Article pubs.acs.org/IECR
Fine Mesh Computational Fluid Dynamics Study on Gas-Fluidization of Geldart A Particles: Homogeneous to Bubbling Bed Priya C. Sande† and Saumi Ray*,‡ †
Department of Chemical Engineering, and ‡Department of Chemistry, Birla Institute of Technology and Science, Pilani, 333031, India S Supporting Information *
ABSTRACT: Gas-fluidization of Geldart A particles was simulated for a domain of lab-scale dimensions. Hydrodynamics of homogeneous regime and transition to bubbling were studied. In this context a detailed fine mesh simulation study is presented for the first time, using the state-of-the-art two-fluid model (TFM). The effect of particle density was investigated. The fine mesh simulations were analyzed for insights into bed transition from homogeneous to bubbling regime and the effect of interparticle forces (IPFs). Simulations reveal that transition to bubbling occurs over a velocity range rather than at a discrete velocity. We propose an index to quantify the effect of IPFs on bed expansion. During homogeneous expansion this IPF index was found to drop exponentially with velocity. It became negligible as bubbling ensued which is in line with the literature. The simulated average bed voidage was found comparable to the corresponding Eulerian−Lagrangian and experimental data.
1. INTRODUCTION According to the well-known powder classification of Geldart,1 group A powders generally have particle diameter and density less than 130 μm and 1400 kg/m3, respectively, but are distinguished from group C powders by their aerateable quality. Their industrial application is very broad and includes fluidized catalytic cracking and fluidized combustion or gassification. Gas−solid fluidization is generally divided into homogeneous and bubbling fluidization. The former is a striking feature of Geldart A particles and is identified by the apparently uniform suspension of particles in the gas-phase without the formation of bubbles and agglomerates which characterize bubbling beds. Homogeneous fluidization is said to occur in the velocity range between minimum fluidization velocity and minimum bubbling velocity. Computational fluid dynamics (CFD) has become a powerful tool to explore the complex hydrodynamics of gas− solid fluidization. It encompasses the Eulerian−Lagrangian approach and the Eulerian−Eulerian approach. The latter is used to elicit the well-known two-fluid model (TFM)2 used in this work in which phases are considered interpenetrating continua. The Eulerian−Lagrangian approach is used to formulate the discrete element method (DEM) and discrete particle model (DPM). The Eulerian−Lagrangian approach is more rigorous than the Eulerian−Eulerian approach but less suitable for a large scale due to its computational intensity.3 Both approaches were used to study bubbling beds4−7 and riser flows.8−10 For riser flows, even industrial scale simulations were attempted using filtered TFM equations.11,12 However, modeling studies on homogeneous expansion and bubbling transition are yet to reach this extent of progress. One reason for this is the dependace of Eulerian−Eulerian models on constitutive relations or closures13 because until date there is no theory that is generally applicable to all flow regimes.14,15 Further, there are well accepted closures for drag and frictional stress available in literature for continuous modeling16−18 but © XXXX American Chemical Society
none for interparticle forces (IPFs), which have been shown to affect homogeneous expansion.19−21 The multiple causes of IPFs are van der Waals interactions, liquid bridges, solid bridge sintering, and electrostatic effects. Among these, the van der Waals forces arising from polarization interactions between molecules are said to dominate in gas−solid fluidization.22 The modeling of van der Waals forces in TFM is as yet unresolved, but DPM simulations have modeled van der Waals forces to investigate the effect of their strength for small theoretical beds.23−25 Unlike continuous modeling, the discrete approach is based on analysis of motion for individual particles, and does not require global assumptions on solids, such as steady-state behavior and/or continuum-based constitutive relations.23 DPM studies reveal homogeneous expansion even in the absence of IPF contribution24,26 showing that IPFs are not solely responsible for homogeneous expansion as was earlier hypothesized.27 For bubbling fluidization, the lack of scale resolution rather than IPF representation was concluded to be the main reason for the failure of TFM,28−30 and hence for bubbling beds at ambient conditions, IPFs are not significant in comparison to the hydrodynamic forces.31 For riser flows, the importance of resolving fine heterogeneous structures or mesoscales (e.g., clusters and streamers formed in turbulent regime) for accurate predictions of bed variables such as interphase drag, particle fluctuations energy, effective particle-phase pressure and viscosities is well established in the literature.13,32,33 In rapid flow regimes, the EMMS approach and filtered approach12,34 have made strides in developing subgrid models, to account for the effects of unresolved mesoscales in coarse mesh simulations. Subgrid models for dense flows are yet to be developed, and fine mesh Received: September 23, 2015 Revised: January 6, 2016 Accepted: February 5, 2016
A
DOI: 10.1021/acs.iecr.5b03565 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Article
Industrial & Engineering Chemistry Research
