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International Journal of Multiphase Flow 37 (2011) 802–811
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International Journal of Multiphase Flow
journal homepage: www.elsevier .com/locate / i jmulflow
Foam flowing vertically upwards in pipes through expansions and contractions
Xueliang Li a,⇑, Xinting Wang a, Geoffrey M. Evans a, Paul Stevenson b
a Centre for Advanced Particle Processing, University of Newcastle, Callaghan, NSW 2308, Australiab Department of Chemical and Materials Engineering, University of Auckland, Auckland 1010, New Zealand
a r t i c l e i n f o a b s t r a c t
Article history:Received 18 October 2010Received in revised form 12 January 2011Accepted 15 February 2011Available online 26 February 2011
Keywords:Pneumatic foamContractionExpansionLiquid fractionDrift-flux
0301-9322/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.ijmultiphaseflow.2011.02.008
⇑ Corresponding author. Present address: DepartmeEngineering, University of Auckland, 20 SymondsZealand. Tel.: +64 (0)21 898 534; fax: +64 (0)9 373 7
E-mail address: [email protected] (X. Li).
The drift-flux analysis of one-dimensional two-phase flow of Wallis (Wallis, G.B., 1969. One-dimensionaltwo-phase flow. McGraw-Hill Book Company, New York.) is utilised for the first time to model the behav-iour of pneumatic foam flowing vertically through an expansion or an contraction. It is demonstratedthat, although a sudden contraction of flow area decreases the liquid fraction, it does not affect the vol-umetric liquid over-flow rate. It is also demonstrated that a sudden expansion of flow area decreases boththe liquid fraction and the volumetric liquid over-flow rate. The liquid fraction of a foam stabilised by2.92 g/L sodium dodecyl sulphate (SDS) solution flowing through a sudden contraction or expansionwas measured by an improved pressure gradient method. The results were found to be consistent withthe theoretical analysis. This study has implications for foam fractionation device design, optimisationand process intensification.
� 2011 Elsevier Ltd. All rights reserved.
1. Introduction
If both phases in a two-phase flow are fluid, the interface be-tween the two phases can adopt various geometries. The descrip-tion of the two-phase flow can be simplified by the classificationof ‘‘flow regimes’’ based on the distribution pattern of the interface.For instance, in vertical gas–liquid flows, the following flow re-gimes are generally accepted (in an order of increasing gas flowrate): Bubble flow, slug (or plug) flow, churn flow, annular flowand wispy-annular flow (Butterworth and Hewitt, 1977; Wallis,1969). Note that another frequently encountered gas–liquid two-phase flow, the pneumatic foam, is excluded from these regimes.Pneumatic foam is widely used in froth flotation and foam fraction-ation, as well as other processes. Foam fractionation is a process toenrich surface active materials, such as proteins, from aqueoussolution by adsorbing the surface active species onto the surfaceof bubbles. A conventional foam fractionation column consists oftwo distinct sections, namely the liquid pool and the foam bed,both of which are gas–liquid two-phase flow. Drift-flux analysisof the flow in the liquid pool was considered in the original workof Wallis (1969) and further exploited by other workers (Pal andMasliyah, 1989; Pilon and Viskanta, 2004; Stevenson et al.,2008a; Vandenberghe et al., 2005; Yianatos et al., 1988). Most ofthe later researchers had also used the same flow characteristiccurve (i.e., the graphical representation of the drift-flux as a
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nt of Chemical and MaterialsStreet, Auckland 1142, New463.
function of phase fraction) for both the bubbly liquid and thefoam/froth, whereas Stevenson et al. (2008a) briefly discussed thatthere should be different curves for the foam and the bubbly liquid.Nevertheless, studies of the flow characteristics of pneumatic foamby utilising the drift-flux model are still rare.
It is not surprising that, because the drift-flux model has beenlargely overlooked by the foam fractionation community, the rela-tive velocity between the gas and the liquid has been overlooked,too. This is evident from the design of foam fractionation appara-tus. Unlike froth flotation, where a froth launder is almost univer-sally used (see e.g. Vandenberghe et al., 2005 and Stevenson et al.,2008a), the foam fractionation community has employed diversemethods of foam collection. Table 1 gives some typical arrange-ments for foam collection that have been described in the litera-ture. Most of these methods involve a combination ofcontraction, expansion and bending of the flow. However, thequestion has never been asked by the authors as to whether thefoam collection methods affect the properties of the foam itself.However, Li et al. (2010) have found that a successive contractionand expansion of flow area results in a dryer foam, which can beexploited for process intensification in foam fractionation.
The first publication related to constrained foam flow was ashort research note by Calvert (1988) who experimentally investi-gated the foam flow through different types of valves. It was foundthat the valves caused different amounts of foam collapse, eventhough the overall pressure loss induced by the valves was thesame. Foam flow through sudden expansion and contractions inboth vertical (Deshpande and Barigou, 2001) or horizontal (Alouiand Madani, 2007, 2008; Dollet, 2010) or inclined (Bertho et al.,2006) channels was experimentally studied and the affect of the
Table 1Foam collecting methods used in application-orientated foam fractionation research.
