Comparison of computational fluid dynamics (CFD) and pressure drop correlations in laminar flow regime for packed bed reactors and columns

    Comparison of computational fluid dynamics (CFD) and pressure drop correlations in laminar flow regime for packed bed reactors and columns Andraˇz Pavliˇsiˇc, Rok Ceglar, Andrej Pohar, Blaˇz Likozar PII: DOI: Reference:

S0032-5910(18)30030-5 doi:10.1016/j.powtec.2018.01.029 PTEC 13115

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Powder Technology

Received date: Revised date: Accepted date:

2 August 2017 5 January 2018 13 January 2018

Please cite this article as: Andraˇz Pavliˇsiˇc, Rok Ceglar, Andrej Pohar, Blaˇz Likozar, Comparison of computational fluid dynamics (CFD) and pressure drop correlations in laminar flow regime for packed bed reactors and columns, Powder Technology (2018), doi:10.1016/j.powtec.2018.01.029

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Comparison of computational fluid dynamics (CFD)

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for packed bed reactors and columns

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and pressure drop correlations in laminar flow regime

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Andraž Pavlišič*, Rok Ceglar, Andrej Pohar, Blaž Likozar

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Department of Catalysis and Chemical Reaction Engineering, National Institute of Chemistry,

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ABSTRACT

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Hajdrihova 19, 1001 Ljubljana, Slovenia

Empirical correlation (EC) equations are still of a great designing importance for industrial plant construction. They are an indispensable modelling tool for engineers, reducing the time to find the optimal operating conditions. Nonetheless, numerical method complexity and product yield optimisation are advancing. Computational fluid dynamics (CFD) is thus nowadays applicable for optimizing chemical reactors. In contrast to EC, CFD acknowledges specific vessel geometry, where local physical and chemical phenomena, contributing to apparent catalytic turnover, prevail. Presently, EC and CFD were compared considering the pressure drop predictions within the packed bed columns for spherical, cylindrical, trilobe and quadrilobe particle packing, in order to determine the limits of EC accuracy. 52 configurations

ACCEPTED MANUSCRIPT were simulated and the estimations within EC validity range margins were in agreement with CFD (< 15%), while in extremes (non-negligible entrance and exit patterns), a 70% deviation

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could be exceeded. Furthermore, boundary wall effects were found to be dependent on the

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stacked pellet shape and orientation, and did not necessarily lead to an increase of viscous

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friction loss, relative to the infinitely wide systems, for the column-to-particle diameter ratios, lower than 10, which is contradictory to non-mechanistic relationship models. While the induced pressure difference within realistic fixed beds is elevated due to gas or liquid surface

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interaction and back-mixing, it can also be decreased by channelling/tunnelling, which is why

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the effective net influence should be analysed with CFD simulations, particularly in novel intensified and micronized processes, in which momentum, mass and heat transfer resistances

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are far from bulk medium continuity.

Keywords: computational fluid mechanics; modelling and optimization; laminar flow pattern;

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friction loss correlations; continuous packed-bed reactor; irregular particle packing

* Corresponding author. E-mail address: [email protected] (A. Pavlišič)

ACCEPTED MANUSCRIPT 1. Introduction

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The two critical variables for the design of a packed bed (PB) reactor are usually predicted

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using empirical correlations (EC), the packing void fraction and the pressure drop across PB,

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that were and still are a base for ongoing discussion among researchers [1–6]. Overgeneralization of typical parameters brings a high error possibility and leads to oversized reactors because of the capacity safety factor. The alternative method to handle the expected

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error is to conduct specific pilot plant tests with representative packing and correct those

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parameters [7,8]. However, these studies can be expensive and require a significant effort in investigating all influential parameters, especially if typical parameters are not constants [9].

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In such reactors, local phenomena are dominant, and the homogeneous model using averaged

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EC, that PB reactor designs are based on, is often no longer valid, so adapted general models were proposed [10–13]. The emphasized non-homogeneity in real void fraction consequently

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reflects in a bed-specific fluid flow profile influencing the certainty of bulk calculation with a generalized void fraction value [14–19]. The fluid flow profile is subject to more recent

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studies because of its complexity and difficult experimental setup.

In contrast to correlations and experimental investigations of adaptive parameters, several numerical approaches are available today to investigate the flow within particle packings, mostly based on computational fluid dynamics (CFD). New methods are possible especially because of advances in computer performance. A crucial starting point in performing a reliable CFD simulation is an accurate representation of the actual system with a geometrical model that is as realistic as possible. Although geometry can be drawn for PB reactors with a small number of particles for larger sets, a more complex methodology has to be implemented [20,21]. Three general approaches can be found throughout the literature: creating an artificial

ACCEPTED MANUSCRIPT bed using a digital packing algorithm usually based on the Monte Carlo method [22,23], mimicking a geometry based on MRI (magnetic resonance imaging) or CT (computerized

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tomography) scan of an actual PB reactor [24], and generating a realistic simulation of reactor

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packing using the distinct element method (DEM) approach [6,25–31]. The last method is

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especially popular in recent times because it allows creating a realistic random distribution of the fillers by simulating an actual act of gravitational packing. At the same time it is fast and

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inexpensive (in contrast to MRI).

