Cfd As A Design Tool For Fixed Bed Reactors

Abstract

Particle-resolved CFD simulations provide detailed insights into fixed-bed reactors. One of the remaining challenges is the generation of high-quality computational cells, which is especially challenging near particle-particle contacts. This contribution presents a generalized and automated contact modification independent of the particle shape by introducing a defined gap of 1 % of the particle diameter. This approach is successfully demonstrated for non-spherical cylinders, hollow and multi-hole cylinders, and a novel particle shape in terms of radial void fraction, pressure drop and an illustrative heat transfer study.

1 Introduction

Random structures made from particles with defined geometric shapes are applied in many different fields, e.g., trickle-bed reactors, separation columns, pebble beds, and fixed-bed reactors. The particles can have many different shapes, e.g., spheres, cylinders, rings, multi-hole cylinders with or without external flutes, often depending on the specific application. For catalytic fixed-bed reactors showing strong heat effects due to internal heat sinks/sources caused by reactions coupled with convective and conductive heat transport, the tube-to-particle diameter ratio (N = D/d _pv) is typically small ranging from 2–10. Consequently, wall effects play a significant role for transport phenomena and, hence, for the reactor performance. Under these circumstances, the classical engineering model formulations are questionable, since they involve correlations unsuitable for theses bed arrangements. Particle-resolved computational fluid dynamics (CFD) models are based on the bed structure consisting of individual particles. The flow around them is solved locally and can be coupled with heat and mass transfer into as well as reaction inside or at the surface of the particles. This modeling approach has gained a lot of interest in the last two decades, which is summarized in three major reviews ^1-3 .

The most fundamental requirement of particle-resolved CFD models is a realistic bed structure, which is typically derived from a separate preceding simulation with a discrete element model (DEM) ⁴ or a rigid body model ⁵ . Both models can lead to excellent results in comparison with experimentally derived bed morphology data. The bed structure is then imported into the typical CFD workflow, i.e., pre-processing, solving, post-processing. In order to solve the set of governing equations in the finite volume framework, the computational domain must be discretized/meshed into a finite number of cells. A high-quality mesh is desired, since it leads to a fast and stable convergence of the equation system. On the other hand, bad computational cells can cause divergence and, hence, corrupt results.

A challenge in the meshing process being specific to particle-resolved CFD simulations are particle-particle and particle-wall contacts. Fixed beds of ideal spheres show point contacts only. However, beds of non-spherical particles can have point, line, and area contacts, which increases the complexity of the meshing process. Originally, two different approaches were proposed for spheres: global and local methods. In the global method, all spheres were either shrunk ⁶ or expanded ⁷ by a certain amount of the particle size. Although both methods result in reasonable geometries for meshing, the global modification leads to a severe change in the bed structure and, hence, influences the transport phenomena significantly. The idea of local methods is to modify the bed structure only nearby the contacts. The local bridges method ⁸ introduces a small cylinder replacing the fluid and a small portion of the particles at the contact point. The solid cylinder is assigned with an effective thermal conductivity, where analytical formulas are given only for spheres ⁹ . The local caps method ¹⁰ moves the surface nodes near a contact point to flatten the sphere's surface. Consequently, fluid computational cells are representing the former contact point. If the cap is small enough, the velocity decreases to zero and, hence, only thermal conduction through the stagnant fluid is present as a heat transfer mechanism.

The different approaches were compared for fixed beds of spheres in several studies. Dixon et al. concluded from CFD simulations that the global methods applied for spheres changed the global porosity and, hence, the pressure drop too much ⁹ . Contrarily, the two local methods produced much smaller changes. For heat transfer, the authors preferred the bridges method over the caps method, since the latter omits the direct thermal conduction from one particle to another. A similar study in different unit cell configurations of spheres also recommended the local bridges methods ¹¹ . Both studies highlighted that the bridge diameter should be smaller than 20 % of the particle diameter to gain reasonable results for the macroscopic and local flow as well as heat transfer characteristics. One study extended the investigation to surface reactions on spheres at low Reynolds number conditions ¹² . The authors recommend bridge-to-particle ratios of 0.15–0.3 to avoid skewed cells and preserve the conversion rate.

