1. Introduction
Fluidized beds are one of the most widely used chemical reactors in many industry areas. It has been used mostly for gas-solid reactions as it provides some advantages including its easiness of handling and transport of solids due to fluid-like behavior, intensive mixing of solids leads to a consistent temperature distribution and favorable conditions for heat and mass transfer. [
1]. Despite its advantages, it has also some drawbacks which heavily relies upon then complexity of hydrodynamics behavior and the difficulty in scaling up the reactor [
1,
2]. Historically, empirical methodologies derived from laboratory-scale investigations of fluidized-bed reactors have been instrumental in calculating key operational parameters, such as minimum fluidization velocity, bubble dynamics, particle attrition rates, and the axial distribution of solid mass fractions [
3].
Numerical simulations offer a shortcut to circumvent the extensive process involved in designing and executing experiments, enabling rapid assessment of relevant fields at the industrial scale [
4]. One of the numerical methods widely used is so-called Computational fluid dynamics (CFD). Several CFD models exist for simulating gas-solid flows, including the two-fluid model, discrete-particle model, hybrid model, and direct numerical simulations. The two-fluid method, particularly its most utilized form, the Eulerian-Eulerian (E-E) approach, considers both phases as interpenetrating continua. Closure relations based on the kinetic theory of granular flows (KTGF) help describe the solid phase with gas-like properties, considering the particles’ motion influenced by interactions within the gas phase and with other particles or walls [
5]. In the discrete-particle approach, the fluid phase is treated as a continuum, while the solid phase is considered discrete, tracking individual particles within the computational domain. This method, notably utilized in the Eulerian-Lagrangian (E-L) framework, is recognized for its unique particle collision detection, which can be either stochastic or deterministic, facilitating detailed analysis of particle dynamics [
6]. In the hybrid methods, the two-fluid and discrete-particle are integrated to leverage both models’ benefits. This sub-domain method simplifies the computational domain based on an equilibrium index, optimizing simulation efficiency and accuracy [
7]. In Direct Numerical Simulation (DNS), it solves directly the Navier-Stokes equation aiming to capture the full range of scales and dynamics by using high-resolution grids and accurate numerical schemes. However, DNS is also limited by its high computational cost and low Reynolds numbers [
8].
Over the past 20 years, numerous studies on gas-solid CFD models have been conducted using commercial CFD software, e.g., ANSYS FLUENT [
9], ANSYS CFX [
10], OpenFOAM [
11], MFiX [
12], BARRACUDA Virtual Reactor [
13], CFDEM coupling [
14], COMSOL Multiphysics [
15], STAR-CCM [
16] and EDEM [
17]. Given the fluidized bed’s complex hydrodynamics, researchers focus on examining various parameters that influence gas-solid flow dynamics. Essential factors to consider in numerical models include superficial gas velocity, drag, specularity and restitution coefficients, and frictional viscosity, as these significantly affect the behavior of fluidized beds [
18].
Taghipour et al. [
19] compared hydrodynamics in a fluidized bed through experiments and FLUENT 6.0 simulations. The study used a pseudo-2D experimental setup and a 2D Eulerian-Eulerian model incorporating Gidaspow [
5], Syamlal-O`Brien [
20], and Wen-Yu drag models. Results showed the bed reaching steady state in 3 seconds, with consistent bed expansion and bubble formation. While the drag models closely matched experimental observations, discrepancies in voidage and bed expansion highlighted the need for further CFD model validation.
Cornelissen et al. [
21] investigated the impact of mesh size, time step, and convergence criteria on the accuracy of a liquid-solid fluidized bed model in FLUENT 6.1.22, benchmarking their findings against experimental data. They determined that a convergence criterion of
led to faster, convergent results compared to the standard
. Additionally, they noted that Courant numbers within the range of
to
produced reliable outcomes regardless of mesh size and time step.