formulation with first-order implicit solver was employed. The second order upwind scheme was used for discretization. 2.2. Mesh and Domain Dimensions. To rule out any effect of domain size and ensure realistic simulations, a typical lab-scale 2D domain of 4 cm width (diameter) and 1 m length was used. A mesh as fine as (2−4)dp (i.e., 0.2 mm × 0.2 mm, as dp = 70 μm) was recommended to accurately capture the true interphase drag force and bed expansion resulting thereof.35 However, simulations from a more recent work by the same authors36 revealed that even a mesh size of 0.4 mm × 0.4 mm resolved subgrid structures like voids and bubbles. These get annihilated by coarser mesh sizes such as 4 mm × 4 mm which have hitherto been commonly used in simulation studies.31 Hence a mesh size of 0.4 mm × 0.4 mm (5.7dp) was deemed optimum in light of the large domain size adopted and the prohibitive computational cost entailed by a mesh size of 0.2 mm × 0.2 mm. 2.3. Other Simulation Details. Simulation parameters which were kept constant throughout the study are summarized in Table 1. Changes if any are mentioned wherever applicable
simulations which resolve all subgrid structures are a prerequisite. A short communication by Wang et al., 200935 showed that the homogeneous bed expansion predicted by TFM and DPM simulations were in quantitative agreement when a fine mesh size of (2−4) dp and small time step (1 × 10−6 s) were used. A theoretical 2D domain size (of the order of mm) was used. Clearly, a detailed fine mesh study for a bed of lab-scale dimensions would be highly desirable. TFM was used to predict minimum bubbling velocity for nonfrictional particles,36 but as yet there is no qualitative or visual description of the transition to bubbling. Since DPM simulations investigate small domains, their description of flow appearance or configuration is limited when compared to TFM simulations. Further, to the best of our knowledge, the effect of particle density (ρp) on dense flows has not been investigated using CFD simulations which implement TFM. In this work TFM fine mesh simulations of frictional particles were carried out for a bed of lab-scale size. This work seeks to investigate and analyze homogeneous expansion and transition to bubbling with emphasis on the qualitative or visual description of flow. The effect of particle density was studied. Results and discussions are presented in three parts describing (a) transition to bubbling, (b) effect and significance of IPFs (by comparing TFM simulations with experimental data), (c) comparison with DPM/DEM data. In all cases only ambient temperature and pressure was considered, and the fluidizing powder material was alumina.
Table 1. Summary of Parameters Used in the TFM Simulations
2. SIMULATION METHOD 2.1. Governing Equations. In TFM the averaged equations for conservation of mass and linear momentum are solved for each phase separately.37 Suitable expressions for the gas−solid interactions are required to mathematically close the conservation equations. To model the important interphase drag interaction we used the well accepted correlation of Gidaspow18 which is a combination of the Wen and Yu38 and Ergun39 models. The former is operated when solid volume fraction is 0.2 or less and the latter takes over for greater values. In TFM the granular phase is modeled by kinetic theory in which the equation for solid-phase stress tensor contains shear and bulk viscosity components. Shear viscosity comprises the kinetic component given by the Syamlal−O’Brien model16 and the collision component given by the combined Gidaspow40 and Syamlal−O’Brien model.16 The expression of Lun et al.14 was used to represent bulk viscosity. The initial packed bed solid volume fraction used in this work corresponds to frictional particles.41 Hence, to account for the frictional viscosity, the model of Syamlal−O’Brien16 adopted from Schaeffer42 was used. The Syamlal−O’Brien model is a Coulomb’s law based model found suitable for a dense regime43 but with the drawback that it manifests zero-order-dependence on the rate of deformation.44 The conservation equation for granular temperature18 is also solved in TFM. It requires a closure for the radial distribution function for which the expression of Oshima et al.45 was used. The energy equation was neglected due to isothermal assumption. The equations described in this section constitute the state of the art TFM for gas−solid flows which are commonly used in literature, for example in the report by Loha et al.46 and are presented in Table S1−S4 of Supporting Information. FLUENT 6.3.26 solver implemented the TFM equations by employing the finite volume solution method and PC-SIMPLE algorithm.47 The unsteady state
parameter
magnitude
initial 2D packed bed dimensions (diameter × height) (cm) grid size (mm) particle diameter (μm) gas phase density (kg/m3) gas phase viscosity (kg/m−s) initial packed bed solid volume fraction (εsmax)a solid packing range for frictional flow (εsmin − εsmax)a specularity coefficient for solid-phase wall shear maximum time step (s) residuals tolerance limit
4 × 12 0.4 × 0.4 70 1.138 1.66 × 10−5 0.55 0.53−0.55 0.5 1 × 10−5 1 × 10−3
a
For simulation validation in Figure 2 different values were used for these parameters.
in the results and discussion section. The Dirichlet boundary condition was used for the bottom gas inflow. Velocity vectors in horizontal x direction (always zero) and vertical y direction were defined to simulate gas inlet flow normal to the distributor. Many transient simulations were conducted by varying y velocity magnitude (in whole numbers) from 4 to 12 mm/s. On the left and right walls, the solid-phase shear was defined by the boundary condition of Johnson and Jackson,48 and for the gas-phase the no-slip boundary condition was used. At the gas exit boundary, zero gauge pressure was imposed to model the system open to atmosphere. In the literature, values for the initial packed bed (maximum) solid volume fraction (εsmax) vary from 0.4 to 0.6, and values in this range which are closer to 0.6 were shown to simulate more realistic drag predictions.49 Hence εsmax = 0.55 was chosen, which also corresponds to frictional particles41 for which the frictional viscosity model mentioned in section 2.1 was used. The initial condition for gas velocity in all interior cells was set to 4 mm/s. A time step as low as 1 × 10−8 s was required during the initial 1 to 2 s to establish convergence, after which the time step was gradually increased to 1 × 10−5 s.
