As noted above, O’Rourke’s algorithm assumes that two parcels may collide only if they are in the same continuous-phase cell. This assumption can prevent parcels that are quite close to each other, but not in the same cell, from colliding, although the effect of this error is lessened by allowing some parcels that are farther apart to collide. The overall accuracy of the scheme is second-order in space.
The probability of collision of two particles is derived from the point of view of the larger particle, called the collector particle and identified below with the number 1. The smaller particle is identified in the following derivation with the number 2. The collision calculation is performed in the reference frame of the collector particle.
In this local reference frame, a collision can occur only if the distance between the two
centers of particle 1 and particle 2 in the normal direction to their relative velocity vector
is less than or equal to 
. Using this geometrical condition, a cross-section can be determined through
which the center of particle 2 must pass in order for a collision to occur. The area of this
cross-section is 
. The so-called collision volume can be derived by multiplying the
cross-sectional area by the relative velocity vector 
and the integration time step 
.
The algorithm of O’Rourke uses the collision volume to calculate the probability of a
collision. Rather than calculating whether the center of particle 2 is within the collision
volume, the algorithm calculates the probability of this event. If there is a uniform
probability of the particle 2 being anywhere within a cell of volume 
, then the chance of this particle being within the collision volume is the
ratio of the collision volume to the cell volume. Thus, the probability of the collector
colliding with particle 2 is:
 | (12–482) |
Equation 12–482 can be generalized for parcels, where there are
particles in parcel 1 and 
particles in parcel 2. The collector parcel 1 undergoes a mean expected number
of collisions given by:
 | (12–483) |
The actual number of collisions that the collector experiences is not generally the mean expected number of collisions. The probability distribution of the number of collisions follows a Poisson distribution, according to O’Rourke, which is given by:
 | (12–484) |
where 
is the number of collisions between a collector and other particles.
For each DPM timestep, the mean expected number of collisions is calculated for each pair of
tracked parcels in each cell. To determine whether a collision will occur, the probability of no
collision (
) is computed and compared to a uniformly sampled random number. If the random
number is larger than 
, the two parcels will collide.
Once it is determined that the two parcels collide, the outcome of the collision must be decided. In general, the following collision outcomes are possible:
Bouncing only
Bouncing occurs for any type of collision if coalescence effects are neglected.
Bouncing and coalescence
Coalescence may typically occur for liquid droplets.
Coalescence happens if the particles collide head-on. If the collision is more oblique, the particles bounce off each other.
If coalescence is not considered, the outcome of particle collision is a central
collision. The collision point at which the two colliding particles touch is determined
randomly as a point of intersection of the line connecting the centroids of the two colliding
particles and their surfaces. In the frame of reference of the collector particle, the
collision direction is described by the basis vector pointing in the direction of the line
connecting the collector particle centroid and the collision point. Therefore, derived from the
laws of momentum and energy conservation, the post-collisional velocity of the particles in
direction is computed by the following equation [221]:
 | (12–485) |
where 
and 
are the masses of particles 1 and 2, respectively, and 
is the restitution coefficient. 
= 1 in ideal collisions, while 
= 0 in plastic collisions. Ansys Fluent assumes perfectly elastic collisions and
uses a coefficient of restitution 
= 1. Note that only the velocities in the 
direction are changed.
In the reference frame being used here, the probability of coalescence can be related to the offset of the collector particle center and the trajectory of the smaller particle 2. The critical offset is a function of the collisional Weber number and the relative radii of the collector and the smaller particle.
The critical offset is calculated by O’Rourke using the expression
 | (12–486) |
where 
is a function of 
, defined as
 | (12–487) |
The value of the actual collision parameter 
is 
, where 
is a random number between 0 and 1. The calculated value of 
is compared to 
, and if 
, the result of the collision is coalescence. Equation 12–484 gives the number of smaller particles that coalesce with the
collector. The properties of the coalesced particles are found from the basic conservation
laws.
In the case of a grazing collision, the new velocities are calculated based on
conservation of momentum and kinetic energy. It is assumed that some fraction of the kinetic
energy of the particles is lost to viscous dissipation and angular momentum generation. This
fraction is related to 
, the collision offset parameter. Using assumed forms for the energy loss,
O’Rourke derived the following expression for the new velocity:
 | (12–488) |
This relation is used for each of the components of velocity. No other particle properties are altered in grazing collisions.