The situation for non-spherical particles is much more complex. In order to modify contact points, lines, and areas differently, they have to be identified at best automatically. For full cylinders, an automation detection algorithm was developed based on geometrical considerations ¹³ . Wehinger et al. used these considerations to develop a method to apply local caps and bridges in a fixed bed of cylinders ¹⁴ . However, there is no general formula for the effective thermal conductivity of the bridges available for beds of non-spherical particles. Since this parameter can have a large impact towards the local heat transfer, the authors recommend using the local caps method to avoid this choice. However, for particle shapes that are not spherical or cylindrical, defining all possible contact configurations based on centroid position and particle orientation is illusory. Other studies ¹⁵ , ¹⁶ applied the shrink-wrap, or wrapping, method on the bed structure, which leads to a continuous solid matrix between several particles. Whereas the essential hydrodynamic behavior is captured in these studies, the appropriate radial heat transfer is uncertain.

In this contribution, we present a novel local caps method to modify individually particle-particle contacts independent of the particle shape. The method is based on particle expansion, local Boolean intersection and subtraction. The resulting small section/gap between neighboring particles is small enough to reduce the local velocity close to zero and large enough to guarantee the insertion of high-quality cells. This novel method is demonstrated for low tube-to-particle diameter ratio beds of non-equilateral cylinders, hollow cylinders, multi-hole particles with outer structure and a more complex particle shape. The influence of the local contact modification is quantified in terms of radial void fraction, cell quality, pressure drop and heat transfer studies.

2 Methods

2.1 Particle Shapes and Fixed-Bed Structure Generation

Tab. 1 summarizes the investigated particle shapes and tube dimensions according to experimental studies from literature. We follow the fixed-bed generation procedure with DEM, which was presented in greater detail elsewhere ⁴ , ¹⁷ , ¹⁸ . In brief, particles are injected with a constant flow rate on random positions on a plane inside the reactor tube. Due to gravity, the particles fall to the bottom of the tube and collide with the wall and/or other particles described with the linear spring model. Slowly, the tube is filled with particles and so the bed is generated. If all of the particles reach a threshold minimum velocity, this transient DEM simulation is stopped, and the resulting bed morphology is converted into a CAD description and handed over to the consecutive CFD simulation workflow. The resulting beds for four of the different particle shapes are shown in Fig. 1.

Table 1. Investigated particle shapes and fixed-bed dimensions.

Bed tag	Particle shape	Particle diameter [mm]	Particle height [mm]	Equivalent sphere diameter d pv [mm]	Inner hole diameter [mm]	Tube diameter [mm]	Number of particles	D/d pv	Bed porosity Exp.	Bed porosity DEM	Ref.
C	Cylinder	8	4	7.26	–	51	460	7.02	0.36	0.34	19
A3	Cylinder	4	12	6.66	–	23.16	238	3.47	0.43	0.44	19
HC4	Hollow cylinder	12.7	12.7	13.4	6.0	103	584	7.69	0.48	0.49	20
MHC	Multi-hole cylinder	12	17	13.8	4 × 3	100	780	7.24	–	0.60	21
65	Unconv. shape	–	35	71.3	–	400	65	5.61	–	0.73	–

Resulting fixed-bed structures for the different particle shapes. For dimensions, see Tab. 1.

2.2 Local Contact Modification and Meshing

Fig. 2 shows the workflow for particle-resolved CFD simulations including an illustration for the contact detection and modification. Contacts, or in extreme but rare situations overlaps, are detected automatically, if neighboring surfaces A and B are closer than a threshold value (see upper picture in Fig. 2). In the next step, particle B is inflated by 1 %d _pv resulting in B*. In the Appendix, results and discussion about the choice of the gap size can be found. Then, a Boolean operation subtracts the inflated particle B* from particle A, which results in the modified particle A'. Consequently, a local cap is created with a uniform thickness between the original particle B and the modified particle A'. In the next step the modified particle A' replaces the original particle A in the bed. This procedure is carried out consecutively for all occurring contacts, i.e., point, line, and area contacts, respectively, resulting in an overlap and contact free bed. Please notice that the overlap in Fig. 2 is untypically large due to an illustrative purpose. In the subsequent step, the resulting modified geometry is meshed with typically two prism layers at particle and reactor walls and polyhedral cells in the bulk solid and fluid region following the recommendations in one of our previous studies ²² . The resulting meshes here consist of approx. 5–10 million cells or around 15 000 cells per resolved particle, where 2/3 are used for the fluid and 1/3 to resolve the solid particles. Obviously, the number of cells increases with the complexity and, therefore, the required resolution of the particle.

