pipe or horizontal pipe), gravitational acceleration affects the fluid in flow direction (vertical setup) or perpendicular to the flow direction (horizontal setup). Figure 2d shows the specific heat flow as a function of superficial velocity for both foams with gravity in flow direction, perpendicular to the flow as well as neglected gravity. In the investigated superficial velocity range (0.1-2 m s −1 ), no significant influence of the consideration of gravity could be found. The reason probably lies in forced convection (i.e., pronounced velocity) being a lot more substantial for the overall convection than natural convection (i.e., effect of gravity). To our knowledge, correlations for the Grashof and Rayleigh numbers for this temperature distribution together with the velocity and pressure do not exist. Thus, this result could not have been anticipated by observing dimensionless numbers alone.
The analyzed heat flows for elevated velocities, changing fluid type, influence of pressure and gravity show the influence of the fluid properties on the overall heat transport in open-cell foam carriers. Additionally, Figure 3 shows the corresponding solid temperature rises of the cases shown in Figure 2. For brevity, we only show maximum temperature rises of the solid (instead of additional solid mean and fluid max/mean temperature rises) as they represent the entire solid temperature distribution in foams satisfactory [19]. In Figure 3, it can be seen that the curves of temperature rises of the two structures converge in the convection dominant regime. This behavior happens regardless of the applied pressure or superficial velocity (see panels a and c). Hence, at certain velocities (>3 m s −1 ) and pressure (>7 bar), the temperature increase seems to be become independent of the foam geometry.
ChemEngineering 2020, 4, x FOR PEER REVIEW 6 of 11 difference in computed heat flows does not change distinctively, although a small difference between air and hydrogen on the one side and methane and carbon dioxide on the other side becomes apparent. The small differences between the gas types can be explained through the interplay of gas viscosity, density, heat capacity and thermal conductivity which change over temperature for each gas individually. The order of thermal conductivities for the gases at 1 bar from highest to lowest is air, carbon dioxide, methane and hydrogen which is not the same order as in Panel b.
The influence of the pressure on the simulated heat flows at v = 0.5 m s −1 is shown in Figure 2c. Again, most of the literature investigated heat transport phenomena in foams only at 1 bar absolute pressure [14,35]. With increasing pressure, the heat transport switches from conduction dominated to convection dominated for both foams. At a pressure of 6 bar, both foams dominantly remove heat over convection and hence lose their advantage over standard pellets. The consideration of the actual working pressure thus influences the performance of the catalyst carriers tremendously. Structures that are designed to work efficiently, i.e., conduction dominated, at 1 bar, might already lose their efficiency at 2.5 bar (compare KC2 in panel c). The general effect is obvious, as both gas density and thus the ability of the fluid to cool down the foam via convection increase with pressure.
Often, CFD models omit gravitational acceleration [14,18,19]. Depending on the reactor setup (vertical pipe or horizontal pipe), gravitational acceleration affects the fluid in flow direction (vertical setup) or perpendicular to the flow direction (horizontal setup). Figure 2d shows the specific heat flow as a function of superficial velocity for both foams with gravity in flow direction, perpendicular to the flow as well as neglected gravity. In the investigated superficial velocity range (0.1-2 m s −1 ), no significant influence of the consideration of gravity could be found. The reason probably lies in forced convection (i.e., pronounced velocity) being a lot more substantial for the overall convection than natural convection (i.e., effect of gravity). To our knowledge, correlations for the Grashof and Rayleigh numbers for this temperature distribution together with the velocity and pressure do not exist. Thus, this result could not have been anticipated by observing dimensionless numbers alone.
The analyzed heat flows for elevated velocities, changing fluid type, influence of pressure and gravity show the influence of the fluid properties on the overall heat transport in open-cell foam carriers. Additionally, Figure 3 shows the corresponding solid temperature rises of the cases shown in Figure 2. For brevity, we only show maximum temperature rises of the solid (instead of additional solid mean and fluid max/mean temperature rises) as they represent the entire solid temperature distribution in foams satisfactory [19]. In Figure 3, it can be seen that the curves of temperature rises of the two structures converge in the convection dominant regime. This behavior happens regardless of the applied pressure or superficial velocity (see panels a and c). Hence, at certain velocities (>3 m s −1 ) and pressure (>7 bar), the temperature increase seems to be become independent of the foam geometry.  For superficial velocities lower or equal than 1.5 m s −1 , the influence of the fluid type on the temperature rise varies for the two structures (see Figure 3b). The effect of the fluid type on the temperature rise increases with convective heat transport. Therefore, KC2 has a more pronounced difference (starting from 0.3 m s −1 ) than KC1 (difference obvious starting from 1.5 m s −1 ). This behavior could be expected as the fluid type influences the convective heat transport and becomes more important when the overall contribution of convection increases. To conclude, the fluid type seems to be less influential in the dominant conduction area where a structured foam reactor should be operated. Equivalently to the heat flows, no significant effect of gravity on the temperature rise could be found (Figure 3d).

