On the prediction of particle collision behavior in coarse-grained and resolved systems

Abstract The discrete element method (DEM) simulates granular processes and detects inter-particle collisions during the simulation. Detection of collision helps researchers to study the occurrence of particulate mechanisms such as aggregation, breakage, etc. DEM demands high computational costs in simulating industrial-level systems, as it involves an enormous number of particles. DEM coarse-grained model can help to overcome this high computational cost issue. However, the frequency and probability of collisions for different particle size classes may change when coarser particles are introduced. This study introduces a new mathematical formulation, namely the collision dependency function (CDF), which predicts the probability of collisions between different particle classes for systems containing resolved and coarse-grained particles. The CDF is extracted by executing one DEM simulation consisting of number-based uniformly distributed particles. Furthermore, a new optimized scheme is used inside the DEM to store the collision data efficiently. Finally, the collision probabilities between size classes obtained from DEM simulations are compared successfully against their counterparts calculated from the developed model for verification.


Introduction
The processing of granular materials is crucial in several industries, like agriculture, pharmaceutical, metallurgical, etc. Flow and mixing during the transport of granular materials are significant phenomena.During any such process, the particles collide, and one may observe mechanisms like the aggregation of smaller particles to form bigger aggregates and the breakage of larger particles into smaller fragments.The estimation of the collision behavior between different particle classes needs to be handled with utmost priority to model such a problem (Das, B€ uck, and Kumar 2020;Das and Kumar 2021).Collision probability can provide information about how and to what extent the particles' size will change due to granular mechanisms such as aggregation and breakage (Barrasso and Ramachandran 2015;Barrasso et al. 2015).Various aspects of mixing phenomena, including the overall frequency of collisions, have been studied experimentally (Buchholtz, P€ oschel, and Tillemans 1995;Prigozhin and Kalman 1998;Schneider et al. 2011).However, it is tough to get information about collisions between individual particles experimentally.Several studies exist in the literature (Buffi ere and Moletta 2000; Tan et al. 2004;Aguilar-Corona, Zenit, and Masbernat 2011;Jiang et al. 2017) where researchers tried to estimate the collision frequency of particles theoretically or experimentally and established correlations.However, none of these studies attempted to predict the collision frequency and collision probability between different classes of particles for different particle size distributions (PSD).The current study aims to tackle this issue using a mathematical formulation.
There are Lagrangian and Eulerian approaches to simulate such granular flow behavior.In the Eulerian approach, the granular matter is treated as a continuum (Gidaspow 1994;Tian et al. 2018).This method cannot investigate the effects of particles' size and their polydispersity.Additionally, in this approach, the extraction of collision kinetics between different particles during simulation is not possible (Senapati and Dash 2021).Hence, this method is ineffective in investigating the collision behavior of the granular systems.As the granular particles are discrete, a Lagrangian approach, known as the Discrete Element Method (DEM), is widely used to simulate the granular flow in any system (Cundall and Strack 1979;Zhu et al. 2008).Each particle is tracked using Newton's second law of motion in DEM, and contact and body forces are accounted accordingly.Since DEM tracks each particle during the simulation, it is easier to study the collision frequency and probability of the system (Norouzi et al. 2016).Several open-source and commercial software [i.e., LIGGGHTS-Public (Kloss 2016), EDEM (Li et al. 2011), Blaze-DEM (Govender et al. 2015), MUSEN (Dosta and Skorych 2020)] are available to simulate granular materials using DEM.
In the present study, LIGGGHTS-Public (Kloss 2016) is used for all simulations.
In general, a regular industry-scale system may contain billions of particles.Since individual particles are tracked throughout the DEM simulation, each particle's information must be stored during the simulation to update their locations, velocities, etc.Clearly, DEM simulation demands a high computational cost and time to simulate an industryscale system with resolved particles (Sakai 2016).There are significant advancements in computer hardware in recent times, and DEM simulation can be done using parallelization techniques such as MPI (Message Processing Interface) and OpenMP (Ka cianauskas et al. 2010).But, the problem of high computational time gets aggravated even further when the simulation of the complex geometry of industry scale has too many mesh elements.Although graphical processing units (GPU) can be used for parallelization to manage computational cost, they are highly expensive (Tiscar et al. 2021).Researchers are keen to execute such extensive simulations in regular CPUs.
