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Start-up transients in a pneumatic foam.

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ASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING
Asia-Pac. J. Chem. Eng. 2011; 6: 613–623
Published online 8 June 2010 in Wiley Online Library
(wileyonlinelibrary.com) DOI:10.1002/apj.456
Research article
Start-up transients in a pneumatic foam
Ryan Shaw,1 Geoffrey M. Evans1 and Paul Stevenson2 *
1
2
Centre for Advanced Particle Processing, University of Newcastle, Callaghan, NSW 2308, Australia
Department of Chemical and Materials Engineering, University of Auckland, 20 Symonds Street, Auckland 1010, New Zealand
Received 15 November 2009; Revised 28 March 2010; Accepted 28 March 2010
ABSTRACT: In the current work, transient features of initiation in a gas-liquid pneumatic foam are investigated by
measuring the evolution of volumetric liquid fraction as a function of height within the column. The addition of wash
water to a flotation froth is only effective when the foam liquid fraction has reached a steady state. This makes start-up
transients in pneumatic foam worthy of study. For the conditions adopted in the experiments, an approximately steady
state was achieved after typically 500s, but there was significant fluctuation in liquid fraction after this time. In general,
three possible regimes in the start-up transient (induction, growth and evolution) have been identified and a tentative
mathematical model has been described for the last two. However, because it has been demonstrated that the method
of obtaining bubble size distributions by analysing images taken through the column wall is deficient, no comparison
of these models with the data has been attempted.  2010 Curtin University of Technology and John Wiley & Sons,
Ltd.
KEYWORDS: foam; aqueous solutions; columns; bubbles; gas–liquid systems; flotation
INTRODUCTION
A pneumatic froth is created by sparging gas bubbles
into a pool of liquid containing a surfactant such that
it rises continuously up a column and overflows from
the top. Such froths are encountered in the processes
of froth flotation and foam fractionation. Liquid is
routinely added to the surface of a foam as either
1. washwater to the surface of a flotation froth in
order to aid rejection of gangue material from the
concentrate stream, or
2. a source of external reflux liquor to engender multiple equilibrium stages in foam fractionation.
Stevenson et al .[1] in 2009 have shown experimentally (by employing magnetic resonance imaging) and
theoretically (via a stability analysis) that liquid added
to an immature foam has a greater tendency to travel
upwards in the column, while liquid added to a mature
foam has a greater tendency to travel downwards in the
column. This has important implications for the operation of flotation and foam fractionation devices, for
operations since if washwater is added to an immature froth the additional liquid reports directly to the
*Correspondence to: Paul Stevenson, Department of Chemical and
Materials Engineering, University of Auckland, 20 Symonds Street,
Auckland 1010, New Zealand.
E-mail: paul.stevenson@newcastle.edu.au
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
Curtin University is a trademark of Curtin University of Technology
concentrate stream and does not aid rejection of the
gangue material.
Stevenson[2] also presented a method estimating the
steady-state liquid fraction and liquid fraction in a
pneumatic foam, given the bubble size distribution is
known as a function of height in the column; the
hydrodynamic state of the pneumatic foam is crucially
dependent upon this bubble size distribution and how
this bubble size distribution varies as a function of
position within the column.
However, because the steady-state condition of pneumatic foam is so strongly dependent on bubble size distribution, Stevenson et al .[1] showed that it was almost
impossible to achieve reproducibility and postulated
that this was due to the variation in bubble size distribution caused by unknown factors.
The behaviour of the foam during the start-up transient period is even more complicated. Barbian et al .[3]
investigated the start-up transient of columns of mineralised foam by measuring the evolution of foam depth,
and suggested that such information could illuminate
flotation performance. Neethling et al .[4] also measured
the depth of growing foam and suggested a model to
describe the process. As discussed in more detail below,
it is possible that their model may be compromised by
the adoption of a mechanism of surface bursting that
does not exhibit dependency on the ambient/freeboard
humidity gradient.
In this study, we take different approaches to both
measuring and modelling the initial transient effects
614
R. SHAW, G. M. EVANS AND P. STEVENSON
in a foam column. Rather than simply measuring the
height of the free surface of the foam as it rises up
the column, we measure how the foam evolves by
measuring the volumetric liquid fraction as a function
of position and time in the column; this is achieved
by measuring differential pressures at various positions
up the column using membrane-piezo crystal pressure
sensors, and checking that the wall shear stress effect
is negligible in the experiment. This method has the
advantage that it reveals information about the foam
transient during the three regimes of its behaviour:
1. The regime in which, although gas is being sparged
to the column, a tenacious layer of foam is only
established after a finite induction time.
