A Fundamental Investigation of Gas/Solid Heat and Mass Transfer in Structured Catalysts Based on Periodic Open Cellular Structures (POCS)

In this work, we investigate the gas–solid heat and mass transfer in catalytically activated periodic open cellular structures, which are considered a promising solution for intensification of catalytic processes limited by external transport, aiming at the derivation of suitable correlations. Computational fluid dynamics is employed to investigate the Tetrakaidekahedral and Diamond lattice structures. The influence of the morphological features and flow conditions on the external transport properties is assessed. The strut diameter is an adequate characteristic length for the formulation of heat and mass transfer correlations; accordingly, a power-law dependence of the Sherwood number to the Reynolds number between 0.33 and 0.67 was found according to the flow regimes in the range 1–128 of the Reynolds number. An additional −1.5-order dependence on the porosity is found. The formulated correlations are in good agreement with the simulation results and allow for the accurate evaluation of the external transfer coefficients for POCS.


developed a geometrical model for open-cell foams on the basis of the TKKD
unit cell with struts with variable cross section between the node and the middle of the ligament. The approach can be extended to model the ideal TKKD unit cell with circular struts with constant cross section. The TKKD unit cell porosity and specific surface area can be expressed as follows: = 1 − 6 + 12 (S1) where Vnode and Vstrut indicate the volume of a single node and of a single strut respectively and, likewise, Snode and Sstrut indicate the exposed area of a node and of a strut. It is worth to emphasize that the struts junction in the node must be modelled in detail.
In each node, four struts converge, and the angle between the struts corresponds to the tetrahedral or Plateau angle: The node volume is computed by considering the node as a sphere cut by the four struts. The node diameter can be expressed as a function of the strut diameter and of the tetrahedral angle according to Eq. (S4): The struts remove from the node four spherical caps with height: Figure S1. Representation of the TKKD circular node. The height of the cup, the strut diameter, the node diameter and the angle between the struts are indicated.
Therefore, the volume of the node is given by the volume of a sphere of diameter dn, minus four times the volume of the spherical cap: In complete analogy, the surface of the node is given by the surface of the sphere of diameter dn, minus four times the surface of a spherical cap: The volume of the strut is equal to the volume of a cylinder having diameter ds and length ls. As the struts merge in the node, the contribution of the junction must be subtracted from the volume of the cylinder. This contribution is given by two times the volume of a cylinder with diameter ds and length equal to the thickness of the node, i.e. − h as shown in Figure S1. Hence, the volume of the strut can be written as follows: The surface of the strut can be derived along the same lines: Finally, the volume of the TKKD unit cell is given by the edge length: Eqs. (S1), (S4), (S8) and (S10) can be combined to obtain the following expression: In complete analogy, Eqs. (S2), (S7), (S9), (S10) can be combined to give: The analytical model is validated against the geometry of virtually generated structures. For this purpose, CAD models are generated by merging solid primitives, e.g., cylinders for struts and spheres for nodes, with suitable discretization to obtain a smooth surface and avoid any surface irregularity which may affect the properties evaluation. Hence, virtually generated models provide the most accurate data, comparable to the properties evaluated on real manufactured samples. Figure S2 shows two parity plots where the geometrical properties evaluated on CAD models are compared to the predictions of the TKKD geometrical model expressed by Eqs. (S11), (S12), proving the high accuracy of the derived model in the range of void fractions of interest for engineering applications (i.e. 0.7 < ε < 0.95). The porosities evaluated on the CAD models ( Figure S2 (a)) superimpose the nominal void fraction line with a maximum error attaining to 0.5% for ε = 0.7, whereas the specific surface areas evaluated on the CAD models ( Figure S2

Mesh generation
The definition of a proper meshing procedure is crucial to accurately describe the convective transport properties inside POCS. The computational domains are generated starting from a CAD file by means of snappyHexMesh in-built utility of the OpenFOAM framework. The CAD generation is performed by using OpenSCAD, an open-source script-based software. The snappyHexMesh utility operates starting from a uniform background mesh, refines the region of the computational domain around the CAD surface using the cut-cell approach according to the user settings and lastly snaps the mesh surface on the CAD file. Figure S3 shows a typical computational domain, made of 3 unit S7 cells along the streamwise direction and 1 unit cell along the transversal one. The implementation of periodic boundary conditions requires corresponding inlet and outlet sections of the domains, separated by a distance equal to an integer multiple of the cell size, and corresponding side walls, two by two coupled and separated by a distance equal to an integer multiple of the cell size as well. To implement cyclic boundary conditions, snappyHexMesh requires not only the domain side boundaries to be topologically matching, but also to be symmetrical planes. In this work, the TKKD and the Diamond unit cell POCS have been considered. Whereas each TKKD unit cell side boundary already constitutes a symmetrical plane ( Figure S4(a)), this condition does not hold for the Diamond unit cell. Instead, the symmetry planes can be identified as any of the planes parallel to the diagonals of the unit cell cross section ( Figure S4(b)). The element represented in Figure S4(b), having cross section size equal to √2dc, retains all the geometrical relevant features such as porosity and wetted surface area. Furthermore, the Diamond structure may be generated from its periodic repetition.
Hence, it is considered as the minimum sized domain in the REV assessment performed in Section 4.1.

