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Langmuir 1995,11, 1381-1391
A Network Model of Static Foam Drainage M. Gururaj, R. Kumar,? and K. S. Gandhi" Department of Chemical Engineering, Indian Institute of Science, Bangalore 560012, India Received June 6, 1994. In Final Form: January 30, 1995@ On the basis of a more realistic tetrakaidecahedral structure of foam bubbles, a network model of static foam drainage has been developed. The model considers the foam to be made up of films and Plateau borders. The films drain into the adjacent Plateau borders, which in turn form a network through which the liquid moves from the foam to the liquid pool. From the structure, a unit flow cell was found, which constitutes the foam when stacked together both horizontally and vertically. Symmetry in the unit flow cell indicates that the flow analysis of a part of it can be employed to obtain the drainage for the whole foam. Material balance equations have been written for each segment of this subsection, ensuring connectivity, and solved with the appropriate boundary and initial conditions. The calculated rates of drainage, when compared with the available experimental results, indicate that the model predicts the experimental results well.
1. Introduction Foam beds have been extensively investigated both as gas-liquid contactors and as separation devices. The performance of these devices depends strongly on the liquid holdup associated with the foam. The liquid holdup varies with height for continuous columns and with time as well for batch foams. A number of attempts exist in the literature for predicting the holdup values for both continuous and batch columns. Though the earlier drainage expressions were empirical, later workers have tried to develop models taking into account the actual structure ofthe foam. For low holdups, typical of foams, the gas bubbles are distorted and are polyhedral in shape. This divides the liquid into films and Plateau borders. Haas and Johnson' were the first to attach separate roles to films and Plateau borders during the drainage process. They showed that flow through the films due to gravity was negligible. Instead films drain into adjacent Plateau borders due to capillary suction. The Plateau borders in turn form a network, through which the liquid flows due to gravity. This mechanism of foam drainage has been accepted by most of the later investigators, though the details of the various models differ. As the number ofplateau borders, their sizes, and their inclinations are required during modeling, the shape of the bubbles in the foam has to be obtained. Most of the investigators have assumed the bubbles to be pentagonal dodecahedral in shape. Different shapes have been assumed for Plateau borders by different investigators. Thus Haas and Johnson' assumed the Plateau border to be circular whereas Desai and Kumar2 assumed it to be triangular. Similarly a number of investigators (Ho et Miles et al.,* Haas and Johnson') have treated the Plateau border walls to be rigid whereas others (Desai and Kumar2)have treated them as partially mobile, the mobility being decided by the surface viscosity of the system. The rate of drainage strongly depends on the
* Author to whom correspondence should be addressed. +Also at Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore, India. Abstract published inAduanceACSAbstracts, March 15,1995. (1)Haas, P. A.; Johnson, H.F. Ind. Eng. Chem. Fundam. 1967, 6, 225-233. (2) Desai, D.; Kumar, R. Chem. Eng. Sci. 1982,37, 1361-1370. (3) Ho, G. E.;Muller, R. L.;Prince, H.R. G. InDistillation; Institution of Chemical Engineers: London, 1969; p 2.10. (4) Miles, G. D.; Schedlovsky,Ross,J . J. Phys. Chem. 1945,49,93107. @