2. The regime where the free surface rises in the
column before it reaches the column lip.
3. The regime where the foam is overflowing the top of
the column, but the liquid fraction within the column
is still approaching steady state.
It is important to investigate this third regime of the
transient, since it is only when the overflowing foam
has achieved steady state that the addition of washwater
travels downwards in the column. In addition, bubble
size distributions as a function of height in the column
are reported. A model for the start-up transient in the
foam column is presented, although the model depends
upon a variable that is inherently unpredictable and, at
present, unmeasurable – the bubble size distribution.
It should be noted at this juncture, that we do not
suggest that observations of a two-phase demineralised
froth stabilised by a relatively high surfactant concentration will directly replicate the behaviour of a flotation
froth that is largely stabilised by attached particles. Following the spirit of many previous investigators, we
choose to conduct experiments on a demineralised froth,
because the system is simple and qualitatively illuminates the behaviour of flotation froths. In addition, this
work is directly relevant to the start-up of systems that
do employ demineralised froth, such as foam fractionation, and biological broth aeration processes.
EXPERIMENTAL
A schematic representation of the experimental apparatus is shown in Fig. 1. Pneumatic foams were generated
by sparging air from a main via a rotameter through a
sintered glass frit. The overflowing foam was collected
in a launder vessel and returned to a liquid reservoir
to ensure that no change in total solution concentration
occurred due to foam fractionation; liquid was supplied
to the base of the column from the reservoir via a peristaltic pump. The column was constructed from Perspex
and had a rectangular internal cross section of dimensions 70 × 80 mm and the height of the column above
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
Asia-Pacific Journal of Chemical Engineering
the interface was 1150 mm. The interface between the
pulp (i.e. the bubbly liquid) and foam was maintained
at a fixed position by introducing a tube returning to the
reservoir at this position, and maintenance of a larger
liquid influx than gas influx guaranteed the fixed position of this interface.
Membrane-piezo crystal pressure sensors were
located at positions 5, 10, 15, 20, 30, 50, 60, 70, 80 and
90 cm, above the pulp/foam interface. The piezo gauge
pressure sensors CTEM70025GY4 were sourced from
Sensortechnics GmbH and had a 0- to 25-mbar range
with a 4- to 20-mA output. The sensors were powered
by a 20-V variable power supply unit and connected to
the data-logger via an external shunt to enable the unit to
resolve the varying output current reference to precision
100- resistors. The data-logger unit was a Datataker
DT80. The pressure sensors were calibrated to provide a
response to the mass of water causing the current output.
Baseline atmospheric (empty) and full column (100%
liquid fraction) values were obtained. Liquid fractions,
ε, as a function of height in the column were inferred by
measuring the differential pressure between two adjacent sensors; P located at a distance of x from each
other, given the density of the solution ρ and calculated
using the equation:
ε=
1 P
ρg x
(1)
Thus, the average liquid fraction at a point midway
between the participating sensors is calculated and
liquid fraction is reported at a position intermediate
between those sensors. Equation (1) is valid only if the
shear stress imparted on the foam by the walls of the
column is negligible. Proof that this indeed was the case
is given in Appendix A.
Foam tended to accumulate at the top of the column;
this was particularly problematic with lower gas rates
because lower gas rates create foams of lower liquid
fraction and therefore higher viscosity. This accumulation was exacerbated by the further drainage of the
accumulated foam, further increasing the viscosity of
the foam in the launder. The accumulation was ameliorated by spraying the top of the foam with liquid
from the reservoir at such an angle so as to fluidise the
laundering foam and cause flow into the system reservoir, thus returning the concentrated surfactant foam
and replenishing the depleted reservoir concentration.
The spray was directed at such an angle to ensure that
the wash did not drain through the column height.
Bubble size distributions were measured by taking
digital photographs of a 2 × 2 cm window located
at positions of 10, 30, 50 and 70 cm above the
pulp/foam interface. Photographs were taken using a
Canon EOS450D DSLR camera using an 18- to 35-mm
lens. Because of a lack of image contrast, each bubble
(approximately 200 per image) was sized manually
Asia-Pac. J. Chem. Eng. 2011; 6: 613–623
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
START-UP TRANSIENTS IN A PNEUMATIC FOAM
Figure 1. Schematic representation of the experimental apparatus.
using the OPTIMAS software package, calibrated using
a metric rule in each image.
The laboratory temperature was in the range 21–
25 ◦ C and the relative humidity was 65–80%.
