EMMS drag model for simulating a gas–solid fluidized bed of geldart B particles: Effect of bed model parameters and polydisperity
Graphical abstract
Introduction
Many industries such as chemical, pharmaceutical, energy, food, and mineral processing use gas–solid fluidized beds to carry out multi-phase or catalytic reactions in different regimes like slugging, bubbling, and turbulence. The feed particles in many of these systems have the sizes and densities of Geldart B particles (Kunii & Levenspiel, 1991). In these gas–solid fluidized beds, gas is passed through an initially kept or continuously fed solid granular material that is often the reaction catalyst or the reactant. The gas passes through this material at high enough velocities to suspend the particles and cause them to behave like a fluid. Accurately predicting bed behavior such as expansion is crucial for designing and scaling gas–solid fluidized-bed reactors (FBRs) (Geldart, 2004). Many experimental and simulation studies have been conducted on fluidization, and some have obtained empirical correlations for design (Hepbasli, 1998). However, knowledge of gas–solid fluidization is incomplete because the fluidization of such beds is affected by many factors such as gas and particle types, bed geometry and scale, distributor type, and operating conditions. Obtaining a universal correlation is nearly impossible (Levenspiel, 2008) because almost every gas–solid fluidization system is specific. Therefore, it is useful to carry out a thorough study and evaluate a given correlation applied to a specific gas–solid system.
With enhanced computational resources in recent years, first-principle computational fluid dynamics (CFD) tools have been increasingly used to supplement or replace expensive actual experiments, or to conduct effective simulations of complex multiphase phenomena in gas–solid fluidized systems. Several approaches have been applied depending on the system and scale (van der Hoef, van Sint Annaland, Deen, & Kuipers, 2008), two of which, the Eulerian‒Eulerian (EE) and Eulerian‒Lagrangian (EL), are the most common. In the EE approach, both the gas and solid are considered interpenetrating continuum phases where integral balances of continuity, momentum, and energy are defined with appropriate boundary and jump conditions at interfaces. In the EL approach, only the gas is considered to be a continuum phase while each solid particle is assumed to follow Newtonian motion; this requires much higher computational cost. The particles can also be modeled as soft or hard spheres (Hoomans, Kuipers, Briels, & van Swaaij, 1996; Tsuji, Kawaguchi, & Tanaka, 1993) for their collision behavior. Although some reduction in computational cost is possible by modeling groups of particles as single particles (Pawar, 2014), the EL method still has a huge computational cost when dealing with systems of over a million particles. Hence, the EL framework is not usually chosen for simulating gas–solid fluidized beds, even dense laboratory-scale beds. The EE approach is preferred in such cases (Kumar & Natarajan, 2009). In the EE approach, the kinetic theory of granular flow (KTGF) is generally applied to the solid phase if it is assumed to have granular properties with defined viscosity and normal stress. This can then be called the Eulerian-granular (EG) approach (Benyahia, Arastoopour, Knowlton, & Massah, 2000). Similarly to the EE approach, the EG framework treats both the gas and solid with a set of continuity equations with respective volume fractions and phase interactions defined by appropriate mass-, momentum-, and energy-exchange models. The EE and EG approaches are sometimes called two-fluid models (TFMs) because they assume continuity of both phases (Hong, Chen, Wang, & Li, 2016; Yang, Wang, Ge, Wang, & Li, 2004).
TFM simulation of a gas–solid fluidized bed requires the calculation mesh to be as fine as a few particle diameters (Andrews, Loezos, & Sundaresan, 2005; Wang, van der Hoef, & Kuipers, 2009). A fine mesh does not need many computational cells for a small-scale fluidized bed but does for an industrial-scale bed. A coarse mesh is usually used to reduce the number of computational cells for an industrial-scale fluidized bed. However, using a homogeneous drag model like the Gidaspow model in a coarse-mesh EE simulation leads to an excessively uniform flow field and overestimated drag force (Shah, Myöhänen, Kallio, Ritvanen, & Hyppänen, 2015). This is because the phases in the EE approach are assumed to be homogeneous within a computational cell, which may not be the reality if a particular particle cluster is smaller than that cell. The gas can still pass through such a cell, which the homogenous drag model cannot take into account. This leads to information loss and an overestimated drag force among other possible inaccuracies. Therefore, either the drag model is corrected in a coarse-mesh simulation to account for sub-grid structures, or energy-minimization multi-scale (EMMS) approach is applied with appropriate closure models to address sub-grid structures and cluster formation (Agrawal, Loezos, Syamlal, & Sundaresan, 2001; Benyahia, 2009; Yang et al., 2004; Zhang, Lu, Wang, & Li, 2010). By incorporating structure-dependent drag coefficients calculated from the EMMS model, one can well describe dynamic formation and dissolution of heterogeneous structures in gas–solid flow in a fluidized bed and significantly improve TFM accuracy (Wang and Li, 2007, Yang et al., 2004).
