Discrete Quadrature S-Gamma Phase Interaction Models Reference

Table 1. Discrete Quadrature S-Gamma Phase Interaction Models Reference
Model Names	S-Gamma Breakup
	S-Gamma Coalescence
	S-Gamma Entrainment (With multiple flow regime phase interaction only)
Theory	See S-Gamma Phase Interactions.
Provided By	[phase interaction] > Models > Optional Models
Example Node Path	[phase interaction] > Models > S-Gamma Breakup
Requires	S-Gamma Breakup and S-Gamma Coalescence models require a Continuous-Dispersed or Multiple Flow Regime phase interaction. S-Gamma Entrainment require a Multiple Flow Regime phase interaction. The dispersed phase must have the Discrete Quadrature S-Gamma model activated. See S-Gamma Model Reference.
Properties	None.
Activates	Model Controls (child nodes)	For S-Gamma Breakup: S-Gamma Breakup Rate See S-Gamma Breakup Rate Properties. S-Gamma Number of Fragments See S-Gamma Number of Fragments Properties. S-Gamma Variance of Fragments See S-Gamma Variance of Fragments Properties. For S-Gamma Coalescence: S-Gamma Collision Rate See S-Gamma Collision Rate Properties. S-Gamma Coalescence Efficiency See S-Gamma Coalescence Efficiency Properties. For S-Gamma Entrainment: S-Gamma Entrainment Properties See S-Gamma Entrainment Properties. S-Gamma Phase Entrainment Rate See S-Gamma Phase Entrainment Rate Properties. S-Gamma Phase Entrainment Diameter See S-Gamma Phase Entrainment Diameter Properties. S-Gamma Gas Entrainment Method Parameters See S-Gamma Gas Entrainment Method Parameters Properties.
	Field Functions	See Field Functions.

S-Gamma Breakup Rate Properties

The breakup rate $G (a)$ is such that the probability that a particle of size $a$ is broken during a time interval $d t$ is $G (a) d t$ .

You can specify the breakup rate as a constant or field function, or one of the following methods. These methods are the same as the corresponding methods in the AMUSIG model.

The constant and field function methods for breakup imply that particles of all sizes have an equal probability of breaking apart. The other methods are size-selective.

Method	Corresponding Method Node
Coulaloglou and Eskin (for turbulent breakup) / Coulaloglou and Eskin Shear (for laminar breakup) This model predicts a broader size distribution than the other models, and is suitable for modeling emulsion formation (water in oil).	For turbulent breakup: Coulaloglou and Eskin Cg The calibration constant. This value is $C_{g}$ in Eqn. (2246). The default is 1.0. WeCrit The critical Weber number. This value is ${W e}_{c r}$ in Eqn. (2246). The default is 0.5. For laminar breakup: Coulaloglou and Eskin Shear Cg The calibration constant. This value is $C_{g}$ in Eqn. (2250). The default is 1.0. Coulaloglou and Eskin Shear > Critical Capillary Number The critical capillary number ${C a}_{c r}$ 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 This parameter is $C_{1}$ in Eqn. (2249). High Viscosity Ratio Exponent Pre-factor This parameter is $C_{2}$ in Eqn. (2249). Low Viscosity Ratio Exponent This parameter is ${E X P}_{1}$ in Eqn. (2249). High Viscosity Ratio Exponent This parameter is ${E X P}_{2}$ in Eqn. (2249). Maximum Viscosity Ratio This parameter is $λ_{*}$ in Eqn. (2249).
Power Law 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 $d_{0}$ .	Power Law Characteristic diameter This parameter is $d_{0}$ in Eqn. (2244). The default value is 0.001 m. Prefactor This parameter is $C$ in Eqn. (2244). The default is a constant value of 0.0 /s. Exponent This parameter is $a$ in Eqn. (2244). The default is a constant value of 1.0.
Tsouris and Tavlarides (for turbulent breakup) / Tsouris and Tavlarides Shear (for laminar breakup) This model predicts that any droplet can be broken (there is no minimum diameter), but the breakup probability decreases exponentially with droplet diameter.	For turbulent breakup: Tsouris and Tavlarides Cg The calibration constant. This value is $C_{g}$ in Eqn. (2246). The default is 0.25. WeCrit The critical Weber number. This value is ${W e}_{c r}$ in Eqn. (2246). The default is 1.0. For laminar breakup: Tsouris and Tavlarides Shear Tsouris and Tavlarides Shear > Critical Capillary Number The properties are the same as for the Coulaloglou and Eskin Shear method.
Kocamustafaogullari Applies to turbulent breakup only. This model is suitable for modeling the break-up of liquid droplets in continuous gas.	Kocamustafaogullari 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.