The evolution of six liquid fraction profiles are
reported herein: The foam was sparged at gas superficial velocities of 15, 24 and 30 mm/s for foams
stabilised by 1.01 × 10−3 M and 6.13 × 10−3 M of
98% purity sodium dodecyl sulphate (SDS). The critical micelle concentration of SDS was approximately
8.33 × 10−3 M.[5]
RESULTS AND DISCUSSION
Data for the evolution of liquid fraction of foam stabilised by SDS at a concentration of 6.13 × 10−3 M as
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
a function of height for superficial gas velocities of 15,
24 and 30 mm/s are shown in Figs 2–5, respectively.
Figure 2 shows that the liquid fractions up the column
appear to have attained a steady state at approximately
300–400 s, although there is significant variation of
liquid fraction around a mean value. It is possible that
the pulsating nature of the peristaltic pumps may have
contributed to this behaviour. It is also possible that this
noise is due to the direct contact between the discrete
structure of the foam and the membrane of the pressure
sensor, as the aperture of the pressure sensor is not
significantly larger than the majority of the bubbles. The
liquid fraction nominally at 10 cm above the bubbly
liquid foam interface is seen to increase almost as soon
as the gas supply is commenced, reaching equilibrium
at around 80 s. The liquid fraction at higher locations
starts to increase from zero at progressively later times
Asia-Pac. J. Chem. Eng. 2011; 6: 613–623
DOI: 10.1002/apj
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R. SHAW, G. M. EVANS AND P. STEVENSON
Asia-Pacific Journal of Chemical Engineering
Figure 5. Cumulative probability distribution for jg =
15 mm/s at a concentration of 6.13 × 10−3 M.
Figure 2. Evolution of liquid fraction in the column with jg =
15 mm/s at a concentration of 6.13 × 10−3 M. This figure is
available in colour online at www.apjChemEng.com.
Figure 3. Evolution of liquid fraction in the column with jg =
24 mm/s at a concentration of 6.13 × 10−3 M. This figure is
available in colour online at www.apjChemEng.com.
Figure 4. Evolution of liquid fraction in the column with jg =
30 mm/s at a concentration of 6.13 × 10−3 M. This figure is
available in colour online at www.apjChemEng.com.
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
after the gas is switched on, corresponding to the
time at which the top of the foam layer reaches that
particular height. The liquid fraction at the highest
point probed (i.e. 75 cm above the interface) reaches
approximately its equilibrium value last, and is dryer
than other locations. Generally, the higher is the foam
in the column, the dryer it becomes. This is caused by
two physical effects that are described in the steadystate theory of rising foam,[2] which may be applied to
the current experiments if it is assumed that the systems
are at a pseudo steady state:
1. Capillary effects mean that the foam local to the
bubbly liquid interface is wetter, but, in the absence
of changes in bubble size distribution due to coalescence and Ostwald ripening, the liquid fraction
approaches asymptotic value.
2. If the bubble sizes increase as a function of height,
then the foam gets dryer.
Unexpectedly, it is seen that the liquid fraction does
not decrease monotonically with height. This is possibly
due to the fact that, as coalescence occurs either on
the surface of the foam or on the free layer, liquid is
liberated from the bubbles as they burst, as explained
above, and it either propagates downwards, if steady
state is attained, or upwards in an immature foam.
It is possible that such effects are being observed
within this foam, and have not found resolution before
500 s.
The 6.13 × 10−3 M solution was retained, but gas
rate increased to 24 mm/s for further experiments,
the results of which are shown in Fig. 3. The results
were unexpected. A small liquid fraction was registered throughout the column almost immediately on
commencement of sparging, but significant amounts
of liquid did not manifest anywhere in the column
until approximately 100 s. This observation is not what
would have been expected after inspection of the results
gained at the lower flow rate in Fig. 2. However, foam
was again first registered at the lowest measurement
position, and last measured at the highest position. A
Asia-Pac. J. Chem. Eng. 2011; 6: 613–623
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
START-UP TRANSIENTS IN A PNEUMATIC FOAM
similar result was observed when the gas rate was
increased further to 30 mm/s (Fig. 4), although the
‘induction’ time for significant foam to appear was
lower at approximately 50 s.
All other things being equal, it is expected that
the equilibrium liquid fraction should increase approximately as the square of the gas rate.[2] However,
by inspection of the equilibrium liquid fractions in
Figs 2–4, this approximate relationship clearly does not
hold.
