A bubble-based drag model at the local-grid level is proposed to simulate gas-solid flows in bubbling fluidized beds of Geldart A particles. In this model, five balance equations are derived from the mass and the momentum conservation. This set of equations along with necessary correlations for bubble diameter and voidage of emulsion phase is solved to obtain seven local structural parameters (
Gas-solid fluidized bed provides thorough mixing, high heat, and mass transfers between gas and solid phases. Due to these reasons, bubbling beds are widely used in industrial applications involving coal gasification, fluid-catalytic-cracking (FCC), and mineral processing [
In the CFD studies, Eulerian approach or the so-called two-fluid model (TFM) has been widely used to simulate industrial-scale reactors due to the limited computational resources [
Another practical way, to account for the effects of these meso-scale structures, is to employ a modified drag model which is established on the basis of heterogeneous structures [
In present work, in light of the above mentioned limitations, a novel bubble-based drag model based on local-grid information is developed to deal with the impact of heterogeneity on the drag force. Firstly, within each computational cell, heterogeneous structures are resolved into the discrete bubble phase, the continuous emulsion phase, and their interphase. Then the structural parameters within each cell are computed by solving seven equations, involving mass conservation, force balance, and reliable correlations for the bubble diameter and emulsion voidage. Next the bubble-based drag coefficient is derived from the resolved structural parameters. Finally, this new drag coefficient is incorporated into the two-fluid model to simulate the hydrodynamics of Geldart A particles in a bubbling fluidized bed. Therefore, the current work can be considered as a tuning process where subgrid corrections are incorporated based on local grid information if the system was inhomogeneous. Comparison of simulation results to available experimental data is also presented for validation of this new drag model.
For a bubbling fluidized bed, the gas-solid flow can be resolved into the bubble and the emulsion phases. The discrete bubbles are surrounded by the emulsion phase. Consequently, the local flow within each computational cell can also be divided into three subsystems: the emulsion phase, the bubble phase, and the interphase, as shown in Figure
Resolution of local heterogeneity within each computational cell for bubbling fluidization.
For solving seven structure parameters, seven independent equations, involving mass conservation, force balance and empirical correlations, are employed as follows.
For specified local information (
For each computational cell, the local information ( Calculate Calculate Calculate Calculate Calculate Calculate Calculate Calculate
Computation scheme of solving the bubble-based drag model.
In this work, the two-fluid model (TFM) is adopted to simulate bubbling fluidization. The CFD software, FLUENT® 6.3.26, was used as fluid solver to carry out all simulations. The governing equations of the two-fluid model for gas-solid flows are summarized as follows.
Gas phase: Solid phase:
Gas phase: Solid phase:
Stress-strain tensor: Gas phase: Solid phase: Solid phase pressure: Solid phase viscosity: Solid phase bulk viscosity: Radial distribution function: Diffusion coefficient of granular energy: Collision energy dissipation:
In the TFM, the continuity equations for the gas and the particle phases are given as
A lab-scale bubbling fluidized bed was selected to validate this new drag model, as shown in Figure
Summary of the physical properties of gas and particles.
Gas density (kg/m3) | 1.225 |
Particle density (kg/m3) | 1780 |
Gas viscosity (Pa·s) | 1.7894 × 10−5 |
Particle diameter (m) | 6.5 × 10−5 |
Minimum fluidization velocity (m/s) | 0.003 |
Schematic diagram of simulated bubbling fluidized bed with Geldart A particles.
Hexahedral mesh with a grid size of 1 cm was used in the present simulations. The mesh-independence for the current gas-solid system has already been carried out in the literature [
Simulation settings used in FLUENT solver.
Time | Unsteady, first-order implicit |
Viscous | Laminar |
Particle-particle coefficient restitution | 0.9 |
Pressure-velocity coupling | Phase-coupled SIMPLE |
Momentum discretization | Second-order upwind |
Volume fraction discretization | Quick |
Granular temperature | Algebraic KTGF |
Granular viscosity | Gidaspow |
Granular bulk viscosity | Lun et al. |
Frictional viscosity | Schaeffer |
Angle of internal friction | 30° |
Solids pressure | Lun et al. |
Frictional pressure | Based KTGF |
Radial distribution | Lun et al. |
Specularity coefficient | 0.6 |
Close packing density | 0.63 |
Time step | 0.0005 s |
Max iterations per time step | 30 |
Initial static bed height | 1.2 m |
Initial bed packing fraction | 0.6 |
Inlet gas velocity | 0.2 m/s |
Figure
Instantaneous snapshots of solid concentration in bubbling fluidized bed (blue and red denote the dilute and dense flow regions, resp.).
Figure
Time series plot of instantaneous solid concentration near the wall and the center regions at two different heights: (a)
Figure
Time series plot of instantaneous axial particle velocity near the wall and the center regions at two different heights: (a)
Figure
Comparison of experimental and simulation results for axial solid profiles.
Figure
Comparison of experimental and simulation results for radial solid concentration profiles at four heights of the bed: (a)
A novel bubble-based drag model for bubbling fluidized beds has been proposed in this work based on the description of heterogeneity within each computational cell. In this model, the effect of meso-scale bubbles on the drag force has been taken into account. Seven structural parameters have to be determined by solving a set of balance equations, involving mass conservation, force balance, and reliable correlations for predicting of bubble diameter and emulsion voidage. Then, a bubble-based drag coefficient has to be derived directly from local structure parameters. It is worthwhile to note that the slip velocity (or the gas and the solid velocities) plays an important role in addition to the local voidage in our model. For validation, this new drag was incorporated into the two-fluid model to simulate the hydrodynamic of Geldart A particles in a bubbling fluidized bed. The simulated results, using this new drag model, show a qualitative agreement with the experiments. More works are, however, required to verify and revise this new model.
Bubble acceleration (m/s2)
Drag coefficient of multi-bubble
Drag coefficient for single bubble
Bubble diameter (m)
Particle diameter (m)
Reactor diameter (m)
Elastic coefficient
Drag force (N)
Gravitational acceleration (m/s2)
Pressure (Pa)
Reynolds number
Real velocity (m/s)
Operating gas velocity (m/s)
Superficial minimum fluidization velocity (m/s)
Superficial slip velocity between bubble and emulsion phases (m/s).
Drag coefficient (kg/m3·s)
Volume fraction of bubbles
Voidage (or particle concentration)
Bulk viscosity (Pa·s)
Viscosity (Pa·s)
Density (kg/m3)
Stress tensor (Pa).
Bubble
Emulsion phase
Gas phase
Particle
Solid phase
Interphase.
The authors declare that they have no competing interests.
Special thanks are due to Professor Sumin Zhou and Xiaojuan Chen for their valuable discussions and helpful suggestions in the revised manuscript. This work is financially supported by the National Natural Science Foundation of China under Grant no. 21406081, the Foundation of Jiangsu Provincial Engineering Laboratory for Advanced Materials of Salt Chemical Industry under Grant no. 201501, Innovation Project of Jiangsu Province under Grant no. SJZZ15_0200, and the Foundation of Huaiyin Institute of Technology, Grant no. HGC1411.