Workflow for computational fluid dynamics modeling of fixed-bed reactors packed with metal foam pellets: Hydrodynamics

In recent years, the catalyst pellets made of open-cell metallic foams have been identified as a promising alternative in fixed-bed reactors. A reliable modeling tool is necessary to investigate the suitability of different foam properties and the shapes of foam pellets. In this article, a workflow for a detailed computational fluid dynamics (CFD) model is presented, which aims to study the flow characteristics in the slender packed beds made of metal foam pellets. The CFD model accounts for the actual random packing structure and the fluid flow throughout the interstitial regions is fully resolved, whereas flow through the porous foam pellets is represented by the closure equations for the porous media model. The bed structure is generated using rigid body dynamics (RBD) and the influence of the catalyst loading method is also considered. The mean bed voidage and the pressure drop predicted by the simulations show good agreement with the experimental data.


| INTRODUCTION
An open-cell metal foam is a solid cellular structure with a network of inter-connected pores; characterized by high porosity, high specific surface area, and global thermal conductivity, it is felicitous for a wide range of applications. 1 In recent decades, the suitability of metallic foams as catalyst supports has been investigated by many researchers in terms of pressure drop, heat and mass transfer characteristics, as well as chemical reactions. 2,3 Meanwhile, the advancement in manufacturing technology has induced the production of pure metallic foams (e.g., Ni, Fe, and Cu) and the alloy compositions (e.g., NiFeCrAl, NiCrAl, Inconel 625, and FeCrAl) in an economical way as well as being tailored for specific applications. 4,5 Moreover, the foam sheets can be shaped as drop-in-pellets type for the application in fixed-bed reactors. Figure 1 depicts a few shapes of metal foam pellet realized by Alantum Europe GmbH, Germany, in using a patented manufacturing process. 6 This design flexibility allows the customization of metal foam pellets to meet the requirements of many industrial processes, where the conventional ceramic pellets have failed to satisfy the entire process demands. Dixon 7 has identified that the major requirements in large-volume applications like methane steam reforming are of incompatible nature.
To feed methane at higher flow rates upon maintaining lower pressure drop prompts the use of large particles, whereas smaller-sized particles are beneficial from a reaction perspective, as they provide higher surface area per unit volume of the bed. Furthermore, special attention is needed in the design of fixed-bed reactors to ensure an adequate transport of energy from the reactor wall to the center core of the bed or vice versa. Hence, a low tube-to-particle diameter ratio, N = D/d p < 10, is the preferred choice for the processes which require heating or cooling of the reactor. 8 Walther et al. 9 have pointed out that the radial heat transfer performance of a fixed-bed reactor can be enhanced by the use of suitable alloyed metal foam pellets, which also ensure lower pressure drop. Thus, the adoption of metal foam pellets in the fixedbed reactor is a process intensification approach, since the reactor efficiency is improved, and the operation cost can be reduced.
However, the tortuous inner structure of metallic foam pellets added with considerable internal flow causes a complex flow field compared with conventional nonporous pellets. A thorough understanding of the fluid dynamics of the packing structure composed of metal foam pellets is very important to harness its potential. Indeed, detailed studies pertaining to flow characteristics are scarce. Kolaczkowski et al. 10 carried out experiments to measure the pressure drop in a slender-tube fixed-bed reactor made of slab-shaped and cubic metal foam pellets. A significant reduction in pressure drop was observed in comparison to solid counterparts. They also devised a method to determine the proportion of fluid that flows through the pellets, which was quantified up to 38%. This flow distribution is dependent on the arrangement of foam pellets in the reactor column, pellet size or shape, and the foam morphological parameters, mainly cell size, and porosity. Since experimental studies of this kind are time-consuming, reliable modeling tools which provide design and optimization space are necessary to support the development of suitable foam pellet shapes.
The particle-resolved computational fluid dynamics (CFD) approach, which accounts for the actual bed geometry, has been proved as relevant to study the fixed-bed reactors with low N-value. 11 At N < 10, the influence of the local bed structure is crucial for the local transport phenomena, which can be well predicted with this detailed modeling approach-see the review articles of Dixon and Partopour, 12 and Jurtz et al. 13 One of the major challenges in the particle-resolved CFD approach is to generate a representative fixed-bed structure. The methods generally used to create packing geometry are tomography scans 14 and by computational methods, [15][16][17][18][19] in which Discrete Element Method (DEM) 20,21 and the Rigid Body Dynamics (RBD) 22 Blender. They reported that both methods predict the bed voidage accurately, whereas a more precise prediction of particle orientation in the synthetic beds made of cylinders and hollow cylinders was observed in Blender simulation than in STAR-CCM+, and the latter uses gluedsphere 27,28 method to model nonspherical particles. In accordance with the previous works, it is confirmed that open-source software Blender integrated with Bullet physics library 29 is robust to create synthetic fixed-bed structures for the use in particle-resolved CFD approach.
On the other hand, the high porous nature of the foam materials poses additional challenges in the CFD modeling of the fixed-bed composed of metal foam pellets. It comprises a problem of bi-disperse porous media, 30 where the fluid has two flow paths: through and around the porous particles. Pore-scale CFD simulations have been used by many researchers to study the detailed flow characteristics inside the open-cell foam structures, in which the pore-scale geometry is created by either imaging techniques or using idealized foam patterns like kelvin cell, Weaire-Phelan, or random structure. [31][32][33][34][35] However, this type of detailed CFD simulation is not practically feasible in the packing structures composed of thousands of foam pellets. To address these kinds of problems, multiscale modeling approaches need to be adopted.
Recently, Wehinger et al. 36 have presented a CFD approach, in which the established particle-resolved CFD model is integrated with pseudohomogeneous porous medium model to account for the transport phenomena inside the foam pellets. The authors used DEM approach to generate the fixed-bed structure. The pressure drop CFD simulations were compared with the experimental data provided by Kolaczkowski et al. 10 and good agreement for medium flow conditions was reported.
Along with this, an illustrative heat transfer study highlighted the potential of metal foam pellets in fixed-bed reactors.
In this work, a workflow for an adapted particle-resolved CFD approach is presented to study the flow characteristics in the fixedbed made of metal foam pellets and validate it against the experimental data of bed voidage and pressure drop. The proposed CFD workflow is similar to Wehinger et al. 36 ; instead of DEM, RBD approach generates the different fixed-bed structures. The inclusion of RBD approach provides more flexibility to model the packing structures with any arbitrary catalyst particle shapes and even allows mimicking the catalyst loading strategies akin to commercial settings, with F I G U R E 1 Metal foam pellets manufactured by Alantum Europe GmbH, Germany [Color figure can be viewed at wileyonlinelibrary.com] low computational efforts. The CFD model is validated against experimental void fraction and pressure drop data for the ceramic Raschig rings and the cylindrical metal foam pellets. It should be noted that the inner structures of the foam pellets are not resolved, whereas flow through the pellets is considered by appropriate closure equations corresponding to the porous media approach. The bed geometry was created using open-source software Blender and the CFD simulations were carried out with Simcenter STAR-CCM+ from Siemens Industry Software Inc. Figure 2 illustrates the proposed CFD workflow. The main objective of this contribution is to develop an adequate CFD model to assist the design of the optimal shape of metal foam pellet.