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The objective of this study is to numerically obtain pressure drops for spherical and arbitrary particles in various PB configurations in the laminar regime with the aid of CFD. The obtained results for different geometries and flow rates are then compared with

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experimentally derived correlations. Finally, the velocity and pressure fields calculated by CFD are further on used to interpret deviation between CFD and EC pressure drop

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2. Methods

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predictions.

2.1. Empirical equation background

Although many EC can be found in the literature for granular (packed beds) and cellular (open-cell foams) porous media, their understanding and formulation almost exclusively fall on the two historically very important equations. While Darcy’s law was very convenient for describing Newtonian fluids in the laminar regime, it failed to anticipate inertia forces when high Reynolds number were applied. Therefore an Austrian scientist P. Forchheimer (1901) in his work “Wasserbewegung durch Boden”, investigated fluid flow through porous media in

ACCEPTED MANUSCRIPT the high velocity regime [32]. He proposed a correction of Darcy’s equation by incorporating an inertia term via the kinetic energy of the fluid. The Forchheimer equation for the pressure

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drop normalised by the length of the packed bed is given as follows:

[1]

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 p  2  wf  f wf . L k1 k2

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Where μ, ρf and wf are dynamic viscosity, density and superficial velocity of the fluid, and k1 and k2 are the Darcian and non-Darcian permeabilities, respectively. The superficial velocity

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is the hypothetical velocity, which the fluid would have if the channel was without packing. In contrast, the local velocity is the actual velocity of the fluid at any given position inside the

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reactor.

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The most common interpretation of permeabilities in the Forchheimer equation was given by

 S V2  solid (1   )2

and k2 

3

 SV solid (1   )

,

[2]

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k1 

3

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Ergun and Orning in 1949 [33]:

where ε and SV-solid are the open porosity and the specific surface area with respect to the solid volume, and α and β are Ergun parameters for viscous and inertial terms, respectively. By inserting Eq. 2 into Eq. 1 and substituting SV-solid by particle diameter dp, based on the geometry of a sphere dp = 6/ SV-solid, we obtain the equation commonly known as the Ergun equation [1]:

p (1   )2  w f (1   )  w f ,  AErgun  BErgun 3 2 L  dp 3 dp 2

[3]

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AErgun and BErgun being coefficients for viscous and inertial terms, respectively. The relation

* ) 6  1.75 , 8

[4]

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AErgun  62  (2 *)62 150 and BErgun   6  (

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between viscous and inertial terms in Eq. 2 and 3 are expressed by:

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α* and β* being experimentally derived parameters that are closely related to the tortuous flow path of the fluid inside the porous media and therefore very important for understanding

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the differences between CFD and EC pressure drop predictions, which will be discussed in more detail in continuation. In addition, tortuosity factors which are included inside α* and β*

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parameters, and could be determined theoretically or experimentally, are one of the reasons

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that Ergun equation fails to predict the pressure drop more accurately, unless spherical

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particles and a column-to-particle diameter ratio higher than 10, are being used. Therefore, many alternations of the Ergun equation are being proposed to broaden the usage of EC, mainly correcting the Ergun equation by taking into consideration the influence of the

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confining walls and particle shape. Not going too much into details, some EC are presented in Tab. 1, which can be considered best suited for predicting pressure drops for systems used in this work according to literature and CFD results (the full list of EC can be found in the supplementary Tab. S1). Their advantages and drawbacks will be discussed later in the paper.

ACCEPTED MANUSCRIPT 2.2. Packing

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A variety of codes can be found using DEM approach mainly due to great interest of the video

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games programmers. However, those codes (physics engines) are optimized for fast

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calculation speed, only mimicking certain physical processes, rather than describing the real physical world. Nevertheless, due to today’s high computer power, physics engines for games and after effects in the film industry involving as much physics as possible inside the code are

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preferable. Therefore, they excite interest also for scientific purposes [37]. One of the

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programs that uses the benefits of a physics engine is software Blender (Blender Foundation), which was the tool of choice for recent scientific studies [38–40]. It is based on a large

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collection of codes, contained in the Bullet Physics Library, used to manage the dynamics of

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rigid bodies by solving the Newton-Euler equations for both translational and rotational motions. Furthermore, joints and contacts are processed using relevant constraint/contact

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equations. The most important advantage is that particle interactions are based on their actual

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surface mesh, so it is quite easy to use very complex shapes in a simulation.

