Comments on “Liquid Film Atomization on Wall Edges-Separation Criterion and Droplets Formation Model”

Gubaidullin, Askar

doi:10.1115/1.2721078

In an intriguing paper (1), Maroteaux, Llory, Le Coz, and Habchi presented a separation criterion for a liquid film from sharp edges in a high-speed air flow. According to their model, the film of thickness

h_{f}

and velocity

U_{f}

separates from a sharp edge of angle

α

if

α > α_{crit}

⁠. The relation obtained for the critical angle is

α_{crit} = \frac{U_{f}}{ω_{\max} h_{f}} \log {(\frac{δ}{δ_{0}})}_{crit}

1

where

δ ∕ δ_{0}

is the amplitude ratio of the final to the initial perturbation of the film surface. When the wave amplitude reaches a critical value, the film stripping from an edge occurs. The critical value

{(δ ∕ δ_{0})}_{crit}

is set equal to 20 as the best fit for their experimental data. The frequency

ω_{\max}

is defined as the most unstable perturbation growth rate that causes the film separation. This maximum frequency is computed from the dispersion relation of Jain and Ruckenstein (JR) (2). The results of 12 tests with dodecane film flowing on springboard or straight step are reported. The geometrical edge angle

α

is equal to

135 \deg

for all tests. The maximum film thickness

h_{f}

is measured while the film velocity

U_{f}

is estimated. The fact of stripping is established from the visual observations. If the critical angle, computed from Eq. , takes values that are inferior to

135 \deg

⁠, the theory assumes to predict stripping. The experimental data are summarized in Table 1.

Table 1

Summary of experiments¹

No.	$h_{f, \max}$ $(μ m)$	$U_{g}$ (m/s)	$U_{f}$ (m/s)	$α_{crit}^{0}$ (from (1))	Re	We	Experimental observation
1	17	80	2.3	⋯	21.7	4.9	no stripping
2	23	80	3.2	128	40.8	6.5	some droplets
3	28	80	3.8	94	59.0	7.8	established stripping
4	21	60	2.5	174	29.1	3.3	rare droplets
5	29	60	2.9	118	46.7	4.5	some droplets
6	38	60	2.6	87	54.8	6.0	established stripping
7	26	80	2.05	⋯	29.6	7.6	no stripping
8	26	80	2.4	158	34.6	7.5	rare droplets
9	32	80	3.2	99	56.8	9.1	established stripping
10	34	60	1.8	165	34.0	5.5	no stripping
11	37	60	1.8	160	37.0	6.0	no stripping
12	41	60	2.25	115	51.2	6.6	established stripping

No.	$h_{f, \max}$ $(μ m)$	$U_{g}$ (m/s)	$U_{f}$ (m/s)	$α_{crit}^{0}$ (from (1))	Re	We	Experimental observation
1	17	80	2.3	⋯	21.7	4.9	no stripping
2	23	80	3.2	128	40.8	6.5	some droplets
3	28	80	3.8	94	59.0	7.8	established stripping
4	21	60	2.5	174	29.1	3.3	rare droplets
5	29	60	2.9	118	46.7	4.5	some droplets
6	38	60	2.6	87	54.8	6.0	established stripping
7	26	80	2.05	⋯	29.6	7.6	no stripping
8	26	80	2.4	158	34.6	7.5	rare droplets
9	32	80	3.2	99	56.8	9.1	established stripping
10	34	60	1.8	165	34.0	5.5	no stripping
11	37	60	1.8	160	37.0	6.0	no stripping
12	41	60	2.25	115	51.2	6.6	established stripping

Reference 1.

Let us analyze this separation criterion. First, consider the JR study: JR investigated the rupture mechanism for stagnant ultrathin (less than a nanometer) films on a solid surface. In their case, the instability arises due to molecular interaction forces between the film and the surrounding fluid or gas. For the case of the negligible surrounding gas viscosity and the absence of the surface active agent, the dispersion relation is expressed as

ω = - (\frac{σ_{eff}}{2 μ h_{0}}) \times [\frac{(k h_{0}) \sinh (k h_{0}) \cosh (k h_{0}) - {(k h_{0})}^{2}}{\cosh^{2} (k h_{0}) + {(k h_{0})}^{2}}]

2

Here,

h_{0}

is the mean film thickness,

k

is a wave number, and

σ_{eff}

is the effective surface tension, which is defined as the sum of the surface tension

σ

and a term due to London/van der Waals interaction

σ_{eff} = σ + \frac{1}{k^{2}} {(\frac{\partial Φ}{\partial h})}_{h = h_{0}}

3

The film is unstable to small perturbations if

ω > 0

⁠, i.e., when

σ_{eff} < 0

⁠. Critical thickness, below which the water film breaks up, is found to be about

10^{- 2} microns

and films thicker than that critical value are stable.