| Pellet characteristics
Two types of catalyst carriers were considered in this study: ceramic Raschig rings and cylindrical metal foam pellets made of NiCrAl alloy (71% Ni, 19% Cr, and 10% Al). The characteristic length scale preferred to study the transport phenomena in a fixed-bed reactor is the particle diameter, d p . In order to compare geometrically different particles, the usual approach is to define equivalent particle diameter in terms of a sphere of equivalent specific surface area, d p,e = (6V p /A p ). Here, V p and A p are the volume and surface area of a particle. Accordingly, the particle Reynolds number is given by Equation (1), where v s is the superficial velocity and μ is the dynamic viscosity of the fluid medium.
Based on the concept of the hydraulic radius, the Reynolds number complement to the bed structure is dependent on the mean bed voidage, ε b , and is related to particle Reynolds number as in Equation (2), which is generally termed as the modified Reynolds number or the bed Reynolds number. 37,38 Re  volume and the outer surface area are used, neglecting the particle porosity, ε. In addition, the porosity of the foam pellets mentioned in Table 1 does not take into account the micro-voids on the strut surfaces, as they are hardly accessible to fluid flow. 39 The pellet samples are not wash-coated, as the present study focuses only on flow behavior under cold flow and nonreactive conditions. Furthermore, wash-coating processes of metal foam pellets might increase the strut thickness, which in turn reduces the overall particle porosity and influences the hydrodynamic quantities as well. 10