To resolve the high computational time issue related to DEM, coarse-grained (CG) method is one of the promising techniques (Bierwisch et al. 2009;Sakai and Koshizuka 2009;Radl et al. 2011).In the coarse-graining method, a group of particles is replaced by a bigger particle with some necessary adjustments in the particles' properties to capture similar results as the resolved system.Thus, coarse-graining can reduce the number of particles, computational cost, and simulation time (Sakai et al. 2010).Different researchers have used different particles' property scale-up techniques and introduced different CG ratios in their simulations (Bierwisch et al. 2009;Sakai and Koshizuka 2009;Nasato et al. 2015;Chan and Washino 2018).The CG ratio is the ratio of the diameter of CG particles and resolved particles.Thus, the CG ratio l means that l 3 resolved particles are clubbed into one CG particle.If the resolved system has polydisperse particles, the size of each class will be scaled by l in the CG system while conserving the total mass in the system.Contact forces on CG particles are modeled by different researchers (Bierwisch et al. 2009;Sakai and Koshizuka 2009) while conserving the total kinetic energy.Sakai and Koshizuka (2009) used the Linear Spring Dash-pot model and used l 3 (where l is the CG ratio) scaling of the contact forces for CG particles.The authors used the CG ratio up to 3 and compared the results in a pneumatic channel flow simulation.The same strategy was further implemented in fluidized beds (Sakai et al. 2010;Hilton and Cleary 2014).These models also suggested l 3 scaling for the drag force due to the presence of fluid in the system.Another approach present in the literature is similarity models (B€ orner, B€ uck, and Tsotsas 2017).According to the principle of similarity in fluid mechanics, some dimensionless values are maintained constant across system sizes in this model.The results from such CFD-DEM CG simulations were compared with experimental results and they agreed well.Nasato et al. (2015) used the non-dimensional technique for the governing equation of motion to obtain the correct scaling laws for different contact models.They proved that the stiffness and damping coefficients are to be scaled by l 2 and l, respectively while using the Linear Spring Dash-pot model.The authors further proved that the Hertz-Mindlin contact model is scale-independent.Thus, when CG particles are introduced into the system using the Hertz-Mindlin contact model, there is no need to scale the particle properties or contact forces in the CG particles in any system as the nondimensionalized governing equation of contact force remains unaltered.In recent times, the augmented CG method is also applied in the fluidized bed simulations (Lin et al. 2020) where reduced particle stiffness model has been employed.The CG method has been executed in heat-and mass-transfer-enabled dense gas-solid reacting flow by Wang and Shen (2022).The simulation results from the CG system agreed well with the resolved simulation and experiment results (Wang and Shen 2022).The CG method has been also employed successfully on non-spherical particles with a hybrid drag force model in a fluidized bed by Lu et al. (2020).De et al. (2022) introduced a new multi-level coarse-graining technique where different CG particles are used dynamically in different parts of the simulation system.
Although researchers have been trying to understand the behavior of CG particles for more than 20 years (Andrews and O'Rourke 1996), the coarse-graining technique still requires more attention and proper analysis to predict the actual behavior of the resolved system.The collision between particles is a major phenomenon to be taken care of in the granular simulations.If the distance between two particles is less or equal to the sum of their radii, DEM registers it as one collision between those two particles.In an industrial system, aggregation and breakage happen due to the collisions between particles.This phenomenon leads to a change in the PSD of the simulation system.Similarly, particles in the corresponding CG system will also aggregate and break due to the particulate mechanisms.Consequently, the collision frequency of CG particles can be different from the resolved particles.For example, for CG ratio 2, eight resolved particles are clubbed into one.Hence, 16 resolved particles participate in the collision where only two CG particles collide with each other.Consequently, there will be more collisions in the resolved system compared to the CG system.Sakai et al. (2010) assumed that when the binary collisions of the CG particles occur, there are l 3 binary collisions between resolved particles.Nevertheless, the assumption mentioned earlier is not well-proven and cannot be appropriate for such simulations.No study can be found in the literature where researchers predicted the probability of collisions between different particle classes without performing the resolved simulation.Furthermore, to the authors' best knowledge, no research article discussed the correlation between the collision probabilities of resolved and CG particle systems.Thus, there is an exigency to bridge this gap of information on collision data between systems involving CG and resolved particles.
This work used a general mathematical formula for the efficient prediction of the probability of collisions between different particle size classes for particulate systems.The collision matrix containing collision probabilities between different size classes of particles can help reflect proper aggregation and breakage rates in a CG system.A general collision dependency function (CDF) is introduced to predict the collision probability between different particle classes.Section 2 presents a brief description of the DEM and the collision detection technique.Section 3 focuses on developing the mathematical model to predict the probability of collisions.The developed model is thoroughly tested and verified for several test simulations in Section 4. Finally concluding remarks are provided in Section 5.

