Flooding in a Vertically Rising GasâLiquid Foam - Industrial

Article pubs.acs.org/IECR

Flooding in a Vertically Rising Gas−Liquid Foam Xinting Wang,† Geoffrey M. Evans,*,† and Paul Stevenson‡ †

Centre for Advanced Particle Processing, University of Newcastle, Callaghan, NSW 2308, Australia State Key Laboratory for Heavy Oil Processing, China University of Petroleum, Qingdao 266555, Shandong, China

‡

ABSTRACT: The maximum liquid fraction that stable pneumatic foam can support is studied in this paper in order to illuminate change of flow regime from turbulent foam flow to the “emulsion regime” in which the distinct interface between the foam layer and the bubbly liquid is lost; this regime transition is nominated as “flooding”. With further increases in gas flow rate, large (or “gross”) bubbles can appear and rise in the column, the existence of which depends on the absolute pressure at the column bottom. Experiments are carried out using foam stabilized by sodium dodecyl sulfate, and the liquid fraction of both the foam layer and the bubbly liquid are measured using a pressure gradient method. The maximum liquid fraction is found to be at a critical point when the interface between the foam and the bubbly liquid layer disappears. The predicted maximum liquid fraction in the foam is given by ε* = (n − 1)/(n + 1), where n is an adjustable constant that depends upon the characteristics of the gas− liquid interface, and the predictions have been experimentally verified.

1. INTRODUCTION Columns of vertical pneumatic foam (i.e., those which continuously rise in a column because gas is sparged to a liquid pool beneath) are central to the processes of column flotation1 and foam fractionation.2 In addition, pneumatic foams are used to enhance gas absorption in gas−liquid contacting processes.3 For the latter application, it has been experimentally demonstrated that a foam of higher liquid fraction favors mass transfer, but if the gas rate is too high, a change in flow regime can cause a reduction in performance.4,5 This work addresses what the maximum liquid fraction of foam is before the flow regime changes and develops a theoretical model, albeit empirical with respect to liquid drainage rate, to predict the hydrodynamic conditions for the regime change. The liquid fraction of pneumatic foam for a given system with a particular bubble size distribution is, among other parameters, a function of the gas rate. In a column with constant cross-sectional area, the liquid fraction generally increases with gas flow rate. Three flow regimes had been identified by Hoffer and Rubin6 based on observations of the velocity profile of the foam bubbles. These are (1) plug flow, in which the bubbles rise in a foam column with uniform velocity so that the velocity profile is flat, (2) turbulent flow, in which bubbles move relative to each other in structures reminiscent of turbulent eddies, and (3) bubbly liquid flow, in which the foam no longer exists as a separate “phase” because the liquid fraction is uniform in the whole column, with occasional very large bubbles (diameters typically around 20 mm) rising quickly through the column. However, their work was limited to a qualitative description of experimental results and lacked any mechanistic explanation. In this paper, we focus upon the change between regimes 2 and 3 that was described by Hoffer and Robin.6 At relatively low gas flow rates, the foam layer, as shown schematically in Figure 1, has a lower liquid fraction than the liquid fraction in the bubbly liquid beneath it, and therefore, there exists a distinct interface between the foam and the bubbly liquid, as shown in the photograph of Figure 2. When © 2014 American Chemical Society

Figure 1. Schematic representation of a foam column showing the layer of pneumatic foam above the bubbly liquid.

Figure 2. Photograph of the interface between the pneumatic foam layer and the bubbly liquid in the “plug flow” regime of Hoffer and Rubin6

the gas flow rate is increased, the liquid fraction in the bubbly liquid decreases, while that of the foam simultaneously Received: Revised: Accepted: Published: 6150

January 17, 2013 February 3, 2014 March 10, 2014 March 10, 2014 dx.doi.org/10.1021/ie4001844 | Ind. Eng. Chem. Res. 2014, 53, 6150−6156

