## 22.11.7 Numerics of the Discrete Phase Model

The underlying physics of the Discrete Phase Model is described by ordinary differential equations (ODE) as opposed to the continuous flow which is expressed in the form of partial differential equations (PDE). Therefore, the Discrete Phase Model uses its own numerical mechanisms and discretization schemes, which are completely different from other numerics used in FLUENT.

The Numerics tab gives you control over the numerical schemes for particle tracking as well as solutions of heat and mass equations (Figure  22.11.4).

Numerics for Tracking of the Particles

To solve equations of motion for the particles, the following numerical schemes are available:

implicit   uses an implicit Euler integration of Equation  22.2-1 which is unconditionally stable for all particle relaxation times.

trapezoidal   uses a semi-implicit trapezoidal integration.

analytic   uses an analytical integration of Equation  22.2-1 where the forces are held constant during the integration.

runge-kutta   facilitates a 5th order Runge Kutta scheme derived by Cash and Karp [ 50].

You can either choose a single tracking scheme, or switch between higher order and lower order tracking schemes using an automated selection based on the accuracy to be achieved and the stability range of each scheme. In addition, you can control how accurately the equations need to be solved.

Accuracy Control   enables the solution of equations of motion within a specified tolerance. This is done by computing the error of the integration step and reducing the integration step if the error is too large. If the error is within the given tolerance, the integration step will also be increased in the next steps.

Tolerance   is the maximum relative error which has to be achieved by the tracking procedure. Based on the numerical scheme, different methods are used to estimate the relative error. The implemented Runge-Kutta scheme uses an embedded error control mechanism. The error of the other schemes is computed by comparing the result of the integration step with the outcome of a two step procedure with half the step size.

Max. Refinements   is the maximum number of step size refinements in one single integration step. If this number is exceeded the integration will be conducted with the last refined integration step size.

Automated Tracking Scheme Selection   provides a mechanism to switch in an automated fashion between numerically stable lower order schemes and higher order schemes, which are stable only in a limited range. In situations where the particle is far from hydrodynamic equilibrium, an accurate solution can be achieved very quickly with a higher order scheme, since these schemes need less step refinements for a certain tolerance. When the particle reaches hydrodynamic equilibrium, the higher order schemes become inefficient since their step length is limited to a stable range. In this case, the mechanism switches to a stable lower order scheme and facilitates larger integration steps.

 This mechanism is only available when Accuracy Control is enabled.

Higher Order Scheme   can be chosen from the group consisting of trapezoidal and runge-kutta scheme.

Lower Order Scheme   consists of implicit and the exponential analytic integration scheme.

Tracking Scheme   is selectable only if Automated Tracking Scheme Selection is switched off. You can choose any of the tracking schemes. You also can combine each of the tracking schemes with Accuracy Control.

Including Coupled Heat-Mass Solution Effects on the Particles

By default, the solution of the particle heat and mass equations are solved in a segregated manner. If you enable the Coupled Heat-Mass Solution option, FLUENT will solve this pair of equations using a stiff, coupled ODE solver with error tolerance control. The increased accuracy, however, comes at the expense of increased computational expense.

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