General concept
Simulation by discrete element method (DEM) involves tracking the movement of every particle in the simulation system with the help of Newton's second law of motion (Cundall and Strack 1979).DEM models particle-particle and particle-wall collisions.A collision is detected for a particle when the overlap between particles or particle-wall is non-zero during impact.The governing equations are solved by considering the particle-particle and particle-wall interactions.The presence of any other uniform or non-uniform applied field forces (i.e., body forces, electrostatic forces, etc.) is also accounted.The governing translational and rotational equations of motion are given in the following equations: where, the variables m i , x i , f i , I i , x i and T i are the mass, position, force, inertia tensor, rotational velocity, and torque of the i-th particle, respectively.For spherical particles with radius R i , the inertia is given as The force f i includes all body forces and contact forces.In the current study, the Hertz-Mindlin contact model (Norouzi et al. 2016) has been employed, where contact forces are scale independent (Nasato et al. 2015) when CG particles are introduced.The contact force on each particle has two components, namely normal (f n, ij ) and tangential (f t, ij ) forces.The normal and tangential contact forces are calculated using the following equations: Here, j n , j t are the normal and tangential stiffness coefficients; g n , g t are the normal and tangential damping coefficients, respectively.Moreover, d n, ij ¼ d n, ij n, where d n, ij is the normal overlap between i-th and j-th particles and n is the unit normal vector connecting the center of i-th and j-th particles if they are in contact.d _ n, ij is the normal relative velocity acting along the vector connecting the center of the i-th and j-th particles.Similarly, d t, ij and d _ t, ij are the tangential overlap and relative tangential velocity between i-th and j-th particle.The tangential force is truncated using Coulomb's law which is f t, ij lf n, ij , where l is the sliding friction.The total torque acting on the i-th particle consists of tangential torque T t, ij and rotational torque T r, ij : The tangential torque, T t, ij , is generated by the tangential component of the contact forces acting on the i-th particle.The rotational torque T r, ij originated from the normal component of the contact forces on the i-th particle.Once the values of all forces and torques acting on the particles are known, the governing Equations ( 1a) and (1b) are solved under appropriate boundary conditions.Equations ( 1a) and (1b) are solved numerically using the Verlet integration method (Kruggel-Emden et al. 2008) to get the updated position and velocity of each particle for the current iteration step.In the Hertz-Mindlin contact model, the stiffness and damping coefficients depend on the overlap between the colliding particles.The stiffness and damping coefficients in normal and tangential directions are provided in the following equations: Here, e, R eff , m eff , E eff , and G eff are the coefficient of restitution, effective radius, mass, Young's modulus, and shear modulus of the particle, respectively.
The overlap between two particles due to collision must be calculated to obtain the contact forces.Thus, the neighbor information of all particles involved in the simulation is available in the "neighlist" array in LIGGGHTS-Public.However, this array does not inform us about the overlapped particles or keep any backup about the collision on previous time steps.It stores all possible neighbors for each particle only.Thus, a few lines of new code are employed in "compute_contact_atom.cpp"file in the source folder "src" of LIGGGHTS-Public to obtain and save collision data.For running the simulations, the changed source files and the relevant input scripts are available on GitHub .The details of the detection and registration of collisions are discussed in the following subsection.