CFD workflow and illustration of contact detection and modification.

The whole workflow is fully automated via a Java script in such a way that all 8 steps in Fig. 2 are executed consecutively inside Simcenter STAR-CCM+ after specifying all relevant parameters. The total elapsed time or wall time obviously varies with the number and geometric complexity of the particles. To give a rough estimated based on bed #C, the DEM part needs around 30 min, contact detection and gap creation 2 min, meshing approximately 1 h and solving flow and temperature around 3 h. Therefore, this workflow enables a very efficient way to study particle-resolved fixed beds and to design application-specific new particle shapes.

2.3 Computational Fluid Dynamics

2.3.1 Governing Equations

The set of governing equations consists of conservation of total mass, momentum, and conservation of energy in terms of specific enthalpy. For brevity reasons, the formulation shown here is for the laminar and steady-state case. The reader is referred to the nomenclature at the end of this manuscript. A detailed overview of CFD modeling of fixed-bed reactors in the turbulent regime can be found elsewhere, e.g., ¹ , ² .

Conservation of mass:

(1)

Conservation of momentum:

(2)

The stress tensor T is written as:

(3)

Conservation of energy in terms of specific enthalpy h:

(4)

With the diffusive heat transport j given by:

(5)

Since only thermal conduction is considered inside the solid, the energy equation there reads:

(6)

The set of governing equations is solved with the commercial CAE software package Simcenter STAR-CCM+ by Siemens ²³ .

2.3.2 Boundary Conditions, Material Properties, and Solver Settings

The geometrical description is shown in Tab. 1, whereas Tab. 2 summarizes the most important material properties and boundary conditions for the DEM and CFD simulation. For the DEM simulation, material properties of glass were used to mimic typical catalytic fixed-bed material. In the CFD simulations, a cool air stream is heated up inside of the fixed bed while the hot wall temperature is set to a constant value. Velocity inlet and pressure outlet are set as boundary conditions. Steady-state CFD simulations with Realizable k–ε RANS turbulence model are solved with the segregated fluid enthalpy solver implemented in STAR-CCM+.

Table 2. Material properties and boundary conditions.

DEM simulation
Particle density [kg m−3]	2500
Particle Poisson's ratio	0.235
Particle Young's modulus [Pa]	7.58 · 1010
Static friction factor	0.5
Rolling friction	0.001
Normal restitution	0.5
Tangential restitution	0.5
CFD simulation
Ideal gas	air
Inlet temperature [K]	298.0
Wall temperature [K]	383.0
Pressure outlet [Pa]	101 325
Fluid thermal conductivity [W m−1K−1]	0.02606
Solid thermal conductivity [W m−1K−1]	1.4
Solid density [kg m−3]	2500
Solid specific heat [kJ kg−1K−1]	1007.02

3 Results and Discussion

3.1 Bed Morphology and Local Velocity

Fig. 3 shows in the upper subfigures (A, C) the radial void fraction (ε _r) as a function of non-dimensional wall distance ((R–r)/d _pv) for non-equilateral cylinders (bed tags #C and #A3). For both particle shapes, the radial void fraction profiles of the synthetically generation beds agree very well with the experimental data from Benyahia ¹⁹ . Interesting to notice for bed #C is the local maximum at approx. 0.6 on the x-axis, which represents the fraction of particles standing. Farer away from the wall, the void fraction oscillates around the mean bed void fraction of 0.338. The bed #A3 shows two local maxima. Notice that the equivalent spherical diameter of these non-equilateral cylinders is somewhere in-between the particle diameter and particle height (see Tab. 1). In the lower subfigures (B, D), the normalized interstitial velocity is shown, which follows the radial bed void fraction. There are no experimental data of the velocity available for these particle shapes.