Blender was used to realistically simulate the packing of particles into the reactor for different PB configurations. The reactor was represented by a simple tube with a diameter of 30 mm and a height of 50, 80 and 140 mm, depending on the size of the filler particles. Spherical and arbitrary particles (cylinders, trilobes and quadrilobes) randomly orientated in a container above the tube were allowed to fall freely under the gravitational pull into the reactor to obtain packings with a column-to-particle diameter ratio (λ) of 10, 8, 4 and 2. Additionally, for arbitrary fillers, the ratio between particle height (hp) to equivalent disk diameter (dDS) were varied between 1–4 (see Fig. 1), ending up with 52 different geometries for CFD to study. Number of particles in packed bed reactor for λ = 10, 8, 4 and 2 were approximately

ACCEPTED MANUSCRIPT 800, 400, 95 and 25, respectively.

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2.3. Numerical setup

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The computational setup was designed to highly resemble the geometries used in the industry

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(e.g. trilobes and quadrilobes are very common for petroleum processing). Furthermore, the shapes of filler particles were chosen on the basis of the experimental data that could be found in literature. For spheres, cylinders, trilobes and quadrilobes, beds with a height (hb) of 30 mm

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for λ = 10 & 8, 60 mm for λ = 4 and 120 mm for λ = 2 were prepared with DEM approach

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described above. Water was forced into the tubular reactor (D = 30 mm), firstly, through a preconditioning zone, where the full parabolic velocity profile was established, and secondly,

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through the bed (see Fig. 2). Boundary conditions consisted of superficial velocity at the inlet,

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ambient pressure at the outlet and the no-slip condition at the reactor and particles walls. For each 52 geometries, five superficial velocities were chosen 0.01, 0.005, 0.001, 0.0005 and

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0.0001 m/s, majority being in the range of laminar flow (10 < Re*; for some extreme cases Re* numbers fall into lower region of laminar-inertial flow but still far away from turbulent

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regime). Simulations were conducted with the steady-state incompressible solver for the laminar flow which can be found in the OpenFOAM package, with the second order scheme (the physical properties of the fluid are listed in Tab. 2). For cases that Re* exceed value of 10 turbulent modelling was applied, however, pressure drops predictions were within 1% to laminar model solver values. Solution was considered to be converged when the residuals dropped for 5 orders of magnitude for both the velocity and the pressure field.

After each geometry simulation, the tortuosity (τ) at wf = 0.01 m/s was calculated as the ratio between the total winding path and the direct path of the streamlines. Approximately 1000 streamlines were evenly distributed across the reactor inlet in order for τ to be statistically

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3. Results and Discussion

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3.1. Grid independency study

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Another important step of the pre-processing phase is meshing the simulation domain so that the created mesh is later on used in the finite volume CFD code. The mesh generating utility

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snappyHexMesh, included in the open-source package OpenFOAM, was used. It generates a 3-dimensional mesh containing hexahedral cells that are automatically split to fit the

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triangulated surface geometries of Stereolithography (STL) format. The mesh approximately

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conforms to the surface by iteratively refining a starting mesh and morphing the resulting split-hex mesh to the surface to obtain an unstructured mesh consisting of hexahedral (bulk)

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and polyhedral cells at the walls. Previous researches indicate that this tool is an effective way of meshing the domain [38,41–44]. Because of its nature, no change of geometry posterior to

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the packing simulation by enlarging or shrinking particles is needed [45], so the porosity is not affected except for small bridges between the particles.

To obtain grid independent results, the mesh was refined until the steady state local velocity profile was converged (Fig. 3). Firstly, a coarse mesh was created which was gradually refined with the density profile of the nodes to be finer towards the reactor/particles walls. Secondly, bridges between particles where allowed to be additionally refined. To reach convergence, at least 10 million elements were needed and 2 additional refinements at bridges between particles. Typically the ratio between particle diameter and length of the mesh cells near the particle surface was 70 (thickness of the elements layer next to surface was  0.03

ACCEPTED MANUSCRIPT mm). At this conditions mean local velocity deviation relative to the finest mesh ( RD )

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dropped to 1% (Fig. 3).

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3.2. CFD validation

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Although CFD has been on the rise for more than half a century, only recent decade computers allow us to solve more complex 3-D geometries. Because of that it is very

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important to validate CFD results. To this end, pressure drops obtained by CFD calculations were compared to experiments found in literature for the widely-studied example of PB

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reactor packed with spheres. Due to great interest of this set up from the chemical engineer

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community, it can be used as the benchmark and therefore for CFD validation. Furthermore, this case is discussed in great detail, so the ability of CFD to resolve local phenomena can

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contribute to a better understanding of a given problematic. On Fig. 4 CFD results are compared with EC pressure drop predictions. For EC, rather than experimental points, fits are

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presented on the graph. While for λ = 10 all EC give similar predictions as CFD, lower λ could lead to a large mean relative deviation (MRD; Eq. 5) with respect to EC fit:

MRD 

1 N

N

 i 1

pi ,calc  pi ,EC pi ,EC

100 .

