Comparison of numerical approaches to model FCC particles in gas–solid bubbling fluidized bed

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Comparison of numerical approaches to model FCC particles in gas-solid bubbling fluidized bed S. Vashisth, A.H. Ahmadi Motlagh, S. Tebianian, M. Salcudean, J.R. Grace

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Received date: 30 December 2014 Revised date: 24 April 2015 Accepted date: 1 May 2015 Cite this article as: S. Vashisth, A.H. Ahmadi Motlagh, S. Tebianian, M. Salcudean, J.R. Grace, Comparison of numerical approaches to model FCC particles in gas-solid bubbling fluidized bed, Chemical Engineering Science, http: //dx.doi.org/10.1016/j.ces.2015.05.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Comparison of numerical approaches to model FCC particles in gas-solid bubbling fluidized bed S. Vashistha,*, A. H. Ahmadi Motlagha, S. Tebianiana, M. Salcudeanb and J.R. Gracea a

Department of Chemical and Biological Engineering, University of British Columbia, Vancouver, Canada V6T 1Z3 b

Department of Mechanical Engineering, University of British Columbia, Vancouver, Canada V6T 1Z4

Abstract A comparative study has been carried out to the determine the ability of computational fluid dynamics (CFD) and computational particle fluid dynamics (CPFD) codes to predict the hydrodynamics of FCC particles in a bubbling fluidized bed. Simulations were conducted in both 2-dimensional (2-D) and 3-dimensional (3-D) CFD configuration and in a 3-D CPFD model. The new structure based force-balance (FB) model of Ahmadi Motlagh et al. (2014) was incorporated in CFD to simulate the behavior of FCC in the bubbling fluidization regime. This model modifies the conventional drag correlation by considering the effect of interparticle forces on the formation of agglomerates inside the bed, updating the drag calculations by replacing the particle diameter in the Wen and Yu correlation by an agglomerate diameter. The effects of superficial gas velocity, particle diameter, particle-particle restitution coefficient and specularity coefficient on voidage are examined. The sensitivity of different gas-solid drag closures in CFD are tested and compared. Experimental time-average axial and radial voidage profiles were simulated with varying degrees of agreement with experimental data obtained using different advanced experimental techniques in the “Traveling fluidized bed” (Dubrawski et al., 2013; Tebianian et al., 2014). Both the FB and CPFD models were successful in resolving key issues in 3-D, whereas 2-D models tended to seriously underpredict particle volume fraction, especially near the wall.

Keywords: Fluidized Bed; Computational Fluid Dynamics; Computational Particle Fluid Dynamics, Voidage, Validation * Corresponding author: Tel : +1-604-822-2482; E-mail: [email protected]


 

1. Introduction Many engineering applications involve multiphase flow. Understanding and modeling the physical mechanisms underlying multiphase flows is extremely challenging due to the interaction between different phases and the interplay of mechanisms. Fluidization of fluidgranular phase is one such area that involves complicated configurations and phenomena such as bubbles, agglomerates and electrostatic charging, with profound impact on macroscopic flow properties. Resolving such multiphase flows to the extent of length and time scale and accounting for inter-phase and intra-phase coupling are challenging. This topic has drawn considerable attention of experimentalists and modelers for several decades, and progress has been made as computational facilities and simulation models increase in sophistication. Bubbling fluidized beds (BFB) are inherently heterogeneous in nature, featuring bubbles/slugs or agglomerates/clusters of particles. Computational modeling efforts in BFB are broadly conducted using two approaches: (a) Eulerian-Eulerian (E-E) and (b) Eulerian-Lagrangian (E-L). In the E-E approach, both continuous and granular phases are treated as fully inter-penetrating continua. The kinetic theory of granular flow is employed to describe the interaction between the particles. Geldart A particles, also defined as ‘aeratable’ particles, have small mean particle diameter (< 120 µm), low particle density (<~1400 kg/m3) and hence are easy to fluidize. However, realistic prediction of Geldart A particles using the Eulerian method is very challenging due to the presence of mesoscale structures like bubble/clusters. The E-E approach gives reasonable results for larger particles, Geldart groups B and D, with varying degrees of model validation (e.g. Ding and Gidaspow, 1990; van Wachem et al., 1999, 1998; Pain et al., 2001; Patil et al., 2005; Wang, 2009). Fluid-particle drag is a crucial aspect of governing equations in determining the predictive accuracy of E-E models. Li and Kwauk (2001, 2003), Beeststra et al. (2006), Ma et al. (2006) reported that standard E-E models over-estimate the drag coefficient by neglecting the effects of the mesoscale structure. Interparticle cohesive forces for Group A particles strongly influence fluidization behaviour (Massimilla and Donsi, 1976) and hence must be incorporated to capture realistic dynamics. To achieve this, the existing drag models can be modified based on various approaches such as empirical correlations, structurebased methods, scaling factor method and multi-scale methods.  

Lagrangian models solve equations of motion for each particle, taking into account particleparticle collisions and the forces acting on particles. The computational particle fluid dynamics (CPFD) approach considers granular particles as computational parcels, with particles of identical density, volume and velocity, located at a specific position lumped together. It is possible to control the number of particles in each parcel. The fluid phase is described by mass and momentum conservation equations based on the multiphase particle-in-cell (MP-PIC) method (Andrews and Rourke, 1996) including strong coupling to the particle phase. CPFD has the capability to introduce the particle size distribution (PSD). Thus, assuming a constant particle diameter may lead to over-prediction of bed expansion (Grace and Sun, 1991). In CFD, a secondary phase with a particle size distribution can be accounted for by assigning a separate phase for each particle diameter. Also, coupling of a population balance method (PBM) with E-E method in CFD allows for different particle sizes. However, solving as many separate continuity and momentum equations as there are phases or PMB are limited by their complexity due to computing power. Despite the differences in the CFD and CPFD frameworks, modeling parameters such as solidphase viscosity, solid stress modulus, restitution coefficient and gas-solid drag closure are crucial in both for resolving meso-structures, such as formation of cluster streamers and particle agglomerates. Alternative models have focused on various aspects of inter-phase momentum transfer, as summarized in Table 1. The main objective of this paper is to demonstrate the capabilities and limitations of CFD and CPFD in terms of predicting the bubbling fluidized bed hydrodynamics, accuracy of estimation compared with experimental data, demand for scale resolution to capture the meso-scale structures, model applicability and computational power requirement. It is of considerable interest to investigate different models given the status of the development of modeling techniques. The simulation conditions were selected based on experimental measurements reported by Dubrawski et al. (2013) and Tebianian et al. (2014) for direct comparison. Dubrawski et al. (2013) conducted extensive experiments in a gas-solid “travelling fluidized bed” using different intrusive (optical probes) and non-intrusive techniques (electrical capacitance tomography, x-ray computed tomography, radioactive particle tracking, dynamic differential pressure measurements) to determine local voidage, bed expansion and pressure  

fluctuations. The experimental dataset is unique and comprehensive. The non-invasive techniques are difficult to implement and require expensive and complex data analysis, whereas the invasive methods interfere with the flow pattern in the reactor. This work was further extended by Tebianian et al. (2014) who applied another invasive measurement technique, borescopic imaging, to determine the voidage in the same “traveling fluidized bed”. Details of these measurement techniques are provided by Dubrawski et al. (2013) and Tebianian et al. (2014). The observed differences among their measurements provide an indication of the accuracy and range of variation of the experimental results. The resulting experimental data are then directly useful for comparing different modeling approaches to predict fluidized bed hydrodynamics. A brief comparison of voidage measurement using different modeling methods was also presented by Tebianian et al. (2014) and is greatly extended in the present work. A detailed study considering the particle pressure, particle viscosity, and cohesive forces is highly desirable/ However, our main focus in this paper is to investigate the influence of drag models on voidage prediction and bed expansion

2. Approaches to model granular flow in bubbling fluidized bed (BFB) The bubbling fluidized bed can be assigned to two regimes: (a) viscous regime where the kinetic and collisional stresses are dominant. (b) frictional regime where friction between particles or between the wall and particles, is important. Modeling of such granular flow associated with the different regimes is challenging due to phenomena occurring at different length and time scales. In this work, we focus on the CFD twofluid model (TFM) (Fluent, 1966) incorporating widely-used drag closures to predict the effect of underlying mechanisms on BFB hydrodynamics. A new promising force-balance drag model introduced by Ahmadi Motlagh et al. (2014) is further explored, as well as the CPFD code (Andrews and Rourke, 1996) which adopts the computational parcel concept.