Fixed bed of non-equilateral cylinders: bed #C and #A3. (A) and (C) radial void fractions, (B) and (D) radial specific velocity.

In Fig. 4, the same data types are shown for hollow cylinders #HC4 (A, B) and multi-hole cylinders #MHC (C, D), where no experimental data is available. For the hollow cylinders, a good agreement between DEM and experimental data from Caulkin et al. ²⁰ can be achieved. In contrast to the solid cylinder, the velocity does not follow the entire radial void fraction. Since the pressure drop inside the smaller inner holes, the majority of the flow goes through the interstitial void between the particles. This can be recognized for the bed #HC4 and #MHC. In the Supporting Information, an additional unconventional particle shape #65 is presented and discussed.

Fixed bed of hollow cylinders bed #HC4 and multi-hole cylinders #MHC. (A) and (C) radial void fractions, (B) and (D) radial specific velocity.

3.2 Pressure Drop

The specific pressure drop Δp/L over the particle Reynolds number (Re _pv) is shown in Fig. 5 for the four different particle shapes. The simulated pressure drop data is compared to the Eisfeld-Schnitzlein (ES) correlation ²⁴ for #A3 and #MHC (cylinders), and Nemec-Levec (NL) correlation ²⁵ for #HC4 (hollow cylinders). For the ES correlation, the constants for solid cylinders are used. In the NL correlation, the Sonntag factor m is set, as recommended, to 0.2, which means that only 20 % of the volume flow goes through the inner holes of the hollow cylinders. The agreement between simulated pressure drop and correlations is excellent. Interestingly, the pressure drop of the multi-hole cylinders #MHC and the hollow cylinders #HC4 coincide.

Specific pressure drop over particle Reynolds number for different particle shapes, see Tab. 1. ES corr.: Eisfeld-Schnitzlein correlation [no.], NL corr.: Nemec-Levec correlation [no.].

3.3 Illustrative Flow Field and Heat Transfer

In Fig. 6, a few results are shown on two cross sections parallel to the tube axis. The velocity normalized with the superficial velocity (Fig. 6A–C) show clearly the non-homogenous distribution with some peak values roughly 10 times higher than the superficial velocity. These velocity peaks are limited to a few spots. For reference case #A3 depicted in Fig. 6A the maximum value is slightly higher than for case #MHC in Fig. 6B and 6C, which is caused by the lower global void fraction. Interestingly, in many of the holes a relatively high velocity of approx. 5 times the inlet velocity is found.

With inlet velocity normalized velocity and temperature plots on cross sections normal to x- and y-axis: (A) and (D) packing #A3, v _in = 1 m s⁻¹, (B) and (E) packing #MHC, v_in = 1.5 m s⁻¹, (C) and (F) packing #MHC, v _in = 4.51 m s⁻¹.

Comparing the same bed based on #MHC particles but for different inlet velocities clearly shows the self-similarity of the flow field, no significant difference can be found between the case with an inlet velocity of v _in = 1.5 m s⁻¹ (Fig. 6B) and with an inlet velocity of v _in = 4.51 m s⁻¹ (Fig. 6C). Both findings are in good agreement with published data ² , ¹⁰ .

The cross sections in Fig. 6D–F show the temperature distribution based on a heated wall with constant temperature T = 383 K. Comparing the same fixed bed but with different inlet velocities shows the expected result, i.e., with lower inlet velocity and longer residence time the gas flow leaves the bed with a higher average temperature and the temperature along the wall reaches higher values. It can also be seen that a few particles especially close to the wall have a significant higher temperature than the surrounding indicating that the conductive heat transport through stagnant gas and through the solid plays a significant role in distributing heat towards the center of the tube. A more detailed study on heat transfer and on the effect of the gaps is submitted for publication.