Inspection of the bubble size distributions for the
three gas rates at an SDS concentration of 6.13 ×
10−3 M (Figs 2–4), and the mean values extracted
from the bubble size distributions as a function of
height (Tables 1–3) shows that all other things are certainly not equal. The bubble size distributions obtained
via analysis of images taken through the column
wall show significant variation from one gas rate to
another. In addition, because Ostwald ripening is largely
insignificant in pneumatic foam, the mean bubble size
must either stay constant as a function of height or
increase monotonically due to bubble coalescence and
gas expansion. No mechanism exists for the mean
bubble size to decease with respect to an increasing
height in the foam. However, our data for mean bubble size as a function of height does in some cases,
decrease. From this observation, we conclude that our
method for measuring bubble size distribution is inadequate and the images taken through the column wall
do not yield distributions representative of the bulk
of the foam. Such concerns are not new, Cheng and
Lemlich[6] have appraised the various mechanisms by
which error can be introduced. One of these mechanisms is that small bubbles can ‘wedge’ big bubbles
away from the wall, thereby decreasing the mean bubble size observed, and it is possible that this is the
mechanism at play in our experiments. Moreover, It
has not yet been established which type of mean bubble size is representative of foam drainage (and therefore the behaviour of pneumatic foam), so the attempt
to use data for wall bubble size distributions as a
model in pneumatic foam behaviour is futile (Figs 6
and 7).
A set of experiments identical to those described
above, except for those that showed a decrease in concentration of SDS to 1.01 × 10−3 M, were conducted.
Table 1. Mean bubble diameters as a function of height
for jg = 15 mm/s at a concentration of 6.13 × 10−3 M.
Arithmetic
mean (mm)
Harmonic mean (mm)
RMS (mm)
10 cm
30 cm
50 cm
70 cm
0.49
0.36
0.73
0.48
0.28
1.08
0.51
0.38
0.81
0.61
0.32
1.58
RMS, root mean square.
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
Table 2. Mean bubble diameters as a function of height
for jg = 24 mm/s at a concentration of 6.13 × 10−3 M.
Arithmetic mean (mm)
10 cm
30 cm
50 cm
70 cm
Harmonic mean (mm)
RMS (mm)
0.87
0.40
1.43
0.57
0.30
1.00
0.54
0.39
1.00
1.33
1.01
1.60
RMS, root mean square.
Table 3. Mean bubble diameters as a function of height
for jg = 30 mm/s at a concentration of 6.13 × 10−3 M.
Arithmetic mean (mm)
10 cm
30 cm
50 cm
70 cm
Harmonic mean (mm)
RMS (mm)
1.35
0.59
1.95
0.70
0.33
1.31
0.76
0.42
0.34
0.63
0.49
0.81
RMS, root mean square.
Figure 6. Cumulative probability distribution for jg =
24 mm/s at a concentration of 6.13 × 10−3 M.
The liquid fraction plots for the three gas rates are
given in Figs 8–10, and the cumulative bubble size
distributions are shown in Figs 11–13, with the three
types of mean bubble size shown as a function of
height in Tables 4–6. There is a similar problem with
the mean bubble sizes not monotonically decreasing
as a function of height as seen in the 6.13 × 10−3 M
data.
The same general features are seen in the 1.01 ×
10−3 M data as were observed in the 6.13 × 0−3 M
data. An induction period is observed at the highest gas rate, and the foam towards the bottom of the
column evolves more rapidly than the foam at the
top. However, there is generally a greater difference
between the liquid fractions at the top and at the bottom. According to the theory of pneumatic foam,[2]
this is consistent with bubble coalescence within the
bulk of the foam as they rise up the column, although
such coalescence is not apparent from the measurements of bubble size distribution as a function of
Asia-Pac. J. Chem. Eng. 2011; 6: 613–623
DOI: 10.1002/apj
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R. SHAW, G. M. EVANS AND P. STEVENSON
Asia-Pacific Journal of Chemical Engineering
Figure 7. Cumulative probability distribution for jg =
30 mm/s at a concentration of 6.13 × 10−3 M.
Figure 10.
Figure 8. Evolution of liquid fraction in the column with jg =
Figure 11. Cumulative probability distribution for jg =
15 mm/s at a concentration of 1.01 × 10−3 M.
−3
15 mm/s at a concentration of 1.01 × 10 M. This figure is
available in colour online at www.apjChemEng.com.
Evolution of liquid fraction in the column with jg = 30 mm/s at a concentration of 1.01 ×
10−3 M. This figure is available in colour online at
www.apjChemEng.com.
Figure 12. Cumulative probability distribution for jg =
24 mm/s at a concentration of 6.13 × 10−3 M.
Figure 9. Evolution of liquid fraction in the column with jg =
24 mm/s at a concentration of 1.01 × 10−3 M. This figure
is available in colour online at www.apjChemEng.com.
height; confirmation that the method for bubble size
measurement is indeed flawed. Certainly, one would
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
expect bubble coalescence within both the bulk of the
foam and at the free surface in addition to bursting at
the free surface due to a reduction in surfactant concentration and the associated reduction in foam film
stability.