Industrial & Engineering Chemistry Research

Article

increases, until the interface becomes indistinct and finally disappears, as shown in Figure 3. We refer to this phenomenon

parameters that describe foam drainage9 such that the superficial liquid drainage velocity, jd, made dimensionless in the form of a Stokes number, Sk, may be empirically expressed as a power law function of the liquid fraction only μjd Sk = = mε n 2 ρgrb (2) It has been demonstrated9 that in the limiting case of channeldominated foam drainage10 that m = 0.0018 and n = 1.92. The first term on the right-hand side of eq 1 is the product of the liquid fraction and the bubble velocity, and the second term is the drainage superficial velocity, given by eq 2, in a Lagrangian reference moving with the bubbles. In Figure 5, assuming constant bubble size along the height of the column, eq 1 is used to plot the liquid superficial velocity

Figure 3. Photograph of the foam after the flooding condition has been surpassed, i.e., the distinct interface between bubbly liquid and foam layers has vanished.

as “flooding” in the current work, which is, however, different from how the term is used in other multiphase systems such as, for instance, liquid−liquid extraction (see Thornton7 for example) or in distillation columns. At gas flow rates above the flooding condition, there no longer exists a distinct foam phase, as it is commonly understood, and instead, the same condition exists throughout the height of the column (which is sometimes referred to as an “emulsion”). As the gas rate continues to increase beyond the flooding condition, three phenomena, which are discussed in the Results section herein, can be observed. The gross bubbles, which are schematically portrayed in Figure 4 and observed by Hoffer and Rubin,6 appear in one of these phenomena.

Figure 5. Liquid superficial velocity versus liquid fraction, as described by eq 1, for the conditions of ρ = 1077 kg m−3, μ = 2.68 mPa s, n = 2.72, rb = 1 mm, g = 9.81 m s−2, and jg* = 39.7 mm.s−1.

versus the liquid fraction for the conditions of ρ = 1077 kg m−3, μ = 2.68 mPa s, n = 2.72, rb = 1 mm, and g = 9.81 m.s−2, which are pertinent to some of the experiments described herein, for various values of jg. For relatively low gas rates, the curve exhibits a maximum, and the maximum represents the equilibrium condition of the pneumatic foam.8 As the gas rate increases, the pneumatic foam becomes wetter and the liquid superficial velocity increases. However, there exists a critical gas rate, jg* at which the curve no longer exhibits a maximum, and this occurs when the first point of inflection is coincident with the point at with the gradient is zero. The critical gas rate jg* is given by the following expression8

Figure 4. Schematic representation of “gross bubbles” rising in a foam column.

Consequently, we define the foam as the upper phase above the interface in the vertical pneumatic foam column. However, at a critical gas flow rate, and corresponding to a maximum liquid fraction of the foam phase, flooding occurs where the bubbly liquid becomes indistinct from the foam. In this paper, the conditions required for the onset of flooding will be theoretically and experimentally investigated.

jg* =

ρgrb2 ⎛ 2 ⎞2 ⎛ n − 1 ⎞n − 1 ⎟ ⎜ ⎟ mn⎜ ⎝n + 2⎠ ⎝ n + 1⎠ μ

and there is a corresponding liquid fraction, ε*, of n−1 ε* = n+1

2. THEORY Consider a pneumatic foam column, as shown schematically in Figure 1, that continuously rises up a column because gas at a superficial velocity of jg that creates bubbles of radius rb is sparged to the bottom. The dynamic viscosity and density of the liquid are μ and ρ, respectively. The liquid superficial velocity, jf, is given by the following expression8 εjg ρgrb 2 n − mε jf = 1−ε μ (1)

(3)

(4)

We assert that this is, in fact, the maximum liquid fraction that a pneumatic foam can attain. If the gas rate increases beyond jg*, the foam exhibits one of three different behaviors that depends upon the absolute pressure at the column bottom, as will be demonstrated experimentally herein. As the gas rate increases, the liquid fraction of the pneumatic foam layer increases, but the liquid fraction of the bubbly liquid layer beneath actually decreases. It is our contention that when the gas rate approaches jg* the liquid fractions of the two layers converge, and the interface between them becomes indistinct. Thus, this is the point at which the interface between the two

where g is the acceleration due to gravity, ε is the volumetric liquid fraction of the foam, m and n are dimensionless 6151

dx.doi.org/10.1021/ie4001844 | Ind. Eng. Chem. Res. 2014, 53, 6150−6156


Article

Table 1. Physical Properties and Drainage Parameters of Solutions Used in Forced Experiments solution

glycerol conc. [w/w %]

temperature [°C]