2.1. CFD - TFM with Force-balance (FB) drag model using Ansys-Fluent The governing equations for the TFM are detailed in Table 2. The solid stress in the momentum equations is closed by the kinetic theory of granular flow. Another important element requiring closure in the TFM is inter-phase interaction. Several semi-empirical gas-solid closure relations  

are available and widely used, despite large uncertainty, significantly affecting the overall predicted hydrodynamics of the fluidized bed. The interaction between the particles and the continuous gas phase are described by a drag closure. Table 3 lists the five drag closure options tested in this paper. The presence of dominating cohesive forces favours formation of agglomerates in Geldart A particles and thus affects the bed expansion. It is important to consider cohesion alongside drag models for accurate prediction. The new force-balance (FB) sub-grid-scale model developed by Ahmadi Motlagh et al. (2014) analyzes the balance among cohesive (van der Waals), drag, gravity and buoyancy forces acting on particles inside a fluidized bed. It can predict formation of agglomerates inside the bed, updating the drag calculations by applying a correction factor, obtained from the balance of forces, to the conventional drag model by Wen and Yu (1966). The FB model incorporates the effect of particle topology (in particular particle asperities) in estimating inter-particle forces. The analysis of force vectors and movement of agglomerates inside the bed are also considered. The equations used in the FB model are summarized in Table 4. Predictions from simulations with different drag models such as Wen and Yu (1966), EMMS model of Hong et al. (2013), the EMMS of Shi et al. (2011) and the Wen-Yu/Ergun approach (Gidaspow, 1994) are compared, and discrepancies are discussed. 2.2 Eulerian-Lagragian model using CPFD The CPFD numerical approach is used to solve the transient fluid and particle mass, momentum and energy equations in 3D. The gas phase is modeled as a continuum and the granular phase as individual Lagrangian particles. The MP-PIC method (Andrews and Rourke, 1996) uses the concept of computational parcel (collection of particles having the same properties: density, volume and velocity). It calculates the particle stress gradient on the grid and then interpolates it to discrete particles. The governing equation and details of the MP-PIC model are summarized in Table 5. A key advantage of the CPFD approach over current CFD models is the inclusion of effects of the size distribution of particles. This distribution influences the momentum exchange term between the particulate and fluid phases. Particles are implicitly coupled to the fluid phase through interphase drag. The mass and momentum equations are solved for the continuous fluid  

phase, and a Liouville equation is solved for the particle phase to predict the spatial distribution of particles with different velocities and sizes. An isotropic solids stress model (Harris and Crighton, 1994) that depends on the particle volume fraction and packing fraction is then applied to model the average collisional force. Karimipour and Pugsley (2012) reported that CPFD was able to predict reasonable bed expansion without modifying existing drag models.

3. Model set-up and parameters 3.1 Experimental details for model set-up All the experimental data in this paper were obtained from Dubrawski et al. (2013) and Tebianian et al. (2014), where the detailed local and average voidages inside a bubbling fluidized bed of fluid catalytic cracking (FCC) catalyst particles are reported for several gas velocities (Ug = 0.3 to 0.5 m/s). These experiments were carried out in the ‘travelling fluidized bed’ in a column 0.96 m long x 0.133 m inside diameter in the dense bed section (static bed height, Ho = 0.80 m), containing FCC particles. Table 6 summarizes the particle properties and operating conditions used in the simulations. The main focus of the experimental work was to compare the performance of several non-invasive measurement techniques (electrical capacitance tomography, X-ray computed tomography, radioactive particle tracking) with invasive techniques (optical probes, borescopy) for voidage measurement. A schematic of the computational domain depicting also the experimental set-up is shown in Figure 1.

3.2 Boundary and initial conditions For CFD, a uniform velocity was specified at the inlet. A constant atmospheric pressure was imposed at the outlet. A no-slip boundary condition was used for the gas phase, i.e. zero velocity relative to the wall. The partial slip boundary condition (Johnson and Jackson, 1987) was adopted forl the solid phase at the wall, with three different specularity coefficients, as listed in Table 6. A specularity coefficient of zero signifies free slip of particles at the wall, whereas a value of 1 indicates that the particles stick to the walls. First and second order upwind discretization schemes were used for volume fraction and momentum terms, respectively. Time steps of 0.0001 and 20 iterations per time step were required to achieve full numerical convergence. A convergence criterion of 10−5 for each scaled residual component was fixed for

 

the relative error between successive iterations. The phase-coupled PC-SIMPLE algorithm was utilized for pressure-velocity coupling. CPFD incorporated the same particle size distribution (PSD) for FCC as in the experiments (Dubrawski et al., 2013), whereas a constant particle diameter equal to the Sauter mean experimental value was adopted in the CFD simulations. For widely distributed particle sizes, simplifying the PSD by a mono-disperse particle of the Sauter mean particle size can lead to considerable errors and unrealistic gas-solid flow characteristics (Niemi, 2012; Wang et al., 2014). The PSD influences the calculated momentum exchange between the particulate and fluid phases. The collision of particles is modeled by a particle stress model. A non-uniform velocity inlet, consistent with the experimental perforated distributor plate (49 perforations on a circular pattern) was specified at the bottom of the bed. An initial constant pressure of 105 Pa was applied at the outlet, with no particle exit condition. Particle-to-wall interactions were dealt with normalto-wall and tangent-to-wall momentum retentions and diffuse bounce, which were set as default values of 0.3, 0.99 and 0 respectively. A detailed investigation is required to check the sensitivity of these parameters, not within the scope of current work. CPFD has a self-adapting time step scheme, CFL (Courant-Friedrichs-Lewy) algorithm (Courant et al., 1928) that was utilized to speed up the execution, unlike CFD. It automatically determines the required time-step based on mesh and the computational parcel for the simulations. An initial time step of 0.002 s was specified; however, the final averaging time step was reduced to 0.0011 s for accurate prediction. CFD would require a much smaller calculation time step, of the order of 0.0001 s for correct prediction. It lacks the ability to adjust the time step automatically.

Liang et al., (2014)

performed time step sensitivity tests of CPFD model using ∆t = 0.0001, 0.0005 and 0.002 s for a 2-D fluidized bed and found that smaller time steps did not improve the predicted instantaneous particle velocity and bubble possibility. However, Li et al. (2010) reported time step as a crucial factor and recommend smaller time steps of the order of 10-4 s in CFD simulations.