Asia-Pac. J. Chem. Eng. 2011; 6: 613–623
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
START-UP TRANSIENTS IN A PNEUMATIC FOAM
foam behaviour is contingent upon having reliable
measurements of the bubble size distribution within
the bulk of the foam, and we have demonstrated that
there are currently no reliable methods of obtaining
such measurements. As a consequence, it can be merely
hoped that these hypotheses will be tested in the future
when and if experimental methods for measuring the
bubble size distribution in the bulk of pneumatic foam
have been developed.
Regime 1: induction
Figure 13. Cumulative probability distribution for jg =
30 mm/s at a concentration of 1.01 × 10−3 M.
Table 4. Mean bubble diameters as a function of height
for jg = 15 mm/s at a concentration of 6.13 × 10−3 M.
Arithmetic mean (mm)
10 cm
30 cm
50 cm
70 cm
Harmonic mean (mm)
RMS (mm)
2.29
0.48
4.76
0.66
0.24
1.60
0.70
0.26
2.11
0.53
0.25
1.70
RMS, root mean square.
Table 5. Mean bubble diameters as a function of height
for jg = 24 mm/s at a concentration of 6.13 × 10−3 M.
Arithmetic mean (mm)
10 cm
30 cm
50 cm
70 cm
Harmonic mean (mm)
RMS (mm)
0.65
0.38
1.43
0.57
0.34
1.17
0.33
0.28
0.36
0.72
0.36
1.78
RMS, root mean square.
Table 6. Mean bubble diameter as a function of height
for jg = 30 mm/s at a concentration of 1.01 × 10−3 M.
Arithmetic mean (mm)
10 cm
30 cm
50 cm
70 cm
Harmonic mean (mm)
RMS (mm)
1.18
0.35
2.27
0.88
0.23
1.58
1.79
1.10
1.98
2.00
0.37
3.30
RMS, root mean square.
TENTATIVE MATHEMATICAL DESCRIPTION
OF THE TRANSIENT
The three regimes of the start-up transient effects in
a column of pneumatic foam were outlined in the
introduction. In this section, we make a preliminary
attempt to mathematically describe these phases of the
transient effect. However, we immediately note that
successful implementation of any theories that describe
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
The first regime of the transient, the induction period,
is the regime we know the least about. From the
experimental data, there does not appear to be any
systematic dependency on a system parameter, and is
therefore likely to be stochastic in character.
Pilon and Viskanta[7] suggested an empirical correlation for estimating the system properties required for
the formation of a tenacious foam; this is unlikely to
be of value in aqueous systems considered herein. Li
et al .[8] have shown that the equilibrium condition of
non-overflowing pneumatic foam is dependent upon the
humidity gradient in the column freeboard; it is probable that this freeboard humidity will be a critical factor
in determining when the foam starts to grow. Factors
other than the properties of the foaming liquid itself,
bubble size and gas rate, probably include the amount
of particulate species in the air and the background
mechanical vibrations that influence the stability of the
foam. However, because the length of the induction
period appears to be governed by a probabilistic (rather
than mechanistic) process, the formulation of a mathematical description is, at present, out of reach.
Neither Pilon and Viskanta[7] or Pilon et al .[10] in
their observation of the growth phase of foam, observed
an induction time.
Regime 2: growing column of foam
Let the average liquid fraction in a column of pneumatic
foam be given by:
1
ε=
H
H
ε(x )dx
(2)
0
where H is the distance between the bubbly liquid
interface and the top surface of the foam. The liquid
flux entering the foam, jf , is entirely accumulated in the
foam layer, so we write:
∂ε
∂H
∂(H ε)
=ε
+H
= jf
∂t
∂t
∂t
(3)
Asia-Pac. J. Chem. Eng. 2011; 6: 613–623
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R. SHAW, G. M. EVANS AND P. STEVENSON
Asia-Pacific Journal of Chemical Engineering
We assume that some bubbles burst at the free surface
of the foam but that there is no internal coalescence,
thereby replicating the assumption of Neethling et al .[4] .
Now, a fraction φ of the gas sparged to the column
is liberated to the atmosphere due to surface bursting.
Thus, by considering a dynamic balance on the gas
phase, we write:
∂ε
∂H
∂[H (1 − ε)]
= (1 − ε)
−H
= jg (1 − φ) (4)
∂t
∂t
∂t
thus by adding Eqns (3) and (4) we find that:
∂H
= jf + (1 − φ)jg
∂t
(5)
Note that this is entirely independent of the liquid
profile or the time derivative thereof. Equation (5)
could be written by inspection on realising that the
sum of the liquid entering the foam and the gas that
remains unliberated at the surface must contribute an
increasingly deep foam layer. In the analysis of Barbian
et al .,[3] a fraction β is defined that is the proportion of
air remaining in the froth (i.e. it is complimentary to
φ). For precisely the same physical system considered
herein, they write [their Eqn (4)] that:
∂H
= βjg = (1 − φ)jg
∂t
(6)
which omits the contribution of the liquid flux to the
dynamic mass balance, which is approximately valid
only for dry, yet stable, foams.