μ [mPa s]

ρ [kgm‑3]

1 2 3 4 5

0 24 31 38 45

19.8 20.0 19.0 20.3 19.0

1.05 2.09 2.68 3.19 4.62

1000 1056 1077 1092 1103

layers corresponds to the flooding condition and the attainment of the maximum liquid fraction in the foam. In order to prove the theory expounded above, the foam drainage parameters m and n must be measured experimentally, and these measurements are described in the next section.

m [−] 0.0269 0.0327 0.0411 0.0364 0.0514

± ± ± ± ±

0.0079 0.0120 0.0097 0.0138 0.0134

n [−] 2.57 2.69 2.72 2.45 2.89

± ± ± ± ±

0.15 0.25 0.15 0.22 0.16

transparent wall of the column, and the bubble size is measured using the image processing software Optimas 6.5. At least 50 bubbles are measured in each photograph. The automatized method of forced drainage that employs pressure gradient measurements developed by Li et al.12 is used in this study. Forced drainage data for all five solutions are shown in Figure 7, and a power-law relationship between the dimensionless

3. PRELIMINARY MEASUREMENTS The conditions of foam flooding are investigated for foam created using humidified air as the gas phase and four solutions (Table 1) of glycerol (BP, DomHealth, New Zealand) in distilled water so that the dynamic viscosity and density of the interstitial liquid can be adjusted. The foam is stabilized by adding 2.92 g/L (i.e., 20% above the critical micelle concentration) sodium dodecyl sulfate (SDS) (90% purity, Sigma-Aldrich, Germany). The temperatures of the system during the forced drainage experiments vary from 19.0−20.0 °C, and the temperatures of the system during the overflowing pneumatic foam experiments vary from 20.9−23.6 °C among different runs. Because the experiments are carried out in a relatively large column without precise temperature control, the viscosity of the solutions at experiment temperatures are calculated using predetermined viscosity versus temperature calibration curves for each solution. The calibration curves are prepared using a Canon−Fenske viscometer (size 75), and temperature covers a range from 18.8 to 38.4 °C, as shown in Figure 6; a wider temperature range than actually observed in the drainage experiments is given for completeness.

Figure 7. μjd/ρgr2bversus ε for the solutions with different glycerol concentrations as indicated in the legend. The linear nature of the data when plotted on a double-logarithmic axes demonstrates that the power-law drainage expression of eq 2 is appropriate functional form to fit the data. The error bars represent the standard deviations of the dimensionless group plotted on the ordinate.

group μjd/ρgr2b and ε by nonlinear least-squares regression (performed in Mathematica 6.0, Wolfram Research, Inc., U.S.A.) gives values of the drainage parameters m and n as listed in Table 1, along with 95% confidence intervals. The dynamic viscosities in the table have been compared with Dorsey’s13 tabulated values for aqueous solutions of glycerol as a function of temperature, and there is no significant difference.

4. FOAM FLOODING EXPERIMENTS 4.1. Methods and Materials. The apparatus used for the overflowing pneumatic foam experiments use to identify the flooding condition is shown schematically in Figure 8. The foam column of a circular cross-section consisted of a Perspex (Plexiglas) base (I.D. = 100 mm, height = 250 mm), a pipe (I.D. = 52 mm, height = 1250 mm), and a froth launder to collect the foam (I.D. = 150 mm, height = 300 mm) on the top. The froth launder is covered by a lid to minimize evaporation because it has been shown14 that surface stability, and therefore the hydrodynamic foam condition, is dependent upon evaporation rate. A vent from the top of the froth launder maintains the pressure at the top of the column at one atmosphere. Two pair of pressure sensors (i.e., four in total), S1−S4, (CTEM70070GY4, SensorTechnics, Germany) are used to

Figure 6. Viscosity versus temperature calibration curves for the solutions with different glycerol mass concentrations, as indicated in the legend. The curves are trinomial regressions.

