To apply this procedure, is rewritten using Equation 2–8 and Equation 3–5:

(6–1)

(6–2)

In this last equation, is the momentum exchange coefficient between solid and fluid phase, defined as:

(6–3)

where is the drag coefficient, calculated using the correlations presented in section Drag force and is the area of the particle projected in the force direction.

Since in the classic finite volume method the fluid velocity is considered constant within a cell, the first term on the right hand side of the Equation 6–2 can be subdivided into two terms and written as:

(6–4)

This expression can be further rewritten, splitting the interaction forces into an explicit term, A, and an implicit term, B, that is a function of the fluid velocity on the current time step:

(6–5)

MOMENTUM SOURCE APPROACHES

Sources until 2023 R2 release

The standard momentum sources approach (2023R2) computes the momentum sources on each direction using the following approach:

(6–6)

This can be re-written as:

(6–7)

Here the source term is clearly split into the explicit portion, the momentum exchange coefficient multiplied by the averaged particle velocity, and the implicit portion, given by the momentum exchange coefficient multiplied by the fluid velocity.

In this approach, as can be seen in the time notation superscripts, the momentum exchange coefficient, and actually all fluid forces over the particles, that were converted in the momentum sources for the CFD solver, were computed by Rocky using the fluid velocities of the previous time steps (the reason why the momentum exchange coefficient has the t-1 superscript).

When using the form show above, replacing the previous time step fluid velocity by the current time step fluid velocity, increases the coupling between the solvers (stabilizing the coupling non-linear problem, and also stabilizing Fluent's linear system solution, by using the linearization of source terms), but allows for imbalances in cases where the solution in Fluent has strong variations over time.

As the CFD simulation in coupling cases usually apply small time steps, this difference, and consequently the imbalance caused by this approach, is expected to be small. Although, with increasing time step on the CFD side, or in a better view point, increasing the ratio of CFD time step to DEM time step, is expected to increase this error.

Sources from 2024 R1 release forward

The new (beta) momentum sources approach (2024R1) computes applies a different equation to compute the momentum sources on Fluent, given:

(6–8)

Here, the main difference is that the usage of the previous time step velocity on the fluid side (seen in the term ), forces the momentum source on Fluent side to exactly match the accumulated/averaged value of fluid-particle forces computed by Rocky.

The downside of this approach is that it significantly reduces the coupling in the solutions, causing the non-linear problem to be more strict in terms of the CFD to DEM time step ratio to achieve numerical stability. The usual criteria of keeping the ratio < 1000 or 500 might not work with this source term approach, with smaller ratios being necessary torun the same simulation.

The heat exchange between particles and fluid is calculated in Rocky and sent to Fluent via a source term. This source term is linearized to enhance stability on the CFD side of the simulation.

The expression for the heat transfer rate between particle and fluid is:

(6–9)

Equation 6–10 can be split into two terms following the general linearized source term on fluent [44] at section Linearized Source Terms:

(6–10)

Then, the following equivalences arise from the source term linearization:

(6–11)