Periodic boundary conditions for 3D flows
Periodic boundary conditions were first introduced by Patankar et al. 3 to model incompressible flows in ducts having streamwise periodic changing cross sections, simplified as 2D cases. The methodology is hereby extended to model reacting compressible flows in the three-dimensional space.
Periodic boundary conditions are implemented considering a computational domain constituted by an integer number of unit cells with size dc. Letting z be the streamwise direction and x, y the transverse coordinates, the domain can be topologically described by X1(y,z) ≤ x ≤ X2(y,z) and Y1(x,z) ≤ y ≤ Y2(x,z), with X1, X2, Y1 and Y2 representing the coordinates of the domain side boundaries and the solid wall boundaries (see Figure S5).
To ensure a net mass flow along the axial direction, it is necessary that the pressure field decreases along the axial direction. Therefore, the pressure field does not follow the same periodic condition expressed for the velocity field. However, the pressure in cross sections situated at a distance equal to dc necessarily have the same shape and differ from a constant value due to the periodic profile. In this view, the pressure drop across each cell is constant as reported in Eq. (S20). where C is the pressure drop across a unit cell. The periodic fully developed flow profile can either be simulated by imposing a driving force, that is the pressure gradient C given by Eq. (S20), which generates the motion of the fluid at a corresponding Reynolds number a priori unknown, or by imposing a fixed mass flowrate (fixed Reynolds number) and determining the pressure gradient afterwards. In our work, the second approach is adopted.
In the fully developed flow, the description of the chemical species mass concentration profiles would require the introduction of a source term to account for the reactants and products consumption and generation. However, the mass concentration profiles in cross sections situated at a distance equal to integer multiples of dc necessarily assume the same shape and differ by a scaling factor which accounts for the reaction. Introducing the normalized mass concentration: where ωi B (ꝏ) represents the species mass concentration at the thermodynamic equilibrium.
The periodicity condition of fully developed species concentration profile requires identical shapes of the normalized mass concentration profiles at successive locations separated by the distance dc, and thus the periodicity condition may be expressed as: * ( , , ) = * ( , , + ) = * ( , , In the analysis of the convective heat transfer, a prescribed wall temperature condition is imposed to achieve a complete analogy to the convective mass transfer case. The periodicity condition for the temperature profile in the analysis of convective heat transfer is achieved along the same lines discussed for the mass transfer.

S11
The normalized dimensionless temperature is employed as scaling factor: * ( , , ) where Tw represents the prescribed wall temperature, reached at the thermodynamic equilibrium.
In the fully developed profile, the shape of the normalized dimensionless temperature profiles at successive locations separated by the distance dc is identical. The periodicity condition can be

Mesh convergence analysis
In the mesh convergence analysis, the snappyHexMesh parameters have been fixed, whereas several background mesh resolutions have been tested. In this view, the snappyHexMesh parameters have been prescribed with surface level of refinement equal to 4 to achieve a high quality of refinement along the POCS surface to accurately describe the gas-to-solid transport process. can be ascribed to the thinner boundary layer at higher velocities which requires additional cells to be fully resolved. The same extensive analysis has been performed for the Diamond unit cell POCS as well, however the result is reported for the most stringent condition at the highest Reynolds number Re = 128. In this condition, the grid independence is reached for a ratio between the cell size and the S13 mesh cell size dc/δ = 20. It is worth noticing that the TKKD unit cell has thinner struts than the Diamond unit cell at same porosity, and therefore the TKKD geometry requires higher superficial velocity than the Diamond geometry at same Reynolds number. This factor plays a key-relevant role in the higher computational requirement of the TKKD geometry, as a higher resolution is required to accurately describe the flow characteristics and more precisely the boundary layer. Due to the low Reynolds numbers considered, prism layers were not added close to the wall surfaces. Despite this, the mesh near wall refinement allows to reach y + <1 in each flow condition to accurately describe the transport process mainly occurring in the boundary layer. The grid independence analysis has been carried out on the representative elementary volumes (REV) described in Section 4.1.