angle the Plateau borders make with the vertical. Both the concept of fixed angle (Haas and Johnson') and random angles (Leonard and Lemlich5)have been employed. Desai and Kumar6 have divided the Plateau borders into two categories, viz., the nearly horizontal and the nearly vertical Plateau borders, and have assigned different roles to the two. The films are assumed to drain into all the Plateau borders equally. The horizontal Plateau borders receive the liquid from the films and drain it into the vertical Plateau borders. Vertical Plateau borders receive liquid from films and horizontal Plateau borders as well as the vertical Platea borders above and discharge it into the vertical Plateau borders below them. This concept has been used by Desai and Kumar6 for analyzing the drainage of semibatch foam and more recently by Ramani et aL7 to predict the drainage in static foams. A further simplified model, but one that includes Plateau border pressure gradient, for static foam has been presented by Narsimhan.8 All these models use a continuum approach even though Plateau borders are of finite size. Recently Bisperink et aL9 have considered how the variation with time of bubble size distributions in foams can be used to monitor the progress of drainage, coalescence,and Ostwald ripening. But simultaneous consideration of all these phenomena is quite complex, and no quantitative analysis of the phenomena has been presented by them. Recently Bhakta and Khilar'O have attempted to avoid the continuum approach and calculated drainage rates by taking individual Plateau borders into account. They used a two-dimensional hexagonal network of Plateau borders, which was as employed by Princen'l and Khan and Armstrong12 during their investigations on foam rheology. They assumed that the two-dimensional network is adequate for a fair representation of the drainage of foams. As Bhakta and Khilar'O have not made a comparison with any existing data, it is difficult to decide whether the two-dimensional network models are realistic or not. The use of equal sized hexagons to form the network is another uncertain feature of the model. (5) Leonard, R. A.; Lemlich, R. AlChE J. 1965, 11, 18-25. (6) Desai, D.; Kumar, R. Chem. Eng. Sci. 1983,38, 1525-1534. (7) Ramani, M.; Kumar, R.; Gandhi, K. Chem. Eng. Sci. 1992,48(3), 455-465. (8)Narsimhan, G. J.Food Eng. 1991,14, 139-165. (9) Bisaerink. C.: RanteltaD.A.: Prins. A. Adv. Colloid InterfaceSci. i ~ S 2 , 3 8 , ~ i 3 - 3 2'. (10)Bhakta, A. R.; Khilar, K. C. Langmuir 1991, 7, 1827-1832. (11) Princen, H.M. J. Colloid Interface Sci. 1983, 91, 160-175. (12) Khan,S. A.; Armstrong, R. C. J. Non-Newtonian Fluid Mech. 1986,22, 1-22. _
I
I
0743-746319512411-1381$09.00100 1995 American Chemical Society
Gururaj et al.
1382 Langmuir, Vol. 11, No. 4, 1995 A criticism similar to the latter is relevant to the models that take three-dimensional structure into account, since as mentioned earlier the bubble structure in foam has been assumed to be pentagonal dodecahedral by most of the investigators. This structure has been adapted in spite of it not being a space filling one. Williams13 has shown that foam is better represented by /3 tetraidecahedron, a 14-sided polyhedron which has two quadrilateral, eight pentagonal, and four hexagonal faces. This shape also satisfies Plateau's laws and is space filling. This structure also generates a three-dimensional network of Plateau borders, through which drainage can occur due to gravity. In the present work, an attempt has been made to evaluate drainage rates through a three-dimensional network generated through the /3 tetrakaidecahedron structure. The calculated drainage rates have been compared with the experimental data available in the literature. The foams considered here are composed of only two phases, viz., liquid and gas. Solids can also be present in some situations, and these can influence the drainage phenomena. The present work does not address the influence of the presence of solid particles in foams. 2. The -posed
Figure 1. The /Itetrakaidecahedron.