3.3. Data collection and analysis: 2-D and 3-D CFD simulations were carried out in the Ornicus server of Canada’s West Grid computer cluster using the commercial software, Ansys-Fluent 15. The FB drag model applied  

user-defined functions in the TFM. The computational domain was created and meshed in Ansys - Workbench. The 3-D CPFD calculations were performed on a Dell server with 8 cores using Barracuda software. The 3-D geometry for CPFD was first defined by Autodesk-Inventor and then meshed by a built-in module in Barracuda software. The voidage profiles, velocity distribution, drag function and pressure fluctuations were recorded at five heights: z = 0.24, 0.40, 0.56, 0.72 and 0.88 m. In order to obtain the radial area-averaged data, 9 circular rings of equal area were defined at each cross-section. The voidage was then recorded and averaged for each ring. Simulations lasted for 15 s of real time for CFD and 25 s for CPFD to obtain a steady state, with the time-average variables obtained over the last 10 s.

4. Results and Discussion

4.1. 2-D CFD simulation analysis 4.1.1. Effect of mesh size Mesh size is a crucial parameter in TFM simulations. For Geldart A particles, a coarser mesh may lead to over-estimation of bed expansion. Mesh size of the order of 3dp is recommended (Wang et al., 2014). Five different mesh numbers (1030; 2480; 3900; 6993; 9600) were tested in the present study for mesh independence of the FB model, and a noticeable effect was observed on hydrodynamics. Figure 2 compares the predicted time-averaged voidage profiles as a function of (a) radial location and (b) height, for different mesh numbers. The voidage profile for the coarser meshes (2480 and 3900) follow the same pattern without significant differences. This is clearly seen in Fig. 2(c) where refining the mesh from 1030 to 9600 cells not only resulted in contraction of the bed, but also to prediction of smaller bubbles. Tebianian et al. (2014) experimentally observed a combination of bubbles and slugs for the same operating conditions. After studying the issue in depth, it was observed that when the agglomerate size exceeds the mesh sizes, which is the case for >3900 cells, the convection mechanism in FB model considers movement of agglomerates inside the bed. This results in propagation of large agglomerates from some computational cells to a major portion of the bed, leading to suppression of drag to very low values over the entire bed and to unrealistic prediction of bubble sizes and more dense regions appearing in the bed. After some seconds of fluidization, this effect could fade away, but  

it was significant enough to influence the results. The effect is carried on to the next time-steps, irrespective of how long the simulations were continued. On the contrary, when the agglomerate sizes fell within the mesh size, such behavior was not seen in the bed, and agglomerates moved freely move inside the bed with little variation in size. Hence, refining the mesh beyond a certain point does not result in more accurate predictions in 2-D simulations of the FB model. In order to select the most appropriate mesh size, both qualitative and quantitative analysis should be considered. Combining the results of Fig. 2 and considering the expected flow behavior for the operating conditions studied here, we selected 2480 cells as the appropriate mesh size for 2D simulations. Mesh studies were also conducted for both 2-D and 3-D simulations with the EMMS models. By comparing voidage and particle velocity profiles and taking computational time into consideration, 8178 cells (compared to 2480 Cells for the FB model) for 2D studies and 5256 cells (compared to 3528 for the FB model) for 3-D studies were selected as most appropriate for the EMMS-Shi and EMMS-Hong models, respectively.

4.1.2. Qualitative analysis of particle volume fraction Figure 3 provides snapshots of instantaneous particle volume fraction predicted by the FB model. At Ug = 0.4 m/s, a heterogeneous gas-solid system is observed, with agglomerates forming and dissolving dynamically. The particles aggregate, form clusters, and then are carried away by the upward-moving gas. The formation of aggregates is well captured by the incorporation of the particle-particle force-balance concept in the FB model, unlike the conventional Wen and Yu drag model. Figure 4 presents a qualitative comparison of particle volume fraction prediction employing different drag models at t = 15 s based on (a) the Wen and Yu (1966) model, (b) the EMMS of Shi et al. (2011), (c) the EMMS of Hong et al. (2013), and (d) the FB model (Ahmadi Motlagh et al., 2014). The results in Fig. 4 shows solid streams leaving the system and significant overestimation of bed expansion using Wen and Yu (1966) model. This behaviour is also reported in similar studies by Zimmermann and Taghipour (2005). Overestimation of the drag force is the likely reason for such flow pattern predictions, which is a common behaviour using conventional drag models on simluations of Geldart A particles (Wang et al., 2009). The EMMS-Shi drag model,

 

which is a bubble-based EMMS version, predicts formation of very small bubbles, leading to overestimation of bed expansion. The likely reason is the operating conditions that are studied in this work which falls in a boundary region between the bubbling and slugging fluidization regimes. The EMMS-Shi model is claimed to obtain satisfactory results for low-velocity conditions (Hong et al., 2013) which probably fall below the operating condition region studied in this work. The EMMS-Hong model, which is a further extension of EMMS-Shi, unifies the approach used by EMMS-Shi and a structure-dependent multi-fluid model (SFM), proposed by Hong et al. (2012), to enable considering different cluster structures in the bed (Hong et al., 2013). The EMMS-Hong model gives good agreement with the experimental observations of bubbles and particle clusters. However, bed expansion is slightly over-estimated which is probably an indication of overestimation of drag force by the model. The FB model is able to capture the aggregate formation and realistic bed expansion compared to the experimental data of Dubrawski et al. (2013).

4.1.3. Effect of specularity coefficient (φ ) and gas velocity (Ug) Figure 5 shows time-averaged voidage profiles for Ug = 0.3 and 0.4 m/s with dp = 103 µm at t = 15 s for specularity coefficients of φ = 0.001, 0.01 and 0.1. The experimental time-mean data from different measurement techniques (Dubrawski et al., 2013; Tebianian et al., 2014) were averaged at each location and 95% confidence limits are plotted in Figure 5. Both experimental and model predictions were folded over due to the symmetry of the system with respect to the centerline at heights of 0.24, 0.40, 0.56 m. The simulated and measured voidage is higher in the central regime than near the wall. The contours in Figure 5 depict radial profiles of the particle volume fraction at three heights. It is clearly visible that the particles are more concentrated towards the walls and then descend along the column walls. The predicted results fall within the range of the experimental data. No significant difference was observed by varying the specularity coefficient over the range of 0.001 to 0.1 for either the FB or the EMMS-Hong model. φ = 0.001 was employed for all further simulations. This is contrary to the results reported by Li et al. (2010) for Geldart B particles which show the likely effect of particle size on flow behaviour near the wall boundary. Further study of this issue is highly desirable.


 

Figure 6 shows symmetrical behaviour of voidage profiles for all heights above the distributor at Ug = 0.3 and 0.4 m/s. With increasing height, the voidage increased at all radial positions, especially in the central region, due to expansion of the bed. The voidage increased with increasing gas velocity at each level. The higher time-mean voidage in the central region reflects the passage of bubbles. At the walls, higher concentrations of particles are predicted, consistent with the experimental observations. As a result, particles tend to ascend in the core of the bed and descend close to the wall The significant difference between the experimental results and predictions near the walls is due to neglecting the wall effects in a 2-D geometry. It is especially important in small-diameter columns, as in the present study (Dc = 0.133 m). Also, in the CFD simulations incorporating the effect of the distributor may resolve uncertainties near the wall.