Now, we make the simplifying assumption that the
column is at pseudo steady state (i.e. any effects of
bubble bursting to the hydrodynamic condition of the
foam propagate instantaneously) so that the same liquid
flux is seen everywhere in the column. Thus, the liquid
flux, jf , by application of Stevenson’s[2] hydrodynamic
theory of rising foam in which the liquid fraction at the
top of the column, εt , is calculated by first numerically
solving:
µjg
= εt n−1 (1 − εt )2
(7)
mnρgrb 2
then computing, one obtains the liquid flux.
jf =
εt jg
ρgrb 2
mεt n
−
1 − εt
µ
(8)
The development of a method of estimating φ
remains incomplete. There are two approaches for making this estimation:
1. A mechanistic model for the bursting of foams at the
free surface of the foam.
Such an approach was taken by Neethling et al .[4]
in their model of start-up transients. They proposed
a deterministic method of predicting when bubbles
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
would coalesce by assuming that films fail when enough
force is applied to a surface to overcome the disjoining pressure. Predictions of the evolution of foam
height (their Fig. 9) were in excellent agreement with
their experimental data. However, we think that this
agreement was possibly serendipitous because[1] our
group has experimentally shown that the evolution
of foam height (and therefore surface bursting) is
strongly governed by the humidity gradient in the
column[8] and[2] the investigations of steady-state pneumatic foam of Ireland[9] have shown that coalescence is, in fact, is a probabilistic effect, which has
attributed the observed coalescence event to mechanical perturbations (i.e. a mechanism totally unlike that
assumed by Neethling et al .). Because of these confounding factors, we do not think it is currently possible
to propose a reliable mechanistic model for surface
coalescence.
2. Inference of ø from observations of bubble diameter
Imagine a stylised foam consisting of a monodisperse
bubbles of radius rb . The bubbles coalesce at the very
top surface so that they have a rbt . The number of
bubbles in a plane within the foam, perpendicular to
the direction of travel, is described by r b 2 , and the
number of bubbles located at the surface is described by
r bt 2 . Thus, the fraction that burst at the surface (thereby
liberating the gas within to the atmosphere) is:
φ =1−
rb
rbt
2
(9)
This approach can be used in an a posteriori fashion
to calculate the evolution of foam layer depth only if
the bubble distribution is monodisperse and measurable.
Given the uncertainty in measuring the bubble size,
however, the application of all expressions proposed
herein is of questionable utility.
Measurements of the evolution of the height of a
column of foam, such as the experiments of Neethling
et al .[4] can, through application of Eqn (9), lead to
inference of φ and it is possible that this may lead to
the development of reliable correlation for the a priori
estimation of φ in the future.
We have measured the liquid fraction as a function
of height in the column, so it is appropriate to explore
the implications for liquid profile due to this theoretical
framework.
Because we make the assumption of the pseudo
steady state, we may adapt the steady-state model[2] to
give a differential equation for the liquid fraction:
pρgrbt ε1+q
∂ε
=
∂x
qσ
for
0≤x ≤H
εjg
µ
− jf
−1
1−ε
ρgrbt 2 mεn
(10)
Asia-Pac. J. Chem. Eng. 2011; 6: 613–623
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
START-UP TRANSIENTS IN A PNEUMATIC FOAM
where σ is the equilibrium surface tension, and the
boundary condition:
ε|x =0 =
n −1
n +1
(11)
Equation (10) is developed from a steady-state mass
balance including a capillarity term in the drainage
equation, and the boundary condition gives the maximum liquid fraction that a pneumatic foam can attain;
see Ref. [2] for a derivation. p and q are dimensionless
constants such that:
r
= pεq
(12)
rb
where r is the radius of curvature of the Plateau border
walls. There have been several sets of values for p and q
proposed in the literature (see Ref. [2] for a survey) but
p = 1.28 and q = 0.46 are tentatively preferred since
these values are said to be valid over a wide range of
liquid fractions.