Network
In a polyhedral foam, the liquid is divided into films and Plateau borders. The Plateau borders form a network through which the liquid flows down. The number of Plateau borders in the network and the inclination of each Plateau border with the vertical are decided by the actual shape of the foam bubble. The pentagonal dodecahedral shape normally assumed for bubbles while developing the continuum models, though useful for engineering analysis of foams, has three drawbacks. Firstly, it is not space filling. When stacked together, the bubbles leave 3% space unfilled, whereas actual foams can have liquid holdup values well below 1%.Secondly, it does not satisfj. Plateau's laws exactly. Thirdly, this structure would predict all films to be pentagonal, whereas in actual foams, hexagonal and quadrilateral films have been reported by Matzke14J5and co-workers16to be present to an extent of 40%. Thus, there is need to employ an alternative structure, which would at least overcome some of the limitations of the pentagonal dodecahedron. Lord Kelvin" suggested the minimal tetrakaidecahedron as the space filling polyhedron having the smallest surface to volume ratio. However, this shape has only quadrilateral and hexagonal faces, whereas a preponderance of pentagonal faces is found in actual foams. Thus, this structure cannot be used to describe foam bubbles. Williamsla Wncen and Levinsonla have suggested that B tetrakaidecahedron is a more likely shape to be formed in nature. Ross and PrestIghave discussed all three shapes and have found that each one has some weakness. The main weakness ofp tetrakaidecahedron is that it does not have the minimum surface area to volume ratio. However, it is the structure which displays many features observed in foams. This shape is shown in Figure 1. I t has 14 faces, out of which two are quadrilateral, four hexagonal, and the remaining eight pentagonal. Its surfaces are curved. A cluster of bubbles having this shape has a body (13) Williams, R. E.Science 1968,161,276-277. (14) Matzke, E. P m . NatZ. Acad. Sci. U.SA. 1945,31,281-289. (15) Matzke, E. Am. J . Bot. 1946,33 (58), 58-80. (16) Matzke, E.; Nestler, J. Am. J . Bot. 1946,33,130-144. (17) Kelvin, L. P m . R. Soc. Landon 1894,55, 1. (18) Princen, H. M.; Levins0n.J. CoZZoid Interface Sci. 1987,120(1), 172-17.5. (19) Ross, S.; Prest, H . Colloids Interfaces 1986,21, 179-192.
Figure2. Model of foam constructedusingtetragona packing of tetrakaidecahedral bubbles. centered tetragonal lattice,13which is symmetric but not isotropic. It represents foam more realistically because it has 14 faces, which is very close to the experimentally observed value of 13.7 in bulk foams,14-16though it does not satisfy Plateau's laws exactly. Further, the average number of sides per face given by this structure is 5.143, which is in close agreement with the observed value of 5.196.13 Thus, in the present investigation, the foam bubbles are assumed to be having tetrakaidecahedral shape, because this shape yields the number of sides per bubble and fractions of various shapes of films, which resemble closely with those measured experimentally. As the orientation of the bubbles in the foam can influence drainage, it is necessary to decide upon the stacking arrangement. This was arrived a t by observing the first layer of bubbles at the liquid pool-foam interface. The bubbles were found to be oblate and closely packed. Such a configuration can be obtained when the /3 tetrakaidecahedrons are arranged in such a way that four pentagonal faces along with four triangular sections of the hexagons form the top portion of the first layer. With this as the starting layer, a vertical stacking arrangement of bubbles following body centered tetragonal lattice was considered appropriate to represent the foam column. A perspective view of such a stacking is shown in Figure 2. The symmetries inherent in the structure have been exploited to define a unit flow cell, the drainage through which permits the evaluation of the overall drainage rate.
Langmuir, Vol. 11, No. 4, 1995 1383
A Network Model of Static Foam Drainage
A
Layer N - 1
Layer N
A LayerN+l
Figure 4. A unit flow cell. A typical column can be formed by stacking one flow cell atop another. The unit flow cell can be easily seen in Figure 3 also.
