16.4.9 Interfacial Area Concentration

Interfacial area concentration is defined as the interfacial area between two phases per unit mixture volume. This is an important parameter for predicting mass, momentum and energy transfers through the interface between the phases. In two-fluid flow systems, one discrete (particles) and one continuous, the size and its distribution of the discrete phase or particles can change rapidly due to growth (mass transfer between phases), expansion due to pressure changes, coalescence, breakage and/or nucleation mechanisms. The Population Balance model ideally captures this phenomenon, but is computationally expensive since several transport equations need to be solved using moment methods, or more if the discrete method is used. The interfacial area concentration model uses a single transport equation per secondary phase and is specific to bubbly flows in liquid at this stage.

The transport equation for the interfacial area concentration can be written as

where $\chi_p$ is the interfacial area concentration ( $m^2/m^3$), and $\alpha_g$ is the gas volume fraction. The first two terms on the right hand side of Equation  16.4-27 are of gas bubble expansion due to compressibility and mass transfer (phase change). $\dot{m}_g$ is the mass transfer rate into the gas phase per unit mixture volume ( $kg/m^3/s$). $S_{RC}$ and $S_{WE}$ are the coalescence sink terms due to random collision and wake entrainment, respectively. $S_{TI}$ is the breakage source term due to turbulent impact.

Two sets of models, the Hibiki-Ishii model [ 129] and the Ishii-Kim model [ 281, 139], exist for those source and sink terms for the interfacial area concentration, which are based on the works of Ishii et al. [ 129, 281]. According to their study, the mechanisms of interactions can be summarized in five categories:

• Coalescence due to random collision driven by turbulence.

• Breakage due to the impact of turbulent eddies.

• Coalescence due to wake entrainment.

• Shearing-off of small bubbles from large cap bubbles.

• Breakage of large cap bubbles due to flow instability on the bubble surface.

In ANSYS FLUENT, only the first three effects will be considered.

Hibiki-Ishii Model

$$S_{RC} = -\frac{1}{3 \phi}(\frac{\alpha_g}{\chi_p})^2 f_c n_b \lambda_c$$ (16.4-28)

$$= -(\frac{\alpha_g}{\chi_p})^2 \frac{\Gamma_c {\alpha_g}^2 \epsilon... ...a_g)}exp(-K_c \frac{{d_b}^{5/6} {\rho_f}^{1/2} {\epsilon}^{1/3}}{\sigma^{1/2}})$$

$$= -\frac{\Gamma_c}{\psi^{11/3}} \frac{\epsilon^{1/3}}{(\alpha_{g ma... ...\rho_f}^{1/2} {\epsilon}^{1/3}}{\sigma^{1/2}}{(\frac{\alpha_g}{\chi_p})}^{5/6}]$$

where $f_c$, $\lambda_c$ and $n_b$ are the frequency of particle/bubble collision, the efficiency of coalescence from the collision, and the number of particles per unit mixture volume, respectively. The averaged size of the particle/bubble $d_b$ is assumed to be calculated as

and

where $f_B$, $\lambda_B$ and $n_e$ are the frequency of collision between particles/bubbles and turbulent eddies of the primary phase, the efficiency of breakage from the impact, and the number of turbulent eddies per unit mixture volume, respectively. In Equation  16.4-31

The experimental adjustable coefficients are given as follows:

$\Gamma_C = 0.188$; $K_C = 0.129$; $\Gamma_B = 0.264$; $K_B = 1.37$ .

The shape factor $\psi$ is given as 6 and $\phi$ as $\frac{1}{36 \pi}$ for spherical particles/bubbles. There is no model for $S_{WE}$ in the Hibiki-Ishii formulation.

Ishii-Kim Model

$$S_{RC} = -\frac{1}{3 \phi}(\frac{\alpha_g}{\chi_p})^2 C_{RC}[\frac{{n_b}^2... ... {d_b}^2}{{\alpha_{g max}}^{1/3}({\alpha_{g max}}^{1/3} - {\alpha_{g}}^{1/3})}]$$ (16.4-33)

$$[1-exp(-C \frac{{\alpha_{g max}}^{1/3}{\alpha_{g}}^{1/3}}{{\alpha_{g max}}^{1/3} - {\alpha_{g}}^{1/3}})]$$

$$= -\frac{1}{3 \pi} C_{RC} u_t {\chi_p}^2 [\frac{1}{{\alpha_{g max}}... ...g max}}^{1/3}{\alpha_{g}}^{1/3}}{{\alpha_{g max}}^{1/3} - {\alpha_{g}}^{1/3}})]$$

$$S_{TI} = \frac{1}{3 \phi}(\frac{\alpha_g}{\chi_p})^2 C_{TI}(\frac{n_b u_t}{d_b})(1-\frac{We_{cr}}{We})^{1/2}exp(-\frac{We_{cr}}{We})$$ (16.4-35)

$$= \frac{1}{18}C_{TI} u_t \frac{{\chi_p}^2}{\alpha_g}(1-\frac{We_{cr}}{We})^{1/2}exp(-\frac{We_{cr}}{We})$$

where the mean bubble fluctuating velocity, $u_t$ , is given by $\epsilon^{1/3} {d_b}^{1/3}$. The bubble terminal velocity, $u_r$ , is a function of the bubble diameter and local time-averaged void fraction.

where $\mu_f$ is the molecular viscosity of the fluid phase, g is the gravitational acceleration and $\sigma$ is the interfacial tension. In this model, when the Weber number, $We$, is less than the critical Weber number, $We_{cr}$, the breakage rate equals zero, i.e. $S_{TI} = 0$. The coefficients used are given as follows [ 139]:

$C_{RC}$	=	0.004
$C_{WE}$	=	0.002
$C_{TI}$	=	0.085
$C$	=	3.0
$We_{cr}$	=	6.0
$\alpha_{g_max}$	=	0.75

Currently, this model is only suitable for two-phase flow regimes, one phase being gas and another liquid, i.e. bubbly column applications. However, you can always use UDFs to include your own interfacial area concentration models, which can apply to other flow regimes.

See the separate UDF Manual for details.