S-Gamma Number of Fragments Properties

The number of fragments ( $n_{f}$ ), together with the variance of fragments ( $σ$ ), provides information about the size distribution of the fragments after a breakup event. In the AMUSIG model, the Fragments Size Distribution method plays a similar role.

By default, binary breakup is assumed: each particle is split into two fragments ( $n_{f} = 2$ ).

You can specify a constant or field function.

S-Gamma Variance of Fragments Properties

The variance describes the spread of sizes of the fragments. This value can be expressed using the Sauter mean diameter and the volume-based diameter of the fragments as $σ = \ln (\frac{d_{32}}{d_{30}})$ .

$σ = 0$ implies that the particle is split into equal fragments. By default $σ = 0.025$ .

You can specify a constant or a field function. No physical models for $σ$ are provided.

When you specify values for $n_{f}$ and $σ$ , you should ensure that $\exp (σ) < n_{f}^{\frac{1}{3}}$ ; otherwise the breakup leads to an increase of $d_{32}$ .

For a fragments size distribution $f (\frac{v^{'}}{v_{0}})$ , where $v_{0}$ is the volume of the parent particle and $v^{'}$ is the volume of the fragment, the variance is calculated as:

Figure 1. EQUATION_DISPLAY

σ = - \ln (n_{f}^{\frac{2}{3}} \int_{0}^{1} f (x) x^{\frac{2}{3}} d x)

(301)

For example, for a parabolic fragments size distribution with shape parameter $β$ :

Figure 2. EQUATION_DISPLAY

σ = - \ln (2^{\frac{2}{3}} \cdot \frac{63 + 3 β}{110})

(302)

S-Gamma Collision Rate Properties

The collision rate $K (a_{1}, a_{2})$ is such that the probability of two particles of size $a_{1}$ and $a_{2}$ colliding during a time interval $d t$ is $K (a_{1}, a_{2}) d t$ .

You can specify a constant or field function, or the Turbulent method.

Method	Corresponding Method Node
Turbulent This method applies to turbulent coalescence only and is the same as the Turbulent Collision Rate Model that is implemented in the AMUSIG model.	Turbulent Cg The calibration constant that determines 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.

S-Gamma Coalescence Efficiency Properties

The coalescence efficiency $λ (a_{1}, a_{2})$ is the probability that two particles of size $a_{1}$ and $a_{2}$ merge after a collision.

You can specify a constant or field function, or one of the following methods.

Method	Corresponding Method Node
Luo Applies to Turbulent coalescence only. The contact time due to the turbulent fluctuations is compared to the deformation time of the particle. This model assumes that a high contact time and a short deformation time (that is, high surface tension) make the coalescence more probable. This method is the same as the corresponding method in the AMUSIG model.	Luo 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 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. This method is the same as the corresponding method in the AMUSIG model.	O'Rourke 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: $G 0$ $G 1$ $G 2$ $G 3$
Coulaloglou and Tsouris Applies to Turbulent coalescence only.	Coulaloglou and Tsouris Prefactor This value is $C$ in Eqn. (2226). The default value is 2.0e12 [ $m^{2}$ ].

S-Gamma Entrainment Properties

Modeling Option: Selects which phase will be modeled for the entrainment.

S-Gamma Phase Entrainment Rate Properties

The entrainment rate $N (d_{e})$ which is the number of new gas/liquid particles that are formed due to entrainment.

You can specify a field function for the liquid /gas entrainment rate or one of the following methods for gas entrainment rate.

