Particles

The Particle statistics provide time-resolved quantities describing the state, transport, interaction, and evolution of particle populations in the simulation. These outputs are useful for monitoring particle inventory, injection and removal mechanisms, breakup and coalescence behavior, interphase transfer, and particle size evolution over time. A unique statistics file is created for each particle family present in a model. The output is a tab-separated ASCII .txt file named Particles_{DynamicName}.txt, where the dynamic name corresponds to the name of the particle family in the Model Tree. Each row in the output file corresponds to a new statistics output time and is appended at the Statistics Output Write Interval.

These statistics generally fall into a few categories:

• Population Statistics: These describe how many particles or parcels are present and how much total particle volume or area exists in the system.

• Source and Sink Statistics: These describe how particles enter or leave the system through injections, outlets, sinks, surface removal, UDF logic, or cutoff-based deletion.

• Interaction Statistics: These describe particle breakup, coalescence, collisions, and particle-particle thermal contact behavior.

• Transfer Statistics: These describe heat and mass transfer between particles and the surrounding fluid.

• Size Statistics: These describe how the particle size distribution evolves using different weighting conventions.

Statistics Table

The index table below shows the statistics that can appear in the Particles output file. Within this table, each statistic corresponds to a column in the output table that evolves with the time column.

Statistics	Units	Details	When Appears
Time	s	simulation time
Breakup Rate	events/s	rate of breakup events not including number scaling
Coalesce Rate	events/s	rate of coalesce events not including number scaling
Colliding Particles Count	Dimensionless	number of particles colliding not including number scaling
Condition Injection Rate	number/s	rate of injection from condition
Count	Dimensionless	total number of particles including number scaling
Custom Particle Variable Mean	[dynamic]	mean custom particle variable value
Custom Particle Variable Rate	[dynamic]	rate of change for custom particle variable in particle-based reaction
Fluid Heat Transfer Rate	W	total heat transfer between particles and fluid, positive indicates heat transfer from fluid to particles
Heat of Reaction	W	total heat generation for particle-based reaction
Inlet/Outlet Removal Rate	number/s	rate of particles leaving inlet/outlet not including number scaling
Inlet/Outlet Removal Volumetric Rate	m^3/s	volumetric rate of particles leaving inlet/outlet
kLa	1/s	total kLa
Mean Diameter Area-Weighted	m	mean diameter weighted by area D[3,2] accounting for number of particles contained in each parcel, for non-spherical particles the sphere-equivalent volume is used
Mean Diameter Number-Weighted	m	mean diameter weighted by number D[1,0] accounting for number of particles contained in each parcel, for non-spherical particles the sphere-equivalent volume is used
Mean Diameter Volume-Weighted	m	mean diameter weighted by volume D[4,3] accounting for number of particles contained in each parcel, for non-spherical particles the sphere-equivalent volume is used
Mean Parcel Diameter	m	mean parcel diameter, for non-spherical particles the sphere-equivalent volume is used
Mean Particle-Particle Heat Flow	W	mean thermal conduction heat flow for particle-particle contacts, particles that do not contact any other particles count as 0 bringing down the mean
Mean Particle-Particle Resistance	K/W	mean thermal conduction resistance for particle-particle contacts, particles that do not contact any other particles count as 0 bringing down the mean
Modeled Breakup Rate	events/s	rate of breakup events
Modeled Coalesce Rate	events/s	rate of coalesce events
Parcel Count	Dimensionless	total number of parcels
Particle Sink Removal Rate	number/s	rate of particles leaving particle sink not including number scaling
Particle Sink Removal Volumetric Rate	m^3/s	volumetric rate of particles leaving particle sink
Region Feed Deficit	number	number of DEM particles that were not successfully fed into system
Region Feed Deficit	m^3	volume of DEM particles that were not successfully fed into system
Region Injection Rate	number/s	particle injection rate
Region Injection Volumetric Rate	m^3/s	volumetric particle injection rate; calculated at reference pressure, does not account for bubble pressure model if active
Small Particle Removal Rate	number/s	rate of particles being removed because their diameter goes below the cutoff value not including number scaling
Small Particle Removal Volumetric Rate	m^3/s	volumetric rate of particles being removed because their diameter goes below the cutoff value
Surface Removal Rate	number/s	rate of particles being removed because they come in contact with gas phase or free slip boundary condition not including number scaling
Surface Removal Volumetric Rate	m^3/s	volumetric rate of particles being removed because they come in contact with gas phase or free slip boundary condition
Total Area	m^2	total surface area of particles
Total Volume	m^3	total volume of all particles
Total Volume Depressurized	m^3	total volume of all particles depressurizing any bubbles to reference pressure if bubble pressure model is used
UDF Removal Rate	number/s	rate of particles being removed by UDF
UDF Removal Volumetric Rate	m^3/s	volumetric rate of particles being removed by UDF

Usage and Interpretation

The overall particle inventory is characterized by Parcel Count, Count, Total Volume, and Total Area. The simulation explicitly tracks parcels, each of which represents a collection of physical particles through a number scaling factor. All reported quantities are computed over parcels, with number scaling used to recover physical particle statistics. Parcel Count is the number of computational parcels in the system, while Count is the total number of physical particles represented by those parcels. When the particle number scale is 1, each parcel corresponds to a single physical particle, and Parcel Count and Count are identical.