Researchers System Foam collecting methodDescription Schematic diagram
Crofcheck and Gillette (2003) Bovine serum albumin (BSA), water, nitrogen gas Foam rise through the column, enters anarrowed pipe and spills over the sides into aannular trough (foam collection cup)
Brown et al. (1999) Beta-casein, water, air Column bends 180� downwards at the upperend
Jeong et al. (2003) Protein from Mimosa pudica L., seed, water, air Top of the column is closed, a side port isopened near the top of the column
Saleh and Hossain (2001) Mixture contains BSA, lactoferrin, water, air Column bends 90� at the upper end, with anexpansion
Du et al. (2000) Egg albumin and BSA, water, air Foam over-flows from the top of the column Not given
Maruyama et al. (2007) Egg albumin, water, nitrogen gas Foam is aspirated out of the column Not given
Burapatana et al. (2005) Cellulase with the assistance of some detergents, water, air Foam collection chamber is fitted to the top ofthe column.
X. Li et al. / International Journal of Multiphase Flow 37 (2011) 802–811 803
changes of flow geometry on pressure loss, bubble size and liquidfraction was reported. However, these authors specifically studiedthe foam rheology, while no mechanistic explanation of thechanges in liquid fraction across the sudden expansion or contrac-tion was given. In the case of foam fractionation, however, it is theliquid flux that is of more importance, as this is directly related tothe enrichment and recovery of the objective species.
In this paper, the one-dimensional (1D) drift-flux model is uti-lised in conjunction with the hydrodynamic theory of rising foam(Stevenson, 2007) to describe the behaviour of a foam flowing up-wards in a vertical pipe with either an expansion or a contractionof flow area. We recognise that in 1D space the concept of expansionor contraction does not exist. However, it is pertinent to assume thatbeyond the vicinity of expansion or contraction, the flow of the foamis in 1D. Although we do not attempt to exploit the effect of changesin flow area on the actual velocity profile of the foam in the currentstudy, we will demonstrate, in Appendix B, that the current analysisis independent of the velocity profile in the expansion or contractionregion. This has also been demonstrated by experimental results (Liet al., 2010) showing that as long as there is a sufficient distance be-tween the foam/liquid interface, a contraction does not affect thevolumetric liquid flux. The implications of this analysis on foam frac-tionation process and device design will also be discussed.
2. Theory
2.1. Drift-flux model
Following the notation of Stevenson et al. (2008a), we define thesuperficial gas velocity, jg, and the superficial liquid velocity, jf, as:
Jg ¼Qg
ACand jf ¼
Q f
ACð1Þ
where Qg and Qf are the volumetric flow rates of the gas andliquid, respectively. AC is the cross-sectional area of the vessel.
The absolute velocities of the two phases in an Eulerian referenceframe, Vg and Vf, can be expressed as functions of the superficialvelocities and the corresponding volumetric phase fractions, i.e.,
Vg ¼jg
aand V f ¼
jf
eð2Þ
where a is the volumetric gas fraction and e � 1 � a is the volumet-ric liquid fraction. The absolute velocity of the two-phase flow, in anEulerian reference frame, is:
V ¼Q g þ Q f
ACð3Þ
The drift velocity of the gas phase, Vgf, is defined as the velocityof the gas phase in a reference frame moving at V (relative to thestationary observer), i.e.,
Vgf ¼ Vg � V ð4Þ
Substituting Eqs. (1) and (2) into Eq. (4) and rearranging gives
aVgf ¼ ð1� aÞjg � ajf ð5Þ
The term aVgf, which will now be assigned the symbol jgf, iswidely known as the drift-flux. Replacing aVgf with jgf in Eq. (5) gives
jgf ¼ ð1� aÞjg � ajf ð6Þ
From Eq. (2), the slip velocity between the gas and the liquidphases, Vs, can be calculated,
V s ¼ Vg � V f ¼jg
a� jf
1� að7Þ
Combining Eqs. (6) and (7) gives
jgf ¼ að1� aÞV s ð8Þ
From Eq. (8) it can be seen that the drift-flux is proportional tothe slip velocity. For the bubbly flow regime, Wallis (1969) gave afunctional form of Vs for a solid–liquid two-phase system. By
0.0 0.2 0.4 0.6 0.8 1.0
-1
0
1
2
characteristic curve
j gf(m
m/s
)
α
foamwet bubble assembly
operating line
-1
0
1
2
jd
Fig. 1. Characteristic drift-flux curve compared with experimental data fromRouyer et al. (2010). q = 1000 kg/m3, g = 9.8 m/s2, rb = 0.13 mm, l = 1 mPa s,m = 0.0698 and n = 2.73.