Method	Corresponding Method Node
Ma (Bubble Surface Energy) This model accounts for the surface entrainment bubble-size spectrum by connecting bubble entrainment with turbulent dissipation rate at the free surface/interface. As such, it applies to turbulent viscous regime only. This bubble entrainment model requires gravity, surface tension, and large interface detection models to be activated.	Ma (Bubble Surface Energy) Calibration Factor The calibration constant that is used to determine the bubble entrainment rate probability. This value is $c_{b}$ in Eqn. (2235). The default value is 0.2.
Yu (Scale Separation) This model accounts for the surface entrainment bubble-size spectrum by using two size regimes separated by a scale radius. The model relates the energy required for bubble entrainment/formation and available turbulent kinetic energy in the interface. As such, it applies to turbulent viscous regime only. This bubble entrainment model requires gravity, surface tension, and large interface detection models to be activated.	Yu (Scale Separation) Calibration Factor The calibration constant that is used to determine the bubble entrainment rate probability. This value is $c_{b}$ in Eqn. (2234). The default value is 0.2.

Method

Corresponding Method Node

Ma (Bubble Surface Energy)

This model accounts for the surface entrainment bubble-size spectrum by connecting bubble entrainment with turbulent dissipation rate at the free surface/interface. As such, it applies to turbulent viscous regime only.

This bubble entrainment model requires gravity, surface tension, and large interface detection models to be activated.

Ma (Bubble Surface Energy)

Calibration Factor: The calibration constant that is used to determine the bubble entrainment rate probability. This value is $c_{b}$ in Eqn. (2235). The default value is 0.2.

Yu (Scale Separation)

This model accounts for the surface entrainment bubble-size spectrum by using two size regimes separated by a scale radius. The model relates the energy required for bubble entrainment/formation and available turbulent kinetic energy in the interface. As such, it applies to turbulent viscous regime only.

This bubble entrainment model requires gravity, surface tension, and large interface detection models to be activated.

Yu (Scale Separation)

Calibration Factor: The calibration constant that is used to determine the bubble entrainment rate probability. This value is $c_{b}$ in Eqn. (2234). The default value is 0.2.

S-Gamma Phase Entrainment Diameter Properties

The parameters for the phase entrainment diameter properties.

You can specify a field function or Limited Hinze.

Method	Corresponding Method Node
Limited Hinze This is a method for modeling droplet/bubble formation diameter bubble formation diameter as in Eqn. (2232).	None.

S-Gamma Gas Entrainment Method Parameters Properties

The parameters for the critical turbulent dissipation rate for bubble creation.

Turbulence Weber Number: The lower bound of the Weber number , ${W e}_{l}$ in Eqn. (2228) used in identifying the critical turbulent dissipation rate for the gas entrainment diameter. This property accounts for the effects of surface tension in the gas entrainment diameter.

Turbulence Froude Number: The lower bound of the Froude number, ${F r}_{u}$ in Eqn. (2228) used in defining the critical turbulent dissipation rate for the gas entrainment diameter. This property accounts for the effects of gravity in the gas entrainment diameter.

Field Functions

The following field function is made available to the simulation when the S-Gamma Breakup model is used and the Interaction Source Storage Retained property is activated in the S-Gamma solver:

Discrete Quadrature Breakup Rate of [phase interaction]: The breakup rate $G (a)$ , as described in Laminar and Turbulent Breakup Models.

The following field function is made available to the simulation when the S-Gamma Coalescence model is used and the Interaction Source Storage Retained property is activated in the S-Gamma solver:

Discrete Quadrature Coalescence Rate of [phase interaction]: The coalescence rate is the product of the collision rate and the coalescence efficiency, as described in Coalescence Models.

The following field functions are made available to the simulation when the S-Gamma Entrainment model is used:

Entrainment Diameter of [dispersed phase] of [phase interaction]: The entrainment droplet/bubble diameter.

Entrainment Rate of [dispersed phase] of [phase interaction]: The entrainment rate in Eqn. (2236).

The following field function is made available to the simulation when the Limited Hinze property is selected for the S-Gamma Phase Entrainment Diameter:

Hinze Entrainment Diameter of [dispersed phase] of [phase interaction]: The Hinze diameter in Eqn. (2232).