Collision detection and registration
Collision detection is one of the most time-consuming steps in DEM simulation as it involves iterations of the pairwise search of particles.The collisions between particles are detected to calculate the position and contact forces of each particle in the current time-step.While calculating the number of collisions between particles, repeated entries may appear.For example, two static particles that are in contact with each other will be counted as collisions at each time step.We need to employ a better strategy that can detect these duplicate collisions and ignore them.Figure 1 illustrates such a situation and tells us how to avoid repeated counting of collisions.Figure 1 shows particles with no collisions, fresh collisions, and repeated collisions as grey, green, and red, respectively.For example, let there be four particles in the system at any random instance as shown in Figure 1.Let these four particles are moving freely and are not in contact with each other at t ¼ t 1 .At time t ¼ t 2 , let the first particle is in contact with the second and third particles, and the fourth particle has no contact with the other particles.Thus, two new collisions are to be counted between the first particle and its adjacent second and third particles.On the next time instance at t ¼ t 3 , let the third particle detach from the first particle and collide with the fourth particle while the first and second particles are continuously in touch.Clearly, the contact between the first and second particles should not be treated as a new collision.Only a new collision happened between the third and fourth particles.
Although several researchers employed contact detection mechanisms at every time-step (Barrasso, Tamrakar, and Ramachandran 2014; Barrasso and Ramachandran 2015; Barrasso et al. 2015;Baba et al. 2021), repeated collisions were not tackled.Reinhold and Briesen (2012) attempted to change this by adding a list of overlooked particles corresponding to each particle.The present study has adopted a new approach to remove the repeated collision data and distinguish new collisions periodically.In this approach, the information of individual collision for each particle is stored in a "neighbor" matrix at each iteration.Each non-zero entry in the i-th row of this neighbor matrix gives the id of the particles in contact with the i-th particle at that instance.This collision data tell which particle is in contact with whom.The neighbor matrix from the previous iteration is compared with the neighbor matrix in the current iteration.Consequently, the new collisions can be detected, counted, and stored in a "collision" matrix.This collision matrix gives us the number of collisions between two identical or different-size classes.When a new collision is detected, the radius of the two colliding particles is noted, and the collision matrix gets updated accordingly.Thus, the framework ignores the repetition in collision data registration efficiently.
The particles in the simulation are soft in nature and deform during collisions.Furthermore, the small DEM timestep length results in continuous particle contact, meaning that collisions between particles span multiple time steps (Mishra 2003).To reduce the computation time of DEM simulation, collision detection can be performed periodically.This allows for the recording of collisions between different particle classes at fixed intervals.The time interval between collision registration steps is called the period of collision registration (PCR) in the current study.Low PCR demands high computational costs, while a high PCR may result in miscounting and under-prediction of collisions.As a result, the optimization of PCR is crucial.In Section 4, PCR has been discussed with examples, and optimized PCR has been set.
A portion of code was written inside a source file of LIGGGHTS-Public (Kloss 2010) to accomplish this approach.The command "compute contact/atom" in LIGGGHTS-Public calculates the instantaneous number of contacts for each atom periodically.The aforesaid command reaches "compute_contact_atom.cpp"file to execute its purpose.Hence, a few new lines of code have been inserted into that CPP file so the current code can be triggered periodically.The details of the computer system used in the DEM simulation are given in Table 1.The flowchart of contact detection during the DEM simulation process executed in this study is presented in Figure 2.

Mathematical model to predict collision probability between particles
The collision between particles can be volume-dependent in nature.In order to quantify this volume-dependency in particle collisions, let us introduce a collision dependency function (CDF) and denote it as K(x, y).The function K expresses the number-normalized volume dependency in particle collisions.The CDF helps us explore the selection probability of particles with sizes x and y for the collision events.According to its construction, CDF is time-independent.That means if the CDF is known beforehand for a particulate system, CDF remains invariant throughout the process.Additionally, the selection rate of particles for the collision events also depends on the availability of concerned particle classes in the system.For example, if corresponding to some particle sizes x and y, although the value of the CDF is positive, there exist no particles in the system with size x or y, then there should not be any collisions in the system.
From the above discussion, it is certain that particles in the system are selected for the collision events depending on the collision dependency function K(x, y) and the presence of particles of the concerned sizes x and y.If there exist N x and N y number of particles of sizes x and y, respectively, the probability of collisions between particles with x and y sizes can be expressed as On the other hand, the computation of the collision probability between two different sizes of particles is easy from DEM results.Firstly, the DEM simulation runs for a sample period of s DEM s, and during this process, the collision detection mechanism works.At every PCR, a new neighbor matrix is created, and it is compared with the previous neighbor matrix.This comparison gives the number of new collision(s) that took place.With the help of this information, the resultant collision matrix is updated after every PCR.At the end of the simulation, the final collision matrix is obtained.In the final collision matrix, the (x, y)-th entry of this collision matrix represents the total number of collisions between x-th and y-th size of particles.Correspondingly, one can easily calculate the collision frequency between particles of sizes x and y as f c ðx, yÞ: Then the DEM-extracted probability of collisions can be calculated as Theoretically, both the formulations (4) and ( 5) of the probability of collisions should provide the same results for any particle population.Then merging the ideas of Equations ( 4) and ( 5), and executing a DEM simulation for a sample period with number based uniformly distributed particles of all probable volumes (i.e., N x ¼ N y ¼ constant), we get Since the fraction on the right side of Equation ( 3) is constant, so the equation can be reduced to where c is a constant and f uni c ðx, yÞ is the collision frequency between particle of sizes x and y for uniformly distributed particles.Additionally, since the function K(x, y) can be found in both the numerator component and denominator component of Equation ( 4), it renders the constant's (c) value insignificant.Accordingly, the formulation (4) reduces to Finally, using Equation ( 8), one can easily predict the probability of collisions between different particle classes for any particle population, including resolved and CG particles, without investing a high computational cost.The predicted collision probability can be helpful in the population balance modeling of the aggregation and breakage process while approximating the collision rate between different particle classes.