| Experimental setup
A suitable experimental setup for pressure drop measurements was realized and is schematically depicted in  that were aligned circumferentially at an angle of 120 apart (see Figure 4). To quantify pressure drop in the bed segments, the corresponding pressure taps at axial locations were connected across the differential pressure transducers (DP/1, DP/2, DP/3, and DP/4-Series 616D 0-20 in w.c). Subsequently, the pressure drop over fourbed segments, each of 400 mm length, was recorded. The differential pressure transducers were provided by Dwyer Instruments, Inc., United States, and the rated accuracy was ±0.25% FS at 25 C. Nitrogen was discharged through the outlet at the very bottom of the reactor column and the outlet pressure was monitored by a pressure transducer (PT/Outlet). Additionally, a differential transducer (DP-TM-AS-dP/0-100 kPa) from Techmark GmbH, Germany, with an accuracy of 0.4% FS was used to monitor the pressure drop across the whole reactor, that is, from inlet to the outlet.

| Synthetic bed generation
In this study, the bed structure was simulated with a rigid body model incorporated in the Bullet physics library, 29 which is integrated with the animation software Blender. A detailed description of using Blender for the realization of packing structures has already been addressed. [24][25][26] As a starting point, the pellet geometry and the reactor tube were created by the triangular surface meshes. Next, the particles were positioned at the top of the container and allowed to fall freely into the reactor tube. The simulation was proceeded by the physics engine upon resolving the gravity and the interaction forces in discrete time steps. Particle-particle and particle-wall collisions were resolved with the appropriate values of friction and restitution coefficients. Table 2  suggested that the static friction coefficient can be used as an adjusting parameter to merge the effects of the loading device, artificial tamping, and the particle surface characteristics. In this way, a filling strategy with less computational effort can be adopted in synthetic bed generation with an adjusted static friction coefficient.
In this study, three filling strategies to generate synthetic beds were compared. A parametric study on friction coefficient was also carried out on each filling strategy. The final selection of the filling strategy and the friction coefficient was carried out by comparing the mean bed voidage of the synthetic bed to the experimental bed and the required computational time. The details are elaborated in Section 3.2.

| CFD
The flow conditions pertinent to single-phase and turbulent flow were simulated by solving Reynolds-Averaged mass and momentum equa- Animation frames 1000-1500 media model was adopted to mimic flow though the pellets and the corresponding pressure loss. When the flow enters a porous medium, the physical velocity increases due to the reduction in the area available for the flow. The velocity rise is dependent on porosity, ε, which is defined as the ratio of volume available for the fluid flow V free to total volume V, ε = (V free /V). As a result, the physical velocity, v in a porous medium is related to the superficial velocity as v s = εv, which disregards the skeleton of the porous medium and assumes that only the fluid passes through the given cross-sectional area.
The mass and momentum conservation equations in a porous medium based on physical velocity can be formulated as: where P v and P i are viscous and inertial resistance tensors, respectively. For flow through homogeneous porous media and at very low fluid velocity, the pressure drop is balanced by the viscous shear stress and is linearly proportional to v s . When the fluid velocity increases, the inertial force starts to contribute and the pressure drop is proportional to v 2 s . The most widely accepted correlation for predicting the specific pressure drop in an isotropic granular media was suggested by Ergun 43 as: The major difficulty in its formulation is to re-define the equivalent particle diameter in terms of the foam morphological parameters such as the strut diameter, d s and the cell size, Ø. A simplistic geometric model, in which a direct analogy between the foam structure and a bed of spherical particles, formulated based on a cubic cell, was proposed by Lacroix et al. 39 In this cubic cell model, the foam structure is represented by solid cylindrical filaments connected in 3D as a regular cubic lattice, where the edges symbolize the foam struts. As a result, a relation between strut diameter and particle diameter was developed by equating the specific surface area of the cubic cell to specific surface area of bed of spherical particles as given by Equation (6). The strut diameter is calculated from the cubic cell model based on the porosity, ε and pore diameter a as shown in Equation (7) and Equation (8), where the pore diameter, a, is approximated based on the cell size, Ø.

