Adaptive Multiple Size-Group Model Reference

Table 1. Adaptive Multiple Size-Group Model Reference
Theory	See Adaptive Multiple Size-Group (AMUSIG) Model.
Provided By	[phase] > Models > Particle Size Distribution
Example Node Path	[phase] > Models > Adaptive Multiple Size-Group
Requires	An Eulerian Multiphase simulation with the following models activated: Material: Multiphase, Multiphase Interaction (Selected automatically) Multiphase Model: Eulerian Multiphase (EMP), Gradients (Selected automatically) Viscous Regime: Laminar or Turbulent A minimum of two phases: one continuous phase and one dispersed phase. This model is used only in dispersed phases. In the dispersed phase, the following phase models activated: Optional Models: Particle Size Distribution Particle Size Distribution: Adaptive Multiple Size-Group The Adaptive Multiple Size-Group model has some limitations and restrictions that you must consider when setting up your simulation. See Limitations and Restrictions.
Properties	Key properties are: Number Of Size-Groups Velocity Option See Adaptive Multiple Size-Group Properties.
Activates	Physics Models	When the dispersed phase in a Continuous-Dispersed phase interaction has the Adaptive Multiple Size-Group model activated, the following optional model becomes available: Fixed Bins See AMUSIG Fixed Bins Model Reference. The following phase interaction models become available. For a laminar viscous regime: Laminar Breakup Laminar Coalescence For a turbulent viscous regime: Turbulent Breakup Turbulent Coalescence See AMUSIG Phase Interaction Model Family Reference.
	Model Controls (child nodes)	Number Density Turbulent Prandtl Number See Model Controls.
	Initial Conditions	For the dispersed phase only: Particle Size Distribution. See Initial Conditions.
	Boundary Inputs	For inlets and pressure boundaries: Particle Size Distribution. See Boundary Settings.
	Solvers	Multiple Size-Group solver. See Multiple Size-Group Solver Properties.
	Reports	Surface Average Diameter, Volume Average Diameter, Surface Moments, Volume Moments. See Adaptive Multiple Size-Group Model Reports.
	Field Functions	See Adaptive Multiple Size-Group Model Field Functions.

Adaptive Multiple Size-Group Model Properties

Number Of Size-Groups

One size-group can be sufficient for basic studies, such as the effect of depth on mean bubble size. Breakup and coalescence interactions require at least two size-groups for a meaningful analysis. However, the recommended minimum is three size-groups. Breakup and coalescence simulations are not allowed with one size-group.

The number of groups cannot be changed during iterations or time steps. If you change the number of size-groups, all of the fields for all of the size-groups are automatically reinitialized.

Velocity Option

Specifies whether the particle size-groups travel at a single shared velocity or individual velocities for each group.

One-Speed
Used where only the mean size of the particle influences the flow.
Multi-Speed
Allows each size group to move with a different velocity and direction. This option is used to study separation by particle size, or secondary currents that are induced by break-up and coalescence in a poly-disperse flow.

Minimum Particle Diameter

Maximum Particle Diameter

The minimum and maximum values control clipping when the Group Diameter is updated. These limits state what is physically plausible, but normally they are not invoked and do not require adjustment.

Size Distribution Diffusion Limiter

Controls the spurious dissipation rate when treating diffusion under high gradients of particle diameters on a coarse grid. Its value varies from 0 (which gives robustness at the expense of maximum spurious dissipation) to a high value (>> 0.5) (improving accuracy) to

\infty

for maximum accuracy. The default value is 0.5. This setting becomes irrelevant on a fine mesh where the gradient is well-resolved.

If there are different results for different values (for example, $λ_{D} = 1.0$ and $λ_{D} = 2.0$ ), it means that a particle undergoes an intensive breakup/coalescence process when it passes through a single cell. To resolve the equations properly, use a finer mesh.

For more information on spurious dissipation, see [460].

Inter-Group Redistribution Rate

Inter-group redistribution is used to maintain approximately equal volume fractions of each of the different size-groups in each cell. The redistribution algorithm has a strong impact on the performance of the AMUSIG method. If the redistribution rate is too high, the values of the size-group volume fractions, and their accompanying residuals, can become oscillatory.

The default inter-group redistribution rate of 0.1 is sufficient for most of applications, although smaller values can be required in some cases. In the most severe situations, an inter-group redistribution rate of 0.01 is recommended. In such severe cases, use an under-relaxation factor ramp of 100 iterations for the Volume Fraction solver.

In a fully-converged case, the results are independent of the inter-group redistribution rate.