10
Figure 3. A perspective of a typical vertical column of foam. If it is joined to its reflection on a plane perpendicular to the half Plateau borders (e.g., 2, 10)and the process repeated, a sheet of flow cells is obtained. If these sheets are stacked on top of each other and joined, foam is formed. When stacked in this fashion, six edges (Plateau borders) of the total 36 edges of each of the /? tetrakaidecahedron make an angle of 0" with the vertical, six others go", and the remaining 24 make an angle of 60". Another possible stacking arrangement can be obtained by rotating the arrangement of Figure 2 by d 2 around an axis perpendicular to the paper. That arrangement was also analyzed in the present work. As the results obtained by the two arrangements are very close to each other, the analysis of the arrangement shown in Figure 2 alone is presented. 2.1. Symmetry in the Network. A physical model of the foam, based on the /? tetrakaidecahedron, was built (Figure 2). The network so formed possessed symmetries, which could be exploited for analyzing the flow of liquid in it. Thus the network was examined to find repeatable structures which would represent foam and flow through it, quantitatively. A column of foam films and Plateau borders could be identified which, when joined with mirror reflections of it, could generate the foam structure. As far as flow of liquid is concerned, the columns therefore could be taken to be independent of each other with no net exchange of liquid from one adjacent column to another. It should be emphasized that in reality there will be an exchange of liquid from one column to another, but the net mass exchanged would be zero. The column identified is shown in Figure 3. This column would be mathematically treated as independent from and identical to the other columns which together represent the foam. The column thus formed was further examined to identify a repeatable unit flow cell. It should consist of a group of connected Plateau borders and films, which when stacked one atop another without staggering and in the same orientation will generate the representative column. This will simplify the analysis of the problem, since the flow in the column can be visualized as the flow from one unit flow cell to the one below it. One such flow cell, derived from the column shown in Figure 3, is shown in Figure 4. Four such unit flow cells can be joined to form a bubble as shown in Figure 5. These flow cells are a set of films and Plateau borders which when put together can represent the flow in foam. Twenty-four Plateau borders, eight pentagonal films, four hexagonal films, and two quadrilateral films can be assigned to the unit flow cell shown in Figure 4. Four pentagonal films and one quadrilateral film are entirely contained in the cell itself. The rest are what it shares
Figure 5. A foam bubble constructed out of unit flow cells. A few hexagons, pentagons, and quadrilaterals are marked h, p, and s, respectively.
II
I1 61116
9
IO
9 9
9 10
Figure 6. Front view of a unit flow cell. The top view shows the 4-fold symmetry. with other unit flow cells. Similarly, while 22 Plateau borders are wholly contained in the cell, the other two are shared by it with other cells. The number of Plateau borders in a foam bubble is 12,with four pentagonal films, two hexagonal films, and one quadrilateral film. The unit flow cell shown in Figure 4 thus effectively represents two bubbles. The unit flow cell chosen for the analysis was examined for further symmetries. There were two planes of symmetry in the direction parallel to gravity. The symmetry is shown in the Figure 6. This helps simplify the problem by reducing the number of variables to be kept track of. 2.2. Connectivity in the Network. The Plateau borders of a cell were identified by assigning numbers to them. The first Plateau border of the layer is the top Plateau border of the layer. Subsequent Plateau borders were numbered as shown in Figure 6. The numbering is shown in Figure 3 as well. It may be noted that the 4-fold symmetry has allowed us to classify all the Plateau borders appearing in the unit flow cell into only ten categories. It
Gururaj et al.