Li et al. [
22] explored how solid phase boundary conditions affect simulations by adjusting specularity and particle-wall restitution coefficients in both 2D and 3D Eulerian-Eulerian models using MFIX software, comparing results to a pseudo 2D experiment. They discovered that the restitution coefficient’s impact was minimal compared to the specularity coefficient, with specularity coefficients under 0.05 leading to notable deviations in solid flow near walls and altering circulation patterns. While both 2D and 3D simulations captured the general flow behavior, 3D simulations aligned more closely with experimental data, emphasizing the significance of wall effects in pseudo 2D experiments for CFD model validation.
Herzog et al. [
23] conducted a comparative analysis using MFIX, OpenFOAM versions 1.6/2.0, and FLUENT 6.3 against experimental findings by Taghipour, focusing on a 2D Eulerian-Eulerian using Gidaspow and Syamlal O`Brien drag models. The study revealed that FLUENT and MFIX closely matched experimental data for both drag models, whereas OpenFOAM’s predictions were not satisfactory, suggesting OpenFOAM requires further refinement and validation for fluidization prediction.
Shi et al. [
18] carried out the research to align 2D and 3D models with Taghipour et al. [
19] experimental findings, utilizing Eulerian-Eulerian frameworks in FLUENT 16.2. Their investigation spanned dimensions, flow regimes, boundary conditions, mesh granularity, and drag models. Optimal grid size emerged as 18 times the particle diameter for closest experimental alignment, with finer meshes diminishing accuracy. Minor variances were observed within specific specularity and restitution coefficient ranges, deeming them suitable. Turbulence’s impact was minimal within the bed but influenced gas flow above, recommending laminar models for non-reactive, isothermal beds and RANS
models otherwise. Syamlal-O`Brien drag models, particularly with Johnson-Jackson boundary conditions and turbulent RANS models, showed superior experimental congruence. They noted 3D models matched experimental data more closely than 2D, cautioning that certain 2D model parameter combinations might mislead due to unrealistic assumptions. Consequently, they advocated for 3D validations, reserving 2D analyses for pre-validated model sensitivity studies.
In the current study, the fluidized bed columns from Taghipour et al. [
19] were analyzed in both 2D and 3D simulations using three different CFD software codes: ANSYS Fluent 2023 R3, MFiX 23.1.1, and OpenFOAM 10. Simulations in MFiX and OpenFOAM were realized as part of the MSc thesis of Daun [
24]. The current work extends the analysis presented by [
24], and compares results further with new simulations realized in ANSYS Fluent. This investigation included the examination of two drag models, Syamlal-O’Brien and Gidaspow, across varied superficial gas velocities. Additionally, the study explored the effects of varying restitution and specularity coefficients. Simulations employed a Eulerian-Eulerian framework underpinned by the kinetic theory of granular flow (KTGF) aiming to highlight and address the subtleties inherent in inter-software comparisons.
4. Results
The results, derived from various simulations, are delineated in terms of bed expansion ratio (
), pressure drop, solid volume fraction contour plots, solid velocity at
m, and voidage at
m. These results were obtained using three distinct CFD codes and evaluated across two drag models. The bed expansion ratio and pressure drop data encompass all examined superficial velocities: 0.03, 0.1, 0.38, 0.46, and 0.51 m/s, corresponding to 0.5, 1.5, 6, 7, and 8 times
, respectively. In contrast, the solid volume fraction contour plots specifically showcase three selected superficial velocities: 0.2, 0.38, and 0.46 m/s, at times of 0.4 s, 1 s, and 3 s. Additionally, the analyses of solid velocity and voidage at
m encapsulate the effects of variations across all two drag models, and the three CFD codes at 0.38 m/s of superficial gas velocity. The simulation outcomes, including bed expansion ratio, pressure drop, and voidage, were evaluated against the experimental results from Taghipour et al. [
19], thereby assessing the simulations’ accuracy by using the formula for standard deviation as elucidated in Shi et al. [
18].
4.1. Bed Expansion Ratio and Pressure Drop
The expansion of the bed was assessed by charting the time-averaged solid fraction at the bed’s center vertically. A criterion was established where a solid fraction value below 1% indicated the extent of bed expansion. This procedure was applied across various superficial gas velocities to observe how bed expansion varied with changes in inlet velocity. The summary of the results of bed expansion and pressure drop are summarized in
Table 1.