804 X. Li et al. / International Journal of Multiphase Flow 37 (2011) 802–811
drawing an analogy to Wallis expression, the following equationfor a gas–liquid system was developed and used, empirically, to de-scribe the two-phase flow in the liquid pool and in the froth sectionof a flotation cell by some researchers (Pal and Masliyah, 1989;Vandenberghe et al., 2005):
jgf ¼ V tað1� aÞz ð9Þ
where Vt is the terminal velocity of a single bubble rising in an infi-nite continuum of the liquid and z is an adjustable index.1 However,Wallis pointed out that Eq. (9) is only valid if the fluid dynamic dragis balanced by buoyancy force and particle–particle (or bubble–bubble, in case of bubbly flow and foam) forces are insignificant. Itis true that there is not direct interaction among the bubbles in abubbly flow (such as that in the bubbly liquid in a foam fractionationcolumn), but the adjacent bubbles in a foam are directly in contactwith each other and there is a normal force between them. Howeversmall the normal force may be, it is the main driving force for the riseof the foam and it is always a significant factor. Besides, the stressstatus at the liquid/gas interface in the foam is apparently differentto that at the surface of an isolated bubble, therefore the use of Eq.(9) in foam is yet to be justified. In the present study, we adoptthe drainage equation developed by Stevenson (2006), which hasproven utility in pneumatic foam.
By carrying out dimensional analysis, Stevenson (2006) sug-gested an expression for the superficial liquid drainage velocityin a foam, jd, which is,
jd ¼qgr2
b
lmen ð10Þ
where q and l are the density and dynamic viscosity of the liquidphase, respectively; g is the acceleration due to gravity, rb is a rep-resentative bubble radius, m and n are two adjustable constants.Using Eq. (10), the slip velocity between the gas and the liquidphases in a foam can be expressed as
V s ¼jd
e¼ qgr2
b
lmen�1 ¼ qgr2
b
lmð1� aÞn�1 ð11Þ
Substituting Eq. (11) into Eq. (8) gives
jgf ¼qgr2
b
lmað1� aÞn ð12Þ
A plot of jgf versus a gives the characteristic curve of the two-phase system. Fig. 1 shows the characteristic curve for the systeminvestigated by Rouyer et al. (2010). They obtained experimentaldata for the relationship between the liquid fraction and liquidsuperficial velocity in a counter-current gas–liquid two-phaseflow. The experiment was designed in such a way that the bubbleswere stopped from rising by a plate which covered the wholecross-section of the column. Liquid was added to the bubbleassembly from the top through the pores (which are smaller thanthe bubble diameter) in the plate and the superficial liquid velocitywas controlled by a pump. The system property parameters used toprepare Fig. 1 were q = 1000 kg/m3, g = 9.8 m/s2, rb = 0.13 mm,l = 1 mPa s, m = 0.0698 and n = 2.73.
Each point on the characteristic curve represents possible solu-tion to Eq. (6), and a straight line passing through such a point iscalled the operating line, which is determined by the operatingconditions: jg, jf and a. This is explained in detail in Fig. 4.1 inthe work of Wallis (1969), and Stevenson et al. (2008a) describesthe application of Wallis method to foam flow. The operating linesfor the experiments of Rouyer et al. (2010) should pass through theorigin on the jgf versus a graph, because the superficial gas velocity
1 Usually the exponent is reported as n, but z is used here instead because n isotherwise assigned in Eq. (10).
is zero. The intercept at a = 1 gives the superficial liquid velocity.Here, the liquid volumetric flux is defined as positive in the direc-tion of gravity; therefore a downwards liquid flow is denoted by apositive intercept and vice versa. It can be seen that amongst thethree parameters, jg, jf and a, only two can be manipulated inde-pendently and the third is determined by the characteristic curve.This is why the drift-flux analysis is particularly convenient as ameans of analysing two-phase flow problems. It can also be seenfrom Fig. 1 that, at high a values, i.e. at low liquid fraction, theexperimental data points exhibit good agreement with the charac-teristic curve, whereas as a decreases from about 0.67, the datapoints start to deviate from the characteristic curve. The conditionat which the deviation occurs was when the bubble bed was ‘fullyfluidised’, i.e., the liquid fraction was so high that the bubblyassembly was broken into separated bubbles and it could no longerbe considered as a foam (Rouyer et al., 2010). Note that thisthreshold is not the same as the stability threshold of rising foam(co-current gas–liquid flow) described by Stevenson (2007) whichwill be discussed later in this paper. Fig. 1 shows that, not onlydoes Eq. (12) have utility for illustrating the phase fraction and fluxof a pneumatic foam, it can also specify the condition under whichfluidisation of the foam bed occurs.
2.2. Drift-flux analysis of rising foam in a straight column
Before proceeding to the drift-flux analysis of foam flowthrough contractions and expansions, it is pertinent to reviewthe drift-flux analysis of pneumatic foam of Stevenson et al.(2008a). Using the same expression for superficial liquid drainagerate and by invoking stability analysis, Stevenson (2007) proposedthe hydrodynamic theory of rising foam which consists mainly ofthe following two points:
(1) For a foam with a specific bubble size, there exists a maxi-mum gas flow rate, j�g, above which there cannot exist a sta-ble foam, i.e., the flow regime changes at this gas ratethreshold. The equilibrium liquid superficial velocity at thisgas rate, j�f , is the maximum liquid rate before the flowregime changes.