1384 Langmuir, Vol. 11, No. 4, 1995
8k 1
10 Figure 7. One-fourth of a unit flow cell. Net work selected for analysis.
k 1 2 3 4 5
6 7 8 9 10
#k
egl 0 eg112 ,Og1/2 0
Table 1. The Table of Coefficients receives drains sk pk Hk liquid from liquid into 1 1 0 0 1 1 1 0
egl 0 eglI2 @gi12 o 0 1
0
2 2 2 2 0
2 2 2 2
2 0 1 1 0 2 0 1 1 0
9, 10 ofprevious layer 3 doesnot receive 3 1,2 4 3 6 doesnot receive 6
4,5
a
doesnotreceive 6, 7
8 9 1of next layer 1of next layer
a
doesnot receive
is also to be noted that the second and tenth Plateau borders are shared equally by adjacent columns. Thus only half of each of them is assigned to the unit flow cell. The unit flow cell contains only three types of films, quadrilateral, pentagonal, and hexagonal. These were identified by f s , f,, and fi, respectively. For analyzing the drainage phenomenon, it is necessary to know the connectivity between the various films and Plateau borders. In general, three films meet to form a Plateau border. It is only these films that drain into the particular Plateau border. Let S be the number of quadrilateral films, H the number of the hexagonal films, and P the number of pentagonal films which form the Plateau border. For each Plateau border, the number S, H , and P are shown in the Table 1. E.g., one square film and two hexagonal films drain into the first Plateau border. One of the four symmetric parts of the unit flow cell is displayed in Figure 7. As seen from it, the Plateau borders are connected to each other forming a network. The amount of liquid that flows out of a Plateau border will be due to the combined effect of gravity and the pressure difference across its length. The total input of liquid to any Plateau border (k)from other Plateau borders (j,1, ...I will be the sum of the amount of liquid drained into the Plateau border ( k ) due to flows out of Plateau borders (i, 1, ...). Hence the connectivities of the Plateau borders are also needed and are listed in the Table 1. 2.3. Structural Parameters. As mentioned earlier, the dimensions of the Plateau border and films are fixed once the shape ofthe bubble is chosen. From the published data (Princen and Levinsonla)on the minimal tetrakaidecahedron, the linear dimensions of the Plateau borders were found. These dimensions are the same in the /3 tetrakaidecahedron also. The various structural parameters are related to R,the radius of a sphere whose volume is the same as that of the bubble. These are area of films ((i) the area of the (ii) the area of the quadrilateral face is Af, = 0.5687R2; pentagonal face is Af = R2;(iii) the area of the hexagonal face is Afh= 1.293R2pand length of a Plateau border (all Plateau borders are of equal length, and the length is given by 1 = 0.72R).
The gravitational force is also a driving force for the drainage through a Plateau border, and depends on the angle the Plateau border makes with the vertical. If this angle is 8 and if 1 is the length of the Plateau border, the body force per unit area, 4, is given by egg2 cos 8. Clearly Plateau borders 1 and 6 are aligned with gravity while Plateau borders 2,5, 7,and 9 are horizontal. Thus for these, the value of q5 is QgZand 0, respectively. Measurements on the model indicated that all the other Plateau borders were inclined at nearly the same angle, it being n13. The angles were measured from the physical model built for the purpose: (i)The angle made by the Plateau borders 1 and 6 is 0". (ii) The angle made by the Plateau borders 2,5,7, and 10 is 90".(iii) The angle made by the Plateau borders 3,4,8, and 9 is 60".From the angle given above, 4 for the Plateau borders 3,4,8, and 9 is ~ g l l 2 The . values of q5 for the various Plateau borders are listed in Table 1. 2.4. Notation. In the course of the analysis, it would be necessary to refer to properties of films and Plateau borders belonging to different unit flow cells. The following notation was adopted to denote these. The unit flow cells were numbered so that the topmost flow cell in contact with gas formed the first cell and the last unit flow cell was the one which is at the foam-liquid pool interface. The various properties of films and Plateau borders belonging to different layers of unit flow cells were distinguished as follows. To every property, a superscript and two subscripts were added. The subscripts pb and f denoted whether property is that of a Plateau border or a film. In the case of a Plateau border its number was added as one more subscript to indicate the particular Plateau border being considered. The superscript indicated the layer to which the Plateau border or film belonged. Thus P9,,3refers to the pressure of the third Plateau border of [he ninth layer. In general the references are f, i for the films, (pb, k ) for Plateau borders, and n for layer. 3. Drainage Process The liquid in the foam is divided between the films and the Plateau borders. As mentioned earlier, films drain into Plateau border due to the pressure differencebetween the films and Plateau border. We assume here that films drain equally into all the Plateau borders surrounding it; e.g., each of the four Plateau borders that frame a quadrilateral film will receive one-fourth of the liquid draining from this film. The liquid flows down the Plateau borders due to gravity. In addition, when the liquid holdup in a Plateau border, and hence its radius of curvature, is different from that in its neighbors, there will be a pressure difference between them and flow can also be created due to this. The equations for describing drainage of films and flow through the Plateau borders are discussed below. 3.1. Drainage of Films. All the existing drainage models assume the thinning rate of the polygonal films to be equal to the thinning rate of a circular film of same thickness and area. This makes it possible to use existing analyses of film thinning. The same approximation was adopted here as well. Films drain due to the lower pressure in the Plateau borders. For the purposes of drainage, the pressure gradient sensed by all films in a layer was assumed to be the Plateau border pressure corresponding to the first Plateau border of that layer. This is not a n unreasonable assumption since the variation in the holdup in a single layer will not be large. If this assumption is not made, the film will see different pressures around its circumference, and drainage under such conditions will not allow the two surfaces of the films to be parallel. Thus, the assumption was made to focus on the network analysis and avoid further complications.