Maroteaux et al. have employed this relation by substituting the body force

Φ

by the force due to “normal acceleration” equal to

Φ = - Δ ρ h_{f} a

⁠. This acceleration is expressed as

a = \frac{U_{f}^{2}}{R}

4

with the radius estimated as

R = h_{f} (\frac{π}{α} + 1)

We will not present the analysis of the arguments that lead to formulas 1,3,4—even if those seem to be controversial—and will only point out a few apparent inconsistencies. Consider the experiment when dodecane film negotiates the edge without stripping: $h_{f} = 17 microns$ ⁠, $α ≅ 2.4$ $(135 \deg)$ ⁠, $U_{f} = 2.3 m ∕ s$ (see Table 1, test No.1). Then, Eq. 4 results in the unrealistic value for the acceleration, $a ≅ 1.35 \times 10^{5} m ∕ s$ ⁠. In fact, it represents the expression for the centripetal acceleration of an element that moves with a constant velocity magnitude along the arc of radius $R$ ⁠. In Ref. 1 this acceleration is denoted as a “normal acceleration” directed “towards the gas,” i.e., outwards. As is well known, the centripetal acceleration is directed inwards or to the center, as its name suggests. Obviously, it does not represent correctly the acceleration of the film. Hence, the force $Φ = - Δ ρ h_{f} a$ does not have much of physical sense especially if employed in the dispersion relation derived for stagnant nanofilms. Indeed, a transposition of JR results to this very different physical phenomenon is questionable.

Furthermore, the values for maximum wave-growth frequency $ω_{\max} ≅ 64 \times 10^{3} s^{- 1}$ and the corresponding wave number $k ≅ 39 \times 10^{3} m^{- 1}$ are not realistic either. This frequency corresponds to the ultrasound range that is hardly characteristic for this case. Nevertheless, the critical angle computed from Maroteaux’s formula is $α_{crit} = 364 > 135 \deg$ ⁠, and the theory corresponds to the experiment. Finally, note that the angle of an edge has not been varied in the tests. Thus, the separation criterion based on the critical angle does not seem to be appropriate.

The film stripping is a very complex phenomenon that may depend on the film flow rate, surface tension, viscosity, gravity, wavy motion, interfacial shear, surface wettability, and geometry (slope inclination, edge angle). Simple physical considerations lead to an assumption that the film stripping occurs when the disruptive aerodynamic forces dominates over the adhesive capillary forces. The film inertia and its viscosity are important as well. These effects are represented by the Weber and Reynolds numbers, which are defined as

We = \frac{ρ_{g} {(U_{g} - U_{f})}^{2} h_{f}}{σ}

and

Re = \frac{ρ_{f} U_{f} h_{f}}{μ_{f}}

respectively. The estimated values for each test of We and Re are presented in Table 1. Unfortunately, mean values of the film thickness and velocity have not been measured in (1). Note that those are more appropriate to define the dimensionless parameters. The dodecane physical properties are taken to be

μ_{f} = 1.35 \times 10^{- 3} kg ∕ m s

⁠,

ρ_{f} = 749 kg ∕ m^{3}

⁠,

σ = 0.025 N ∕ m

(not specified in (1)) and the density of air is

ρ_{g} = 1.2 kg ∕ m^{3}

⁠. It can be seen that a zone of stripping corresponds to the elevated values of Re, We, and is clearly separated from that where no stripping occurs in (We, Re) coordinates (see Fig.1). These considerations, however, by no means represent a complete dimensional analysis of the phenomenon. More detailed and accurate experimental data obtained for fluids with varied physical properties are desired to establish a reliable correlation. It should be noted that this approach is somewhat similar to that developed by Nigmatulin and co-workers in their study of the droplet entrainment in annular flows (3).

Figure 1

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Experimental data plotted as a function of the Weber and Reynolds numbers

Furthermore, the separation may strongly depend on the wave characteristics. Thus, accurate experimental measurements of those are required. Additionally, under certain conditions the shear-driven film motion is governed by large amplitude nonlinear waves. The linear stability analysis cannot be applied in such a case.

1.

Maroteaux

,

F.

,

Llory

,

D.

,

Le Coz

,

J.-F.

, and

Habchi

,

C.

, 2002, “

Liquid Film Atomization on Wall Edges-Separation Criterion and Droplets Formation Model

,”

ASME J. Fluids Eng.

0098-2202,

124

, pp.

565

–

575

.

Crossref

2.

Jain

,

R. E.

, and

Ruckenstein

,

E.

, 1976, “

Stability of Stagnant Viscous Films on a Solid Surface

,”

J. Colloid Interface Sci.

0021-9797,

54

(

1

), pp.

108

–

116

.

Crossref

3.

Nigmatulin

,

R. I.

, 1991,

Dynamics of Multiphase Media

,

Hemisphere

, Vol.

2

.

by American Society of Mechanical Engineers

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