3.4 Mesh Quality and Solver Convergence

The main objective of this investigation is to introduce an automated and generalized workflow to generate and mesh fixed beds of any arbitrary shape. As stated earlier, the mesh quality is a critical parameter to achieve a reliable convergence. In Simcenter STAR-CCM+, the mesh quality can be quantified by several metrics like 'Cell Quality', 'Aspect Ratio' and 'Skewness Angle'. Other parameters like 'Face Validity', 'Chevron Quality' and 'Cell Warpage' are available but not discussed here because relevant cells were not detected for the investigated cases. Notice that other software might have different metrics.

In Fig. 7 the cumulated number of cells for different cell metrics of the full mesh of bed #MHC are shown. The different colors indicate the region where the cells are located: white are the solid cells in the particles, black are the fluid cells around the particles and in gray the cells in the extrusion upstream or downstream of the bed.

Histograms of different metrics to estimate cell and mesh quality for multi-hole cylinder #MHC. (A) Cell aspect ratio, (B) cell skewness, (C) cell quality for the four computational regions (fluid, particles, extrusion up- and downstream).

Fig. 7A shows the cell aspect ratio. Most of the cells have a value close to one, which means that the extend in each direction is almost the same. But even an aspect ratio of 0.05 is tolerable, since these high aspect ratio cells are found predominantly in the flat gaps where gradients are typically small.

The cell skewness angle is shown in Fig. 7B. The angle is defined between the face normal vector of a cell and the vector connecting the centroids of both cells. An angle of zero indicates a perfectly orthogonal mesh, while skewness angles above 90° typically result in solver convergence issues. For the generated mesh most of the cells have a maximum skewness angle around 25° and the number of cells with higher values decreases rapidly and no cells are detected with a skewness angle above 85°.

Fig. 7C shows the cell quality distribution. Cell quality is not only a function of the relative geometric distribution of the cell centroids of the face neighbor cells, but also of the orientation of the cell faces. A cell quality of 1.0 is considered perfect, a cube or a perfect polyhedron is an example of such a perfect cell. The plot shows that most of the cells have a value of larger than 0.3 indicating a reasonably good mesh. But even the few cells with a low quality are acceptable as long as they are not affecting convergence.

Fig. 8 gives a visual overview for the cell quality distribution on a detail of a meshed bed structure. Fig. 8A shows a cross section through the original CAD representation of the DEM results with a line contact between two cylindrical particles and an overlap between two particles. After the gap generation step described in Fig. 2, the resulting CAD representation is depicted in Fig. 8B. Finally, Fig. 8C shows the cell quality distribution in that area. It can be clearly seen that most of the cells have a value between 0.5 and 1. The whitish cells are mostly located on the solid side and a few cells can be found close to and in the gap, which is related to the transition and cannot be avoided.

(A) and (B) show a cross section through the CAD representation before and after the gap generation, respectively. (C) shows the distribution of the cell quality in this cross section.

As a practical proof of the quality of the resulting mesh, all investigates cases irrespectively of the inlet velocity and the particle shape converge within 500–1000 iterations.

4 Conclusions

One of the major challenges in particle-resolved fixed-bed reactor CFD simulations is the generation of a high-quality mesh, especially for non-spherical particles. In this contribution, we presented a generalized and automated workflow to modify particle-particle contacts. The ability of this approach to deal with non-equilateral cylinders, hollow and multi-hole cylinders as well as an unconventional shape was demonstrated. The local introduction of defined gaps of size (1 % of the equivalent particle diameter) shows no substantial change to the local bed morphology, which was demonstrated with the good agreement of the radial bed void fraction with experimental data as well as pressure drop prediction. The illustrative heat transfer study shows that the velocity decreases in the gap and leads to predominant thermal conduction through the stagnant gas. However, additional local heat and mass transfer experiments in such fixed-bed structures are needed to finally validate the model, e.g., in terms of the gaps size, spatial discretization of gaps, etc.

As a conclusion, this generalized approach can be used for all kinds of particle shapes, which makes it an invaluable tool for the particle-resolved CFD simulation framework to calculate transport phenomena in fixed beds. For high aspect ratio or arbitrary particle shapes, e.g., rock fractions, tablets including coating, etc., we recommend choosing the gap size carefully and based on thorough sensitivity studies towards physical quantities. A feasible starting point is to set the gap size to 1 %d _pv. However, for moving particles coupled with the surrounding flow in particulate systems, e.g., sandpile formation, fluidized beds, spouted beds, blast furnaces, etc., we recommend using the CFD-DEM modeling approach reviewed thoroughly elsewhere ²⁶ .