A Network Model of Static Foam Drainage
Langmuir, Vol. 11, No. 4, 1995 1385
The Reynolds equationz0for the rate of thinning of an i type of film of the nth layer is
where the reference pressure in the bubble is assumed to be zero. The Reynolds equation is being used anticipating that the surfactant concentrations in the experiments would be such that the interfaces can be treated as immobile. It has been shown that drainage of films is more complex, particularly when the film thickness is of the order of 100 nm. For example, Nikolov and Wasan2I have shown that phase changes occur during drainage. They have demonstrated a number of metastable equilibria. However, for these phenomena, the films have to be extremely thin making negligible contribution to the drainage rates. Hence in the present work, such phenomena have not been included and the Reynolds equation has been assumed to be applicable through out. From geometric considerations, it is seen that if di, i = 1 , 2 , 3 , are the different thicknesses of the films forming the Plateau border, the radius of the Plateau border can be approximated by
- 6,
+ 6, + 6,
+
&
$"' +
6, 3
+ 6,Y + O . 6 4 4 a p b
0.322
'pb
(2)
where a,b is the average cross-sectional area of the Plateau border. Usually, the thicknesses of the films forming the Plateau border are quite small, and the above equations can be approximated as
'pb
=
0.322
(3)
Figure 8. Flow through a Plateau border network segment.
border, the term dPldz has been expressed as
dP - (Qgl cos 8 -
dz-
+ hp)
(6)
1
Here AP is the appropriate pressure differefice arrived at as follows. The pressure in the Plateau border defined by eq 4 has been nominally assigned to the top of the Plateau border. Consider the situation shown in Figure 8. With the definition of the pressure adopted, for flow into the kth Plateau border, AP will be given by the difference in the pressures in the j t h and the kth Plateau borders. Combining eqs 4 , 5 , and 6, this flow rate can be calculated. It should be noted that this flow rate is also equal to the outflow from the j t h Plateau border. Similarly for flow out of the kth Plateau border, AP will be given by the difference in the pressures in the kth and the Zth Plateau borders. 3.3. The Drainage Equations. The material balance equations were written for the network described. These equations are given below: Films. Films do not receive any liquid, but only drain into Plateau borders. Thus eq 1 itself constitutes the material balance for a film. Applying that to the i type film in the nth layer we get
The pressure P in the Plateau border will then be
(7) (4) since the reference pressure in the bubbles was assumed
to be zero. Equation 4 was used in eq 1to calculate the rate of drainage for films. 3.2. Flow through a Plateau Border. For the flow through the Plateau border, the expression for the average velocity derived by Desai and Kumarz has been used but with one modification. For a Plateau border with immobile walls, their expression reads
These authors had considered flow due to gravity alone. As explained earlier, there could be a gradient in the Plateau border pressure when mechanical equilibrium does not prevail. This pressure difference between a Plateau border and the adjoining ones will depend on the holdup variation with height. Each Plateau border has been assumed t o be of constant cross section throughout its length. Assuming the pressure gradient to be linear over the lengths as small as the length of the Plateau (20)Reynolds, 0.Philos. Trans. R . Soc., London 1888,A177,157234. (21)Nikolov,A.;Wasan, D.J . ColloidInterfaceSci.1989,133,1-13.