Results and discussion
In the previous Section 3, a theoretical claim was made that the collision probability between different types of particles for any PSD of resolved and the CG system can be predicted using the CDF.This can help us to predict the probability of collisions between different particle classes for any given PSD without their corresponding simulations.Only one simulation consisting particles of uniform PSD can help us predict the possible collision probabilities between different size classes of particles.To verify our claim, the simulation of a rotating cylindrical drum is considered.The rotation of the cylindrical drum has been chosen as it is a collisiondriven phenomenon.Rotating drums have been studied more widely in experiments and simulations than other tumbling mixers, indicating the importance of this equipment (Alizadeh, Bertrand, and Chaouki 2014;Barrasso et al. 2015;Zhang et al. 2021).Zhang et al. (2021) showed that the holdup ratio is increased when the number of flights is increased.Clearly, the presence of baffles or flights changes the degree of mixing.The physical details of the drum (Figure 3) and particle properties (Table 2) are mentioned in Subsection 4.1.

Simulation system
The mixing phenomenon inside the rotating cylindrical drum is simulated to verify our claim.A closed-end cylindrical drum with a length of 50 cm and diameter of 25 cm is considered (Figure 3) in the simulation.Eight baffles were created on the inside wall of the cylindrical drum for agitation amongst particles as the presence of baffles influences the mixing phenomena greatly (Zhang et al. 2021).Each baffle has a height of 2.5 cm in the radial direction.The rotational speed remains constant (60 rpm) during all simulations.For each simulation, the particles are generated randomly inside the drum and left under gravity to settle for 1 s of real-time as the total kinetic energy of the particles becomes a steady state.After the settlement of the particles, the drum starts rotating for s DEM ¼ 1 s of real-time.During the rotation of the drum, particles agitate and collide with each other.Thus, collisions are detected and registered periodically during this process.The time step in the DEM simulations was considered to be 10 À5 s, which is 10% of the Rayleigh time (Mishra 2003).Rayleigh time is the time duration needed to transfer energy between two colliding bodies (Li, Xu, and Thornton 2005).It is a measure to estimate the required DEM time step in the simulation to bring accuracy.It is important to note that the total mass of all particles in all the simulations for this study system has been kept constant, which is 3.678 kg.The corresponding PSD for each simulation is also provided for each case.Thus, the number of particles in each size class is easily derivable.

Optimization of PCR
The collision detection mechanism should not be implemented at each time-step, and the reasons are mentioned in Section 2.2.The PCR is optimized to avoid high computational costs while minimizing computational error.3. From Table 3, it is clear that as the value of PCR increases, the computation time required for the simulations decreases, and the error in calculating the amount of collisions increases.It is because if the value of PCR increases, the system runs fewer loops of collision registration, which results in calculations of lesser collisions  and higher computational efficiency.However, we need to choose the optimum PCR value, which provides high computational efficiency and a bounded relative error.To check this, the gradients of "relative error in collisions" and "simulation time" are plotted against the values of PCR, which are illustrated in Figures 4(a,b).Figure 4(a) depicts that the rate of change in the relative error of collisions concerning the base case is strictly increasing and almost linear.However, the rate of change in the computational time gain decreases exponentially and becomes asymptotic.For these reasons, we choose the optimized value of PCR to be 50 Dt DEM for all simulations considered in this manuscript, which is 16 times faster than the base case and the corresponding error in calculating calculations is also in the tolerable range (2.67%).Similar exercises were performed for several other particle size distributions for verification purposes which reflected a similar conclusion.