(2) For a foam with a specific bubble size and at a given gas flowrate below the threshold value given in point (1), there is anequilibrium condition of the foam where the foam has aliquid fraction that is determined by system properties andthe gas rate. This equilibrium condition gives the maximumliquid superficial velocity, je
f , that the foam at the given gasrate can support.
From the above statements it can be readily inferred that theliquid fraction of a rising foam cannot be greater than its
X. Li et al. / International Journal of Multiphase Flow 37 (2011) 802–811 805
corresponding equilibrium value unless water is added to the topof the column. In column flotation this water is referred to aswash-water which is added to the top of the froth to aid the rejec-tion of gangue material from the concentrate stream. Wash-waterhas analogy to external reflux in foam fractionation, where part ofthe product stream is returned to the foam to enhance interfacialadsorption. It can also be inferred that, even though the liquid frac-tion of a foam/froth can be manipulated by the addition of wash-water, the volumetric foamate over-flow rate, (i.e., the amount ofliquid delivered by the foam), will not change. The above charac-teristics of a rising pneumatic foam were interpreted by Stevensonet al. (2008a) using the drift-flux model.
It has been shown, in the work of Stevenson et al. (2008b), thatthe operating line representing an equilibrium-state pneumaticfoam forms a tangent to the characteristic curve. Fig. 2 demon-strates the process of assembling the characteristic curve fromthe operating lines determined by the hydrodynamic theory of ris-ing foam of Stevenson (2007). System properties used here are for2.92 g/L SDS solution: q = 1000 kg/m3, g = 9.8 m/s2, rb = 0.53 mm,l = 1 mPa s, m = 0.016 and n = 2. It can be seen from Fig. 2 thatfor the system in question, at a superficial gas flow rate of13 mm/s with a corresponding gas fraction of 0.67, the liquid fluxreached its maximum possible value. This point represents thethreshold of co-current upwards foam flow (Stevenson, 2007). As
0.0 0.2 0.4 0.6 0.8 1.0
-2
0
2
4
6
8
10
j gf(m
m/s
)
α-2
0
2
4
6
8
10
Fig. 2. Derivation of the jgf versus a curve from the operating lines determined fromthe hydrodynamic theory of rising foam of Stevenson (2007). System propertiesused here are for 2.92 g/L SDS solution: q = 1000 kg/m3, g = 9.8 m/s2, rb = 0.53 mm,l = 1 mPa s, m = 0.016 and n = 2.
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
jf = j
f
e
jf < j
f
e
Dim
ensi
onle
ss li
quid
vel
ocit
y
Dimensionless gas velocity
jf > j
f
e, operation
impossible
Fig. 3. Calculated dimensionless (with respect to the threshold value) liquidvelocity versus gas velocity curve showing various operating conditions of a straightfoam column. System properties are the same as in Fig. 2.
mentioned above, this threshold is not to be confused with the flu-idisation point shown in Fig. 1.
Let the equilibrium liquid superficial velocity at a given gas ratebe je
f , which can be made dimensionless by dividing it by j�f . Thesuperficial gas velocity can also be made dimensionless by dividingit by j�g The dimensionless equilibrium liquid velocity is plottedagainst the dimensionless gas velocity in Fig. 3. It can be seen thatthe equilibrium liquid flux increases exponentially with the super-ficial gas velocity. The area above the curve in Fig. 3 represents theconditions where jf > je
f . Under these conditions, steady-state oper-ation is impossible. The area under the curve represents the condi-tions where jf < je
f . This can be achieved by adding wash-water tothe foam while keeping the superficial gas velocity constant. Aswill be shown below, this can also be achieved by forcing the foamthrough a contraction.
3. Foam flow through vertical contractions and expansions
3.1. Foam flow through a contraction
Fig. 4 shows a schematic diagram of a foam flowing verticallythrough a sudden contraction. The dashed box denotes the controlvolume upon which the material balance will be performed. Thechanges in flow pattern caused by the changes in flow area is con-tained in the control volume and beyond the control volume, flowsare in 1D. Let the cross-sectional area of the column before andafter the contraction be AC and A0C . The contraction ratio, X, is de-fined as
X ¼ A0CAC
ð13Þ
The gas and liquid superficial velocities before the contractionare denoted as jg and jf, respectively. By continuity, the superficialvelocities after the contraction, j0g and j0f , have to satisfy the follow-ing relationship:
j0g ¼jg
X
j0f ¼jf
X
8>><>>:
ð14Þ
because the volumetric flow rates of both phases are constantacross the contraction.
The foam before the contraction should be at the equilibriumcondition and therefore the operating line can be plotted in thesame way as these shown in Fig. 2. j0g and j0f can then be determinedfrom Eq. (14) and the operating line for the flow above the contrac-tion can be plotted from
jgf ¼1X½jgð1� aÞ þ jfa� ð15Þ
as shown in Fig. 5. Other parameters used in Fig. 5 includeq = 1000 kg/m3, g = 9.8 m/s2, rb = 0.53 mm, l = 1 mPa s, m = 0.016
'gj
'fj
gj fj
Fig. 4. Schematic diagram of a foam flowing vertically through a sudden contrac-tion. The dashed box denotes the control volume, which contains the changes inflow area and flow pattern.