3. RESULTS AND DISCUSSION 3.1. Transition from Homogeneous to Bubbling Bed. 3.1.1. Ambiguity in Judging Minimum Bubbling Point. By B
DOI: 10.1021/acs.iecr.5b03565 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Article
Industrial & Engineering Chemistry Research
conditions.56 A similar consideration will be analyzed for TFM fine mesh simulations, in the next section. 3.1.2. A Regime of Transition to Bubbling. In our prior work56 on effect of mesh size on CFD simulations of gasfluidized Geldart A particles, five different mesh sizes were investigated. The triggering of dilute regions (explained in the next paragraph) at minimum bubbling conditions was consistently observed independent of mesh size. Hence in simulation context, the appearance of dilute regions was proposed as a marker to signal the transition to the bubbling regime. In this section we test this premise by examining TFM fine mesh simulations with different particle density (ρp) values. Figure 1 panels a−c present the pseudo-steady-state simulation snapshots for inlet gas velocities in the range 4−12 mm/s and for ρp of 1, 2, and 2.8 g/cm3. For all simulations, the pseudo-steady-state was reached before 20 s; hence this time was chosen to take the snapshots shown in Figure 1a−c. For a particular ρp with increasing gas velocity, the snapshots clearly show the appearance of dilute regions, which can be defined as regions of altered voidage that destroy the homogeneous structure of the bed hitherto maintained. Take for example in Figure 1a, the bed at 4 mm/s is uniformly expanded and the voidage appears largely unvaried. By 6 mm/s there is no appearance of a bubble, designating the regime as homogeneous, but the expansion appears very different from the previous snapshot at 4 mm/s. Turbulent structures or mesoscales appear in the entire crosssection of the bed and become more prominent with increase in velocity. For the case of 2 g/cm3 Figure 1b, a dilute region which appears to have uniform voidage is seen at 8 mm/s, but it does arrest the uniform voidage previously seen at 6 mm/s. For the case of 2.8 g/cm3 Figure 1c, the dilute regions appear along with some bubbles at 10 mm/s only, and before this velocity the uniformity of bed is maintained. Hence dilute regions visibly disrupt the uniform bed, increase its dynamics and thereby create local voidage fluctuations. The gradation of the solid packing scale in Figure 1a−c is so chosen to reveal dilute regions which have a variation of at least 0.017 in solid volume fraction, and a variation less than this is said to represent a uniform bed. The velocity at which these dilute regions persist is designated umb ′ to distinguish from umb, since dilute regions rather than “first simulated bubble” was used as marker. The u′mb values detected for different ρp are given in Table 3, and resemble the trend of umb reported by Wang et al.36 from their fine mesh TFM simulations. They used a visual method for bubble detection in the first 1.5 to 2.5 s of simulation flow time, while we ensured that pseudo-steady-state was reached. We note that the empirical correlation of Abrahamsen and Geldart57 gives no change of umb,e with ρp (eq 2). Snapshots in Figure 1a−c show “bubbles” (threshold solid volume fraction = 0.2) in the bed even for velocity as low as 4 mm/s, but we judged these to be voids rather than the “first bubble”. This distinction was made whenever the homogeneous bed structure was intact; that is, solid volume fraction was nearly uniform (solid volume fraction variation not exceeding 0.017). For all values of ρp in Figure 1a−c, clearly visible multiple bubbles appear around 12 mm/s, and this velocity marks the complete breakout of heterogeneous structures in the bed, giving way to a freely bubbling bed. We designate this velocity as u′b. Observation of clearly defined multiple bubbles at 12 mm/s was also reported from DEM simulations.58 Hence for TFM simulations, transition to bubbling apparently occurs over a velocity range (u′mb−u′b) rather than at a single velocity,
Experiment. The visual method of determining minimum bubbling velocity from experiment (umb,e) relies on sighting the “first obvious bubble” or “first clearly defined bubble” upon slowly increasing inlet gas velocity. This makes it a subjective method as evidenced by the varied umb,e values reported (Table 2) by different authors for similar particle properties. Some Table 2. Various Minimum Bubbling Velocities Reported by Experimental Works and Compared with TFM Simulations authors Abrahamsen and Geldart (1980) Xie and Geldart (1995) Lettieri et al. (2002) our fine mesh TFM simulations
dp (μm)
umb,e (mm/s)
comment on the visual method
70
8.781
empirical correlation based on 48 gas−solid systems
68
5.37
clearly defined bubble appeared at bed surface
71
9.8
70
8 (svf