4.2 3-D simulation analysis 4.2.1. Mesh refinement study The 3-D computational domain with circular cross-section was meshed using a structured grid. Figure 7 (a-b) plots voidage and particle velocity results from the mesh refinement test for 3-D CFD simulation of the bubbling fluidized bed (BFB) for Ug = 0.4 m/s based on the FB drag model. Based on irregularities observed with agglomerate growth that exceeded mesh sizes in 2D simulations, the agglomerate size was constrained by the size of computational cells in 3D simulations. The bed expansion and qualitative shape of bubbles/slugs for different mesh sizes were in agreement with experimental observations for the range of operating conditions. It can be seen that the model predicts similar voidage and particle velocity profiles for the range of mesh sizes considered. The coarsest mesh (2332 cells) deviates compared to other mesh densities at the wall in terms of voidage and at the center for particle velocity. Note that the model predicts reasonable voidage and velocity profiles for mesh numbers of 3528 and 5256. The radial profile of voidage in the 3-D simulation is more consistent with the experimental data in the upper section of the bed and near the wall. Considering the extensive computational requirements for utilizing more refined meshes for the 3-D simulation and based on the results discussed above, 3528 cells were found to be sufficient to obtain the CFD results and were selected for data extraction.

 

The mesh study for the CPFD model is presented in Figure 8. The number of computational parcels or particles increased as the mesh was refined. In the present work, three mesh numbers were studied: 36,000, 140, 000 and 200,000 cells. Increased mesh resolution clearly affected the flow characteristics and improved the simulation results (see Figure 8). However, refining the mesh to 200,000 nearly doubled the computational time without much change in the flow resolution. Hence, 140,000 cells were preferred in subsequent work. Liang et al. (2014) reported similar experiencethat neither refined mesh nor large parcel number could significantly improve the simulated results, while enormously increasing the computational cost.

4.2.2. Qualitative analysis of instantaneous particle concentration using CFD and CPFD Snapshots of particle volume fraction from 3-D CPFD predictions after 5 to 40 s of flow time for FCC particles at Ug = 0.4 m/s are presented in Figure 9, showing the bubble formation process. As fluidization proceeds, the bubbles split and coalesce continuously. The high gas velocity causes formation of large bubbles which are slug-like, but also in transition of the zone to turbulent fluidization, as reported by Zimmermann and Taghipour (2005). The flow field was stable beyond 20 s, with the solid volume fraction distribution thereafter being similar at different times. Particle and gas instantaneous velocity vectors for the granular and gas phases predicted by the FB model in a vertical diametral plane are shown in Figure 10. The macroscopic solid circulation patterns were captured in the simulation, with particles rising in the center due to bubble motion, being dispersed as bubbles broke up and subsequently descended along the wall. Transport of particles due to bubble wakes and drift led to gross circulation patterns, up the centre and down along the wall. CPFD predictions of gas velocity vectors are superimposed on the particle volume fraction in Figure 11. The CPFD simulations predict the gross circulation patterns and formation of gas slugs. Close observation of the vector field imposed by particles reveals the presence of particles trapped inside gas slugs. Also, particles descending along the walls enter the wakes of slugs and are then carried upwards.


 

Figure 12 shows profiles of time-average solid hold-up as a function of height for Ug = 0.4 m/s and Ho =0.8 m. The average solid holdup changes very little over the height interval of 0 to 0.98 m. The actual expanded bed height is about 1.0 to 1.1 m. There is a significant difference between the predictions of the FB and EMMS-Shi models. However, the EMMS-Hong model, specifically designed for BFB, gives results similar to the FB model. The corresponding solid hold-up distributions for CFD and CPFD at t = 15 s are shown in Figure 13. Both approaches show the presence of bubbles and emulsion phase. In the CFD predictions, there are agglomerates at the wall, with large bubbles and slugs passing up the center. The CFD model predicts a higher overall voidage at lower heights than CPFD. More agglomerates were observed at the higher heights of z = 0.72 and 0.88 m (Figure 13). CPFD predicts higher particle volume fraction at z = 0.24, 0.40 and 0.56 m than CFD. Bubbles and agglomerates in fluidized beds strongly influence key features such as heat and mass transfer. A threshold particle volume fraction of 0.2 is used here to delineate the boundaries between bubbles and the dense phase. Based on this threshold, bubble frequencies and dimensions were calculated. Post-processing of the CPFD results in this study applied the same threshold to differentiate bubbles. Iso-volumes were created to visualize particle volume fraction up to 0.2 to represent the bubbles. Figure 14 gives a 3-D visualization of bubble formation, coalescence and break-up, with bubble shapes continuing to change due to the evolving granular flow.

4.2.3. Effect of drag model Two-phase interfacial momentum exchange or drag is a function of the volume fraction of gas, gas density, viscosity and particle diameter. Figure 15 compares 3-D snapshots of solid volume fraction for Ug = 0.4 m/s, φ = 0.1, t = 15 s based on different drag models: (a) Wen and Yu; (b) EMMS-Shi; (c ) EMMS-Hong; (d) FB model; (e) CPFD model. The predicted solid volume fractions for the FB model, CPFD and EMMS-Hong model follow similar trends and fall within the range of experimental data (Figure 16). The solid volume fraction predicted by the FB model is lower at the center compared to the CPFD model, however, the reverse was observed near the wallAlthough the EMMS-Hong results for 3-D cases fall within the range of experimental data, comparing the FB and Hong models, one can see slightly higher predicted voidage (radially or
 

axially) due to higher bed expansion for the Hong model. It is evident from Figure 16 that incorporating the effect of inter-particle forces for Geldart A particles affects the hydrodynamics by comparing with Wen and Yu drag law. It is crucial to take into account the effect of van der Waals forces in addition to drag and particle-particle collisions. The van der Waals forces acting between particles are dominant for the occurrence of homogenization. Deen et al. (2006) reported the formation of particle streamers at a high Hamaker constant (10-20 J). At low Hamaker constants, bubble form due to weak van der Waals forces. Direct measurement of cohesive forces in Geldart A particles, which strongly depend on surface properties, is very difficult (Ye et al., 2005). Up to now, direct experimental measurement of cohesive forces has not been achieved, making it difficult to provide reliable estimates of their magnitudes. The FB model introduces the effect of particle agglomeration on the gas-solid drag force, taking particle asperities into account. Interparticle van der Waals forces are predicted to delay the onset of bubbling and to extend the range of homogeneous fluidization. Quantitative differences remain between different models and experimental results, possibly due to the closure model used for the drag force. Bed expansion data are compared in Table 7. The expanded bed heights show that both the drag model and mesh size are important in predicting bed expansion. The dependency of bed expansion on mesh size was also reported recently by Wang et al. (2009), who found that a very small time step (~ 10-5 s) and a highly refined mesh (based on grid independence testing) were required for correct prediction of bed expansion using the TFM approach. Refining the mesh makes solution of the model exceedingly computer-intensive and time-consuming, thus decreasing the functionality of the model for large-scale applications. The CPFD approach is able to predict the correct bed expansion without resorting to such a highly refined mesh or needing to modify the drag law. Of interest is the close agreement between results obtained by the FB model and CPFD simulations, although the latter does not take into account the inter-particle cohesive forces. A tentative reason was analyzed to be the effect of the PSD. The FCC particles used in this study have a wide PSD (Dubrawski et al, 2013), reaching particle diameters larger than 200 µm. Investigation of the mean agglomerate diameter in the bed using the FB model suggested that in most regions of the dense phase, the time-average agglomerate diameter was around 200 µm. This implies that the size of agglomerates in general does not surpass the maximum particle diameter for the FCC particles studied here. Hence, the
 

effects would be similar to the drag reduction due to presence of particles as large as the upper limits of PSD which is taken into account in CPFD simulations. Further studies considering Geldart A particles with narrower PSDs and comparison of the results with simulations utilizing the FB model are recommended.