0.0 0.2 0.4 0.6 0.8 1.00
4
8
12
j gf(m
m/s
)
α
after contraction
before contraction
0
4
8
12
Fig. 5. Calculated operating lines of a foam flowing through a sudden contractionwhere the flow area is halved (i.e., X = 0.5). System property parameters are thesame as in Fig. 2. Before contraction, jg = 5 mm/s; after contraction, jg = 10 mm/s.
0.010 0.015 0.020 0.025 0.0300.70
0.75
0.80
0.85
0.90
0.95
1.00 After contraction Before contraction
α(-
)
Qg (L/s)
incr
ease
in
gas
frac
tion
Fig. 7. Experimental results showing that the liquid fraction decreases across thecontraction at three different volumetric gas flow rates (0.013, 0.017 and 0.027 L/s).The liquid phase was 2.92 g/L SDS solution and the gas phase was air. Thecontraction ratio X = 0.31.
806 X. Li et al. / International Journal of Multiphase Flow 37 (2011) 802–811
and n = 2. The superficial gas velocity was 5 mm/s before thecontraction and 10 mm/s after the contraction. These parameterswere chosen because they are representative of the experimentsconducted herein, with the values of m and n being those measuredfor 2.92 g/L SDS solution by Stevenson et al. (2007). The detail ofFig. 5 around a! 1 is shown in Fig. 6. It can be seen that the gasfraction of the foam has increased after the contraction, i.e., thereis a decrease in liquid fraction. This conclusion may be surprisingto some researchers in the foam fractionation community since itis common experience that a higher superficial gas velocity meansa higher liquid fraction for a foam in a straight column.
To verify this analysis, a foam fractionation column with a con-traction of flow area was constructed from Perspex (Plexiglass).The internal diameters of the column before and after the contrac-tion were 90 mm and 50 mm respectively (i.e., X = 0.31). The liquidphase was 2.92 g/L SDS solution (which is 20% above the criticalmicelle concentration) and the gas phase was air. Pressure sensorswere fitted onto the wall of the column and the same pressure gra-dient technique used by Shaw et al. (2010) was employed to mea-sure the liquid fraction. The error introduced by wall friction on theliquid fraction measurement was corrected by comparing the pres-sure drop in a rising foam with that in a static foam of the same li-quid fraction (see Appendix A). Bubble size was estimated bytaking photographs through the transparent wall of the column;there was no discernible difference across the contraction. Thegas fractions before and after the contraction at three differentgas rates are shown in Fig. 7. For convenience, volumetric gas flowrate is used instead of superficial gas velocity when preparing Fig. 7because the cross-sectional area changes. Each data point repre-sents the average of five repeated measurements and the relative
0.90 0.95 1.00
-0.5
0.0
0.5
1.0
A characteristic curve
operating line after contraction
j gf(m
m/s
)
α
operating linebefore contraction
A
B
-0.5
0.0
0.5
1.0
increase ingas fraction
Fig. 6. Detail of Fig. 5 showing the gas fraction before (point A) and after thecontraction (point B).
error is typically less than 5%. It can be seen that for all the threetested gas rates (i.e., 0.013, 0.017 and 0.027 L/s), the gas fraction in-creases after the contraction, which agrees with the predictions ofthe analysis described above. Qualitative simulation of the processrequires accurate measurements of bubble size within the bulk ofthe foam, of which no simple method is currently available.
As discussed above, the liquid fraction of a foam can be manip-ulated by adding wash-water, which decreases the net superficialliquid velocity within the foam. It is shown herein that the liquidfraction of the foam can also be manipulated by passing the foamthrough a sudden contraction. Let the equilibrium liquid fraction ofa foam for a given superficial gas velocity be ee, we assert thefollowing:
(1) if wash-water is being added to a foam and the system is atsteady state, jf < je
f but e > ee;(2) if a foam is forced through a sudden contraction and system
is at steady state, after the contraction jf < jef and e < ee.
If the foam is in a straight column and the bottom of the foamlayer is in contact with the bubbly liquid, the condition e < ee rep-resents an un-steady state. By stability analysis, Stevenson (2007)has shown that the foam will gain liquid from the liquid pool be-neath it, and eventually the liquid fraction will increase to ee. Thefoam in the smaller pipe after the contraction is not in directcontact with the liquid pool, therefore it has nowhere to gain extraliquid (as the foam before the contraction has already reached themaximum capacity) therefore it can only stay at e < ee. It can beinferred that, if additional liquid is provided to the bottom of thefoam in the smaller pipe, it will be transported upwards by thefoam and the liquid fraction of the foam will increase. This is ver-ified experimentally by adding liquid to the foam at 5 cm above thecontraction using a capillary tube. Note that, although we still callthe added liquid ‘wash-water’, it is added to the bottom of thefoam rather than the top. The liquid fraction of the foam afterthe contraction was measured. The foam was collected using afroth launder and the delivered liquid superficial velocity, definedas the delivered liquid volumetric flow rate divided by the cross-sectional area of the column, was also measured. The results areshown in Table 2.
