Yearbook 2013/2014 - ehedg

Flow behaviour of liquid jets impinging on vertical walls 67
by the falling film. Soil in this region will be removed by the
action of detergent and lower shear stresses, as reported
by Morison and Thorpe (2002). 1 Under some conditions
the falling film will narrow below the impingent plane and
give poor contact with the soil, as shown in Figure 2 (b). It is
therefore important to be able to predict the transition from
the wide, gravity, film flow behaviour to the narrow, rivulet
regime.
Predicting the film jump
A model for the film jump has recently been developed. 2 This
allows R to be predicted from
¼
⎡ m 3 ⎤
R = 0.276 ⎢ ⎯⎯⎯⎯⎯⎯ ⎥
⎣ μργ(1− cos b) ⎦
(1)
In this equation, m is the jet mass flow rate; μ is the viscosity
of the liquid and ρ is its density; γ is the surface tension, and
β is the contact angle of the liquid on the substrate.
Figure 3 shows good agreement between experimental
data and the model for water on Perspex. Nozzle sizes, d N
,
typical of those used in industrial practice have been tested.
Comparison with other data sets, including those reported
in Morison and Thorpe (2002), are reported in Wilson et al.
(2011) and Wang et al. (2013). 1–3
surfactant molecules had time to collect at the solid/liquid/air
interface, gave poor agreement with the measured values.
This indicates that dynamic surface tension effects are
important in these flows.
A second important finding reported in Wang et al. (2013)
is that at higher flow rates and with larger nozzles, R was
independent of the nature of the substrate. This indicates
a change in the phenomena controlling the flow pattern at
higher flow rates from one controlled mainly by interfacial
forces to one where fluid inertia become important. Using
a contact angle of 90° in Equation (1) gave reasonable
predictions for R in these cases.
Predicting the film width
The relationship between W and R cannot be obtained using
the simple models behind Equation (1). Measurements of
W (= 2Rc in Figure 3) indicate that Rc ≈ 2R at lower flow
rates and approaches Rc ≈ 4/3R at higher flow rates. These
empirical results allow W to be estimated.
Falling film flow patterns
The tendency to exhibit gravity or rivulet flow in the region
below the impingement plane has been found to follow the
criterion given by Hartley and Murgatroyd (1964) for the
stability of wide falling liquid films. 4 This says that film will be
stable if the wetting rate, defined as m/W, is larger than the
critical value given by
m
⁄ W
≥ 1.69 (μρ/g) 0.2 [γ(1− cos b)] 0.6 (2)
Here g is the acceleration due to gravity. Equation (2) holds
for vertical surfaces; for inclined walls, g is modified to
account for the angle of slope.
We have found that Equation (2) gives reasonable predictions
of the transition from the gravity to rivulet regimes for these
falling films. Equation (2) has also been found to apply for
solutions containing a surfactant, using the values of γ and
b obtained from equilibrium contact angle measurements.
Surfactants which promote wetting on the soil or substrate
will give smaller values of b and therefore, from Equation
(2), reduce the flow rate required to avoid rivulet formation
(as W is less sensitive to surfactant content).
Figure 3. Comparison of experimental measurements of R with
values predicted from Equation (1) for water on Perspex for
different nozzle sizes and temperatures. Reproduced from Wang et
al. (2013) with permission.
Equation (1) shows that R is larger for liquids with a small
contact angle (i.e., ones that wet the surface) and for liquids
with a lower surface tension. Surfactants are often added
to promote wetting and change the contact angle. Recent
work has demonstrated that the effect of surfactants on
R comes mainly through their influence on the surface
tension. 3 Predictions of R using Equation (1) using contact
angles measured under equilibrium conditions, where the
Figure 4 shows an example for water jets impinging on a
vertical glass wall. Solid symbols indicate that the falling film
exhibited gravity flow, while open symbols denote rivulet flow
behaviour.
The data are plotted in terms of the Eötvös number, a
dimensionless width, and a dimensionless flow rate, F, given
by
and
Eo = ρgW 2 ⁄ γ (3)
F = ρgm 2 ⁄ γμ2 (4)