The EMMS model was originally formulated to describe heterogeneity in circulating fluidized beds. This approach characterizes the solid-phase distribution as including a dense-cluster phase and dilute-dispersed phase with an interphase between them under a stability condition (Li & Kwauk, 1994). The EMMS model used by Zhang et al. (2010) and Lu et al. (2013) introduces a correction factor for the gas–solid drag force based on the volume fraction and slip velocity. Other researchers have filtered the results of fine-mesh simulations to derive closure correlations (Igci, Andrews, Sundaresan, Pannala, & O’Brien, 2008; Shah, Myöhänen, Kallio, Ritvanen et al., 2015). Drag models formulated in the EMMS approach have been shown useful for coarse-grid simulation of industrial-scale fluidized beds and Geldart A particles in small-scale fluidized beds (Shah, Myöhänen, Kallio, & Hyppänen, 2015; Shah, Myöhänen, Kallio, Ritvanen et al., 2015; van der Hoef et al., 2008). However, it is still unclear whether these models can make better predictions for Geldart B and D particles than a homogenous drag model like Gidaspow’s. This is what our study investigates in detail through a series of simulations involving parameters and assumptions of an FBR model.
The drag model or correlation for the gas–solid momentum exchange, particle–particle restitution coefficient, and particle–wall specularity coefficient considerably affects the local solid volume fraction in fluidized beds. While drag correlations and correction factors have been formulated via the EMMS approach that could be used in a coarse mesh, these corrected drag models have rarely been used with a fine mesh to evaluate their predictions. Therefore, we study the effects of gas–solid fluidization parameters within the EMMS drag model. We compare predictions of the EMMS drag model, which is heterogeneous, with those of the Gidaspow drag model, which is homogeneous, for a laboratory-scale gas–solid fluidized bed using a fine mesh. Some groups have used the EMMS approach to develop models (Li et al., 1999, Wang and Li, 2007, Yang et al., 2004), while others have used it to apply models (Shah, Myöhänen, Kallio, Ritvanen et al., 2015). We study the effects of varying restitution coefficient, specularity coefficient, and polydispersity condition of Geldart B particles. A cold model gas–solid polysilicon FBR process is also simulated.
Section snippets
Governing equations and constitutive laws
We model gas–solid flow in a fluidized bed using the TFM approach with a gas as the primary phase and a solid as the secondary phase. Both phases are considered interpenetrating continua sharing the flow volume. Consequently, the flow volume at any particular time has volume fractions of gas and of solid:
The KTGF is used to describe the solid stress, and the drag model is used to calculate the interaction force between the gas and solid phases. We model dense gas–solid flows in
Geometry, model parameters, and simulation settings
We used the laboratory-scale pseudo-two-dimensional (2D) fluidized bed proposed by Kumar and Natarajan (2009) in their experimental study. It has a height of 1.0 m, width of 0.28 m, and depth of 0.025 m. We used a 2D CFD model that ignores the effect of small depth. A three-dimensional model considering the small depth might give better quantitative predictions, but 2D is less computationally demanding.
This geometry was first discretized into sufficiently small grids (˜0.003 m) using GAMBIT to
Model validation
Fig. 2, Fig. 3 compare experimental bed-expansion and pressure-drop data with the predictions of the Gidaspow (1994), EMMS-I, and EMMS-II drag models (Li et al., 1999, Yang et al., 2003). The predicted bed expansions are reasonable for all three drag models in the range of superficial gas velocities studied, 0.12–0.6 m/s. However, the Gidaspow (1994) model predicts higher bed expansion than the EMMS models do. This agrees with previous findings (Ghadirian and Arastoopour, 2016, van der Hoef et
Conclusions
In this work, we simulated gas–solid fluidized beds with Geldart B particles to compare the predictions of homogeneous and heterogeneous drag models considering different model parameters, fluidization regimes, and polydisperity conditions. We compared the predictions of the homogeneous Gidaspow model (Gidaspow, 1994) and heterogeneous EMMS models of Li et al. (1999) and Yang et al. (2003) with existing experimental pressure-drop and bed-expansion data. There was no significant difference in
Conflict of interest
The authors declare that they have no conflict of interest.
Acknowledgements
This research was supported by Korea Electric Power Corporation (Grant number: R18XA06-14) and supported by the Human Resources Development (No. 20184030202070) of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Trade, Industry and Energy.
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