4.2.4. Comparison of 2-D and 3-D models Figure 16 compares the solid volume fraction distribution predicted using the 2-D and 3-D codes. Significant differences occur, with better agreement between the 3-D code predictions and the experimental data, suggesting the need for 3-D simulations for engineering applications. 3-D simulations are, in any cases more realistic than 2-D ones considering wall effects and threedimensional phenomena. When the micromechanics of particle-fluid systems, of bubble size and velocity are the focus, then 3-D simulations should be used, as they give more realistic results. 2D models neglect the friction resistance of the walls While 2-D simulations have theses drawbacks, they have the benefit of reduced computational time, and they can be used to predict qualitative behavior. However, great care is needed in interpreting quantitative 2-D predictions.

5. Concluding remarks Hydrodynamic characteristics of bubbling/slugging fluidized beds of FCC particles, such as voidage distribution, particle volume fraction profiles, bed expansion, formation of agglomerates, solid velocity vectors and circulation patterns have been investigated using CFD and CPFD codes. Several drag models were employed in the CFD simulations, and their predictions are compared. It is shown that existing drag models for fluidized beds need parameter tuning to be applied to Geldart A powders due to formation of clusters and corresponding reduction in overall drag. The new force balance (FB) model (Ahmadi Motlagh et al., 2014) is investigated given its capability to model the formation of agglomerates. The application of different drag models significantly affects the flow of the granular phase by influencing the predicted bed expansion and the particle concentration in the dense phase regions of the bed. The FB model is promising enough to warrant further testing and development. The predictive capability of the CPFD method was analyzed for the influence of several crucial modeling parameters, including mesh size, interphase drag models and particle size distribution.
 

Including the real particle size distribution in the CPFD scheme led to better predictions of the overall flow characteristics than assuming mono-disperse particle of mean size. The results show promising predictive capability, without having to resort to modifying the drag model. The FB model and CPFD both capture the gross solid circulation phenomena. The voidage profile and bed expansion predictions are consistent with the experimentally measured data. The salient features of TFM and CPFD, together with their potential and shortcomings, are summarized in Table 8. Neither the FB model nor CPFD needs a highly resolved numerical grid to capture the mesoscale behavior of cluster formation, making these models promising. Acknowledgment The authors are grateful to the Natural Sciences and Engineering Research Council of Canada and to the Canada Research Chairs program for financial support of this work. The authors also acknowledge Prof. C.M. Hrenya, Department of Chemical Engineering, University of Colorado, Boulder for her valuable guidance.

Nomenclature CD

drag coefficient of a single particle [-]

dp

particle diameter [µm]

e

restitution coefficient [-]

Fd

drag force [N]

Fc

collisional force [N]

Fvw

van der Waals force [N]

Fg

gravity [m s-2]

g0

radial distribution function [-]

P

fluid pressure [Pa]

Ps

particle pressure [Pa]




Re

Reynolds number

t

time [s]

v

velocity [m s-1]

umb

minimum bubbling velocity [m s-1]

Greek letters

β

drag coefficient [-]

ε

volume fraction [-]

Θ

granular temperature [m2 s-2]

λ

thermal conductivity [m2 s-2]

µ

viscosity [Pa s]

ρ

density [kg m-3]

τ

stress tensor [Pa] [m s-1]

φ

specularity coefficient [-]

Subscripts g

gas phase

s

solids phase

w

wall

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Ansys-Fluent Users Guide, version 12.0, 2009. Fluent Inc., Lebanon, NH. Beeststra, R., van der Hoef, M.A., Kuipers, J.A.M. 2006. A lattice-Boltzmann simulation study of the drag coefficient of clusters of spheres. Computers & Fluids 35, 966-970. Bokkers, G. A., Laverman, J. A., van Sint Annaland, M., Kuipers, J. A. M. 2006. Modeling of large-scale dense gas-solid bubbling fluidized beds using a novel discrete bubble model. Chemical Engineering Science 61, 5590-5602. Courant, R., Friedrichs, K., Lewy, H. 1928. Über die Partiellen Differenzengleichungen der Mathematischen, Physical Mathmatics Ann. 100, 32-74. Deen, N.G., van Sint Annaland, M., Kuipers, J.A.M. 2006. Detailed computational and experimental fluid dynamics of fluidized beds, Applied Mathematical Modeling 30, 14591471. Ding, J., Gidaspow, D. 1990. A bubbling fluidization model using kinetic theory of granular flow. AIChE Journal 36, 523. Dubrawski, K., Tebianian, S., Bi, H.T., Chaouki, J., Ellis, N., Gerspacher, R., Jafari, R., Kantzas, A., Lim, C., Patience, G.S., Pugsley, T., Qi, M.Z., Zhu, J.X., Grace J.R. 2013. Traveling column for comparison of invasive and non-invasive fluidization voidage measurement techniques. Powder Technology, 235, 203-220. Ergun, S. 1952. Fluid flow through packed columns, Chemical Engineering Progress 48 (2), 8994. Gao, J.; Chang, J.; Xu, C.; Lan, X.; Yang, Y. 2008. CFD simulation of gas solid flow in FCC strippers. Chemical Engineering Science 63, 1827–1841. Gidaspow, D. 1994. Multiphase Flow and Fluidization. Continuum and Kinetic Theory Description. Academic Press, Boston.


 

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Li, J., Kwauk, M., 1994. Particle-fluid two-phase flow: The Energy Minimization Multi-scale method. Metallurgical Industry Press: Beijing, P.R.China. Li, J., Kwauk, M. 2001. Multi-scale nature of complex fluid-particles systems. Industrial Engineering Chemistry and Research 40, 4227-4237. Li, J., Kwauk, M. 2003. Exploring complex systems in chemical engineering: The multi-scale methodology. Chemical Engineering Science 58, 521-535. Li, T., Pougatch, K., Salcudean, M. and Grecov, D. 2008. Numerical simulation of horizontal jet penetration in a three-dimensional fluidized bed, Powder Technology 184, 89-99. Li, T., Grace, J and Bi, Xi. 2010. Study of wall boundary condition in numerical simulations of bubbling fluidized beds, Powder Technology 203, 447–457. Liang, Y., Zhang, Y., Li, T., Lu, C. 2014. A critical validation study on CPFD model in simulating gas-solid bubbling fluidized beds, Powder Technology 263, 121-134. Lu B, Wang W, Li J. 2009. Searching for a mesh-independent sub grid model for CFD simulation of gas–solid riser flows. Chemical Engineering Science 64, 3437–3447. Lu, H., Sun, Q., He, Y., Ding, Y. S. J., Li, X. 2005. Numerical study of particle cluster flow in risers with cluster-based approach. Chemical Engineering Science 60, 6757–6767. Lv, X., Li, H., Zhu, Q. 2014. Simulation of gas–solid flow in 2D/3D bubbling fluidized beds by combining the two-fluid model with structure-based drag model. Chemical Engineering Science 236, 149-157. Ma, J., Ge, W., Wang, X., Wang, J., Li, J. 2006. High-resolution simulation of gas-solid suspension using macro-scale particle methods. Chemical Engineering Science 61, 7096– 7106. Massimilla, L., Donsi, G. 1976. Cohesive forces between particles of fluid-bed catalysts, Powder Technology 15, 253-260.  