The method of forced drainage11 is employed to determine the foam drainage parameters m and n. The column is described in Section 4, in which the foaming liquid is added to the top of a static layer of foam at a certain flow rate and the liquid fraction is measured. A single-pore sparger is used to create a foam of approximately uniform bubble radius of 1.0 ± 0.1 mm. Photographs of the bubbles are taken through the 6152



Article

liquid is also pumped into the column at a specific rate. The foam−liquid interface, if clearly visible, is maintained at a location equidistant of S2 and S3 by changing the outlet elevation of a vented underflow tube. Gas at different superficial velocities is supplied to the column and the liquid fractions are monitored in real time via the data acquisition system, from which the attainment of steady state of the system can be determined. As the gas rate increases, the foam becomes wetter until the foam−liquid interface vanishes. The experiments are repeated using the four different solutions as detailed in Table 1. 4.2. Results. Figures 9−13 show the evolution of liquid fractions in both upper and lower parts of the column as the

Figure 8. Schematic diagram of the foam flooding apparatus.

measure the pressures in both the foam layer and the bubbly liquid. The foam−liquid interface is maintained at a location by changing the outlet elevation of a vented underflow tube under the different gas rates. The liquid fraction is obtained from pressure measurement by using the following relationships that assumes that the gas density is insignificant with respect to the liquid density P − P1 ε= 2 ΔP2,1 (5) for the foam phase and P − P3 ε= 4 ΔP4,3

Figure 9. Liquid fraction in the foam layer and the bubbly liquid under different superficial gas velocities for solution 1; the horizontal line shows the predicted maximum liquid fraction calculated from the forced drainage data via eq 4.

(6)

for the bubbly liquid, where P1−P4 are the measured pressures at sensors 1−4, respectively, and ΔP2,1 and ΔP4,3 are the pressure differences between the corresponding pair of sensors when the column is filled with water only. Note that in eqs 4 and 5, the wall shear stress is neglected. This is valid because when the liquid fraction is greater than about 0.2, the frictional pressure gradient is insignificant when compared to the gravitational pressure gradient.12 Foam is generated pneumatically by sparging compressed air via a calibrated rotameter through a pool of surfactant solution. The gas spargers are porous rubber pipes (also known as “air stones”), which are commonly used in aquaria. Typically, foams with bubble radii, ranging from 0.2 to 0.6 mm, are generated using this kind of sparger during the experiments. The bubble size increases with pore gas velocities (i.e., the velocity of the air through the pore of the sparger). The overflowing foam is collected in a reservoir vessel and destroyed mechanically by stirring with a propeller. The surfactant solution in the same tank is circulated by a peristaltic pump via a buffer tank, which reduces the mechanical noise caused by the pump and measured by the pressure sensors, to the foam column. The foam column is operated in a continuous mode, i.e., gas bubbles are continuously generated from the bottom, and the


superficial gas velocity increases for all the five solutions with different viscosities during the overflowing pneumatic foam experiments. It can be seen that, at lower gas rates, the liquid fraction of the bubbly liquid is much higher than that of the foam layer. Therefore, the interface between those two sections is distinct. As expected, when the gas rate increases, the liquid fraction of the foam phase increases whereas that of the bubbly liquid decreases and the two values approach one another. As the system approaches the flooding point, the interface 6153



Article

The reason why the liquid fractions in the upper and lower parts do not precisely converge in some of the graphs is thought to be because the bubble size distribution is not perfectly uniform with height in the column in some experiments. The horizontal lines in Figures 9−13 represent the maximum liquid fraction calculated from eq 4 using the values of n determined for each solution by employing the method of forced drainage (Table 1). It is shown that the predictions of eq 4 approximate the maximum foam liquid fraction that is experimentally observed. There are three phenomena that could be observed when the superficial gas velocity is further increased beyond the flooding condition: (1) The liquid fraction of the column could be approximately maintained at the maximum value, ε*, by judicious increases in the pressure at the bottom of the column by manipulation of the outlet height of the vented underflow. It is shown in Figure 14 that the liquid superficial velocity of solution 4 through the