Verification
The optimized PCR is obtained in Subsection 4.2.The CDF has to be found to predict collision probability between different size classes of particles.To obtain the CDF, the rotating drum is simulated with uniformly distributed particles (number-based) of all different-sized particles (including resolved and CG particles of CG ratio l).If the resolved system has n size classes, there will be at most 2n types of different-sized particles present in this simulation.It is to be noted that if the resolved system has particle size classes of volumes v 1 , v 2 , v 3 , :::, v n , respectively, then, the corresponding CG system will have particles of volumes l 3 v 1 , l 3 v 2 , l 3 v 3 , :::, l 3 v n , respectively when the CG ratio is l.Thus, the simulation to find the CDF will contain an equal number of particles of volumes v 1 , v 2 , v 3 , :::, v n , l 3 v 1 , l 3 v 2 , l 3 v 3 , :::, l 3 v n , respectively.This simulation helps to find out and store the CDF.The CDF is the collision matrix whose (x, y)-th entry is the total number of collisions between x-th and y-th type of particles.All details of the CDF (K(x, y)) are discussed in Section 3.This CDF will be used to find the collision probability for any system using the formulation 4. The corresponding weights N x and N y have to be updated in Equation ( 4) to predict the collision probability for any PSD.Different PSDs are considered for resolved and CG particles to verify the proposed model.The corresponding collision matrices are calculated from DEM simulations as well.The collision matrices obtained from the DEM simulation are compared against the predictions by our model.Three different test cases have been considered to verify the developed model ( 4).In the first test case, collision probabilities between different particle size classes are predicted for CG and resolved systems separately for different PSDs, and the predicted results are compared against their respective DEM results.The CG particles have a CG ratio of 2 in this test case.The second test case considers a different CG ratio (l ¼ 1.587) for CG particles.In the third test case, a single simulation system considers the mixed combination of resolved and CG particles.Thus, test cases 1 and 2 had either resolved or CG particles in the simulation domain, whereas the third test case constituted both resolved and CG particles in the same simulation system.The details of each test case have been provided in the respective subsections.
In all test cases, system containing particles of volumes 10, 20, 30, 40, and 50 mm 3 was considered as the resolved system.The simulations were run for 1 s in real-time for each case.The total L 1 and L 2 errors normalized with the  total sum of entries of the collision matrix from DEM values are provided in separate tables for each test case.These L 1 and L 2 errors are calculated as (Bhoi et al. 2019;Das et al. 2020): Equations ( 9) and ( 10) help to quantify the prediction errors where P DEM and P Pred are the collision probabilities obtained from DEM and model prediction, respectively.

Test case: 1
The collision probability for resolved and CG systems with CG ratio 2 is predicted in test case 1.The resolved system has particles of volumes 10, 20, 30, 40, and 50 mm 3 : Thus, the corresponding CG system has particles of volumes 80, 160, 240, 320, and 400 mm 3 : The simulation of a rotating drum with uniformly distributed particles of all possible volumes of particles (including resolved and CG particles) is observed to obtain the CDF.The obtained CDF (K(x, y)) is shown in Figure 5.
To assess the applicability of the developed model of CDF, we applied it to particle systems with diverse distributions.In particular, we considered normal, uniform, ascending, and erratic particle size distributions for test case 1, as depicted in Figure 6.For the erratic distribution, the number fractions corresponding to different size classes are chosen arbitrarily.The inclusion of the erratic particle distribution was intended to confirm the generalizability of the model to arbitrary particle distributions.Using Equation (4), collision probabilities were predicted for all such cases.DEM simulations were conducted to obtain the collision probability between different-sized particles in each case for verification purposes.
In Figures 7-10, all the results for different PSDs are plotted against each other for particles with CG ratio 2.      In each figure, the probability of collision between different-sized particles obtained from DEM is displayed on the left (a), and the predicted probability is displayed on the right (b).These results for all cases show a good match between predicted and DEM results.Figure 7 displays the comparison between collision probabilities obtained from DEM and predicted values for number-based uniform PSD.Since the number of particles in each size class is the same, the bigger particles will participate in a higher number of collisions during the simulation than smaller particles.This trend has been seen in Figure 7 in both predicted and DEM results and a good agreement between them can be found.
Similarly, the peaks in the collision probability plots are formed according to weightage (N x and N y ) of different size classes for other PSDs.The prominent peak in the collision probability plot is visible for the particle class having the highest weightage in each distribution.For example, the number of particles of the highest volume (400 mm 3 ) has the maximum number of particles in ascending PSD.Thus, the collision probability shows a strong upward trend with an increase in particles' size, as shown in Figure 9.For the normal distribution, the size class with particle's volume 240 mm 3 has the maximum number of particles.The highest peak in the surface plot of collision probability also corresponds to the size class of the particle's volume 240 mm 3 : This trend is similar in both predicted and DEM results shown in Figure 10.A similar trend can also be visible for erratic PSD in Figure 8.All the results for each PSD show a good agreement with each other.
The combination of all resolved and CG particles' sizes was considered while creating the CDF.Thus the collision probability for the resolved system for different PSDs can also be predicted using that CDF.PSDs were considered the same as CG particles while predicting the collision probabilities for resolved simulation.Thus, the number of each class of particles in the resolved system will be multiplied by 8 (¼ 2 3 ) with respect to the CG simulation system as the CG ratio was 2. The predicted  and DEM results are shown for uniform (Figure S1), erratic (Figure S2), ascending (Figure S3), and normal (Figure S4) distributions in the Supplementary Material.Predicted results for all these PSDs show a good agreement with the results obtained from DEM.The corresponding L 1 and L 2 errors for each case are displayed in Table 4. Judging from the table, it can be claimed that our model's predicted values are accurate enough for CG and resolved systems.The snapshots after 1 s simulation of resolved and CG system are shown in Figure 11.

