Accounting for sub-parcel collisions when using the Coarse Grain Model (CGM)

In a previous post, the advantages of using the Coarse Grain Model (CGM) were discussed, along with the details of the model implemented in Rocky DEM. In general, coarsening the particles reduces the number of particles, decreasing the computational cost of the simulation. 

By using this approach, the particle population is represented by parcels of particles and the energy density in the original and coarse-grained system is similar, i.e, the velocities and temperatures of the parcels are expected to behave similarly to those in the original system.

However, using CGM can underestimate the total energy dissipation of the system as some collisions that happen in the original system are missed when coarsening the particles. Read below to understand why this happens and how to account for this dissipation when using the Coarse Grain Model. 

Missed sub-parcel collisions

In the original system, each individual particle has its own velocity and the relative velocity between particles may lead to collisions. Figure 1 illustrates these interactions: collisions happening between particles from the same group (intra-parcel collisions) are highlighted by orange dotted circles, whereas collisions happening between particles from different groups pass by each other (inter-parcel collisions) and are highlighted by blue dotted circles. 

When scaling the particles using the CGM model, the coarsened particle (also referred to as a parcel) represents groups of the original (smaller) particles and its velocity is given by the average velocity of the original particles. As a result, these sub-parcel collisions are not tracked when using the parcel-based approach, and the corresponding dissipated energy is not accounted for by the model. 

Intra-parcel contacts (marked by orange circles): collisions between particles from the same group. Inter-parcel contacts (marked by blue dotted circles): collisions between particles from different groups.
Figure 1. Intra-parcel contacts (marked by orange circles): collisions between particles from the same group. Inter-parcel contacts (marked by blue dotted circles): collisions between particles from different groups.

How to account for this energy dissipation when using Coarse Grain Modeling?

In Rocky DEM, the user has the option to include an additional energy dissipation to compensate for the sub-parcel collisions that are not captured when coarsening the particles, based on the work of Radl et al. [1]. 

The main challenge of the model is the estimation of the collision rate and inter-parcel stress for every flow regime, ranging from inertial to quasi-static regimes. Moreover, the velocity correction that is introduced by the model needs to be consistent regardless of the parcel size (including the possibility of using a 1-1 simulation) for both dense and dilute systems. 

Essentially, a spherical neighborhood region is defined around each parcel (as shown in Figure 2) and the parcel velocity is corrected at each time step based on the neighboring parcels mass-weighted averaged velocity and on a particle (parcel) response time.

Neighborhood region for computing the average parcel velocity.
Figure 2. Neighborhood region for computing the average parcel velocity.

The list of neighbor particles is updated every time the parcel travels a certain distance, given as a multiplier of the neighborhood radius, rRadl, which in turn is estimated based on the largest particle size. 

The parcel response time term uses the neighboring parcel velocities to estimate the probability of the missed contacts (radial distribution function) and the energy fraction lost in these missed contacts (damping function). The smaller the parcel response time, the faster the parcel velocity goes to the average velocity. 

In addition, as the Radl correction model is not limited to a particular flow regime, the model needs to weight the velocity correction according to the flow condition. In this sense, the parcel response time term includes two important sub-terms: one to turn off the model when the particles are settled or moving on a bed with slow contacts (shut-off function) and one term that reduces the particle response time as the scaling factor increases, as the greater the scaling factor, the more particles per parcel. 

More information about the model (including equations and implementation details) can be found in the Rocky DEM Technical manual. 

When to turn on this model and other limitations

Although in principle this correction should be used in any DEM simulation using the parcel approach [1], these internal collisions happen more often when particles move rapidly close to each other, i.e., the relative velocity between parcels is relevant. Consequently, the dissipated energy is more relevant in non-static dense flows with fast collisions.

Moreover, as the velocity correction includes additional averaging steps, the computational cost increases considerably when this model is turned on. 

Finally, Radl et al. model is not available for multi-GPU simulation in the current Rocky version. That impairs the simulation speed-up by combining more than one GPU card to solve the DEM equations. 

References

[1] Radl, S., Radeke, C., Khinast, J. G., and Sundaresan, S. (2011). Parcel-based approach for the simulation of gas-particle flows. In Proceedings of the 8th International Conference on CFD in the Oil & Gas, Metallurgical and Process Industries., 124–134


Lucilla Almeida

CAE Specialist at ESSS, D.Sc.

Lucilla holds a BE (Chemical) undergraduate degree, an M.Sc. in Chemical Engineering and a Ph.D. in Nuclear engineering from the Federal University of Rio de Janeiro. She joined ESSS in 2008 and has spent 5 years focused on applying CFD to solve common engineering problems in the Oil and Gas industry, dealing with turbulent and multiphase flow simulations. Since 2013, she is an Application Engineer for Rocky DEM Business Unit, supporting users, working on consultancy projects and validating models implemented for the CFD-DEM coupling.


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