Model Controls

The following child node is available for turbulent viscous regimes only.

Number Density Turbulent Prandtl Number: Sets the ratio of kinematic turbulent viscosity and coefficient of turbulent diffusion of the Number Density.

Initial Conditions

Simcenter STAR-CCM+ samples the initial conditions once, and then attempts to approximate the cumulative distribution for each method with a few groups. You set the number of groups with the Number Of Size-Groups property.

Particle Size Distribution

Sets the initial particle size distribution in the dispersed phase.

Method	Corresponding Initial Conditions Nodes
Uniform distribution A top-hat function, uniform within the specified diameter range and zero outside the range.	Maximum Diameter The parameter $d_{\max}$ in Eqn. (2296). Minimum Diameter The parameter $d_{\min}$ in Eqn. (2296).
Rosin-Rammler distribution The cumulative distribution function for the Weibull or Rosin-Rammler distribution.	Scale Parameter The parameter $λ$ in Eqn. (2299). Shape Parameter The parameter $k$ in Eqn. (2299). The larger the value of $k$ the narrower the distribution.
Log-normal distribution The cumulative distribution function for the log-normal distribution.	Mean Log Diameter The parameter $μ$ in Eqn. (2301). Standard Deviation Log Diameter The parameter $σ$ in Eqn. (2301).
User Specified Distribution	User Specified Diameter Distribution A cumulative distribution function (CDF) of particle diameter that you specify. The following method is available. Table (Particle Size CDF) In the Table (Particle Size CDF) child node, specify the following properties: Table The table that contains the user-specified distribution data. Table: Particle Size The column that contains the particle size data. $d_{i}$ in Eqn. (2305). Table: CDF The column that contains the particle size cumulative distribution function data. $F (d)$ in Eqn. (2305).

Boundary Settings

Simcenter STAR-CCM+ resamples the inflow conditions at every iteration, and can contribute to different size-groups depending on the conditions inside the converging flow next to the boundary.

Inlets and Pressure Boundaries

Particle Size Distribution

Sets the particle size distribution at the boundary.

Method	Corresponding Physics Value Nodes
Uniform distribution A top-hat function, uniform within the specified diameter range and zero outside the range.	Maximum Diameter The parameter $d_{\max}$ in Eqn. (2296). Minimum Diameter The parameter $d_{\min}$ in Eqn. (2296).
Rosin-Rammler distribution The cumulative distribution function for the Weibull or Rosin-Rammler distribution.	Scale Parameter The parameter $λ$ in Eqn. (2299). Shape Parameter The parameter $k$ in Eqn. (2299). The larger the value of $k$ the narrower the distribution.
Log-normal distribution The cumulative distribution function for the log-normal distribution.	Mean Log Diameter The parameter $μ$ in Eqn. (2301). Standard Deviation Log Diameter The parameter $σ$ in Eqn. (2301).
User Specified Distribution	User Specified Diameter Distribution A cumulative distribution function (CDF) of particle diameter that you specify. The following method is available. Table (Particle Size CDF) In the Table (Particle Size CDF) child node, specify the following properties: Table The table that contains the user-specified distribution data. Table: Particle Size The column that contains the particle size data. $d_{i}$ in Eqn. (2305). Table: CDF The column that contains the particle size cumulative distribution function data. $F (d)$ in Eqn. (2305).

Multiple Size-Group Solver Properties

The Multiple Size-Group solver relaxation factors can be varied independently of the Volume Fraction solver relaxation factors.

Coalescence rates increase rapidly with particle concentration. Higher concentrations can require additional relaxation, as well as confirmation that the concentration is being modeled realistically with suitable models for turbulence, for turbulence dispersion, and for breakup.

Implicit Under-Relaxation Factor

The default value of 0.5 gives reliable convergence in One-Speed cases.

Keeping the Implicit Under-Relaxation Factor high (but no more than 0.8) helps to keep down the overall number of iterations.

Explicit Under-Relaxation Factor

The default value of 1.0 gives reliable convergence in One-Speed cases.

Some Multi-Speed cases where each size-group can travel in a different direction can require additional relaxation. For example, the Explicit Under-Relaxation Factor might be reduced to a factor of 0.25 for cases with intense breakup.

Particle Size Frozen

When activated, the Group Diameter is not updated, but the Number Density continues to track changes in Volume Fraction so that these three fields remain physically consistent. If adjusting the relaxation factors does not resolve any convergence difficulty, this option can help you to identify the equation that is responsible for the difficulty.