McKeen, T., Pugsley, T. 2003. Simulation and experimental validation of a freely bubbling bed of FCC catalyst, Powder Technol. 129, 139-152. Niemi, T.J. 2012. Particle Size Distribution in CFD Simulation of Gas-particle Flows, (Master Thesis) Aalto University, Finland. Pain, C.C., Mansoorzadeh, S. and de Oliveira, C. R. E. 2001. A study of bubbling and slugging fluidised beds using the two-fluid granular temperature model, International Journal of Multiphase Flow 27, 527-551. Patil, D.J., Sint Annaland, M. van and Kuipers, J.A.M. (2005). Critical comparison of hydrodynamic models for gas-solid fluidized beds - Part II: freely bubbling gas-solid fluidized beds. Chemical Engineering Science, 60(1), 73-84. Shi, Z., Wang, W., & Li, J. 2011. A bubble-based EMMS model for gas–solid bubbling fluidization. Chemical Engineering Science 66(22), 5541–5555. Shuai,W., Huilin, L., Guodong, L., Zhiheng, S., Pengfei,X., Gidaspow, D. 2011. Modeling of cluster structure-dependent drag with Eulerian approach for circulating fluidized beds, Powder Technology 208, 98-110. Tebianian, S., Motlagh, A.H.A., Vashisth, S., Cocco, R.A., Ellis, N., Hays, R., Karri, S.B.R., Grace, J.R. 2014. Extending the comparison of voidage measurement and modeling techniques in fluidized beds, in Proc. 11th International Conference on Fluidized Bed Technology. Ed. J. Li, X.J. Bao and W. Wang. Chemical Industry Press, Bejing 137 – 142. van Wachem, B., Sasic, S. 2008. Derivation, simulation and validation of a cohesive particle flow CFD model. AIChE Journal 54, 9-19.


 

van Wachem, B.G.M., Schouten, J.C., Krishna, R. and van den Bleek, C.M. 1999. Validation of the Eulerian simulated dynamic behaviour of gas-solid fluidized bed. Chemical Engineering Science 54 (13-14), 2141 - 2149. Wang, J. 2009. A review of Eulerian simulation of Geldart A particles in gas-fluidized beds, Industrial Engineering Chemistry and Research 48, 5567-5577. Wang, J., Liu, Y. 2010. EMMS-based Eulerian simulation on the hydrodynamics of a bubbling fluidized bed with FCC particles, Powder Technology 197, 241–246. Wang, S., Hao, Z.H., Lu, H.L., Yang, Y.C., Xu, P.F., Liu, G.D., 2012. Hydrodynamic modeling of particle rotation in bubbling gas-fluidized beds. International Journal of Multiphase Flow 39, 159-178. Wang, Q., Yang, H., Wang, P., Lu, J., Liu, Q., Zhang, H., Wei, L., Zhang, M. 2014. Application of CPFD method in the simulation of a circulating fluidized bed with a loop seal, IDetermination of modeling parameters. Powder Technology 253, 814-821. Wen, C.Y., Yu, Y. H. 1966. Mechanics of fluidization. Chemical Engineering Progress Symposium Series, 62, 100-111. Yang, N., Wang, W., Ge, W., Li, J., 2003. CFD simulation of concurrent upgas solid flow in circulating fluidized beds with structure-dependent drag coefficient. Chemical Engineering Journal 96, 71-80. Ye, M., van der Hoef M.A., Kuipers, J.A.M. 2004. A numerical study of fluidization behavior of Geldart A particles using a discrete particle model. Powder Technology 139(2), 129–39. Ye, M., van der Hoef M.A., Kuipers, J.A.M. 2005. From discrete particle model to a continuous model of Geldart A. Chemical Engineering Research and Design 83(A7), 833-843.

 

Yu, A.B., Xu, B.H., 2003. Particle-scale modelling of gas–solid flow in fluidisation. Journal of Chemical Technology and Biotechnology 78, 111-121. Zhang, D. Z.; Vanderheyden, W. B. 2002. The effects of mesoscopic structures on the macroscopic momentum equations for two-phase flows. International Journal of Multiphase Flow 28, 805-822. Zou Z, Li H Z, Zhu Q S, Wang Y C. 2013. Experimental study and numerical simulation of bubbling fluidized beds with fine particles in two and three dimensions. Industrial Engineering Chemistry and Research. Zimmermann, S., Taghipour, F. 2005. CFD modeling of the hydrodynamics and reaction kinetics of FCC fluidized-bed reactors. Industrial and Engineering Chemistry and Research 44, 9818-9827. 

List of Figures Figure 1. Schematic of experimental column showing computational domain [Ho is the initial bed height] Figure 2. Mesh refinement test for 2D geometry using FB model for FCC at Ug = 0.4 m/s, dp = 103 µm, φ = 0.001 (a) radial voidage profile (b) axial voidage profile (c) solid volume fraction; left figures in each case represents snapshot at t =15 s and right figures are time-average values over 15 s. Figure 3. Instantaneous particle volume fraction of FCC at various times, t = 0.5, 1, 5, 10, 15 s for Ug = 0.4 m/s, dp = 103 µm, φ = 0.001, Ho = 0.80 m. Figure 4. Qualitative comparison of particle volume fraction of FCC at Ug = 0.4 m/s, dp = 103 µm, φ = 0.001, t = 15 s (a) Wen and Yu drag model; (b) EMMS-Shi model; (c ) EMMS-Hong model; (d) FB model. Figure 5. Effect of specularity coefficient on voidage at various heights, z = 0.24, 0.4 and 0.56 m for Ug = 0.4 m/s, dp = 103 mm, t = 15 s Figure 6. Effect of superficial gas velocity on voidage at various heights, z = 0.24, 0.4 and 0.56 m for dp = 103 µm, φ = 0.001, t = 15s.  

Figure 7. Mesh refinement test for 3D FB model at two levels; FCC at Ug = 0.4 m/s, φ = 0.001, t = 15 s (a) voidage; (b) particle velocity Figure 8. Time-average particle volume fraction predictions from mesh refinement test for 3D CPFD geometry for FCC at Ug = 0.4 m/s, t = 40 s (i) 36,000 real cells and 300,000 particle clouds (ii) 140,000 real cells and 1.2 million particle clouds. Figure 9. 3D CPFD results for instantaneous particle volume fraction of FCC at Ug = 0.4 m/s. Figure 10. FB model prediction of instantaneous gas and particle velocity vectors; Ug = 0.4 m/s, t = 15 s. Figure 11. CPFD prediction for gas velocity vectors superimposed by the particle volume fraction shown in color; Ug = 0.4 m/s, t = 15 s. Figure 12. Variation of time - averaged particle volume fraction with height at Ug = 0.4 m/s, t = 15 s, Ho = 0.80 m. Figure 13. Solid volume fraction at different axial locations; Ug = 0.4 m/s, t = 15 s and 40 s predicted by CFD and CPFD. Figure 14. Visual representation of bubble formation in fluidized bed at Ug = 0.4 m/s using CPFD with bubble threshold at solid volume fraction of 0.2. Figure 15. Qualitative analysis of 3D snapshots of solid hold-up for FCC at Ug = 0.4 m/s, Ho = 0.80 m, φ = 0.001, t = 15 s (a) Wen and Yu; (b) EMMS-Shi; (c ) EMMS-Hong; (d) FB model; (e) CPFD. Figure 16. Effect of drag model on radial profile of voidage distribution at Ug = 0.4 m/s, Ho = 0.80 m, t = 15 s.