[5]

As expected the Ergun correlation strongly deviates from CFD calculations, since it does not contain any wall corrections. However, we observe a distinguish jump in MRD at λ = 2 in comparison with MRDs at λ > 2 (Fig. 4, see supplementary Tab. S2). This goes in line with calculated tortuosities at wf = 0.01 m/s where  for λ = 2 is significantly higher than  at λ > 2

ACCEPTED MANUSCRIPT (Fig. 4; see supplementary Tab S7). Because AErgun and BErgun are kept constant in Ergun equation, although τ is significantly higher at those conditions, the Ergun equation gives

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underestimated pressure drops. From Fig. 5 it is evident that streamlines in the case of λ = 2

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are more swirly and therefore need to travel a longer path (even back mixing can be detected)

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in comparison with streamlines when λ = 10 was chosen. On average, the additional path that the streamlines need to travel is prolonged by 5%. The reason for that could be concluded from Fig. 6 where radial porosity distributions are presented. Voidage variation was subject to

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numerous studies since it is well known that it influences the pressure drop at small λ (<10)

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[10–12,46,46–49]. CFD results for radial porosity distributions (RPD) were compared with the experimentally derived correlation by de Klerk which confirms the resemblance to real packing [50]. RPD was found to correlate to streamline disturbances that correspond to higher

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τ at lower λ. In theory the tortuosity for spherical particles is independent of particle size

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which goes in line that we calculated  to be the same for λ > 2 (see supplementary Tab S7).

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However, the repetition pattern of packed spheres is disturbed at the reactor wall (Fig. 6), which leads to an uneven distribution of voidage at low λ. Therefore, a pronounced velocity

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gradient is established near the reactor wall (Fig. 5). This impacts the flow characteristic leading to higher inertia loses that are anticipated in Ergun equation. Furthermore, at low λ, the surface area of the reactor wall with respect to the total surface area is not negligible any more so wall friction, which is not included in the Ergun correlation, additionally increases the error. The geometry of the packing is interrupted at the wall, which causes higher nearwall porosity. This leads to the phenomenon of "flow channelling", in which case the velocity profile reaches a maximum near the wall[51]. Although channelling could further pronounce wall friction it was not detected in our case. This is in agreement with previous findings where channelling occurred only in the case of structured sphere packings [46].

ACCEPTED MANUSCRIPT The correlation proposed by Allen et al. [36], which was chosen to be presented in Fig. 4 since it gives the best overall matching (sphere + arbitrary particles) with CFD, is in good

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agreement with CFD results for λ > 2 (MRD is lower than 6%, see Table S2-5). However, at λ

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= 2 the difference between Allen et al. and CFD prediction of the pressure drop remains high

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as in the case of Ergun (MRD ≈ 75 %). Although Allen et al. correlation was not specifically derived for λ < 10, it gives similar results as EC (e.g. Reichelt correlation [52]) that includes the correction of the wall. However, they treat viscous and inertia forces quite differently in

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the laminar-inertial regime. While Reichelt correlation gives significantly more weight to

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viscous forces, this is not the case for Allen et al. equation. According to CFD it seems that Reichelt equation underestimate inertia forces while Allen et al. overestimate them. Nevertheless, after adding inertia to viscous forces, the pressure drop predictions are very

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much alike in both cases for λ > 2.

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As expected, the lowest deviation between CFD and EC was in case where the wall effect was included inside the correlation. As example of such EC, Reichelt correlation is presented on

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Fig 4 (for other EC with wall correction factors see supplementary Tab. S1). In addition to correction factor M, which was firstly introduced by Mehta and Hawley [49], Reichelt tailored the Ergun equation by adding an empirically derived correction factor Bw. By applying those factors to the Ergun correlation, MRD never exceeds 15% for all λ (Fig. 4).

3.3. Pressure drops for arbitrary particles at λ > 2

As argued many times before in the literature it is highly unlikely that simple capillary models, such as the Ergun equation, would be sufficient to predict the pressure drops for arbitrary particles with great precision using universal constants. To overcome this problem,

ACCEPTED MANUSCRIPT two branches of new EC were developed, where dp was replaced by the equivalent surface volume diameter dSV = 6Vp /Ap . First, the already existing correlations for spherical particles

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were simply tailored by fitting new constants for either each respective geometry or groups of

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geometries with similar shape [11]. Second, correction factors were added to EC and so they

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become shape depended [35]. Alternatively, EC fitted for spherical particles by replacing dp with dSV can be used, however, high errors are expected. Below, one example per each approach is presented and compared with CFD results chosen on the basis of the highest

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relevance with simulated particle geometry according to literature (for comparison between CFD and the rest of EC see Tab. S2-S6). Since it was already shown in the case of spherical

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particles that a high error is induced at λ = 2, unless EC with wall correction is used, overall MRDs on Fig. 7–9 are referring to all λ > 2, however, all data points calculated with CFD are

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presented on the graphs. Due to the high differences between EC and CFD at λ = 2, special care is needed to interpret the deviation between them, which will be discussed in next

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section.