Test case: 2
For the second test case, particles of CG ratio 1.587 are considered.The CG ratio of 1.587 tells that four particles are clubbed into one particle (1:587 3 ¼ 4).The resolved particles have the same volume range as test case 1 whereas the CG system has particles of volume 40, 80, 120, 160, and 200 mm 3 : The particles of volume 40 mm 3 are common in both samples.Thus, while finding out the CDF, the uniform distribution of all nine kinds of particles is considered.The corresponding CDF (K(x, y)) is shown in Figure 12.For the current test case, only two different PSDs were considered: D1 and D2 distributions, where the D1 distribution is arbitrary in nature, and the D2 distribution is a normal distribution where the mean particle volume is 160 mm 3 : The PSDs for the resolved system are displayed in Figure 13.Figures 14 and 15 show the comparison between extracted and predicted collision probability for D1 and D2 distribution for CG particles.Like test case 1, the highest peak of the collision probability plot is reflected for both predicted, and DEM results against the particles of volume 120 mm 3 , which has the highest weightage in the D1 distribution.For the D2 distribution, the peak is against 160 mm 3 , and both results show a similar trend.Thus, these results also reflect good assent and support our claim for a CG ratio of 1.587.For verification purposes, the collision probability of the corresponding resolved system is also predicted and   compared against its DEM result in Figures S5 and S6, respectively for D1 and D2 distributions.These results are also indicative of good agreement and are displayed in the Supplementary Material.The corresponding L 1 and L 2 errors for each case are displayed in Table 5.

Test case: 3
In some situations, instead of coarse-graining all particle size classes, coarse-graining some specific size classes can be a more realistic approach.For example, for a descending distribution (shown in Figure 16(a)), the presence of smaller resolved particles is much higher compared to larger-sized particles.In this case, a combination of CG and resolved particles can be considered in the simulation.In the present case, the first two kinds of particles are replaced with their corresponding CG particles with CG ratio 2. Thus, the new distribution of particles becomes as shown in Figure 16(b).Clearly, the new system of simulation has particles of volume 80, 160, 30, 40, and 50 mm 3 , respectively whereas the   corresponding resolved system has particles of volume 10, 20, 30, 40, and 50 mm 3 , respectively.Changing the corresponding weightage (N x and N y ) in Equation ( 4), the collision probability for such a scenario can also be predicted with reasonable accuracy.The results obtained from DEM and predicted results are shown in Figure 17.The resolved simulation corresponding to this distribution can also be predicted for this system using the same CDF.The DEM and predicted results are shown in Figure 18.Although the number of the first kind of particles is higher than other kinds, the bigger particles will face a higher number of collisions due to their larger volumes.Thus, a genuine peak is not visible against the first kind of particles in both figures.
The corresponding L 1 and L 2 errors for each case are displayed in Table 6.

Conclusion
It is evident that the introduction of CG particles into the system reduces the number of particles significantly and consequently changes the collision frequency and probability between the different classes of particles.Prediction of collision probability from CG system to resolved system or vice-versa is arduous.This study presents a novel model for predicting the collision probabilities between particles of different size classes for resolved and CG systems simultaneously.This method is useful to predict the collision probability for any given PSD from a single DEM simulation of uniformly distributed particles of all possible size classes.A new collision detection technique has been adopted to ignore repeated collisions while calculating collision data.Furthermore, a new collision dependency   Based on the predicted and corresponding DEM results, it can be concluded that the new formulation can predict the collision probabilities of any CG system for different CG ratios and their corresponding resolved systems simultaneously.It is also demonstrated that this model can predict the collision probability for the simulation systems where both resolved and CG particles are considered.This study can help researchers to understand the aggregation or breakage behavior in the resolved and CG systems for different PSDs, surpassing the computational efforts of conducting multiple large-scale DEM simulations.The findings suggest that the method can also be helpful in future PBM-DEM coupling frameworks for the CG system.Collision probabilities can be predicted for different PSDs without running the simulations for both resolved and CG systems.To find the collision dependency function (K(x, y)), the number of uniformly distributed particles must be statistically sufficient.
One of the limitations of the current model is the inability to predict the collision frequency between different particle classes.A proper scaling model is required to predict the total collision frequency of resolved system from CG system and vice-versa.Nevertheless, the developed model is capable of predicting the collision probabilities between any particle classes for resolved and CG systems simultaneously.This work may be considered as an initial but significant step toward building a PBM-DEM coupling framework for the CG system.

