AMUSIG Phase Interaction Model Family Reference

Table 1. AMUSIG Phase Interaction Model Family Reference
Model Names	Interaction Length Scale
	Laminar Breakup
	Laminar Coalescence
	Turbulent Breakup
	Turbulent Coalescence
Theory	See AMUSIG Breakup and Coalescence.
Provided By	[phase interaction] > Models > Optional Models
Example Node Path	[phase interaction] > Models > Turbulent Coalescence
Requires	These model groups are available only for a Continuous-Dispersed phase interaction. The dispersed phase must have the Adaptive Multiple Size-Group model activated. Laminar groups of coalescence and breakup models require the Laminar viscous regime activated in the Physics continuum. Turbulent groups of coalescence and breakup models require the Turbulent viscous regime activated in the Physics continuum. Breakup and coalescence interactions require at least two size-groups for a meaningful analysis. However, the recommended minimum is three size-groups. The number of size-groups is specified as a property of the Adaptive Multiple Size-Group model. Most models also require Surface Tension to be defined under Multiphase Material in the phase interaction.
Properties	Key property for breakup models: Number of Abscissas See Breakup Models. The coalescence models have no properties.
Activates	Physics Models	Laminar Breakup Turbulent Breakup See Breakage Rate Models. Laminar Coalescence Turbulent Coalescence See Coalescence Models.
	Materials	Surface Tension See Material Properties.
	Field Functions	See Field Functions.

Interaction Length Scale Model Properties

Specifies the length scale that is used for interaction area and transfers. Three methods are available:

Adaptive Multiple Size-Group Diameter: The usual and default method for AMUSIG runs. When this option is selected in a multi-speed run, each size-group automatically uses its own group diameter for the drag force and for other interaction models. In a one-speed run, the Sauter or surface-mean diameter is automatically used instead.
Constant and Field Function: These options apply the same scale to all size-groups. They are of little modeling value in a multi-speed run, but they can be used for passive testing of the AMUSIG model, without using the size distribution prediction in other phase interaction models.

Breakup Models

When the Laminar Breakup model or the Turbulent Breakup model is selected, the Multiple Size-Group Breakup model is selected automatically. This model has the following property:

Number of Abscissas: A discretization parameter that is common to the Fragments Particle Size Distribution submodels that are used in any break up calculation. The number of abscissas is independent from the number of groups. The default value is 3. This value can be increased to match the number of groups, but it does not have a strong effect on results or performance.

Breakage Rate Models

The breakup process is modeled in two parts, a Breakage Rate and a Fragments Particle Size Distribution. The preferred model combinations are:

Laminar breakup: Shear Breakage Rate Model with the Fragments and Satellites Daughter Particle Size Distribution.
Turbulent breakup: the Martinez-Bazan or Tsouris and Tavlarides models with their corresponding Fragments Particle Size Distribution submodel. To model the break-up of liquid droplets in continuous gas use the Kocamustafaogullari model. For emulsion formation (water in oil), the Coulaloglou and Eskin model is recommended.

The preferred model combination is activated by default. However, alternative submodel combinations can be used, as described in the following table:

Breakage Rate Model	Available Fragments Particle Size Distribution Submodels
Shear Breakage Rate Model Applies to laminar breakup only. Model properties: Calibration Constant Cg This value is $C_{g}$ inEqn. (2250). The default is 1.0. Model child nodes: Critical Capillary Number This value is ${C a}_{c r}$ in Eqn. (2251). The default is 1.0. You can specify this value as a constant, a field function, or using the Power Law method. The latter defines the critical capillary number as a function of the viscosity ratio between dispersed and continuous phase and provides the following properties: Low Viscosity Ratio Exponent Pre-factor: $C_{1}$ in Eqn. (2249). High Viscosity Ratio Exponent Pre-factor: $C_{2}$ in Eqn. (2249). Low Viscosity Ratio Exponent: ${E X P}_{1}$ in Eqn. (2249). High Viscosity Ratio Exponent: ${E X P}_{2}$ in Eqn. (2249). Maximum Viscosity Ratio: $λ_{*}$ in Eqn. (2249). Capillary Breakup Probability Controls the definition of $f$ in Eqn. (2250). The following methods are available: Coulaloglou and Eskin: defines $f$ as given by Eqn. (2254). Cristini: defines $f$ as given by Eqn. (2252). Tsouris and Tavlarides: defines $f$ as given by Eqn. (2253).	Fragments and Satellites Daughter Particle Size Distribution This model is based on the physical observations of the breakup process: the mother droplet is extended by the flow to form a dumbbell shape, then the neck is ruptured into small satellites. This model assumes that the mother droplet splits into two large fragment droplets of the same size and a number of small satellites. Model properties: Fragments Volume Ratio The ratio of volume of the fragment to volume of the mother droplet. The satellites have uniform size distribution. The default value is: 0.9. Parabolic Fragments Particle Size Distribution An empirical distribution that can be used with any breakup model. Model properties: Shape Parameter The default value is 1.5, which corresponds to the smallest standard deviation of particle sizes. A value of 1.0 provides a uniform distribution, while a value of 0.0 gives the highest possible standard deviation of particle sizes. This distribution is illustrated below:
Coulaloglou and Eskin Breakage Rate Applies to turbulent breakup only. This model predicts a broader size distribution than the other models, and is suitable for modeling emulsion formation (water in oil). Model properties: Calibration Constant Cg This value is $C_{g}$ in Eqn. (2246). The default is 1.0. Critical Weber Number This value is ${W e}_{c r}$ in Eqn. (2246). The default is 0.5.	Parabolic Fragments Particle Size Distribution See the description above.
Kocamustafaogullari Breakage Rate Applies to turbulent breakup only. This model is suitable for modeling the break-up of liquid droplets in continuous gas. Model properties: Calibration Constant B This value is $B_{1}$ in Eqn. (2266). The default is $2 \sqrt{3}$ . Critical Weber Number This value is ${W e}_{c r}$ in Eqn. (2246). The default is 12.0. Ohnesorge Number Prefactor This value is $a$ in Eqn. (2268). The default is 1.5. Ohnesorge Number Exponent This value is $b$ in Eqn. (2268). The default is 0.74.	Parabolic Fragments Particle Size Distribution See the description above.
Martinez-Bazan Breakage Rate Applies to turbulent breakup only. Model properties: Calibration Constant Cg This value is $C_{g}$ in Eqn. (2246). The default is 0.25. Critical Weber Number This value is ${W e}_{c r}$ in Eqn. (2246). The default is 1.0.	Martinez-Bazan Fragments Particle Size Distribution This model assumes that the particle size distribution depends on the turbulent stress that is applied to the droplet or bubble ([515]). Parabolic Fragments Particle Size Distribution See the description above.
Tsouris and Tavlarides Breakage Rate Applies to turbulent breakup only. The model properties are the same as for the Martinez-Bazan Breakage Rate model.	Tsouris and Tavlarides Daughter Particle Size Distribution This model assumes that the energy of the particle size distribution is minimized ([560]). The resulting particle size distribution is bi-modal. Model properties: Diameter Ratio The default value is: 0.2154434621334076. Parabolic Fragments Particle Size Distribution See the description above.
Power Law Breakage Rate Power Law Breakage Rate is a generic model with adjustable parameters for the breakage rate multiplier $K_{B}$ of number density at some particle size $d$ scaled by a characteristic diameter $d_{0}$ . Model properties: Characteristic diameter This parameter is $d_{0}$ in Eqn. (2244). The default value is 0.001 m. Model child nodes: Breakage Prefactor This parameter is $C$ in Eqn. (2244). The default is a constant value of 0.0 /s. Breakage Exponent This parameter is $a$ in Eqn. (2244). The default is a constant value of 1.0.	Parabolic Fragments Particle Size Distribution See the description above.

Coalescence Models

When the Turbulent Coalescence model or the Laminar Coalescence model is selected, the Multiple Size-Group Coalescence model is selected automatically. This model has no properties.

The coalescence process is modeled in two parts, a Collision Rate and a Coalescence Efficiency. The preferred model combinations are:

Laminar coalescence: Laminar Collision Rate with Vinckier-Moldenaers Coalescence Efficiency.
Turbulent coalescence: Turbulent Collision Rate with Luo Coalescence Efficiency or O'Rourke Coalescence Efficiency .