Figure 14. Superficial liquid velocity and liquid fraction in the upper part of the column under different superficial gas velocities for solution 4, beyond the flooding condition as increasing of the gas velocity liquid fraction in the upper part keeps constant (i.e., phenomenon 1).

column increases as the gas velocity increases, but the liquid fraction is almost constant when phenomenon 1 is exhibited. Therefore, for a specific gas velocity, there is a specific liquid superficial velocity required to keep the liquid fraction at the maximum value ε*, so that “gross bubbles” do not occur. (2) If the pressure at the bottom of the column is lower than that required to maintain ε* as the gas rate increases, the liquid fraction of the column reduces significantly, and there are occasional “gross bubbles” (rb ≈ 10 mm) rising very quickly up the column when the liquid fraction dropped below 0.20. Figure 15 shows the superficial liquid velocity of solution 4 and liquid fraction in the foam layer versus gas velocities when phenomenon 2 is exhibited. It is shown that beyond the flooding condition the liquid superficial velocity remains constant as gas velocity increases, despite a reduction in liquid fraction. (3) If the pressure at the bottom of the column is higher than that required to maintain ε*as the gas rate increases, the liquid fraction of the whole column increases. As shown in Figure 16, when phenomenon 3 is exhibited, increasing the gas velocity beyond the flooding condition significantly increases the liquid


becomes indistinct. When the foam-liquid interface finally vanishes the liquid fractions in the foam layer and the liquid pool almost converge and the system becomes homogeneous. 6154



Article

the above phenomena was the absence of the foam−liquid interface. Clearly, the second phenomenon in which gross bubbles rise quickly in the column is practically undesirable in foam contacting operations because they reduce the overall residence time of the gas and the specific surface area. These gross bubbles have also been previously observed by Aguayo and Lemlich,15 Hoffer and Rubin,6 and Stevenson8 under very high gas flow rates. However, the mechanism of formation is not understood. It is speculated herein that the formation of gross bubbles in a pneumatic foam may have analogy to the gross bubbles observed in the two-phase behavior of a gas fluidized bed.16 In his excellent monograph on the stability of fluidized beds, Gibilaro17 has explained that the cause of gross bubbles in fluidized beds lies in the relative velocities of dynamic and kinematic waves of perturbation in the column and whether small perturbations of voidage amplify or attenuate. In simple terms, stable (or homogeneous) fluidization occurs when the velocity of a dynamic wave (i.e., the velocity at which step changes of voidage propagate through the bed) is greater than the velocity of the kinematic wave (i.e., the velocity of pressure fluctuations to propagate through the bed). As the fluid (i.e., the gas or liquid in the continuous phase) superficial velocity increases beyond the minimum fluidization velocity, the bed expands. If at any time during this expansion the velocity of the kinematic wave begins to exceed the velocity of the dynamic wave, then unstable fluidization will begin, and gross bubbles will manifest as the minimum bubbling velocity is exceeded. If the condition that the kinetic wave exhibits a higher velocity than the dynamic wave is met at minimum fluidization, then the minimum bubbling velocity is coincident with the minimum fluidization velocity. In general, liquid fluidized beds exhibit homogeneous fluidization throughout the range of void fractions, whereas gas fluidized beds exhibit bubbling fluidization. However, when fine particles are fluidized by a gas, there is an initial regime in which the bed fluidizes homogenously as the gas rate increases above the minimum fluidization condition until it reaches the minimum bubbling velocity, above which bubbly fluidization manifests. It would appear that the onset of gross bubbles in fluidized beds is related to the onset of gross bubbles in pneumatic columns of gas−liquid foam because the gross bubbles begin to occur under certain circumstances at a defined value of the gas rate (even though the gas phase is continuous in gas fluidization but discontinuous in a pneumatic foam). So, in order to further investigate this possible relationship, it is necessary to determine the velocity of dynamic and kinematic waves in pneumatic foam. The latter is a simple task because it is simply derived by means of a mass balance. However, the velocity of the dynamic wave is a much more problematical because knowledge of the axial component of the interbubble force is required. It has recently been demonstrated18 that the interbubble normal force in a gas−liquid foam is nonzero but undetermined, so calculation of the dynamic wave velocity remains elusive. Finally, it is appropriate to draw attention to the phenomenon of “gushing” that afflicts some bottles of Belgian Trappist beer. Gushing causes some bottles of beer to erupt violently when opened, thereby ejecting beer froth into the air above the bottle, as opposed to the froth gently overflowing from the top. The phenomenon has been qualitatively described by Shokribousjein et al.19 We believe that the gushing occurs because the rate of nucleation of CO2 bubbles