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Latter solution for the prediction of pressure drops in the case of arbitrary particles by EC is shown on Fig. 7, where CFD results are compared with the Ergun correlation. Regarding to experimental data from literature, Ergun equation should give under predicted pressure drops for cylinders and polylobed extrudates (PE), which is in agreement with CFD results [35,36,53]. As expected, overall MRD is rising with the complexity of the geometry, since parasitic drag is dependant of the shape of the particle (see Fig. 7). Naturally, this approach gives the highest error and should be avoided if possible. According to CFD calculations, EC predicted pressure drops for cylinders would be about 30% of the real value and with PE error jumps to ~50% for λ > 2.

ACCEPTED MANUSCRIPT The correlation that was developed to cover the particle shape effect is presented on Fig. 8 along with the CFD results. Nemec&Levec enhanced the Ergun equation by adding to it the

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sphericity factor (Φ) [35]. With this correction, the overall MRD for individual geometry

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(cylinder, trilobe, quadrilobe) is ~13% (see Fig. 8) which is close to the estimated accuracy of

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the correlation (authors stated that the correlation is able to predict the pressure drop within 10%). It should be noted that Φ exponent is different for cylinders and PE, however, after the linearization of Nemec&Levec correlation, both can be presented on the same graph. Even

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though Φ brings EC and CFD simulated pressure drop predictions closer together, deviations (EC vs. CFD) vary with the hp/dDS ratio (ζ, see Tab. S2-5), but not between different geometry

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groups. This suggest that Φ alone is not sufficient enough to smooth the effect of geometry, so another aspect needs to be considered – wetted surface, which will be discussed in next

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paragraph.

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The third option in dealing with pressure drops for arbitrary particles by using correlations is to fit constants for each specific geometry group (Fig. 9). Allen et al. in their work proposed

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an equation where rather than dSV, the total particle volume ∑Vp to surface ∑Sp ratio is used [36]. They stated that ∑Vp / ∑Sp ratio should be understood to refer to the duct size rather than an equivalent particle size (dSV = 6∑Vp / ∑Sp), as used by Ergun. The correlation highly resembles the Kozeny–Carman equation, but they broaden the usage by separately fitting constants not only for spheres but also cylinders, cubes, acorns and crushed rocks [2]. Surprisingly, Allen correlation gave the least deviation with respect to CFD for PE, although according to literature that should be Nemec&Levec. Overall MRD for cylinder beds is ~15 % and ~9 % for trilobe and quadrilobe beds (see Fig. 8 and 9), which is significantly lower for the latter two geometries than in the case of Nemec&Levec correlation. It is worth knowing that Nemec&Levec specifically fit their correlation for PE while constants for cylinders were

ACCEPTED MANUSCRIPT used in the case of Allen equation (since it is the closest geometry to PE). Therefore, we expect that Nemec&Levec would outperform the Allen correlation and the latter would be

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true if we would follow the Allen equation to the letter and use the theoretical volume and

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surface area of the particles. However, geometry parameters were taken from the obtained

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meshes. While the particles volume was always close to the theoretical value (~99%), their surface area could differ from the theoretical for more than 20% in some cases. One may argue that this is the error from the meshing procedure, but it should be noted that in the case

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of spheres, the surface area was always > 95% for all λ in comparison with the theoretical

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value. Hence, we cannot rule out that minor error is introduced during meshing, however, in the case of the arbitrary particles, the majority of the deviation from the theoretical surface area is contributed from another factor. Unlike spheres, arbitrary particles may consist of a

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certain number of flat surfaces. During the packing, those surfaces could overlap and therefore give a lower wetted surface than theoretical values. This affects the skin friction

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drag and consequently the pressure drop of PB reactor. Dolejš&Machač proposed a correlation which included a correction factor for the wetted surface (ω), but ω was obtained

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by fitting the experimental data for a specific geometry and was not measured directly [54]. Unless investigated PB reactor falls on the particle geometry that was already covered in their work, additional experiments of the pressure drop for each particle shape needs to be performed individually. This is very time consuming, and moreover, much simpler correlations could be used if experiments were carried out and fit constants to conform specific particle geometry without any radical decrease of accuracy in comparison with Dolejš&Machač.

