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 615 616 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 617 618 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 DOI: 10.1002/apj 619 620 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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