Total Volume defines the particle hold-up in the system and is calculated as

\[V_{\text{tot}} = \sum_i n_i V_i\]

where \(n_i\) is the number scaling factor and \(V_i\) is the volume of parcel \(i\). The summation is taken over all parcels in the ensemble.

Total Area is particularly important for problems involving interphase transfer or reactions and is calculated as

\[A_{\text{tot}} = \sum_i n_i A_i\]

where \(n_i\) is the number scaling factor and \(A_i\) is the surface area of parcel \(i\). The summation is taken over all parcels in the ensemble. In steady or statistically steady operation, these quantities should fluctuate but not exhibit long-term drift. Persistent growth or decay typically indicates an imbalance in injection, removal, or population dynamics.

Particle injection and removal terms define how the population evolves over time. Injections are captured through quantities such as Condition Injection Rate and Region Injection Rate, while removal occurs through multiple mechanisms including inlet and outlet flow, particle sinks, surface interactions, small-particle cutoff, and UDF-based logic. These terms are used to diagnose why the particle population is increasing, decreasing, or failing to reach expected levels. If particle inventory is lower than expected, removal terms should be checked first. In DEM feeding problems, Region Feed Deficit indicates when requested particles cannot be successfully inserted into the system due to overpacking.

When breakup and coalescence models are active, the corresponding rate terms describe population balance dynamics. Breakup Rate and Coalesce Rate report event frequencies without number scaling when the Breakup or Coalescence Representation is set to Discrete. Modeled Breakup Rate and Modeled Coalesce Rate report effective event frequencies when injection downsampling is active within the Discrete representation. These modeled rates include number scaling and represent the predicted physical breakup and coalescence rates.

Breakup and coalescence statistics should be interpreted together with the diameter statistics. Increasing breakup typically drives a decrease in the number-weighted diameter, while increasing coalescence tends to increase the volume-weighted diameter. In many systems, both processes are active simultaneously, and a dynamic steady state may exist even when both rates remain nonzero.

Particle size evolution is captured through multiple diameter definitions, each representing a different moment of the size distribution. These moments are computed as

\[D[a,b] = \frac{\sum_i n_i d_{i,a}}{\sum_i n_i d_{i,b}}\]

where \(n_i\) is the number scaling factor and \(d_i\) is the diameter of parcel \(i\). The summation is taken over all parcels in the ensemble. The number-weighted diameter \(D[1,0]\), which is the average diameter, is sensitive to small particles. The area-weighted diameter \(D[3,2]\), also called the Sauter mean diameter, is relevant for interfacial processes. The volume-weighted diameter \(D[4,3]\), also called the DeBrouckere mean diameter, is dominated by larger particles and often reflects bulk behavior. These metrics should not be expected to evolve identically. In broad size distributions, for example, the volume-weighted diameter may remain relatively large even as the number-weighted diameter decreases significantly.

Heat and mass transfer statistics quantify coupling between particles and the surrounding fluid. The Fluid Heat Transfer Rate represents the total heat exchange between the fluid and the particle ensemble

\[Q_{\text{tot}} = \sum_i n_i Q_i\]

where \(n_i\) is the number scaling factor and \(Q_{\text{tot}}\) is the heat transfer of parcel \(i\). The summation is taken over all parcels in the ensemble.

The \(kLa\) output represents the total volumetric mass transfer coefficient between the particles and the fluid

\[(k_L a)_{\text{tot}} = \frac{1}{V_f} \sum_i n_i (k_{L,i} A_i)\]

where \(kL,i\) is the convective mass transfer coefficient in the fluid surrounding parcel \(i\), \(A_i\) is the surface area of parcel \(i\), \(n_i\) is the parcel number scale, and \(V_f\) is the total fluid volume. This value is most meaningful when interpreted alongside particle inventory and surface area. For example, an increase in Total Area due to breakup will often result in an increase in \(kLa\), because more interfacial area becomes available for transfer.

When particle–particle thermal contact is modeled, Mean Particle-Particle Heat Flow and Mean Particle-Particle Resistance describe conductive interactions between particles. Because parcels without contacts contribute zero to these averages, the reported values reflect both the strength of conduction and the fraction of particles in contact. Low values may therefore indicate either weak conduction or low contact frequency. Heat of Reaction describes the total rate of heat generation across all particles in the ensemble.

In practice, particle statistics are used to confirm that the model is behaving as intended, determine whether the system has reached steady or statistically steady behavior, and interpret why the particle population is changing over time. A typical workflow is to first examine the inventory terms, then evaluate source and sink balances, next evaluate diameter statistics and breakup/coalescence behavior, and finally assess transfer quantities. This sequence helps distinguish whether observed behavior is driven primarily by transport, population balance physics, or interphase coupling.