Going back to the Allen correlation, if surface areas obtained from meshing procedure are used, not only does the overall MRD outperformed Nemec&Levec correlation, but also

ACCEPTED MANUSCRIPT significantly reduces ζ impact on the pressure drop deviation between CFD and EC (Fig. 10, see Tab. S2-5). This indicates that ω is more relevant to the pressure drop than Φ in the

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laminar regime. However, it is difficult to properly measure the wetted surface directly, so for

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now Φ correction factor is the best choice for predicting the pressure drop with EC using the

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least amount of effort. Nemec&Levec equation is closely followed by Eisfeld&Schnitzlein correlation [55], which gives slightly better results at higher λ (see Tab. S2-6), but CFD data

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are much more scattered around the correlation curve (see Fig. 8 and Fig. S1).

On Fig. 11 are presented typical representation schemes of the results obtained by CFD

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simulation for quadrilobes at wf = 0.01 m/s, λ = 10 and ζ = 4. The velocity field is highly inhomogeneous along the radial axis of the packed bed reactors (to some degree this also true

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for pressure field), which put homogeneity models for solving mass transport with chemical reactions within such reactors into question. Furthermore, from this point of view it is

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surprisingly that EC for pressure drop predictions offer so much accuracy. However, as would be explained in the next section, this is true only for the reactors that contain enough amounts

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of filler particles which consequently statistically averaged the local phenomena effects.

3.4. Pressure drop for arbitrary particles at λ = 2

As mentioned before, EC pressure drop deviation for arbitrary particles increases drastically in comparison with CFD results when λ < 4. While spheres were subject to numerous studies at low λ ratios, arbitrary particles were not given as much attention. Usually findings from spherical particles were taken and applied to other geometries. What does not make the relation between spherical and non-spherical particles straightforward is that spheres cannot

ACCEPTED MANUSCRIPT influence the pressure drop with their orientation. As shown by Pahl in his early work in 1975, the opposite stands for arbitrary particles [56]. While it was known before that the

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porosity function inside the Ergun equation will be inadequate for non-spherical particles,

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engineers were still puzzled, because the pressure drop data were much more scattered than

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anticipated. Pahl attributed this effect to the particle orientation by investigating cylinders and their impact on the pressure drop while varying ζ and the packing procedure. His assumption was later confirmed by several studies [57,58]. In line with previous findings, CFD results of

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the present work also show a significant impact on the pressure drop due to particle orientation. On Fig. 12, mean particle orientations (MPO) of quadrilobes for all combinations

o

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of λ and ζ with corresponding MRDs (CFD vs. Allen) are presented (similar results stand also for other two arbitrary geometries; 90 means that the particles are parallel with their largest

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dimension in the direction of the reactor axis). As expected, the orientations at λ = 2 significantly deviate from the orientations at higher λ, since at these conditions the driving

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force of the reactor wall to orientate the particles vertically is the largest. While there is a distinctive trend of raising the percentage of particles being oriented vertically with raising ζ

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at λ = 2, it gets less pronounced at lower λ and almost reversed at λ = 10. At high λ, the o

dependency of ζ on particle orientation is rather poor, with MPO around 25 . This is in agreement with the Vollmari et al. experiment where they measured MPO for cylinders at ζ = o 3.5 and λ = 15 to be 24 [6]. Next, tortuosity was calculated for all λ at ζ = 4 and wf = 0.01 m/s

(these conditions were chosen since the impact of ζ on particle orientation was the largest). Surprisingly, τ was independent of the shape group and therefore, average τ of all three arbitrary particles are labelled on Fig. 12 (for τ values of individual geometries see Tab. S7). Tortuosities for λ > 2 were fairly close to each other, however, at λ = 2, τ is significantly lower (it drops from ~1.2 to ~1.1), most likely due to preferential orientation (MPO at this o

conditions was 76 for ζ = 4). This may explain why there is such a big difference in the

ACCEPTED MANUSCRIPT pressure drop predictions between CFD and EC at low λ. In addition, tortuosities for all ζ at λ = 2 and wf = 0.01 m/s were calculated. Again, τ was independent of the particle shape group.

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In addition, for all ζ > 1 τ seems to be also independent of ζ. All in all, no distinguishing trend

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could be found between τ and particle shape. On the other hand, MPO seems to have

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influence on the τ which has a significant impact on the pressure drop, since every deviation of those two parameters from values obtained for an infinite wide channel (we can

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approximate this for λ = 10) leads to higher MRD.

Lastly, the influence of confining walls on the pressure drop for arbitrary fillers was

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investigated by plotting modified friction factors vs. λ, as computed from CFD simulations for Rep=2 (Figs. 13–14). Results suggest that high ζ (=4) at low λ (=2) could lead to a reduction

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of the pressure drop (this effect is especially evident for cylindrical geometry). This is contradictory to common belief since the wall effect should increase the pressure drop at these

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conditions. Low tortuosity and high MPO may be the reason that suppresses the effect of the confining wall. However, at low ζ (=1) and λ (=2) this phenomenon disappears and the system

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behaves as normally predicted in the literature (Fig.14).