| Meshing and solution methodology
In order to spatially discretize the computational domain, the meshing process was carried out using the commercial software STAR-CCM+. Polyhedral cells were used in the bulk region and three prism layer cells were considered at the wall of the pellets and the reactor tube. The meshing process in a packed bed is inherently complex due to the presence of the high number of particle-particle and particle-wall contacts, which may lead to local bad cell qualities. To overcome this issue, a meshing procedure by the local cap method was proposed by Eppinger et al. 45

| Validation of synthetic fixed bed structures
The influence of the loading method on the bed structures is investigated by adopting three different filling strategies in Blender simulations and is illustrated in Figure 6: In the second step, each of the loading strategies is subjected to a range of friction coefficient and the mean bed voidage of the generated beds is investigated. Figure 7A depicts the comparison of the calculated mean bed voidage for different cases. The general trend is a linear increase in bed voidage while the friction coefficient increases. This is in line with other numerical studies. 48,49 The friction between the relative contact surfaces is lower at low surface roughness value, which in turn creates denser beds. As far as a comparison of the loading methods is concerned, the Pot-Brush method allows loose beds, whereas the line method results in comparatively denser beds. In Pot-Brush-method, the momentum of the falling particles undergoes reduction upon collision with the brush structure, whereas the momentum gained by the falling particles is higher in the Line-method since there is minimal interaction between the particles on the way into the container. Consequently, the particles may displace more evenly in the bed structure until their momentum dies out. Figure  The practical applicability of any simulation method has been decided also based on the computational power required for its execution. Here, the computational time taken to simulate each loading strategy is estimated based on one CPU (intel Core i7-8700K) and is depicted in Figure 7B. For the Line and Array methods, the total simulation time is less than an hour, while Pot-Brush requires almost double the computational power. This is attributed to the extra work of the physics engine to simulate the pouring process of particles from a pot and to simulate the additional collision of particles with the brush structure. Even though, the time consumed for the RBD simulation using Blender is very much lower than the conventional methods like DEM simulations. 26 Along with the bed voidage, another important physical parameter of the bed structures made of nonspherical particles is the particle orientation. Moreover, the alignment of particles with respect to the flow direction is more significant when the particles are of hollow type like Raschig rings, multi-hole cylinders, and so forth. Therefore, the impact of the loading methods and the friction coefficient on the orientation of Raschig rings are analyzed with the numerically generated beds. Figure 8A-C show the global particle distribution of orientation for the loading methods Line, Array, and Pot-Brush, respectively. Here, the particle orientation is defined as an angle (0 ≤ θ ≤ 90 ) between the vertical axis of the reactor column and the vertical symmetrical axis of an individual Raschig ring.
It is interesting to note that the predominant particle orientation is 80 -90 for all the studied cases, which is in line with other experimental observations. 19 The dependency of particle orientation with friction coefficient is almost similar for all the loading methods. The tendency of particles to align either parallel (0 -10 ) or orthogonal (80 -90 ) to the vertical axis is higher for the friction factors < 0.5. In using the line method, the least particle orientation is in the range and unloading of the same packing. Indeed, the overall characteristics can be maintained somewhat similar, that is, the greater tendency to align either perpendicular (90 ) or parallel (0 ) to the column vertical axis.
As the final step in the validation of the bed structure, a bed from each of the loading strategies is selected, which bears the mean voidage close to the experimental bed, and subjected to pressure drop CFD simulations. Here, it is worthwhile to recall that the mean bed voidage of 0.619 was achieved at friction factors of 0.9, 0.7, and 0.4 for the Line, Array, and Pot-Brush methods, respectively (see Figure 7A). However, the orientation of particles in these beds are not exactly similar as depicted in Figure 9A. The pressure drop simulated by using the numerical bed from each loading strategy is compared with the experimental data for a mass flow rate of 56.5 kg/h and is shown in Figure 9B. An error bar of ±10% is included along with the experimental data to cover up the variation in pressure drop along with the bed segments and other possible instrumentation errors. A very good agreement is observed for all the cases as the deviations are even less than 3%. Hence, it is inferred that the mean bed voidage is the critical parameter in defining the pressure drop characteristics of fixed beds. Furthermore, the effects of catalyst loading device on bed morphology can be included in numerical bed generation by adjusting the RBD parameter: friction coefficient, in a simple loading strategy which is of less computational cost. In this case, Array method is found as efficient from the practical perspective, as it consumes lesser computational time (see Figure 7B) as well as being capable of simulating a bed morphology close to experimental bed.
The Array-method is then extended to generate a representative bed structure for the metal foam pellets. It should be noted that the numerical bed for the metal foam pellets is made of solid cylinders, as flow through the pellets is accounted by the porous media model (see Section 2.4.1). As explained earlier, the friction coefficient is adjusted to achieve a mean bed voidage close to the experimental bed made of metal foam pellets-see Supporting Information.

