Discrete Quadrature S-Gamma Phase Interaction Models Reference

S-Gamma Breakup

The following child nodes are available for the S-Gamma Breakup model:

[First/Second] Dispersed Regime Breakup Rate

The breakup rate $G\left(a\right)$ is such that the probability that a particle of size $a$ is broken during a time interval $dt$ is $G\left(a\right)dt$ .

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

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 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).	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 ${We}_{cr}$ in Eqn. (2246). The default is 0.5.
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 Applies to Turbulent breakup only. This model predicts that any droplet can be broken (there is no minimum diameter), but the breakup probability decreases exponentially with droplet diameter.	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 ${We}_{cr}$ in Eqn. (2246). The default is 1.0.

[First/Second] Dispersed Regime Number of Fragments

The number of fragments $n_f$ , together with the fragments variance $\sigma$ , provides information about the size distribution of the fragments after a breakup event.

By default, binary breakup is assumed: each particle is split into two fragments, that is, $n_f=2$ .

[First/Second] Dispersed Regime Fragments Variance

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 $\sigma =\ln \left(\frac{d_{32}}{d_{30}}\right)$ .

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

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

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

For a fragments size distribution $f\left(\frac{v^{\prime }}{v_0}\right)$ , where $v_0$ is the volume of the parent particle and $v^{\prime }$ is the volume of the fragment, the variance is calculated as:

$$\sigma =-\ln \left(n_f^{\frac{2}{3}}{\int }_0^{1}f\left(x\right)x^{\frac{2}{3}}dx\right)$$

(306)

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

$$\sigma =-\ln (2^{\frac{2}{3}}\cdot \frac{63+3\beta }{110})$$

(307)

S-Gamma Coalescence

The following child nodes are available for the S-Gamma Coalescence model:

[First/Second] Dispersed Regime Collision Rate

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 $dt$ is $K(a_1,a_2)dt$ .

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 for EMP.	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.

[First/Second] Dispersed Regime Coalescence Efficiency

The coalescence efficiency $\lambda (a_1,a_2)$ 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 for EMP.	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.
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

S-Gamma Gas Entrainment Method Parameters

The parameters to calculate the critical turbulent dissipation rate ${ϵ}_{cr}$ in Eqn. (2227). Available only if the entrained phase is a gas phase, that is, if bubbles are created.

Turbulence Weber Number

The lower bound of the Weber number , ${We}_l$ in Eqn. (2228), that is used to calculate the critical turbulent dissipation rate. Accounts for the effects of surface tension when calculating the gas entrainment diameter.

Turbulence Froude Number

The lower bound of the Froude number, ${Fr}_u$ in Eqn. (2228), that is used to calculate the critical turbulent dissipation rate. Accounts for the effects of gravity when calculating 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 $G\left(a\right)$ , 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.

Secondary Phase Entrainment Diameter of [phase interaction]

The entrainment droplet/bubble diameter.

Secondary Phase Entrainment Rate of [phase interaction]

The entrainment rate in Eqn. (2236).

The following field functions are available only for the Unrestricted MMP-MMP phase interaction Flow Regime:

Dispersed Phase Fraction of [phase]

The dispersed phase fraction is the ratio of volume of phase which is dispersed to the volume of total phase in a cell, $k_{phase}=\frac{{\alpha }_{phase,dispersed}}{{\alpha }_{phase}}$ . The values range from 0 to 1.

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).