Assessment of the characteristic length (extended)
As introduced in Section 5.1, multiple choices are possible for the definition of the characteristic length, which in principle may be any geometrical parameter computed according to the equations reported in Section 3.1 - Table 1.
Lämmermann et al. 14 proposed the window diameter as characteristic length for the static liquid holdup in POCS, defined for the TKKD and the Diamond by Eqs (S27) and (S28), respectively: The authors propose the window diameter to account for the effect of the geometrical parameters on the static liquid holdup in the two-phase pressure drop modelling, also aiming at providing a unified approach that can be applied to different POCS unit cells. In this view, the window diameter increases on increasing the cell size at fixed porosity and increasing the porosity at prescribed cell size, following the behavior of the liquid holdup. When using the window diameter as characteristic length, the transport phenomena are modelled considering the fluid as flowing in pores.
Mass transfer simulations results, reinterpreted using the window diameter as characteristic length, are reported in Figure S7. Since the window diameter is linearly dependent on the cell size and the strut diameter, which is linearly dependent on the cell size as well, the Sherwood number is only affected by the porosity, as already discussed in Section 4. The effect of the porosity is thus discussed. Figure S7 (a) shows the Sherwood number against the Reynolds number for the TKKD unit cell. The Sherwood number increases at increasing Reynolds numbers, consistently to what discussed in Section 4. The Sherwood number increases with the porosity, in contrast with the behavior shown in S15 Section 4 considering the strut diameter as characteristic length. Moreover, the Sherwood number is almost constant at Redw > 100, regardless of the porosity, which thus mostly influences the pure laminar regime at low Redw. Figure S7 (b) shows the Sherwood number against the Reynolds number for the Diamond unit cell. As for the TKKD, Shdw increases at increasing Redw and increasing porosity in all the considered flow regimes. Contrarily to the TKKD, the data do not align in the transitional laminar regime, instead, a residual dependency on the porosity is found even at high Redw. Dietrich 5 proposed the hydraulic diameter for foams, defined as: The hydraulic diameter is generally related to the flow phenomena in ducts and honeycombs, i.e.
internal flows. In this view, when considering the hydraulic diameter as characteristic length, the transport phenomena are modelled considering flow in ducts.   Table 1), and thus the effect of the S17 cell size is once again included in the Sherwood number dependency on the Reynolds number, as already discussed in Section 4 for the strut diameter. The effect of the porosity is hereby discussed. Figure S8 (a) shows the Sherwood number against the Reynolds number for the TKKD unit cell. The Sherwood number increases at increasing Reynolds number and increasing porosity, consistently to what previously discussed considering the window diameter as characteristic length and in contrast to what discussed in Section 4 considering the strut diameter. In analogy with the window diameter, the data appear to align at Redh > ∼300 in the nonstationary laminar regime, whether a residual dependency of the Sherwood number on the porosity is shown at lower Redh in the full laminar regime. Figure S8 (b) shows the Sherwood number against the Reynolds number for the Diamond unit cell.
The Sherwood number increases at increasing Redh and increasing porosity. In contrast to the TKKD, the residual dependence of the Shdh on the porosity is evident in all the flow conditions. Moreover, the data appear to collapse on a single line at decreasing porosity.
Because of the non-trivial dependency of the Sherwood number on the porosity and on the Reynolds number, the hydraulic diameter is shown to be a poor descriptor of the gas-solid transfer properties of the examined geometries and is thus discarded.
Reichelt et al. 54 proposed the equivalent sphere diameter, or Sauter diameter for mass transfer in generalized porous media : The equivalent sphere diameter is inherited by the description of transport phenomena in packed beds, for which a sphere-like diameter is generally adopted to formulate heat and mass transfer correlations. The porosity and the specific surface area of POCS are required to define the Sauter diameter. In this work, the porosity and the cell size are assigned, accordingly, the specific surface area is provided by the geometrical model. In this view, by combining the expressions reported in S18 Section 3.1 - Table 1, the Sauter diameter can be defined for the TKKD and the Diamond according to Eq. (S31) and Eq. (S32), respectively: In Eq. (S31) and Eq. (S32), the expressions ψ(ε) and β(ε) are inherited from the geometrical models and are non-linear functions of the porosity only, being the strut diameter linearly dependent on the cell size at prescribed porosity. Noteworthy, ψ(ε) and β(ε) are weak functions of the porosity, varying between 1.5 and 2 for the two considered geometries in the considered porosity range ε = 0.7-0.95, accordingly, a quasi-linear relationship exists between the Sauter diameter and the strut diameter, as already observed for open-cell foams 9 .
It is thus possible to revisit the heat and mass transfer correlations proposed in Section 5.2 considering the Sauter diameter as characteristic length, with different coefficients accounting for the factors ψ(ε) and β(ε). In the case of the TKKD unit cell, the functional form provided by Section 5.2, Eq. 8 is kept in the formulation of the heat and mass transfer correlation: Sh The applicability ranges of the derived correlations are 1 ≤ RedSauter ≤ 200, 0.7 ≤ ε ≤ 0.95 and 1 ≤ dc ≤ 8 mm.  with deviations below ±15% between the correlation and the data, and also proving the suitability of the Sauter diameter and its equivalence to the strut diameter in the description of the gas-solid interphase heat and mass transfer in POCS.