Evaluation of a Dual-Grid Method for Multiscale TFM and CFD-DEM Simulations of Dense Gas-Solid Flows

Simulations of chemical reactors using CFD has become more popular over the last years. Dense gas-solid flow simulations can be achieved through several methods. The most common simulation approaches are the Eulerian-Eulerian, Eulerian-Lagrangian model and the Direct Numerical Simulation (DNS). The Direct Numerical Simulation is the most computationally demanding model as the particle surface is depicted as a boundary during the simulation. Therefore, this model is only suitable for small scale geometries with a small number of particles. With the Eulerian-Lagrangian approach the fluid phase is modeled in a Eulerian way, while the particles are tracked individually. This approach consists of several sub-models, e.g., Multiphase Particle-In-Cell (MP-PIC) [1] or Discrete Element Model [2], often called CFD-DEM. The focus for the Euler-Lagrangian approaches will be on CFD-DEM in this work. This approach is not as computationally demanding as the DNS, but is limited by high numbers of particles. The Eulerian-Eulerian approach is the computationally least demanding method for dense gas-solid simulations compared to the above mentioned models. In this approach the fluid is modeled according to a Eulerian model and the solid phase is regarded as a pseudo fluid and simulated the same way as the fluid. This model is often called Two Fluid Model (TFM) [3]. The particles are not resolved individually, as it is done using CFD-DEM or DNS, but the particles are regarded as a solid phase in a cell with a certain volumetric solid fraction. In order to account for the particulate behavior of the pseudo fluid, i.e. the solid phase, further models have to be considered.

The Kinetic Theory of Granular Flow (KTGF) [4] is widely applied for TFM in dense gas-solid flow simulations. In this theory the particles are modeled as a fluid. Collisions between particles are approximated with statistical approaches. To achieve a high accuracy of the collision forces, a large number of particles in a cell is needed for the calculation of the statistical equations. The finite volume method discretizes the computational domain in three-dimensional cells. To satisfy the above stated conditions of a large number of particles in a cell, the particles have to be either very small compared to the cell size or the grid must be coarsened. Uddin et al. [5] suggest a grid size of 18 times the particle diameter (Geldart B particles) for grid independence. A coarse mesh, however, might not be able to resolve occurring bubbles or in 3D cases the fluid flow accurately, while the particle collisions might be modeled accurately.

Simulations of dense gas-solid flows using the CFD-DEM approach model the fluid as a continuum according to the Eulerian approach, while the solid phase is considered as discrete particles. Each particle is tracked with the equations of Newton’s Second Law of Motion. The coupling between the fluid and solid phase is done with the calculation of cell averaged forces between the fluid and particles. Important parameters influencing these forces are for example the solid phase fraction and the solid phase velocity. Particles which volumes are overlapping multiple cells are troublesome for the calculation of cell averaged values [6], as for example the solid phase fraction is calculated by dividing the total volume of all particles in a cell by the volume of the cell. Most approaches use the center point of a particle to determine if a particle is within a certain cell. At the worst, i.e., if the center point of a spherical particle is in the corner of a cubic cell, a maximum volume fraction of the particle of 87.5 % is outside of the cell but still accounted for the calculation of the cell-averaged solid volume fraction. To minimize the magnitude of error for this issue, the cell-to-particle volume ratio has to be great. In the literature a value of at least two is suggested [7]. Similar to the TFM, the cell size can be scaled up to overcome this problem with the disadvantage of inaccuracies regarding the fluid flow. Also, simulations of smaller particles are preferable according to this. But with smaller particles the number of total particles often increases as well. A larger amount of particles might result to unfeasible computational times as the particles are simulated with a discrete method.

A dual-grid method for TFM and CFD-DEM simulations was developed to overcome the problem of grid resolution for the fluid and solid phase. With this method the fluid phase and the solid phase are calculated on separate computational grids. The fluid grid consists of a finer grid to resolve the flow structure. The solid grid is a coarser grid. CFD-DEM and TFM simulations require a coarser grid to model the particulate phase with a high accuracy. Parameters which are necessary for the calculation of closure terms, e.g., momentum exchange or cell-averaged volume fraction of the solid phase, are mapped between those grids in a separate inner loop of the time-step until convergence is reached for each transient time-step. TFM and CFD-DEM simulations with the dual-grid method were performed with lab-scale and small pilot-scale geometries found in literature. The grid sizes were varied for the fluid and the solid mesh. All simulation results performed with the dual-grid model are compared to conventional TFM and CFD-DEM solvers. Furthermore, the simulation times for all cases are evaluated to determine the computational efficiency of the dual-grid method compared to the conventional solvers.

References

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[7] Z. Zhou et al., J. Fluid Mech. (2010), 661, 482-510.