It can be seen from Table 2 that a decrease in gas fraction (i.e.,increase in liquid fraction) occurred when wash-water was added.The wash-water was added at a rate equal to 50% of the original
Table 2Experimental results of wash-water addition to the foam after the contraction.
Wash-water superficialvelocity (mm/s)
Delivered liquid superficialvelocity (mm/s)
Volumetric gasfraction (–)
Zero 0.0272 0.9700.0136 0.0393 0.953Difference 0.0121 �0.017
0.0 0.2 0.4 0.6 0.8 1.00
4
8
12
normal operation
j gf(m
m/s
)
α
before expansion
after expansion0
4
8
12
Fig. 10. Calculated operating lines for foam flowing vertically through an expansion(X = 2). System properties are the same as in Fig. 5. Before expansion, jg = 10 mm/s.After expansion, jg = 5 mm/s.
X. Li et al. / International Journal of Multiphase Flow 37 (2011) 802–811 807
delivered liquid superficial velocity and nearly all of the addedwash-water was transported out of the column by the foam, prov-ing that the foam was not at equilibrium state.
At this juncture, the question arises as to what would happen tothe foam if the column is contracted to an infinitesimal flow area?Rearranging Eq. (15) and setting X to zero, the expression
ajX!0 ¼jg � Xjgf
jg � jf
�����X!0
¼jjgj
jjgj þ jjf jð16Þ
is obtained, i.e., when the contraction reduces the flow area to zero,the in situ gas fraction of the foam will approach the volumetricflow fraction of the gas phase. This can be graphically demonstratedin Fig. 8. In fact, all the operating lines cross the abscissa at a = |jg|/(|jg| + |jf|) (modulus signs are used here because the gas volumetricflux is defined positive upwards while the liquid volumetric flux isdefined positive downwards. However the in situ phase fraction isdetermined by the absolute value of the volumetric flow rates),which means that they have the same volumetric flow fraction.The consequence of this is that, although the liquid fraction is re-duced by the contraction of flow area, the volumetric liquid over-flow rate does not change. This is similar to when the liquid fractionis increased by the addition of wash-water. In this case, the volu-
0.90 0.95 1.00
-0.5
0.0
0.5
1.0
j gf(m
m/s
)
α
X=0
contra
ction ratio
decrea
sing to
zero
X = 1
-0.5
0.0
0.5
1.0
Fig. 8. Calculated series of operating lines when the flow area reducing to zero afterthe contraction.
'gj
'fj
gj fj
Fig. 9. Schematic diagram of a foam flowing vertically through a sudden expansion.The dashed box denotes the control volume, which contains the changes in flowarea and flow pattern.
metric liquid over-flow rate does not change either. Note that thedifference between the in situ phase fraction and the volumetricflow fraction, which arises because of a finite slip velocity betweenthe gas and liquid phases, has been overlooked by some researchers(see, e.g., Deshpande and Barigou, 2000; Maruyama et al., 2006; Wuet al., 2010), resulting in erroneous interpretation of experimentaldata and problematic modelling.
3.2. Foam flow through an expansion
Fig. 9 shows a schematic diagram of a foam flowing verticallyupwards through a sudden expansion. Similar to the contractioncase, a control volume is drawn to contain the expansion vicinityand beyond the control volume, the foam flow is 1D. After theexpansion, the superficial gas velocity decreases as a consequenceof the increasing flow area due to the expansion. According toFig. 3, the corresponding equilibrium liquid superficial velocity willdecrease. The extra amount of liquid existed in the foam below theexpansion will be liberated to drain down the column. Since thefoam in the larger pipe should be at equilibrium if no wash-wateris added, its operating line can be determined directly from systemproperties and operating conditions. Consider an expansion ratiodefined exactly the same as the contraction ratio, i.e., the ratio ofthe flow area after the expansion to the flow area before the expan-sion, the operating line of the foam before the expansion can bedetermined by Eq. (15) as well. Note that X now is greater thanunity. Fig. 10 shows the two operating lines before and after anexpansion of flow area where superficial gas velocity is reducedfrom 10 mm/s to 5 mm/s, i.e., X = 2. The operating line for a foamat the same superficial gas velocity but in a straight column, issuperimposed. Detail of the operating lines around a! 1 is shownin Fig. 11. It can be seen that the gas fraction of the foam before theexpansion is not only lower than that of the foam after the expan-sion, but it is also lower than when the foam is under normal con-ditions (i.e., without expansion or contraction). This is because theliquid liberated by the foam after expansion effectively acts aswash-water to the foam beneath, which increases the liquid frac-tion and decreases the gas fraction.
This is again verified by experimental results as shown inFig. 12. The column used for performing the contraction experi-ments as described earlier was reconfigured so that the foam canflow vertically upwards from the smaller pipe (internal diame-ter = 50 mm) to the wider section (internal diameter = 90 mm),causing an expansion of flow area with an expansion ratio of 3.2.It can be seen from Fig. 12 that for all the gas flow rates tested(0.0033, 0.0067, 0.010, 0.013, 0.017 L/s), the gas fraction always in-creased after the expansion (i.e., liquid fraction decreases). The gas
0.7 0.8 0.9 1.0
-1
0
1
2
3
4
C
B gas fractionbefore expansion
j gf(m
m/s
)
α
gas fractionafter expansionA
-1
0
1
2
3
4
Fig. 11. Details of Fig. 10 in the foam regime. Points A and C represent theconditions of the foam before and after the expansion, respectively. Point B is anormal operation point.