 

List of Tables Table 1. References using Eulerian-Eulerian and Eulerian-Lagragian fluidized bed models published from 2000 - 2014. Table 2. Governing equations and constitutive correlations in TFM model. Table 3. Five drag closures. Table 4. Equations for four possible collision scenarios. Table 5. Governing equations in CPFD model. Table 6. Input parameters used in the simulation Table 7. Expanded bed height predictions for FCC; φ1 = 0.001; Ho = 0.80 m Table 8. Comparing TFM and CPFD models.

   



 

Table 1. References using Eulerian-Eulerian and Eulerian-Lagragian fluidized bed models published from 2000 - 2014. Authors

Main Features of Model

Remarks

Krishna and van Baten Bubble-emulsion Eulerian model to Properties of the emulsion phase such as (2001) scale-up BFB density and viscosity were set constant. Empirical correlations were used to determine bubble-emulsion interaction force. Zhang and Vanderheyden Eulerian model with scaled drag Corrected drag model by rescaling (2002) correlation parameter obtained from homogeneous fluidization. Yang et al. (2003)

McKeen (2003)

and

Eulerian model with drag correction Extended EMMS approach. Incorporated based on effect of particle clustering structure-dependent drag model into Eulerian model. Pugsley TFM with scaling drag model of Modification attributed to formation of Gibilaro et al. (1985) by a fractional clusters of size smaller than CFD grid constant of 0.25. size.

Yu and Xu (2003); Ye et al. DEM – CFD coupled model (2004)

Zimmermann Taghipour (2005)

Lu et al.(2005)

Incorporated van der Waals force. Found existence of a homogenous regime with the adhesion force.

and Eulerian model with modified Drag modified based on minimum Syamlal O’Brien (1987) drag fluidization velocity. Predicted bed correlation expansion and radial void fraction in good agreement with experiments. Eulerian model with structure-based Concept of gas-cluster modeling was drag extended.

Bokkers et al. (2006) and Discrete Laverman et al. (2007) model

bubble-emulsion

phase Phase properties like density and viscosity were obtained from experiments and bubbles were individually tracked.

Li et al. (2008)

Eulerian model with modified Reported limited range of applicability of Syamlal O’Brien drag correlation Zimmermann & Taghipour (2005) model.

Gao et al. (2008)

Eulerian model with modified particle Replace particle size with the constant representation by cluster cluster size in standard Eulerian model and showed improved predictions. 



van Waschem and Sasic TFM with force-balance model in Force balance model used to predict (2008) BFB cluster size of the emulsion-phase. Lu et al. (2009)

Eulerian model with drag correction Extension of EMMS model for CFBs. based on effect of particle clustering

Wang and Liu (2010)

EMMS based TFM for BFB

Shuai et al. (2011)

TFM with cluster structure-dependent Dynamic formation and dissolution of (CSD) drag model. clusters captured in CFB.

Shi et al. (2011)

Modified EMMS model

Karimipour (2012)

and

Used implicit cluster diameter expression for inter-phase drag relation.

Bubble-based EMMS model to consider effect of heterogeneous structures on inter-phase drag.

Pugsley MP-PIC approach for BFB using Promising predictions without need for CPFD modification of drag model.

Zou et al. (2013)

CFD-Modified balance model

agglomerate

force Predictions consistent with experiments. The model doesn’t take back-mixing effect into account.

Lv et al. (2014)

Modified the conventional drag Solid circulation pattern and axial and correlations by considering the local radial solid concentration profiles well captured. structural parameters.

Liang et al. (2014)

CPFD model

Reported that CPFD can obtain better profiles of solids velocity than TFM. CPFD cannot simulate bubble coalescence phenomena in BFB for dp ~ 700 micron.

BFB – Bubbling fluidized bed; TFM – Two-fluid model; DEM – Discrete element method

 

Table 2. Governing equations and constitutive correlations in TFM model. ∂ ( ρ mε m )

1.

Continuity equations



2.

Momentum conservation equations

 «

∂t

ª ∂ ( ρ g ε g vg ) «¬

Conservation of momentum for solid phase

4.

Granular energy conservation

5.

Stress-strain tensor for  phase m (s: solids and g: gas)

Solid pressure

7.

Radial distribution function

8.

Solid viscosity model:

º + ∇. ( ρ g ε g vg vg ) » = −ε g ∇P + ∇.τ g + β ( vg − vs ) + ε g ρ g g »¼

ª ∂ ( ρ sε s vs ) º + ∇. ( ρ sε s vs vs ) » = −ε s ∇P − ∇Ps + ∇.τ s + β ( vs − vg ) + ε s ρ s g « ∂t ¬ ¼

3.

6.

∂t

C C º § 3 ª ∂ ( ρ sε sθ ) · + ∇. ( ρ sε s vsθ ) » = ¨ − Ps I + τ s ¸ : ∇vs + ∇. ( kθ ∇θ ) − γ θ + ϕ gs 2¬ ∂t ¹ ¼ ©

 «





§ ©

· ¹

2

Ps = ε s ρ sθ + 2 ρ s (1 + e ) ε s2 g 0θ 1 ª § 3º · ε g0 = «1 − ¨ s ¸ » « ¨© ε s , max ¸¹ » ¬ ¼

−1

µ s = µ s ,col + µ s ,kin 1





4 §θ · 2 µ s ,col = ε s ρ s d p g 0 (1 + e ) ¨ ¸ 5 ©π ¹ 2 10 ρ s d p θπ ª 4 º µ s ,kin = 1 + g 0ε s (1 + e ) » 96ε (1 + e ) g «¬ 5 ¼ 0

s

Kinetic term (Gidaspow model): 9.

Solid bulk viscosity



4 §θ · λs = ε s ρ s d p g0 (1 + e ) ¨ ¸ 3 ©π ¹

Table 3. Five drag closures.   

C

τ m = ε m µ m ( ∇.vm + ∇.vmT ) + ε m ¨ λm − µm ¸ ∇.vm I 3



Collision term

+ ∇. ( ρ mε m vm ) = 0

1

2

Wen-Yu (Wen & Yu, 1966)

d 1p = Cd

3 σr µf − µp 8 σg r



Re < 0.5 

0.5 ≤ Re ≤ 1000  Cd = Re > 1000 

Wen-Yu/Ergun combination (Gidaspow, 1994)

24 1 + 0.15 Re 0.687 θ f−2.65  Re

(

)

Cd = 0.44θ f−2.65 



θ p > 0.85θCP

D = Dp2 



θ p < 0.75θCP

D = D1p 



EMMS(Shi, Wang, & Li, 2011)

24 −2.65 θf  Re

Cd =

0.50θ CP ≥ θ p ≥ 0.75θ CP

D=

θ p − 0.85θ ( Dp2 − D1p ) + D1p 0.85θCP − 0.75θ CP

Drag coefficient for gas-solid system:

­ ρg ε s (1 − ε s ) µ g + 1.75ε s u g −us , ε g < ε mf °150 2 εg dp dp ° ° ε g ε s ρ g u g −us −2.65 °3 β = ® CD ε g Hd , ε mf ≤ ε g < ε d dp °4 ° ° 3 C ε g ε s ρ g u g −us ε −2.65 , g ε g > εd °4 D dp ¯ where