Supporting Information

Supporting Information for this article can be found under DOI: https://doi.org/10.1002/cite.202000182.

Acknowledgements

The authors would like to express their sincerely thank to Prof. Dr.-Ing. Matthias Kraume for his constant support during all the years. He has been an excellent adviser. His passion for research and teaching as well as his precise analyzing always inspired us. Open access funding enabled and organized by Projekt DEAL.

Symbols used

c [J kg⁻¹K⁻¹]

specific heat

D [s⁻¹]

rate of deformation

D [m]

tube diameter

d [m]

particle diameter

d _pv [m]

equivalent spherical diameter,

h [J kg⁻¹]

specific enthalpy

h [m]

particle height

I [–]

identity tensor

j [W m⁻²]

diffusive heat transport

k [W m⁻¹K⁻¹]

thermal conductivity

L [m]

bed height

N [–]

tube-to-particle diameter ratio, N =D/d _pv

Nu [–]

Nusselt number

p [Pa]

pressure

Pe [–]

Peclét number,

Pr [–]

Prandtl number,

Re [–]

Reynolds number,

T [Pa]

viscous tress tensor

T [K]

temperature

v [m s⁻¹]

velocity vector

V [m³]

volume

Greek letters

μ [Pa s]

dynamic viscosity

ρ [kg m⁻³]

density

Sub- and Superscripts

e

equivalent

f

fluid

p

particle

s

solid

w

wall

0

superficial

Abbreviations

CFD

computational fluid dynamics

DEM

discrete element method

Appendix

It should be kept in mind that the size of the gap should be chosen in such a way that on the one hand cells of good quality can be created and on the other hand no additional artificial flow channels with significant fluid velocity are formed. In order to create good quality cells, the gap should be as large as possible, while to suppress artificial flow the gap should be as small as possible. This is not that critical for fluid dynamic investigations only but becomes important as soon as heat transfer is considered. The value of 1 % of the equivalent particle diameter d _pv is a reasonable trade-off. However, it should be noted that the actual choice of this gap size depends also on the local surface discretization, on the particle aspect ratio as well as on the local flow field. To illustrate this, a comparison is conducted for particle #A3 for three different gap sizes of 0.5 %, 1 % and 2 %. All other mesh parameters are kept constant including the surface triangulation. Tab. A1 shows that the gap size has almost no effect on the global porosity and the pressure drop. In contrast to this, the number of iterations to reach convergence is affected. A larger gap leads to faster convergence. This can be attributed to a better overall cell quality, mainly in the gaps. Especially for the case with a gap size of 0.5 % and no further surface cell size refinement, the cells have a higher aspect ratio and a lower overall quality requiring more iterations to reach convergence.

Table A1. Comparison of porosity, pressure drop and number of iterations to reach convergence for case #A3 and three different gap sizes.

Gap size	0.5 % d pv	1 % d pv	2% d pv
Porosity	0.4444	0.4444	0.4447
Pressure drop [Pa] for a superficial velocity of x m s−1	291.0	289.7	286.5
Iterations to convergence	1400	800	400

Fig. A1 shows a comparison of the gap between two particles. It can be clearly seen that for a gap size of 0.5 % and 1 % the fluid in the gap is almost stagnant. This is important for investigation with heat transfer to ensure that convective transport in the gap does not influence the result. This is no longer the case for a gap size of 2 %, where a significant fluid velocity in the gap occurs.

Comparison of the fluid velocity for three different gap sizes. Detail from bed #A3 for a superficial velocity of 1.03 m s⁻¹ (Re = 5000).

Supporting Information

Filename	Description
cite202000182-sup-0001-misc_information.pdf311.8 KB	Supplementary Information

Please note: The publisher is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.

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Citing Literature

Cfd As A Design Tool For Fixed Bed Reactors

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