Figure 15. Superficial liquid velocity and liquid fraction in the upper part of the column under different superficial gas velocities for solution 4, beyond the flooding condition as increasing of the gas velocity liquid fraction decreases significantly, and the gross bubbles are observe at gas velocity of 58 mm/s (i.e., phenomenon 2).

Figure 16. Superficial liquid velocity and liquid fraction in the upper part of the column under different superficial gas velocities for solution 4, beyond the flooding condition as increasing of the gas velocity liquid fraction in the upper part slightly increases (i.e., phenomenon 3).

superficial velocity while increasing the liquid fraction only marginally, and no gross bubbles manifest.

5. DISCUSSION The lack of gross bubble production when the flooding condition is surpassed but the pressure at the bottom of the column increases (i.e., phenomena 1 and 3) can be readily explained by the necessity of the multiphase mixture to maintain the gauge pressure that is set by the location of the outlet of the vented underflow. As the pressure increases, the liquid fraction must increase to maintain the same gauge pressure in the column, and therefore, the gross bubbles are prevented from forming because their presence would, in fact, decrease the gauge pressure. Thus, it is shown that increasingly large values of liquid fraction can be obtained without gross bubble formation by judicious manipulation of the pressure at the bottom of the column. The common feature of all three of 6155



Article

(2) Stevenson, P. Foam separations. In Comprehensive Biotechnology, 2nd ed.; Moo-Young, M., Ed.; Elsevier: Amsterdam, The Netherlands, 2011; Vol. 2, pp 715−726. (3) Stevenson, P. Gas−Liquid Mass-Transfer in Foam. In Foam Engineering: Fundamentals and Application; Stevenson, P., Ed.; John Wiley & Sons, Ltd.: New York, 2012; pp 331−349. (4) Helsby, F. W.; Birt, D. C. P. Foam as a medium for gas absorption. Appl. Chem. 1955, 5, 347−352. (5) Metzner, A. B.; Brown, L. F. Mass transfer in foams. Ind. Eng. Chem. 1956, 48, 2040−2045. (6) Hoffer, M. S.; Rubin, E. Flow regimes of stable foams. Ind. Eng. Chem. Fund. 1969, 8, 483−490. (7) Thornton, J. D. Spray liquid−liquid extraction columnsprediction of limiting holdup and flooding rates. Chem. Eng. Sci. 1956, 5, 201−208. (8) Stevenson, P. Hydrodynamic theory of rising foam. Minerals Eng. 2007, 20, 282−289. (9) Stevenson, P. Dimensional analysis of foam drainage. Chem. Eng. Sci. 2006, 61, 4503−4510. (10) Verbist, G.; Weaire, D.; Kraynik, A. M. The foam drainage equation. J. Phys. Cond. Matter. 1996, 8, 3715−3731. (11) Miles, G. D.; Shedlovsky, L.; Ross, J. Foam drainage. Phys. Chem. 1945, 49, 93−107. (12) Li, X.; Wang, X.; Evans, G. M.; Stevenson, P. Foam flowing vertically upwards in pipes through expansions and contractions. Int. J. Mult. Flow. 2011, 37, 802−811. (13) Dorsey, N. E. The Properties of Ordinary Water-Substance; Reinhold Publishing: New York, 1940. (14) Li, X.; Evans, G. M.; Stevenson, P. The effect of environmental humidity on static foam stability. Langmuir 2012, 28, 4060−4068. (15) Aguayo, G. A.; Lemlich, R. Countercurrent foam fractionation at high rates of throughput by means of perforated plate columns. Ind. Eng. Chem. Proc. Des. Dev. 1974, 13, 153−159. (16) Davidson, J. F.; Harrison, D. The behaviour of a continuously bubbling fluidised bed. Chem. Eng. Sci. 1966, 21, 731−738. (17) Gibilaro, L. G. Fluidization-Dynamics; Butterworth Heinemann: Oxford, U.K., 2001. (18) Li, X.; Evans, G. M.; Stevenson, P. Remarks on approaches that relate foam drainage rate to bubble terminal velocity. Int. J. Mult. Flow. 2012, 42, 24−28. (19) Shokribousjein, Z.; Deckers, S. M.; Gebruers, K.; Lorgouilloux, Y.; Baggerman, G.; Verachtert, H.; Delcour, J. A.; Etienne, P.; Rock, J.M.; Michiels, C.; Derdelinckx, G. Hydrophobins, beer foaming and gushing. Cerevisia 2011, 35, 85−101.