Conclusions
This study analyzed the often-neglected influence of fluid properties on heat transport in opencell foam reactors. As pointed out, usually heat transport in open-cell foams is investigated using air at 1 bar pressure absolute.
When foams are designed as catalyst carriers (for, e.g., dynamic CO2 methanation), they should at best remove all heat radially via conduction (here: for working pressure between 4 and 10 bar, the velocity should be lower 0.5 m s −1 ). At elevated velocities even structures with relatively thick struts can shift into the convection dominated regime. Not only for elevated velocities, but especially for pressure higher than 1 bar (usual operation conditions of CO2 methanation 4-10 bar) the shift towards convection being dominant might proceed rapidly. In contrast, the choice of fluid mixture seems not as significant as pressure and velocity ranges. Obviously, this has to be checked for other gases of other reactions or inert gases. Nevertheless, air seems a reasonable fluid to study the general heat transport behavior of new foam designs, as the deviations of thermal fields to other gases are not that severe. This way, fluid mixture equations can be omitted and computational effort can be reduced. Additionally, the reactor orientation (represented through gravity) does not seem to have an impact on the heat transport in foams which is relevant for heterogeneous catalysis.
In conclusion, for the design of structured reactors for highly exothermic gas phase reactions, it is crucial to consider the exact velocity and pressure ranges of the investigated reaction. This is especially important, when the catalyst carriers are designed for dynamic or changing load operation. This type of operation might cause tremendous jumps in superficial velocity, pressure or fluid composition which further might influence the overall heat transport and might cause a thermal runaway or serious harm to the catalyst.  For superficial velocities lower or equal than 1.5 m s −1 , the influence of the fluid type on the temperature rise varies for the two structures (see Figure 3b). The effect of the fluid type on the temperature rise increases with convective heat transport. Therefore, KC2 has a more pronounced difference (starting from 0.3 m s −1 ) than KC1 (difference obvious starting from 1.5 m s −1 ). This behavior could be expected as the fluid type influences the convective heat transport and becomes more important when the overall contribution of convection increases. To conclude, the fluid type seems to be less influential in the dominant conduction area where a structured foam reactor should be operated. Equivalently to the heat flows, no significant effect of gravity on the temperature rise could be found (Figure 3d).

Conclusions
This study analyzed the often-neglected influence of fluid properties on heat transport in open-cell foam reactors. As pointed out, usually heat transport in open-cell foams is investigated using air at 1 bar pressure absolute.
When foams are designed as catalyst carriers (for, e.g., dynamic CO 2 methanation), they should at best remove all heat radially via conduction (here: for working pressure between 4 and 10 bar, the velocity should be lower 0.5 m s −1 ). At elevated velocities even structures with relatively thick struts can shift into the convection dominated regime. Not only for elevated velocities, but especially for pressure higher than 1 bar (usual operation conditions of CO 2 methanation 4-10 bar) the shift towards convection being dominant might proceed rapidly. In contrast, the choice of fluid mixture seems not as significant as pressure and velocity ranges. Obviously, this has to be checked for other gases of other reactions or inert gases. Nevertheless, air seems a reasonable fluid to study the general heat transport behavior of new foam designs, as the deviations of thermal fields to other gases are not that severe. This way, fluid mixture equations can be omitted and computational effort can be reduced. Additionally, the reactor orientation (represented through gravity) does not seem to have an impact on the heat transport in foams which is relevant for heterogeneous catalysis.
In conclusion, for the design of structured reactors for highly exothermic gas phase reactions, it is crucial to consider the exact velocity and pressure ranges of the investigated reaction. This is especially important, when the catalyst carriers are designed for dynamic or changing load operation. This type of operation might cause tremendous jumps in superficial velocity, pressure or fluid composition which further might influence the overall heat transport and might cause a thermal runaway or serious harm to the catalyst.