This option is deactivated by default.

Reconstruction Frozen

When On, Simcenter STAR-CCM+ does not update reconstruction gradients with each iteration, but rather uses gradients from the last iteration in which they were updated. Activate Temporary Storage Retained in conjunction with this property. This property is Off by default.

Reconstruction Zeroed

When On, the solver sets reconstruction gradients to zero at the next iteration. This action means that face values used for upwinding (Eqn. (905)) and for computing cell gradients (Eqn. (917) and Eqn. (918)) become first-order estimates. This property is Off by default. If you turn this property Off after having it On, the solver recomputes the gradients on the next iteration.

Temporary Storage Retained

When On, Simcenter STAR-CCM+ retains additional field data that the solver generates during an iteration. The particular data retained depends on the solver, and becomes available as field functions during subsequent iterations. Off by default.

Adaptive Multiple Size-Group Model Reports

Surface Average Diameter

The Surface Mean Diameter that is calculated using Eqn. (2285) is averaged over the facial area $A_{f}$ of the specified surfaces.

You specify the following properties:

Property	Description
Phase	The dispersed phase.
P	The value $p$ in Eqn. (2284).
Q	The value $q$ in Eqn. (2284).
Parts	This report is available for boundaries and part surfaces, and for boundary-mode and contact-mode interfaces. It is not available for derived parts.
Representation	Volume Mesh or None.

Volume Average Diameter

The Volume Mean Diameter that is calculated using Eqn. (2286) is averaged over the specified region.

You specify the following properties:

Property	Description
Phase	The dispersed phase.
P	The value $p$ in Eqn. (2284).
Q	The value $q$ in Eqn. (2284).
Parts	This report is available for regions and geometry parts. It is not available for derived parts.
Representation	Volume Mesh or None.

Surface Moments

The Surface Moment that is calculated using Eqn. (2289) for the specified surfaces.

You specify the following properties:

Property	Description
Phase	The dispersed phase.
Moments Order	The value $n$ in Eqn. (2289).
Moments Type	Number Density Weighted or Volume Fraction Weighted.
Parts	This report is available for boundaries and part surfaces, and for boundary-mode and contact-mode interfaces. It is not available for derived parts.
Representation	Volume Mesh or None.

Volume Moments

The Volume Moment that is calculated using Eqn. (2290) for the specified region.

You specify the following properties:

Property	Description
Phase	The dispersed phase.
Moments Order	The value $α$ in Eqn. (2290).
Moments Type	Number Density Weighted or Volume Fraction Weighted.
Parts	This report is available for regions and geometry parts. It is not available for derived parts.
Representation	Volume Mesh or None.

Adaptive Multiple Size-Group Model Field Functions

The naming convention is that the size-groups are indexed in order of increasing particle size. However, this naming convention is not always true in all cells until the calculation is well converged.

GroupDiameter of [size group]: $\sqrt[3]{\frac{6 α}{π n}}$ where $α$ is the Volume Fraction and $n$ is the Number Density.
Number Density of [size group]
Number Density of [phase]
Interval Volume Fraction of [phase]: The volume fraction of the phase that is present in size-groups with diameter between the specified minimum and maximum particle size.
Mean Diameter of [phase]: The generic mean diameter that is calculated using Eqn. (2284). You can specify the appropriate values of $p$ and $q$ . The default is $d_{10}$ or the number mean.
Surface Mean Diameter of [phase]: The Surface Mean Diameter that is calculated using Eqn. (2285).
Volume Mean Diameter of [phase]: The Volume Mean Diameter that is calculated using Eqn. (2286).

D[N] Diameter of [phase]

The diameter at which $\begin{matrix} N % \end{matrix}$ of the dispersed phase mass is comprised of smaller size particles, and $N = 10, 20, .., 90$ . The D[N] Diameter of [phase] is calculated based on cumulative distribution data of a particle size distribution, where the X-axis represents the particle size and the Y-axis represents the mass percentage of particles in the sample.

The D50 Diameter of [phase] is the median particle size, that is, the size at which 50% of the particles are smaller. This median particle size divides the distribution into two equal parts by mass.

Under the Multi-Speed option, AMUSIG shows a number of flow and turbulence variables that are computed on size-groups. Some examples are:

Velocity of [size-group]
Volume Fraction of [size-group]
Turbulent Viscosity of [size-group]
Turbulent Viscosity Ratio of [size-group]
Effective Viscosity of [size-group]
Ustar of [size-group]
Wall Shear Stress of [size-group]
Wall Y+ of [size-group]