upon depressurisation of the supersaturated beer causes a gas superficial velocity in the bottleneck greater than that given by eq 3, and the excess gas is manifest as gross bubbles. These gross bubbles rise rapidly in the bottleneck causing the eruption of beer, or gushing, as they leave the bottle.

6. CONCLUSIONS There exists a maximum liquid fraction, ε*, that a pneumatic foam can achieve, and the value can be estimated by Stevenson’s6 expression ε* = (n − 1)/(n + 1), which indicates that the maximum liquid fraction occurs at flooding and depends on the bulk properties of the liquid, as well as the gas− liquid surface properties, upon which n implicitly depends. The value of n could be conveniently obtained by performing forced drainage experiments. When the flooding point is exceeded, the liquid fraction of homogeneous bubbly flow can be manipulated by control of the pressure at the bottom of the column. In addition, when the superficial gas velocity exceeds the value that causes flooding, there are three different phenomena observed under different operating conditions. Beyond the point of flooding, the hydrodynamic condition of the foam is governed by both gas superficial velocity and the pressure at the bottom, whereas before the system reaches the flooding, the hydrodynamic condition is independent of the absolute pressure at the column bottom.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: geoff[email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors acknowledge financial support from the Discovery Projects scheme of the Australian Research Council under Grant DP0878979. They are grateful to Mr. Raymond Hoffman of the Department of Chemical and Materials Engineering, University of Auckland, for his workshop support.

■

NOMENCLATURE g = acceleration due to gravity [m s−2] jf = superficial liquid velocity [m s−1] jg = superficial gas velocity [m s−1] jg* = superficial gas velocity at the flooding condition [m s−1] m = dimensionless number used in eq 1 [−] n = dimensionless index used in eq 1 [−] P = local pressure [Pa] ΔP2,1 = pressure difference between sensors 2 and 1 when the column was filled with liquid [Pa] ΔP4,3 = pressure difference between sensors 4 and 3 when the column was filled with liquid [Pa] rb = characteristic mean bubble radius [m]

Greek

ε = volumetric liquid fraction [−] ε* = critical volumetric liquid fraction at the threshold of foam instability [−] μ = liquid dynamic viscosity [Pa s] ρ = liquid density [kg m−3]

■

REFERENCES

(1) Finch, J. A.; Dobby, G. S. Column Flotation; Pergamon Press: Oxford, U.K., 1990. 6156


Flooding in a Vertically Rising GasâLiquid Foam - Industrial

Recommend Documents

Flooding in a Vertically Rising GasâLiquid Foam - Industrial