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

^{1}

^{2}

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## Abstract

**:**

_{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 efficiently 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 effect of the fluid properties on heat transport under conditions close to the CO

_{2}methanation reaction for two different 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.

## 1. Introduction

_{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, for the methanation reaction, low temperatures are thermodynamically favorable [7]. Therefore, an effective heat removal ensures efficient conversion and protects the catalyst particles from sintering.

_{2}methanation [8]. The reason is that heat is dominantly removed via radial conduction in foam reactors, whereas pellets in fixed-bed reactors remove heat axially via convection [9]. However, only when heat conduction dominates over convection, foams can be advantageous over pellets as pellets generally have a higher axial convective heat transport [8].

^{−1}), solid thermal conductivities (1–200 W m

^{−1}K

^{−1}), and thermal radiation (temperature levels 500–1200 K, solid emissivities 0–1) [15,19]. It was found that the solid thermal conductivity and the superficial velocity are key parameters for designing efficient heat-removing foams for the usage as catalyst carriers [19]. Moreover, for reactor tube diameters below 25 mm and superficial velocities lower than 0.5 m s

^{−1}, the investigated structure (irregular foam) showed conduction being the dominant heat removal mechanism (this parameter range is in relevant order of magnitude for decentralized small scale methanation plants [29,30]). Furthermore, thermal radiation can contribute significantly to the heat transport at conditions relevant for methanation [15].

_{2}as fluids [18]. They found significant changes in computed fluid temperatures and justified the deviation through the six times higher thermal conductivity of He (they also assumed constant fluid properties). Additionally, Bianchi et al. [31] found a significant difference for computed heat transfer of either air or water (liquid).

_{2}+ 4H

_{2}⇄ CH

_{4}+ 2H

_{2}O), however, involves gas mixtures (CO

_{2}, H

_{2}and CH

_{4}), higher velocity ranges (i.e., ≥0.5 m s

^{−1}) and elevated pressure ranges (4–10 bar) [6]. Furthermore, the effect of gravitational acceleration has not been addressed so far in CFD studies of heat transport in open-cell foams. This might be relevant, because reactors can be operated vertically or horizontally leading to a different effective direction of gravity (i.e., natural convection).

_{2}methanation reaction.

## 2. Materials and Methods

_{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].

_{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

_{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.

^{+}wall-treatment [34].

## 3. Results and Discussion

_{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.

^{−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.

_{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.

^{−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.

^{−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.

^{−1}) and pressure (>7 bar), the temperature increase seems to be become independent of the foam geometry.

^{−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).

## 4. Conclusions

_{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.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Latin | |

c_{p} | isobaric heat capacity, J kg^{−1} K^{−1} |

d_{c} | cell diameter, m |

d_{s} | strut diameter, m |

g | gravitational acceleration, 9.81 m s^{−2} |

Q | heat flow, W |

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

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

h | specific enthalpy, J kg^{−1} |

p | pressure, Pa |

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 |

U | velocity, m s^{−1} |

v | superficial velocity, m s^{−1} |

Greek | |

ε_{O} | open porosity, - |

μ | dynamic viscosity, Pa s |

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

ρ | density, kg m^{−3} |

## Appendix A. Depiction of Volume Meshes

## Appendix B. Grid Independence Study

**Figure A2.**Grid independence study. (

**a**) Solid temperature rise from initial 500 K plotted against number of cells; (

**b**) specific heat flow from solid to wall plotted against number of cells. For both used geometries, the second-largest meshes (KC1: 3.1 mio. Cells; KC2 4.1 mio cells) were found to give reasonable results. Conditions: air; p = 1 bar, v = 4 m s

^{−1}.

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**Figure 1.**Illustration of model with boundary conditions. The orange highlighted fluid properties are investigated in this study.

**Figure 2.**Heat flows for fluid property variation (

**a**) Increase in inlet velocity incl. turbulence modelling; (

**b**) Influence of fluid type; (

**c**) Influence of pressure; (

**d**) Influence of gravitational acceleration. Conditions: S = 12.5 W; λ

_{s}= 5 W m

^{−1}K

^{−1}.

**Figure 3.**Solid temperature rises per applied heat source intensity for fluid property variation (

**a**) Increase in inlet velocity incl. turbulence modelling; (

**b**) Influence of fluid type; (

**c**) Influence of pressure; (

**d**) Influence of gravitational acceleration. Conditions: S = 12.5 W; λ

_{s}= 5 W m

^{−1}K

^{−1}.

Parameter, Symbol. | Kelvin Cell 1 (KC1) | Kelvin Cell 2 (KC2) |
---|---|---|

open porosity, ε_{O} | 0.724 | 0.845 |

specific surface area, S_{V} | 1467.8 m^{−1} | 1518.9 m^{−1} |

cell diameter, d_{c} | 1.924 mm | 1.733 mm |

strut diameter, d_{s} | 0.591 mm | 0.35 mm |

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

Fluid dynamic viscosity | µ | Sutherland equation |

Fluid heat capacity | c_{p,f} | polynomial |

Fluid thermal conductivity | λ_{f} | Sutherland equation |

Fluid density Superficial velocity | δ_{f}v | ideal gas law const. (0.1–4 m s ^{−1}) |

Pore Reynolds number (air) Wall/inlet temperature Outlet pressure | $R{e}_{\mathrm{p}}=\frac{v\cdot {d}_{\mathrm{s}}\cdot \rho}{\mu}$ T _{w} = T_{in} p | const. (0.3–76) const. (500 K) const. (1–10 bar) |

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

Solid thermal conductivity | λ_{s} | const. (5 W m^{−1} K^{−1}) |

Solid density Solid heat source | δ_{s}S | const. (3950 kg m^{−3})const. (total: 12.5 W) |

Gravitational acceleration | considered | |

Turbulence Radiation | Realizable k-ε RANS (All y^{+} wall-treatment)neglected [15] |

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**MDPI and ACS Style**

Sinn, C.; Wentrup, J.; Thöming, J.; Pesch, G.R.
Influence of Pressure, Velocity and Fluid Material on Heat Transport in Structured Open-Cell Foam Reactors Investigated Using CFD Simulations. *ChemEngineering* **2020**, *4*, 61.
https://doi.org/10.3390/chemengineering4040061

**AMA Style**

Sinn C, Wentrup J, Thöming J, Pesch GR.
Influence of Pressure, Velocity and Fluid Material on Heat Transport in Structured Open-Cell Foam Reactors Investigated Using CFD Simulations. *ChemEngineering*. 2020; 4(4):61.
https://doi.org/10.3390/chemengineering4040061

**Chicago/Turabian Style**

Sinn, Christoph, Jonas Wentrup, Jorg Thöming, and Georg R. Pesch.
2020. "Influence of Pressure, Velocity and Fluid Material on Heat Transport in Structured Open-Cell Foam Reactors Investigated Using CFD Simulations" *ChemEngineering* 4, no. 4: 61.
https://doi.org/10.3390/chemengineering4040061