The application of Eqn (10) is likely to be problematic. As the top surface of the foam proceeds up the
column the humidity increases, leading to a decrease in
the humidity gradient and this changes the instantaneous
value of φ and therefore rbt . Once the foam reaches the
top of the column and begins to overflow, it is likely
that the bubble size at the top surface will take a finite
time to reach steady state. If internal coalescence occurs,
then the steady-state liquid fraction profile can be estimated by invoking Stevenson’s hydrodymanic theory
of rising foam[2] only if the bubble size is known. An
attempt at modelling the transient behaviour of such
columns may be considered as foolhardy. The unfortunate thing about pneumatic foam is that almost every
aspect of their physical behaviour is strongly dependent
upon their bubble size distribution and we do not yet
have a reliable method to measure this within the bulk
of the foam.
Pilon et al .[10] attempted a mathematical description
of this regime and assumed, like our present analysis,
that there was no change in bubble size due to coalescence or Ostwald ripening in the bulk of the foam.
However, this method has similarity with the later analysis of Barbian et al .[3] in that it only considers gas
flux as a contributor to foam height. Pilon observed the
growth of foams that were not destined to reach the
top of the column and overflow (i.e. they never reached
regime 3), so their experimental observations are not
directly relevant to the current study.
Regime 3: evolution of overflowing foam
When the foam reaches the lip of the column and
begins to overflow, for the assumption that the foam
is in a pseudo steady state (i.e. is isotropic) to hold,
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
there must be neither internal coalescence nor any
Ostwald ripening. As seen in the experimental results
reported, the liquid fraction (especially at the top of the
column) continues to develop over several hundreds of
seconds. This is due to the fact that rates of internal and
surface coalescence change with respect to time, and
therefore foam that is spatially and temporally variant
is present in the column as it starts overflowing the lip.
The physics that describes this process is complex and
dynamic. Once an experimental method of measuring
bubble size distributions within the bulk of foam is
developed, then empirical descriptions of this process
may be possible. However, until then, all that can be
given is a brief qualitative description based upon the
dynamic mass balance of Stevenson et al .[1] :
∂jf ∂ε
∂ε
=−
∂t
∂ε ∂x
(13)
The mass balance reveals that the liquid fraction
at any point in the column continues developing as
long as the spatial gradient of the liquid fraction is
non-zero and the liquid flux changes with the liquid
fraction (i.e. pseudo steady state is not maintained).
However, near the interface between bubbly liquid
and foam, capillarity ensures that the liquid profile
is never spatially invariant. Thus, we find that the
condition for a steady overflowing pneumatic foam is
that the pseudo steady state is achieved throughout
the column (i.e. liquid flux is constant throughout the
column).
The futility of applying a theoretical and predictive
model and testing its validity without a precise and
reliable method of measuring an essential variable – in
this case, the bubble size distribution in the bulk of the
foam column, has been outlined above.
CONCLUSIONS
1. Measurements of the evolution of liquid fraction
profile during the initiation transient phase of a
pneumatic foam have been made by measuring the
pressure in the column as a function of height. It has
been confirmed that this method is valid and that the
contribution of shear stress imparted on the foam by
the pipe wall to the pressure gradient is insignificant
for the experiments reported herein.
2. Three regimes of the transient have been identified.
The first is an apparently stochastic induction period
before a tenacious foam is established. Subsequently,
the foam grows up the column. When it reaches the
lip of the column and starts to overflow, the liquid
fraction in the column continues evolving before an
approximately equilibrium condition is attained.
3. The difference in the liquid fraction between the
top and bottom of the column is greater for
Asia-Pac. J. Chem. Eng. 2011; 6: 613–623
DOI: 10.1002/apj
621
622
R. SHAW, G. M. EVANS AND P. STEVENSON
lower-concentration foaming solutions, and it is proposed that this is due to greater bubble coalescence
than is seen in foams created from solutions with
higher surfactant concentration.
4. By measuring bubble size distributions from images
taken through the column wall, it is apparent that
this method is deficient because it does not yield
bubble sizes that monotonically increase with height.
This means that attempts to model our experimental
observations are futile, coupled with the fact that we
do not know which type of mean correctly represents
foam drainage behaviour.
5. Despite the problems with measuring bubble size
distributions, a tentative model of the second
(growth) regime of the start-up transient is proposed.
However, the first (induction) regime is apparently
stochastic in nature, and a mathematical description
is out of reach. The third (evolution) regime could be
numerically described by the application of Eqn (13)
and a mass balance.
6. What is apparent from this work is that, if models for
the initiation transient of pneumatic foam are ever to
be verified, it is paramount that a reliable method of
measuring bubble size distribution within the bulk
of a pneumatic foam is established.
Acknowledgement
This work was funded under the Discovery Projects
scheme of the Australian Research Council (Project No.
DP0878979).