H d = −0.01354 + 0.00386 exp(6.22859 × ε g ); if ( H d > 1)then( H d = 1)

EMMS(Hong, Shi, Wang, & Li, 2013)

Drag coefficient for gas-solid system:

­ ρg ε s (1 − ε s ) µ g + 1.75ε s u g −us , ε g < ε mf °150 2 εg dp dp ° ° ε g ε s ρ g u g −us −2.65 °3   β = ® CD ε g Hd , ε ≤ ε < ε 4 d mf g d p ° ° ° 3 C ε g ε s ρ g u g −us ε −2.65 , g ε g > εd °4 D dp ¯  



­°72.67959 − 372.71879 × ε g + 639.94137 × ε g2 − 366.87111ε g3 , Hd = ®  2 °¯−0.02369 + 0.01012 × ε g + 0.78978 × ε g , (0.45 < ε g ≤ 0.5517) (0.5517 < ε g ≤ 1) FB (Ahmadi Motlagh, et al., 2014)





ρ g ε g ε s vg − vs −2.65 §1· 3 β = ¨ ¸ . CD εg dp ©N¹ 4



0.687 24 ª ­ º ° C ε Re ¬«1 + 0.15 ( ε g Re ) ¼» Re < 1000 CD = ® (g ) °C 0.44 Re > 1000 ¯

Re =



ρ g ε g v g − vs d p µg

Table 4. Equations for four possible collision scenarios. 0.4

§ § ·· E ¨ 4. d p . ¨ ¸ ¸ 2 ¨ ¸ ¨ π © 2. (1 − ϑ ) ¹ ¸ × 1 + e × ( 3θ )1.2  a1 = + ( ρ p − ρ g ) g 0.052 × πρ p × ¨ ( ) ¸ πρ p d 3p 6 ¨ ¸ ¨¨ ¸¸ © ¹ 

2

a2 = + 0.125π .Cd . (1 − ε g ) ρ g ε g Vg − Vs ε g −2.65

a3 = +



A 24δ 2


 ( Fd + Fg + Fvw = Fc )



a1d a2 + a2 d a + a3 = 0


 ( Fg + Fvw = Fc + Fd )



a1d a2 − a2 d a + a3 = 0



 ( Fd + Fvw = Fc + Fg )



− a1d a2 + a2 d a + a3 = 0


  ( Fd + Fg + Fc = Fvw )



− a1d a2 − a2 d a + a3 = 0

 

Table 5. Governing equations in CPFD model. Continuity equation for fluid

∂ (θ f ρ f ) + ∇. (θ f ρ f u f ) = 0 ∂t

Momentum equation for fluid

∂ (θ f ρ f u f ) + ∇.(θ f ρ f u f u f ) = −∇p − F + θ f ρ f g + ∇.ρ f τ f ∂t

Particle acceleration

du 1 1 = D (u f − u p ) − ∇p + g − ∇τ p dt ρp θpρp

Particle volume fraction

θ fξ

1 = Ωξ

Np

¦Ω

p

n p S pξ

1

S pxξ ( x p ) = 0 xi −1 ≥ x p ≥ xi +1 =1 x p = xi Implicit equations for particle velocity









(

U pn +1 =



1

ρp

C Ppn +1 −

1

θpρp

Cȉ nρρ+1 + g

1 +CtD p

X pn +1 = X np + ǻtU np +1



Particle normal stress

)

U pn +Ct D pU nf +p1 − 

Interphase momentum transfer



Fξn +1 =

1 ȍξ

ª

¦S «« D (U P

IJ=



¬

p

n +1 f ,p

− U pn+1 ) −

º C Ppn +1 » n p m p ρp »¼ 1

Psθ pβ max ª¬(θ CP − θ p ) ε (1 − θ p ) º¼

Table 6. Input parameters used in simulations Geometry Vessel dimension Solid fraction at maximum packing Minimum fluidization velocity

2D and 3D 0.96 m long x 0.133 m id (dense bed section) 1.36m long x 0.190 m id (freeboard section) 0.55 6.06 m/s 






Gas superficial velocity Gas density Bed depth Particle size distribution Sauter mean particle diameter Particle density CFD calculations Grid

0.3, 0.4, 0.5 m/s 1.225 kg/m3 0.80 m As in Dubrawski et al. (2013) 98 µm 1560 kg/m3

Mesh numbers for 2D geometry: 2480, 3900, 6993, 9600 Mesh numbers for 3D geometry: 2332, 3528, 5256, 7176 Uniform velocity inlet; Pressure outlet; Gas wall : no –slip; Solid-wall: partial slip Wen-Yu (Wen and Yu, 1966), EMMS-Shi (Shi et al., 2011), EMMS-Hong (Hong et al., 2013), FB (Ahmadi Motlagh et al., 2014) Laminar 15 s 0.0001 s SIMPLE 0.99 0.001, 0.01, 0.1

Boundary conditions Drag models Gas flow type Simulation time Time step Pressure-velocity coupling Restitution coefficient Specularity coefficient CPFD calculations Grid Total number of clouds Drag models Flow type Flow boundary condition

36,000; 140,000; 200,000 300,000; 1.2 x 106 Wen-Yu/Ergun combination Turbulent (Large eddy simulation) Non-uniform velocity inlet based the perforated distributor plate ( 49 perforations on circular pattern) 100000 Pa 40 s 0.001 Maximum momentum redirection from collision: 40% Normal-to-wall momentum retention: 0.95 Tangent-to-wall momentum retention: 0.99 Diffuse bounce: 0

Pressure boundary condition Simulation time Time step Particle-particle interaction Particle-wall interaction

Table 7. Expanded bed height predictions for FCC; φ1 = 0.001; Ho = 0.80 m Ug, m/s 0.3

Expt, (m) 1.020

FB Model (m) 1.037

EMMS-Hong (m) 



CPFD (m) 1.042

0.4 0.5

1.100 1.165

1.098 1.175

1.120 -

1.112 1.169

Table 8. Comparing TFM and CPFD models TFM

CPFD

The granular material is characterized by mean particle size and density.

Capable of distributions.

Granular phase properties such as granular temperature, solid viscosity and particle interaction are determined based on Kinetic theory of granular media, with empirical values of restitution coefficient and friction coefficient.

Particle-particle interaction is calculated from Harris-Crighton’s model; Particle-wall interaction is determined from normal-to-wall momentum and tangent-to-wall momentum retention and diffuse bounce.

Needs reduction of gas-solid drag to avoid bed expansion.

Predicts reasonable bed expansion without needing any reduction in the gas-solid drag model.

Finer mesh generally prediction accuracy.

the

In addition to mesh size, parcel number is also a crucial parameter. Finer mesh size and large parcel number can greatly increase the computational time.

Smaller time-steps ~ 10-4 or 10-5 s, is needed for better convergence.

An appropriate time step is automatically determined and adjusted by the CFL algorithm

enhances

 

incorporating

particle

size

based on mesh and number of particle-parcels. Limited application to large or industrial sized fluidized beds.



The concept of computational particles and MP-PIC method enables simulation of large numbers of particles in industrial scale fluidized beds.



 

Highlights

Employed CFD and CPFD codes to study hydrodynamics of bubbling FCC fluidized beds.



Voidage distribution, bed expansion and formation of agglomerates are compared.



Hydrodynamic predictions using several drag models in CFD simulations are analyzed.



Critical comparison of CFD, CPFD with experimental data presents their pros/cons.



 

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