4. Conclusions

To conclude, the pressure drops inside PB reactor for spherical and arbitrary particles (cylinders, trilobes and quadrilobes) were investigated by the aid of CFD. Results obtained by CFD were compared with the most common EC and a general agreement exists between them when the packed bed characteristic parameters were inside EC validity. For some extreme

ACCEPTED MANUSCRIPT cases, such as λ = 2, MRD raised over 50% which makes EC quite impractical. By means of tortuosity and particle orientation it was shown that at these conditions, EC are inadequate to

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fully balance those two parameters. Their values are mostly associated with form drag and

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therefore at lower Re number their impact on the pressure drop is less pronounced. In

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addition, MPO and τ seem to be constants for infinitely wide reactors, which in practice means for all λ > 10. Therefore, no special care is needed for EC. Fitting constants to reflect the specific geometry is sufficient that EC give solid predictions. Furthermore, the most

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relevant variable for predicting the pressure drop in the laminar regime was found to be the

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wetted surface. Although skin friction drag dominates at these conditions for higher Re numbers (but still in laminar regime) the pressure drop contribution from form drag is not negligible anymore. The latter is evident when comparison is made between Allen [36] and

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Nemec&Levec [35] EC, where the Allen correlation outperforms Nemec&Levec at lower Re numbers, but vice versa is true for higher Re numbers. This indicates that Φ inside

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Nemec&Levec correlation is more related to form drag and therefore more appropriate for use at higher Re numbers. All in all, it seems that ω, τ and MPO are indispensable parameters for

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pressure drop predictions in case of arbitrary particles if we want to use EC at λ < 4. Unfortunately, all three parameters are rather hard to obtain by experiments so Nemec&Levec correlation remains the best choice for arbitrary particle investigated in this paper. On the other hand, CFD has no problem evaluating those bulk parameters. Even more, it provides a full inside view on the local phenomena which can greatly contribute to a better understanding of PB reactors. Lastly, the described tasks are a first step in building a stable framework for the virtual modelling of PB reactors and will continue in coupling validated fluid flow models with other chemical transport phenomena and reaction kinetics. This will provide a good basis for the unification, gradual supplementation and expansion of the empirical methods for the determination of PB reactor design parameters.

ACCEPTED MANUSCRIPT

shape defining parameter [/]

AErgun

Ergun coefficient for viscous energy losses [/]

Ap

particle surface area [m2]

BErgun

Ergun coefficient for kinetic energy losses [/]

BW

wall correction term [/]

c

shape defining parameter [/]

D

column diameter [m]

dDS

equivalent disk diameter [m]

dp

particle diameter [m]

dSV

equivalent surface volume diameter [m]

f*

modified friction factor [/]

fAllen

Allen modified friction factor [m-1]

fN&L

Nemec&Levec modified friction factor [/]

hp

particle height [m]

L

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hb

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ai

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Nomenclature

bed height [m] length of the packed bed reactor [m]

M

modification factor [/]

N

number of samples [/]

r

distance from reactor axis [m]

R

reactor radius [m]

Re*

Ergun modified Reyndols number [/]

ReAllen

Allen modified Reyndols number [m-1]

ReDuct

modified Reyndols number for duct [/]

ACCEPTED MANUSCRIPT Nemec&Levec modified Reyndols number [/]

Rep

modified Reyndols number for particles [/]

SV-solid

specific surface area with respect to the solid volume [m2]

T

temperature [K]

Vp

particle volume [m3]

wf

superficial velocity [m s-1]

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Greek symbols

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ReN&L

Ergun coefficient for viscous term [/]



Ergun coefficient for inertial term [/]



column-to-particle diameter ratio [/]



tortuosity [/]



wetted surface [m2]



dynamic viscosity [Pa s]

*

Ergun coefficient for viscous term [/]

*

Ergun coefficient for inertial term [/]

pi,calc

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p

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

pressure drop [Pa] pressure drop predicted by CFD calculation [Pa]

pi,EC

pressure drop predicted by EC [Pa]

ε

porosity [/]

ζ

hp/dDS ratio [/]

ρf

density of the fluid [kg m-3]

Φ

sphericity [/]

ΦAllen

packing dimension [m]

ACCEPTED MANUSCRIPT Abbreviations computational fluid dynamics

DEM

distinct element method

EC

empirical correlations

MPO

mean particle orientations

MRD

mean relative deviation

PB

packed bed

RPD

radial porosity distributions

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CFD

Subscripts fluid

i

index

p

particle

b

bed

V-solid

solid volume

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D

f

Acknowledgements

The authors gratefully acknowledge the European Commission, as the herein-presented research work was partially established within the MefCO2 project. This project has received funding through a Sustainable Process Industry through Resources an Energy Efficiency (SPIRE) call under the European Union’s Horizon 2020 research and innovation programme (grant agreement n° 637016). The authors also gratefully acknowledge the Slovenian Research Agency (ARRS) for funding the program P2–0152 and project J2–7319.