NOMENCLATURE
g acceleration due to gravity (m/s2 )
H height of the foam layer (m)
jd superficial liquid drainage velocity relative to the
bubbles (m/s)
jf superficial liquid velocity up the column (m/s)
jg superficial gas velocity (m/s)
m number used in Eqn (6) (dimensionless)
n index used in Eqn (6) (dimensionless)
p constant used in Eqn (10) (dimensionless)
Pdifferential pressure between adjacent sensors (Pa)
q index used in Eqn (10) (dimensionless)
r radius of curvature of Plateau border walls (m)
rb characteristic mean bubble radius (m)
rbt characteristic mean bubble radius on the foam
surface (m)
t time (s)
x distance in the column measure positive upwards
(m)
β fraction of gas retained in the foam
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
Asia-Pacific Journal of Chemical Engineering
ε
εt
µ
ρ
σ
φ
M
volumetric liquid fraction in the foam
liquid fraction at the top of the foam
interstitial liquid dynamic viscosity (Pa s)
interstitial liquid density (kg/m3 )
equilibrium surface tension (N/m)
fraction of gas liberated at the foam surface
concentration – molarity (mol/L, kmol/m3 )
APPENDIX A: EFFECT OF WALL SHEAR
STRESS ON LIQUID FRACTION
MEASUREMENTS
Measurements of pressure gradient have been used to
directly infer the liquid fraction of the foam via Eqn (1).
However, there are, in fact, two sources of pressure
gradient within the column in our experiments:
1. The weight of the liquid within the foam.
2. The shear stress imparted by the column walls onto
the foam.
Indeed, Deshpande and Barigou[11] appear to have
discounted the weight of the liquid in the foam altogether and have used measurements of pressure gradient
to infer the friction factor for foam flow through a vertical pipe. Clearly, their approach is in conflict with
ours since we discount the shear stress contribution to
the pressure gradient, and instead use measurements to
infer the liquid fraction.
In order to assess if either of these approaches is
correct, we have performed an experiment to infer the
liquid fraction immediately after the gas is suddenly
switched off so that the foam instantaneously stops
rising in the column. Figure A1 shows this experiment
for a gas rate of 15 mm/s and an SDS concentration
of 6.13 × 10−3 M. Several on/off cycles were observed
but in the interests of clarity, only one is shown in
Fig. A1.
Liquid fraction inferred from application of
Eqn (1). The gas rate is suddenly switched on (solid black
line) and off (dashed black line). This figure is available in
colour online at www.apjChemEng.com.
Figure A1.
Asia-Pac. J. Chem. Eng. 2011; 6: 613–623
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
If shear stress was a significant contributor to pressure
gradient in our experiments, then one would expect
the inferred liquid fraction to show a step reduction
as the gas is switched off and the shear stress drops
to zero. However, the inferred liquid fraction does not
exhibit such a step change and, instead, gradually falls,
displaying a liquid fraction evolution characteristic of
free drainage of foam. Thus, we can assert that the shear
stress contribution to pressure gradient in our column
was insignificant. This is not to say, of course, that this
is generally true. Indeed, we would expect the shear
stress contribution to become significant for tubes of
relatively small cross-sectional area carrying relatively
dry foam.
START-UP TRANSIENTS IN A PNEUMATIC FOAM
[2] P. Stevenson. Miner. Eng., 2007; 20, 282–289.
[3] N. Barbian, K. Hadler, E. Ventura-Medina, J.J. Cilliers. Miner.
Eng., 2005; 18, 317–324.
[4] S.J. Neethling, H.T. Lee, P. Grassia. Colloids Surf. A, 2005;
263, 184–196.
[5] E. Harlow, D. Lane. Antibodies: A laboratory manual, Cold
Spring Laboratory Press: New York, 1999.
[6] H.C. Cheng, R. Lemlich. Ind. Eng. Chem Fund, 1983; 22,
105–109.
[7] L. Pilon,R. Viskanta. Chem. Eng. Proc., 2004; 43, 149–160.
[8] X. Li, R. Shaw, P. Stevenson. Int. J. Min. Proc., 2010; 94,
14–19.
[9] P.M. Ireland. Chem. Eng. Sci., 2009; 61, 4866–4874.
[10] L. Pilon, A.G. Federov, R. Viskanta. Chem. Eng. Sci., 2002;
57, 977–990.
[11] N.S. Deshpande, M. Barigou. Chem. Eng. Sci., 2000; 55,
4297–4309.
REFERENCES
[1] P. Stevenson, M.D. Mantle, A.B. Tayler, A.J. Sederman.
Chem. Eng. Sci., 2009; 64, 1001–1009.
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
Asia-Pac. J. Chem. Eng. 2011; 6: 613–623
DOI: 10.1002/apj
623
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