| Validation of the CFD model
The validation of the CFD simulation is carried out using the experimental pressure drop data discussed in Section 3.1. Firstly, the ability of particle-resolved CFD approach in predicting the pressure drop of fixed-beds is validated. Figure 10A Figure 10B, the simulated specific pressure drop with 10% reduction in cell size, that is, 1080 μm, is closer to experimental data, with a relative error of 12%-25%.
The process of shaping a required pellet geometry from the foam sheet may also affect the physical characteristics of pellets' outer faces.
It should be noted that the required height or diameter of a foam pellet has been achieved by stacking thin layers of multiple foam sheets and are adhered to them by heat treatment processes. The thickness of a single layer depends on its cell size and is around 3 mm for the case of 1200 μm. In the end, the required pellet shapes have been cut out from the thickened sheet. 5 Upon physical examination of the processed pellets, it is revealed that there are more closed and roughened cells along with the cutting faces as a side-effect of the shaping process (see Figure 10B). This change in foam morphology can cause an additional increase in the pressure drop by reducing the effective porosity. Moreover, the roughened wall surfaces combined with the fine structures of the metal foam pellets might induce additional flow artifacts in the packed bed arrangement that are difficult to reproduce completely in a CFD framework with the porous-media model. To account for these flow artifacts in the CFD model, a correction factor which modifies the original porosity is applied. As depicted in Figure 10B, CFD results with 2.3% reduction in original porosity, that is, 0.87 to 0.85, showing an excellent agreement with the experimental data. Meanwhile, the suitability of the same correction factor for different foam morphologies and pellet shapes is questionable. This embarks further investigation on the pressure drop characteristics of various pellets shapes and foam morphologies and it will be addressed in a future work.

| CFD flow field analysis
A detailed study of the flow field is very important to analyze the performance of a reactor. However, experimental insights are very difficult to achieve. The contour maps of velocity magnitude normalized by the superficial velocity of 3.8 m/s are shown in Figure 11A, However, fluid particles experience different resistances while flowing around and through the particles and it is strongly related to the actual foam morphological parameters, mainly cell size, and porosity.
The significance of particle orientation is also evident from the contour plots of the Raschig ring bed (see Figure 11A). The inner holes are subjected to considerable fluid flow when it is parallel to the flow direction. Despite, the majority of Raschig rings are aligned perpendicular to the flow direction and are exposed to the eddies and the flow obstructions induced by the upstream particles. Therefore, only a fraction of the inner hole regions could support the fluid flow and, subsequently, the effective bed porosity is reduced.
A qualitative estimation of the amount of flow through the pellets is carried out using CFD simulations. To quantify flow through the pellets, a few cross-sectional planes are selected along the bed, where the average flow velocity in the porous pellet region and the corresponding flow area is calculated. Figure 12A illustrates the streamlines injected from the reactor inlet, and the selected planes for mass flow calculation are shown in Figure 12B. It should be highlighted that the illustrated streamlines could not reflect the real flow tortuosity inside the pellets. Figure 12C  We showed that the catalyst loading methods can also be modeled along with synthetic bed generation in using RBD approach integrated in Blender software, with low computational efforts. It should be noted that the mean bed voidage is the critical parameter relevant to the fluid dynamic characteristics of a packing structure.
Therefore, the mean bed voidage of the numerical bed should be in good agreement with the real packing structure. It has been achieved by adjusting one of the RBD parameters, that is, friction coefficient, regardless of the catalyst loading method.
In comparison to the experimental data of Raschig rings, the pressure drop predicted by the established particle-resolved CFD approach shows an excellent agreement. It is also evident that detailed CFD simulations could capture even the intrusive effects of measuring devices like the thermowell. The validated particle CFD model is then adapted to mimic the internal flow through the pellets with appropriate closure equations, which are dependent on foam morphological parameters such as cell size and foam porosity. The pressure drop predicted by the adapted CFD model shows good agreement to experimental data in using a modified porosity, which also considers the impact of closed cells along with the cutting faces of the foam pellets in the effective bed voidage.
The flow distribution in the fixed-beds composed of metal foam pellets is highly dependent on the foam morphology. Therefore, it is recommended to verify the applicability of the proposed CFD workflow in different foam morphologies and pellet shapes. This will be addressed in future works. Additionally, the proposed CFD workflow will be extended to study the heat transfer characteristics and chemical reactions.

ACKNOWLEDGMENT
The authors kindly acknowledge the financial support provided by the

DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available from the corresponding author upon reasonable request.