0.000 0.005 0.010 0.015 0.020
0.5
0.6
0.7
0.8
0.9
1.0 After expansion Before expansion
α( −
)
Qg (L/s)
incr
ease
in
gas
frac
tion
Fig. 12. Experimental results at five different volumetric gas flow rate showing thatthe liquid fraction decreases across an expansion. System properties were the sameas in Fig. 7. The expansion ratio was X = 3.2. The volumetric gas rates were from0.0033 to 0.017 L/s with an increment of 0.0033 L/s.
A BFig. 13. Illustration of: A. a conventional froth launder and B. a new concept inlaunder design.
808 X. Li et al. / International Journal of Multiphase Flow 37 (2011) 802–811
fraction of the foam before the expansion is very close to that of thefoam before contraction (i.e., Fig. 8), indicating that both are atthe same equilibrium condition. Because the internal diameter ofthe column is about 100 times bigger than the diameter of thebubbles and also because the foam stabilised by the 2.92 g/L SDSsolution has very high stability, no noticeable foam coalescencewas observed in our experiments. If the foam is of low stabilityand the column diameter is not significantly bigger than the diam-eter of the bubbles, sudden expansion may cause an increase inbubble size as observed by Deshpande and Barigou (2001).
Table 3Effect of contraction and expansion of flow area on foam behaviour.
Volumetric liquidover-flow rate
Volumetric liquidfraction
Sudden expansion Reduces ReducesSudden contraction No change Reduces
3.3. An example of the application of the analysis on device design
Based on the above analysis, a new design of froth launderwhich may be beneficial to the foam fractionation process is illus-trated in Fig. 13B, as well as the conventional froth launder (A) forcomparison. The new froth launder has a gradual expansion sec-tion, where the superficial gas velocity decreases. According tothe analysis above, when the foam enters the expanding section,it will liberate liquid, which reduces the liquid over-flow rate.The traditional froth launder also causes a sudden expansion offlow area, but much of the liquid will report to the product streambecause the outlet is at a lower position than the top of the actualcolumn, as shown in the graph. Conversely, the outlet of the newfroth launder is located above the top of the column, so the liber-
ated liquid will drain back to the foam beneath, and only the dryfoam will exit the system from the outlet.
Depending on the adsorption isotherm which gives the equilib-rium condition between the surface excess and bulk concentrationof a species that adsorbs to a gas–liquid interface, the liquid liber-ated from the expanding foam may serve as internal reflux to thefoam beneath. Internal reflux is normally a consequence of bubblecoarsening within the bulk foam (Stevenson et al., 2008b), whichenhances interfacial adsorption but has a detrimental effect oninterfacial area. The new froth launder has the potential to engen-der internal reflux without sacrificing surface area.
Therefore, the newly proposed froth launder design has the po-tential to reduce the volumetric flux and enhance interface adsorp-tion at the same time, without impacting the surface area flux. Allof these factors favour enrichment in the foam fractionationprocess.
Another example of utilising the contraction and expansion offlow area for process intensification in foam fractionation is theinsertion of a foam riser tube mounted at the centre of a plate intoa conventional straight column, thus the foam is forced to flowthrough a contraction followed by an expansion (Li et al., 2010).The foam riser device has advantages over the new froth launderas illustrated in Fig. 13 in terms of that it does not increase the sizeof the device and that it can be retro-fitted into existing columns.
4. Conclusions
The stability analysis of a rising pneumatic foam by Stevenson(2007) was used in conjunction with the drift-flux model of Wallis(1969) through continuity considerations to analyse the flowbehaviour of pneumatic foam flowing vertically through an expan-sions or a contraction. It is demonstrated, by theoretical analysisand by experimental results, that both the expansion and contrac-tion of flow area cause an increase in gas fraction of the foam, i.e., adecrease in liquid fraction. However, a contraction does not affectthe delivered liquid fraction, thus has no effect on foam fraction-ation enrichment. The expansion, on the other hand, does reducesthe overall liquid flux, by returning some of the liquid to the foambelow the expansion. The conclusions of this analysis are summa-rised in Table 3. The analysis is independent of the local changes invelocity profile in the vicinity of the contraction or expansion, andit has predictive capability in describing the behaviour of the foam.
X. Li et al. / International Journal of Multiphase Flow 37 (2011) 802–811 809
Acknowledgements
The authors would like to acknowledge financial support fromthe Discovery Projects scheme of the Australian Research Councilunder Grant Number DP0878979. They are grateful to Mr RaymondHoffman of the Department of Chemical and Materials Engineer-ing, University of Auckland, for his workshop support.