Figure 1 .
Figure 1.The graphical representation of new and repeated collisions between particles.Particles with no collision, new collisions, and repeat collisions are displayed with grey, green, and red colors, respectively.

Figure 2 .
Figure 2. Flowchart of the DEM simulation process to count the number of collisions and to prepare the collision matrix.
To optimize the PCR, several simulations were conducted with different values of PCR for number-based uniformly distributed particles with volumes ranging between 10 and 200 mm 3 : To check the accuracy and efficiency of the PCR values, the DEM time step (Dt DEM ) is chosen as the base value of PCR.Apart from this, several simulations were conducted with different values of PCR, such as 10 Dt DEM , 20 Dt DEM , 50 Dt DEM , 80 Dt DEM , 100 Dt DEM , and 200 Dt DEM : The simulation time and the relative error in calculating collisions for each of the mentioned cases are presented in Table

Figure 3 .
Figure 3.The geometry of the rotating drum used in DEM simulation.

Table 2 .
Values of material properties and system information used.Particle properties and system information Values Time for settling the particles 1 s DEM time step (Dt DEM ) 1 Â 10 À5 s Drum diameter

Figure 4 .
Figure 4.The gradient of (a) error percentage, (b) gain in simulation time for different PCR.

Figure 5 .
Figure 5. Test case 1-Extracted values of the CDF function from DEM simulation of uniformly distributed particles.

Figure 6 .
Figure 6.Different size distributions considered for Test case 1 for resolved system.

Figure 7 .
Figure 7. Test case 1-Comparison of the probability of collisions between particle classes obtained from (a) DEM simulation, and (b) developed formulation (4), for CG system (CG ratio 2) with uniformly distributed particles.

Figure 8 .
Figure 8. Test case 1-Comparison of the probability of collisions between particle classes obtained from (a) DEM simulation, and (b) developed formulation (4), for CG system (CG ratio 2) with erratically distributed particles.

Figure 9 .
Figure 9. Test case 1-Comparison of the probability of collisions between particle classes obtained from (a) DEM simulation, and (b) developed formulation (4), for CG system (CG ratio 2) with particles distributed in ascending order.

Figure 10 .
Figure 10.Test case 1-Comparison of the probability of collisions between particle classes obtained from (a) DEM simulation, and (b) developed formulation (4), for CG system (CG ratio 2) with normally distributed particles.

Figure 11 .
Figure 11.Test case 1-Snapshots of the initial and final instances of the system having: (a) resolved particles, (b) CG particles (CG ratio 2) at the start of the drum rotation, and (c) resolved particles, (d) CG particles (CG ratio 2) after 1 s of drum rotation.

Figure 12 .
Figure 12.Test case 2-Extracted values of the CDF function from DEM simulation of uniformly distributed particles.

Figure 13 .
Figure 13.Different size distributions considered for Test case 2.

Figure 14 .
Figure 14.Test case 2-Comparison of the probability of collisions between particle classes obtained from (a) DEM simulation, and (b) developed formulation (4), for CG system (CG ratio 1.587) following D1 distribution.

Figure 15 .
Figure 15.Test case 2-Comparison of the probability of collisions between particle classes obtained from (a) DEM simulation, and (b) developed formulation (4), for CG particles (CG ratio 1.587) following D2 distribution.

Figure 16 .
Figure 16.Test case 3-Size distributions of particles considered in test case 3: (a) the descending distribution for resolved particles only and (b) the modified distribution when the first two kinds of resolved particles are replaced by CG particles (CG ratio 2).

Figure 18 .
Figure 18.Test case 3-Comparison of the probability of collisions between particle classes obtained from (a) DEM simulation, and (b) developed formulation (4), for the system having resolved particles only following descending distribution.

Figure 17 .
Figure 17.Test case 3-Comparison of the probability of collisions between particle classes obtained from (a) DEM simulation, and (b) developed formulation (4), or the system having both resolved and CG particles (CG ratio 2) following descending distribution.

Table 1 .
Information of computer used in the simulation.

Table 3 .
Table for optimizing PCR in the simulations.

Table 4 .
The errors in predicting collision probabilities for different PSDs for test case 1.

Table 5 .
The errors in predicting collision probabilities for different PSDs for test case 2.

Table 6 .
The errors in predicting collision probabilities for different PSDs for test case 3. (CDF) is developed to predict the collision probabilities between different particle size classes in the resolved systems, CG systems, and systems with a mixture of resolved and CG particles for different PSDs.Finally, The predictions of the developed model are compared successfully against the corresponding results obtained from different DEM simulations. function