Edge Stripping

Edge stripping models the break-up of the liquid film over a sharp edge.

Maroteaux et al. [626] proposed a model for the break-up of liquid films over a sharp edge. This model is based on the Rayleigh-Taylor instability and contains the break-up criterion and resulting droplet diameter distribution. However, Gubaidullin [620] has raised questions about the validity of the breakup criterion. Therefore, an alternative criterion is adopted, proposed by Friedrich et al. [619], that is based on a force balance. The droplet diameter distribution still follows Maroteaux.

The model is only applied at edges where the corner angle ( $θ$ in the diagram above) is greater than the user-defined minimum corner angle, $θ_{\min}$ . If a boundary face has multiple sharp edges, stripping occurs over all of them in a weighted manner.

The force ratio $F R$ , which is the ratio of the film momentum flux to the surface tension and gravity force, is:

Figure 1. EQUATION_DISPLAY

F R = \frac{{We}_{f}}{1 + \frac{1}{\sin θ} + {Bo}_{f} \frac{L_{b}}{h_{f} \sin θ}}

(2757)

where ${We}_{f}$ is the film Weber number, ${Bo}_{f}$ is the Bond number, $L_{b}$ is the break-up length, and $h_{f}$ is the film thickness.

The film Weber number is:

Figure 2. EQUATION_DISPLAY

{We}_{f} = \frac{ρ_{f} {v_{f}}^{2} h_{f}}{σ}

(2758)

where $ρ_{f}$ is film density, $v_{f}$ is film velocity projected orthogonally to the stripping edge, $h_{f}$ is film thickness, and $σ$ is surface tension.

The film Bond number is:

Figure 3. EQUATION_DISPLAY

{Bo}_{f} = \frac{ρ_{f} {g_{θ} h}_{f}^{2}}{σ}

(2759)

where $g_{θ}$ is the component of acceleration normal to the downstream wall.

Arai and Hashimoto [613] give the break-up length as:

Figure 4. EQUATION_DISPLAY

L_{b} = 0.0388 {h_{f}}^{0.5} {Re}_{f}^{0.6} {We}_{rel}^{- 0.5}

(2760)

where ${Re}_{f}$ is the film Reynolds number and ${We}_{rel}$ is the relative Weber number.

The film Reynolds number is:

Figure 5. EQUATION_DISPLAY

{Re}_{f} = \frac{ρ_{f} v_{f} h_{f}}{μ_{f}}

(2761)

where $μ_{f}$ is the film dynamic viscosity.

The relative Weber number is:

Figure 6. EQUATION_DISPLAY

{We}_{rel} = \frac{h_{f} ρ {(v_{g} - v_{f})}^{2}}{2 σ}

(2762)

where $ρ$ is the gas density and $v_{g}$ is the component of gas velocity normal to the stripping edge.

The break-up is deemed to occur when $F R > F R_{C}$ , where ${F R}_{C}$ is the user-defined critical force ratio, which is proposed in [619]. The critical force ratio default value is 1. Of the fluid crossing the edge, only a fraction $x_{S}$ separates from the film. This fraction is approximated using the formula below:

Figure 7. EQUATION_DISPLAY

x_{S} = {\begin{matrix} 0 & if F R \leq F R_{C} \\ 0.44 (F R - F R_{C}) & if F R_{c} < F R \leq (F R_{C} + 1.6) \\ 0.057 (F R - F R_{C} - 1.6) + 0.704 & if (F R_{C} + 1.6) < F R \leq ({F R}_{c} + 6.792) \\ 1 & if ({F R}_{C} + 6.792) < F R \end{matrix}

(2763)

which is based on the experimental data in [619].

Note	For edge stripping into a dispersed phase, Eqn. (2764) to Eqn. (2769) are not used. The droplet diameter is specified manually rather than being computed.

For droplet formation, the model that is presented in [626] is adopted, which estimates the parent droplet diameter $D_{d}$ as:

Figure 8. EQUATION_DISPLAY

D_{d} = c_{1} \sqrt{\frac{λ h_{f}}{π}}

(2764)

where $c_{1}$ is a user-defined droplet diameter scale factor. This factor is set to 3.78 by default. The wavelength $λ$ is calculated from the most unstable wavenumber $k$ , which maximizes the growth rate:

Figure 9. EQUATION_DISPLAY

ω = - (\frac{σ - (ρ_{f} - ρ) a / k^{2}}{2 μ_{f} h_{f}}) (\frac{(k h_{f}) \sinh (k h_{f}) \cosh (k h_{f}) - k^{2} h_{f}^{2}}{\cosh^{2} (k h_{f}) + k^{2} h_{f}^{2}})

(2765)

where the acceleration $a$ is computed from:

Figure 10. EQUATION_DISPLAY

a = \frac{v_{f}^{2} θ}{h_{f} (π + θ)}

(2766)

and $λ$ is related to $k$ by $λ = 2 π / k$ .

According to the model, the cumulative droplet size distribution $F (D)$ is a Rosin-Rammler distribution:

Figure 11. EQUATION_DISPLAY

F (D) = 1 - e^{- {(D / X)}^{q}}

(2767)

where $q$ is a user-defined parameter, which is set to 1.5 by default, and:

Figure 12. EQUATION_DISPLAY

X = \frac{D_{d}}{{(3 \ln 10)}^{1 / q}}

(2768)

There are two options for determining the droplet diameter:

Generate droplets with a deterministic diameter calculated as the average of the size distribution that is:

Figure 13. EQUATION_DISPLAY

D = X Γ (1 + \frac{1}{q})

(2769)

where $Γ (x)$ represents the Gamma function.

Generate droplets with a randomized diameter $D$ according to Eqn. (2767).

The droplets are ejected at the film velocity.

Edge Stripping with Flow from Both Sides

For a case where flow comes toward an edge from both sides, the flows are combined into one stream of Lagrangian droplets that conserves mass, momentum, species, and energy. For example, if both flows have different temperatures, the droplet temperature is set to a value between the two. For opposing flows, the stripping fraction $x_{S}$ (Eqn. (2763)) is set to 1 for both sides (which is full stripping). Droplet diameter is computed based on the film properties of the side with the biggest flux. The following diagram illustrates opposing-flow edge stripping.