Appendix A
A.1. Determination of phase fraction in pneumatic foam by pressuregradient measurement
The pressure gradient in a steady vertical two-phase upwardflow is the sum of two components: (1) the hydrostatic pressuredue to gravity of the two phases and (2) the shear stress impartedon the flow by the column wall. In the case of pneumatic foam,both components are functions of the phase fraction. Howeverfor the hydrostatic pressure gradient there is a simple relationship,i.e., dP=dx ¼ qge. Therefore if we can decouple the two compo-nents, we will be able to obtain the liquid fraction. In this study,the effect of the wall shear stress was decoupled by comparingthe pressure gradient in a moving foam to that in a static foam.Note that Deshpande and Barigou (2000, 2001) measured the pres-sure gradient and inferred that it was entirely due to wall shearstress, which is incorrect.
Fig. A1 shows the apparatus used in this study. The apparatuswas similar to that used by Shaw et al. (2010), except that a buffertank was used to isolate the peristaltic pumps from the foam col-umn, so that the mechanical noise caused by the pumps wasminimised.
The evolution of the pressure gradient of a freely draining foamwas first obtained. This was done by making a rising foam, waitingfor steady state to be achieved (which could be ascertained from
riovreseR
Buffer tank
Washing pump
Foamate recirculation
Feeding pump
Foam motion control pump
Fig. A1. Schematic diagram of th
the real-time pressure signal displayed on the computer screen),and then cutting off the liquid and gas supplies simultaneously.The pressure signal was continued to be recorded at a frequencyof 2 Hz, and a typical result is plotted in Fig. A2 (Curve A). The freedrainage took approximately 10 minutes.
A steady-state rising foam was produced again under identicaloperating conditions and the gas and liquid supplies were cut offsimultaneously. However, instead of leaving the foam to drainfreely and unmolested, upwards and downward motion was super-imposed on the foam using the foam motion control pump, with aperiod of approximately 60 s. The pressure signal was recordedduring the operation and a typical cycle is labelled in Fig. A2 (CurveB). Detail of Fig. A2 from t = 300 s to t = 360 s is shown in Fig. A3.The difference in the pressure gradient between the peak valueat the sudden stop and that of the freely draining foam is inter-preted as the consequence of shear stress.
The above procedure was repeated for several different liquiddynamic viscosities and with different speeds of the foam motioncontrol pump. Two sets of results are shown in Fig. A4 as an exam-ple. Unsurprisingly, the effect of wall shear stress is most signifi-cant for dry foam. These results were used to correct the liquidfraction measurements.
Appendix B. Multiple contractions or expansions and foamrheology
The main conclusion of the analysis performed in the currentpaper, expressed prosaically, is that, for a vertical column witheither an expansion or a contraction, the hydrodynamic conditionsin the column are determined by the section of the column wherethe foam has the lowest equilibrium liquid fraction and liquid flux,with the proviso that each section the flow of the foam is fully-developed and liquid is not added or withdrawn from the foam.The assertion is still valid for multiple contractions or expansions.For example, if a foam undergoes two contractions in a series, the
P
Froth launder
Pressure transducer
Pressure transducer
Flow meter
Pressure regulator
Data acquisition system
e experimental equipments.
Pre
ssur
e G
radi
ent
(kP
a/m
)
-1
0
1
2
3
0 120 240 360 480 600 720
Time (s)
Free drainage of static foam
Foam moving up and down
sudden stop
foam starts to rise
foam starts to fall
A
B
Fig. A2. Decoupling the contributions of hydrostatic pressure and wall shear stressto the pressure gradient in the foam by suddenly stopping the foam from rising.
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
300 310 320 330 340 350 360
Pre
ssur
e gr
adie
nt (
kPa/
m)
Time (s)
Free drainage of static foam
Foam moving up and down
total
gravitational
frictional
Fig. A3. Decoupling the frictional pressure gradient and the gravitational pressuregradient.
0
20
40
60
80
0.001 0.01 0.1 1
Con
trib
utio
n of
she
ar s
tres
s (%
)
Liquid fraction of the foam (-)
m = 1 mPa s
m = 2 mPa s
µ = 1 mPa·s
µ = 2 mPa·s
Fig. A4. The contribution of wall shear stress to the total pressure gradient at twodifferent bulk viscosities and a wide range of liquid fractions.
'gj
'fj
gj fj
Fig. A5. The effect of a contraction on the flow pattern of the foam can beapproximated by a series of contractions via the rectangle rule. Only threerectangles are drawn here for clarity.
810 X. Li et al. / International Journal of Multiphase Flow 37 (2011) 802–811
hydrodynamic conditions of the foam in both sections are directlydetermined by the equilibrium states of the foam before thefirst contraction. The hydrodynamic state of the foam after the sec-ond contraction is independent of the foam state after the firstcontraction.
The velocity profile of the foam in the vicinity of an expansionor a contraction can be approximated by a series of contractionsand expansion, as illustrated in Fig. A5. It can now be inferred that
j0f is directly determined by jf via mass balance and it is not affectedby the actual velocity profile in the developing region (which isdependent upon foam rheology). The one-dimensional model thushas predictive capability for the hydrodynamic state of the foam inthe fully-developed flow above the contraction without knowledgeof the nature of the flow in the vicinity of the contraction. The sameconclusion can be drawn in the case of an expansion.
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