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ACCEPTED MANUSCRIPT X. M. Mayerhofer, J. Govaerts, N. Parmentier, H. Jeanmart, L. Helsen, Experimental

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investigation of pressure drop in packed beds of irregular shaped wood particles,

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ACCEPTED MANUSCRIPT Captions Figures

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Figure 1: Scheme of the packing procedure.

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Figure 2: Scheme of the computational setup.

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Figure 3: Cross section local velocity for PB reactor filled with spherical particle of dp=7.5 mm at wf =0.01 m/s.

Figure 4: Pressure drop comparison between CFD and EC for spherical particles.

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Figure 5: Streamlines of PB reactor filled with spheres at wf = 0.01 m/s for λ = 2 (left) and λ = 10 (right) geometries (velocity legend refers to both cases).

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Figure 6: Radial porosities of PB reactor filled with spheres for various λ. Figure 7: Modified friction factor predicted by CFD and Ergun correlation.

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Figure 8: Modified friction factor predicted by CFD and Nemec&Levec correlation. Figure 9: Pressure drop predicted by CFD and Allen et al. [36] correlation.

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Figure 10: MRD variation in dependency of ζ at different λ and particle shape. Error bars represent min and max deviation from mean MRD at specific λ.

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Figure 11: CFD results for quadrilobes at wf = 0.01 m/s, λ = 10 and ζ = 4: (top-left) streamlines coloured by velocity field, (top-right) streamlines coloured by pressure field, (bottom-left) cross section of the PB reactor (upper-half => pressure filed, bottom-half => velocity field) and (bottom- right) computational mesh. Figure 12: Mean particle orientation for quadrilobe for various ζ and λ with corresponding MRD. Additionally, average tortuosities of arbitrary particles are labelled for various ζ and λ. Figure 13: Modified friction factors vs. λ at ζ = 4. Figure 14: Modified friction factors vs. λ at ζ = 1. Tables Table 1: Correlation equations.

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Table 2: Fluid properties.

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ACCEPTED MANUSCRIPT Table 1 Equation for

Wall

Modified friction factor and Reynolds

Constants

Δp / L

correction

sphere 2 (1   ) 2   w f 1    wf 150    1.75  3  f 3 2  dp  dp

sphere 2 (1   )2   w f 1    wf M   M2  3  f  3 2  dp  dp BW

M 1 

4dp

/

; BW  a1 (dp / d )2  a2 

2

6D(1   )

cylinder 2 150 (1   ) 2   w f 1.75 1    f  w f   2  4  3  3 3 3 2  d SV  d SV  

polylobed extrudates

D TE

sphere&cylinder

 

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Ap  1     a1 w 2f a2     Re  Re c    f  2  4  V    3  Duct  p  Duct

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a2=1.5

YES

a2 = 0.88

f N&L  f * 

4

3

 ; Re N&L  Re* 

1

6

polylobed extrudates

AErgun = 150 4

fN&L  f * 2 ; Re N&L  Re* 

2 150 (1   ) 2   w f 1.75 1    f  w f   2  2  3  6 3 5  d SV   d SV 

Allen et al. [36]

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[35]

NO BErgun = 1.75

cylinder

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Nemec &Levec

AErgun = 150

c = 150

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Reichelt [34]

c

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wd Re *  f f p  (1   )

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Ergun [1]

(p / L) 3dp A  Ergun  BErgun (1   )  f w 2f Re*

f *

f N &L 

AErgun Re N & L

 36  Vp2  B Ergun ;    3  Ap

NO 5

1

3 6  Vp  ; d SV   Ap 

sphere

2  f * Re cDuct  Allen f Allen   a1 Re Allen  a2 ; dSV  Re*   Re Allen     dSV  Allen  Re Duct 

BErgun = 1.75

c 1

4  Vp fw   Allen;  Allen   (1   )  Sp

a1 = 172 a2=4.36 c=0.12

NO cylinder a1 = 216 a2=8.8 c=0.12

ACCEPTED MANUSCRIPT Table 2 Abbreviation

Value

T

293.15 K

Viscosity (water)

μ

0.001

Pa∙s

Density (water)

ρf

1000

kg/m3

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Temperature

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Properties

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Graphical abstract

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CFD simulations for 52 different packed bed reactor geometry set-ups were conducted A comprehensive comparison between CFD and empirical correlations (EC) was made EC give solid prediction of pressure drops for column to particle diameter > than 4 Wall influence on the pressure drop at low  (<4) is depended of particle shape Most relevant parameter for pressure drop prediction is wetted surface (laminar flow)

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• • • • •