Collision Rate Models

The preferred model combination is activated by default. However, alternative submodel combinations can be used, as described in the following table:

Collision Rate Model	Available Coalescence Efficiency Submodels
Laminar Collision Rate Applies to Laminar coalescence only.	Film Drainage Coalescence Efficiency This model is provided for expert users. It is a generic coalescence efficiency model that is compatible with any collision rate model. This model has the following properties: Weber Number Exponent Accounts for surface tension. This value is $m$ in Eqn. (2279). The default value is 1.0. Reynolds Number Exponent Accounts for viscosity. This value is $n$ in Eqn. (2279). The default value is 0.5. You can adjust these property values to fit your own experimental data and model the appropriate physics. This model also has the following child node: Coalescence Efficiency Prefactor You can provide the appropriate expression. The default value is 1, meaning that every collision event in the collision rate ends in a successful coalescence of the two particles into one. This value is $C$ in Eqn. (2279). Vinckier-Moldenaers Coalescence Efficiency To calculate the coalescence efficiency, this model compares the following times: the drainage time, that is, the time that is required to squeeze the liquid film between two droplets up to a critical thickness that is controlled by Hc. the contact time, which depends on the shear rate. This model has the following property: Calibration Constant Hc The default value is 1.5 E-8 m. Uniform Coalescence Efficiency The coalescence efficiency is identical for all groups – the probability of coalescence does not depend on the sizes of two colliding particles. This option is used mainly for debugging purposes. This model has no properties, but has the following child node: Coalescence Efficiency You can provide the appropriate expression. The default value is 1, meaning that every collision event in the collision rate ends in a successful coalescence of the two particles into one.
Turbulent Collision Rate Applies to Turbulent coalescence only.	Film Drainage Coalescence Efficiency See the description above. Luo Coalescence Efficiency The contact time due to the turbulent fluctuations is compared to the deformation time of the bubble. This model assumes that a high contact time and a short deformation time (that is, high surface tension) make the coalescence more probable. This model has the following property: Calibration Constant C1 The probability of coalescence once two particles have collided. A higher value reduces the probability of coalescence. This value is $C$ in Eqn. (2269). The default value is 1. O'Rourke Coalescence Efficiency This model accounts for the coalesce of liquid droplets in gas, and selects a collision outcome map with the following outcomes: grazing (stretching) separation, coalescence, and bouncing. The collision is controlled by the ${We}_{coll}$ in Eqn. (2270), for each pair of interacting droplets. The following empirical constants from Eqn. (2272) are exposed with this model: $G 0$ $G 1$ $G 2$ $G 3$ Uniform Coalescence Efficiency See the description above.
Enhanced Power Law Collision Rate Allows you to investigate alternative coagulation kernels K between particles of two sizes $d_{1}$ and $d_{2}$ , scaled by characteristic diameter $d_{0}$ . This model has the following property: Characteristic Diameter The characteristic diameter of the power law breakage rate. This parameter is $d_{0}$ in Eqn. (2283). The default value is 0.001 m. This model has the following child nodes: Collision Prefactor The calibration constant $C$ in Eqn. (2283). This value has dimensions of metres-cubed per second. Collision Exponents A, B, C, and D The exponents $a$ , $b$ , $c$ , and $d$ in Eqn. (2283).	Film Drainage Coalescence Efficiency See the description above. Uniform Coalescence Efficiency See the description above.
Uniform Collision Rate This model, and the corresponding submodel Uniform Coalescence Efficiency are simplified models that allow analytical verification solutions. This model has the following child node: Collision Rate The default value is zero because it is arbitrary. This value has units of $m^{3} / s$ .	Film Drainage Coalescence Efficiency See the description above. Uniform Coalescence Efficiency See the description above.

Material Properties

Surface Tension: See Surface Tension Material Properties.

Field Functions

Breakup Rate of [phase interaction]: The total number of breakup events per second per unit volume. This field function can be used to highlight the flow features that are associated with breakage.
Coalescence Rate of [phase interaction]: The total number of coalescence events per second per unit volume. This field function can be used to highlight the flow features that are associated with coalescence.

Under Multi-Speed AMUSIG, the phase interaction models show a number of additional field functions, such as:

Interaction Length Scale of [phase interaction_size-group]
Interaction Area Density of [phase interaction_size-group]
Eulerian Drag Coefficient of [phase interaction_size-group]