Impingement

Liquid droplets can impinge on a dry wall to form a fluid film or they can impinge on an existing fluid film. In both cases, the mass, the momentum, and the energy of the droplets are transferred to the film.

For the Bai-Gosman and the Bai-Onera wall impingement models, the formation of fluid film from droplets depends on different impingement regimes. The droplets can exhibit behaviors such as adherence, rebound, spread, or splash. Depending on droplet behavior, only part of droplet mass, momentum, and energy might be transferred to the fluid film.

Incident Mass Flux Impingement

This model calculates the mass flux of the droplets that impinge on the wall to form a fluid film, using the inertia of the droplets near the wall. Impingement occurs on the walls at which a shell interface has been created. The liquid phase impinging on the fluid-film interface must have the exact same material components as the fluid-film phase.

The Incident Mass Flux Impingement model is applicable when the dispersed phase is calculated by the Eulerian Multiphase (EMP) model or the Mixture Multiphase (MMP) model.

The mass flux that impinges on the wall (or on an existing fluid film) is calculated as:

$${\dot{m}}_{\mathrm{imp}}=\gamma {\rho }_{\mathrm{imp}}{v}_{\mathrm{imp}}\cdot a{\alpha }_{\mathrm{imp}}$$

(2737)

where:

• $\gamma$ is the impingement efficiency, which specifies the fraction of the flux incident on the surface that impinges on the fluid film. The impingement efficiency can be a value from 0 through 1.
• ${\rho }_{\mathrm{imp}}$ is the density.
• $v_{\mathrm{imp}}$ is the velocity.
• $a$ is the wall area vector (normal to the wall in the outward facing direction).
• ${\alpha }_{\mathrm{imp}}$ is the volume fraction.

The kinematic and dynamic conditions at the interface between the fluid film and the surrounding multiphase fluid (film free surface) are satisfied. The impinging mass flux that is calculated at the interface is used in all of the other transport equations.

Caraghiaur Impingement

The drop deposition rate per unit interfacial area of the liquid film, as defined by Caraghiaur [615], is as follows:

$${\dot{m}}_{\mathrm{d}\mathrm{e}\mathrm{p}}=\varphi \sqrt{\frac{2}{\pi }}{v}_d^{\prime }\cdot {\alpha }_d{\rho }_L$$

(2740)

The term ${\alpha }_d$ is the volume fraction of the droplets and ${\rho }_L$ is the liquid density, so it follows that the term ${\alpha }_d{\rho }_L$ is the droplet concentration. $\varphi$ is the fraction of droplets that hit the wall. The RMS fluctuating velocity of the droplet $v_d^{\prime }$ is calculated as a function of the particle relaxation time ${\tau }_p$ using the relation of Tchen [reflink]:

$$\frac{v_f^{\prime 2}}{v_d^{\prime 2}}=1+\frac{{\tau }_p}{T_L}$$

(2741)

where $v_f^{\prime }$ is the RMS fluctuating velocity of the fluid film and ${\tau }_p$ is the relaxation time.

${\tau }_p$ is a measure of the inertia of the droplet, and is calculated as:

$${\tau }_p=\frac{{\tau }_{p0}}{\psi \left({\mathrm{Re}}_p\right)}$$

(2742)

where:

$$\psi \left({\mathrm{Re}}_p\right)=\{\begin{array}{cc}1+0.15{\mathrm{Re}}_p^{0.687}& {\mathrm{Re}}_p\le {10}^3\\ 0.11{\mathrm{Re}}_p/6& {\mathrm{Re}}_p>{10}^3\end{array}$$

(2743)

$${\mathrm{Re}}_p=\frac{d_d|v_f-v_d|}{{\nu }_f}$$

(2744)

and:

$${\tau }_{p0}=\frac{{\rho }_d{d}_d^2}{18{\rho }_f{\nu }_f}$$

(2745)

The subscripts $d$ and $f$ stand for droplet and fluid respectively.

The variable $T_L$ represents the Lagrangian integral time scale of the fluid following the path of an inertial droplet. $T_L$ is calculated according to the Zaichik model, which is based on the equation of Zaichik and others ([633]):

$$T_L=\frac{4}{3}\frac{k}{C_{0ϵ}}$$

(2746)

where:

$$C_0=\frac{C_{0\infty }{\mathrm{Re}}_{\lambda }}{{\mathrm{Re}}_{\lambda }+C_1}$$

(2747)

$${\mathrm{Re}}_{\lambda }=\sqrt{\frac{15v_f^{\prime 4}}{ϵ\nu }}$$

(2748)

Where $C_{0\infty }=7.0$ , ${\mathrm{Re}}_{\lambda }$ is the Reynolds number calculated for the Taylor Microscale, $C_1=32.0$ , and $ϵ$ is the turbulence dissipation rate.