Figure 2 presents the bed expansions determined using the Syamlal-O`Brien drag model within Fluent, MFiX and OpenFOAM. From this figure, it is evident that all CFD codes forecast minimal bed expansion at lower velocities, with a linear increase in expansion noted for velocities exceeding 0.38 m/s. In 3D scenarios, bed expansion appears reduced compared to analogous 2D situations. Moreover, the figure illustrates that Fluent tends to forecast greater bed expansion compared to MFiX and OpenFOAM.
Figure 3 illustrates the bed expansions observed when employing the Gidaspow drag model in MFiX and OpenFOAM. From this figure, it is apparent that three software packages register bed expansion at all velocities above the minimum fluidization velocity. The expansion appears to increase linearly in relation to the superficial gas velocity. The 3D simulations show a notably lesser degree of bed expansion compared to their 2D counterparts. A comparison with
Figure 2 reveals that the expansions resulting from the Gidaspow model from OpenFOAM and MFiX exceed those observed with the Syamlal-O`Brien model. Surprisingly, Fluent forecasts lower bed expansions than OpenFOAM and MFiX when utilizing the Gidaspow model. Additionally, a contraction in the bed at the lowest gas velocity is noted in Fluent and MFiX with both drag models, a phenomenon not observed in OpenFOAM simulations.
The pressure drop across the bed was assessed at the outlet of the fluidized bed reactor for each tested superficial gas velocity.
Figure 4 and
Figure 5 illustrate a pattern where the pressure drop initially spikes to a peak value before gradually diminishing as the superficial gas velocity increases. It is observed that with the Gidaspow model, the peak pressure drop occurs at lower superficial gas velocities compared to the Syamlal-O`Brien model. Additionally, both figures indicate that MFiX and Fluent estimates a greater pressure drop across all tested superficial gas velocities than OpenFOAM, in both 2D and 3D simulations. A comparison between 2D and 3D results shows that Fluent and OpenFOAM reports a smaller pressure drop in 3D compared to 2D, whereas MFiX exhibits an increase in pressure drop from 2D to 3D simulations. It is also worth mentioning, specifically for Fluent, that the pressure drop at superficial velocities lower than the fluidization velocity tends to predict a very high pressure drop. This observation aligns with the findings from Taghipour et al. [
19], El Ajouri et al. [
33], suggesting that the primary cause could be attributed to the solids not being fully fluidized at these lower velocities, thereby being predominantly influenced by interparticle frictional forces. Such forces are not adequately predicted by the multifluid model used for simulating gas–solid phases in Fluent, which may not incorporate these frictional interactions effectively.
4.2. Solid Velocity and Voidage
Figure 6 displays the time-averaged solid particle velocity in the y-direction at a height of 0.2 meters, with a superficial gas velocity of 0.38 m/s, utilizing both Syamlal-O`Brien drag model in Fluent, MFiX and OpenFOAM. The observed trend across all data lines reveals a downward velocity near the walls transitioning to a parabolic upward velocity near the bed’s center. Notably, Fluent predicts significantly higher velocities than MFiX, particularly near the walls and the reactor’s center. Furthermore, for 3D simulations, velocities were averaged across the z-direction cells to align with the 2D results, offering a comprehensive view of the domain. While MFiX’s 3D outcomes closely resemble its 2D counterparts, OpenFOAM and Fluent markedly diverge, maintaining a relatively constant velocity across the bed as opposed to the parabolic profile observed in 2D simulations. In contrast to the Syamlal-O`Brien drag model,
Figure 7 illustrates the time-averaged solid particle velocity using the Gidaspow drag model, processed similarly to
Figure 6. The outcomes indicate that Fluent and MFiX yield nearly identical predictions, with velocities marginally higher at the bed’s center but displaying a somewhat flattened profile across both 3D and 2D simulations. Conversely, OpenFOAM’s results diverge slightly, showing higher velocities near the walls and at the center of the bed in both 3D and 2D scenarios. Notably, the 3D results in OpenFOAM exhibit a slightly flatter velocity profile compared to their 2D counterparts, suggesting a distinct behavior in spatial velocity distribution when using the Gidaspow model.
Further results, as shown in
Figure 8, compare the time-averaged voidage for 2D and 3D simulations at the same bed height and gas velocity, employing the Syamlal-O’Brien drag model. This figure indicates that Fluent displays the highest voidage in 3D simulations compared to others, presenting an asymmetrical appearance relative to the rest. In OpenFOAM, the 3D simulations show a maximum voidage nearly identical to their 2D counterparts. Conversely, MFiX’s 3D simulations exhibit a zone of low voidage extending from the wall, which sharply increases to a peak significantly greater than that observed in 2D simulations at the center of the bed. On the other hand,
Figure 9 showcases the time-averaged voidage for 2D and 3D simulations at the same bed height and gas velocity, utilizing the Gidaspow drag model. From this figure, it is evident that MFiX’s 3D simulations replicate the behavior observed in the Syamlal-O’Brien case, with voidage sharply increasing at the center of the bed in 3D simulations. For the Fluent simulations, both 2D and 3D present almost identical patterns, though the 3D simulations exhibit slightly higher voidage at the left wall. In OpenFOAM, the 3D results are slightly higher than 2D, maintaining a nearly similar pattern.
4.3. Solid Volume Fraction
The evolution of the fluidized beds and time averaged solid volume fractions with three chosen superficial gas velocities equal to 0.2, 0.38, and 0.46 m/s as the results from three different CFD codes with Syamlal-O`Brien drag model as well as Gidaspow drag model are depicted in
Figure 10 and
Figure 11. Initially, the bed height rose with the formation of bubbles until it stabilized at a constant steady-state height. The solid flow patterns inside the bed are relatively symmetrical for all cases before t < 3 s. The chaotic and transient bubble formation process at 3 seconds indicates that the statistical steady-state condition was achieved. Therefore, it confirmed the previous findings that a simulation duration of 3 seconds is sufficient to attain statistical steady-state behavior across all drag function models [
19,
23].
At the beginning of simulation, Fluent and MFiX display comparable patterns, with OpenFOAM showing slight deviations. As the flow progresses, the patterns of flow and bubble formation for both Syamlal-O’Brien and Gidaspow models remain similar in Fluent and MFiX but differ in OpenFOAM, highlighting that the numerical results of MFiX and Fluent are quite aligned, as also confirmed by Herzog et al. [
23]. A closer examination reveals good agreement in the time-averaged solid volume fraction among all codes. All cases show a lower voidage observed near the wall and at the base of the bed, whereas an increased voidage is noted in the bed’s central area. This pattern of solid volume fraction distribution indicates a higher formation of bubbles in the central region as opposed to the bottom. Specifically, in the Syamlal-O`Brien model, Fluent exhibits a higher voidage and greater bed expansion compared to MFiX and OpenFOAM. Conversely, in the Gidaspow model, OpenFOAM demonstrates larger bed expansion and more voidage. The case studies with three superficial velocities further reveal that MFiX tends to display a more distinct phase border than OpenFOAM and Fluent, where phase interfaces blend more smoothly. This distinction appears to lead to the formation of more, but smaller, bubbles in OpenFOAM, whereas MFiX features fewer but significantly larger bubbles, highlighting the intricate dynamics of fluidized beds as captured by different simulation tools and drag models.
Author Contributions
Conceptualization, K.E.E., P.F.D. and P.K. ; methodology, K.E.E., P.F.D. and P.K.; software, P.F.D. and P.K.; validation, P.F.D. and P.K.; formal analysis, P.F.D. and P.K.; investigation, P.F.D. and P.K.; resources, K.E.E.; data curation, P.F.D. and P.K.; writing—original draft preparation, P.K.; writing—review and editing, K.E.E., P.F.D. and P.K. ; visualization, P.F.D. and P.K.; supervision, K.E.E; project administration, K.E.E.; funding acquisition, K.E.E. All authors have read and agreed to the published version of the manuscript.