Plateau Borders. The material balance for a Plateau border can be written as (Rate of accumulation of liquid in Plateau border) = (rate of inflow of liquid from films) (net amount of liquid exchanged with adjoining Plateau borders)
+
As explained earlier, the flow rate out of kth Plateau border can be calculated by combining eqs 4,5,and 6. Let this be denoted by f&,k. Now the kth Plateau border will receive liquid from other Plateau borders to which it is connected. Let them be j and 1 and these have been identified in Table 1. Since the flow rate out ofjth and lth Plateau border is equal to the flow rate into the kth Plateau border, the material balance can be written as:
where a k j are constants that account for the symmetry of the network. The right-hand side is listed in detail in Table 2. A kth Plateau border can in general be formed with Sh quadrilateral films, P k pentagonal films, and Hk hexagonal films. The amount of liquid that the Plateau border will
Gururaj et al.
1386 Langmuir, Vol. 11, No. 4, 1995 becomes
Table 2. The Flow Relations Plateau border 1 2
&in
Qout
~
3
4 5
For the ninth Plateau border of the last layer U , the pressure drop for the liquid flow will be
6
7 8 9
49
10
get from quadrilateral films is sk times one-fourth of the amount of liquid drained by one quadrilateral film. Then the amount of liquid given by the quadrilateral films to kth Plateau border will be
+ p;,9
- Ppoal
where PPl is the pressure in the liquid pool. In formulation of all the previous equations, the pressure ofthe bubble was taken as the reference pressure. Hence the pressure of the pool will be less than the pressure in the bubble by a factor of 2alR where R is the radius of the bubble. The flow of liquid into the pool, will therefore be given by
dq~
sk ---
4
dt
Expressions are similarly obtained for pentagonal films and hexagonal films. Therefore the amount of liquid the films give to any kth Plateau border can be written as
Ppool= -201R
Now the overall mass balance can be written for all types of Plateau borders. 3.4. Nondimensionalization. In order to minimize the number of parameters to be studied in the simulation, eqs 7 to 11 were nondimensionalized. The nondimensional variables have been defined as -
The flow out of the kth Plateau border will be
k = 1, 3 , 4 , 6 , 8 , 9 The value of 1 for each k can be found from Table 1. The completely horizontal Plateau borders 2,10,5,and 7 have to be treated slightly differently. Consider the first two. These are shared symmetrically by two columns. Thus it is necessary to assign the pressure of these Plateau borders to the middle of the Plateau border. Hence only half of the output of these is received by Plateau borders of any one column into which they discharge. Therefore, for these Plateau borders, we define 4ib,k to be equal to half of the total flow out of them. Thus we can write
t = tlz
(15)
P = VIV
(16)
P = PI@
(17)
= cos(p)/2
(18)
and gk
In the above z is a time scale for a fluid particle to travel through a Plateau border, v is the amount of liquid in a full Plateau border at time t = 0, and CP. is the drop in the gravity potential over one whole length of the Plateau border in the direction of gravity. The characteristic scales are then given by (19) @ = egl
k = 2, 5 , 7 , 10 In the above equations, Y b , k represents the volume of the Plateau border. Now consider Plateau borders 5 and 7. These discharge equal amounts into two of sixth and four of eighth type of Plateau borders, respectively. The situation is therefore exactly similar to Plateau borders 2 and 10. Hence these two were also treated by considering only half of their volume, and thus eq 11 is applicable for them as well. The ninth and tenth Plateau borders of any layer will have its flow influenced by the pressure in the first Plateau border of the next layer. For these Plateau borders the expression for the flow out will be still given by eq 10 but for being replaced by P$;. The first Plateau border of the top layer does not receive any liquid from top. Hence, the mass balance equation
cb,l
(14)
(20)
= k(413 (21) In terms of nondimensional variables, the film draining eq 7 becomes
(22) where Kf,,i’sare given by
Kf,i=
(23) The values of the Kf,i for the various shapes of films in the foam are: For quadrilateral film
K , , = 0.241
(24)
Kf,p= 2.19
(25)
For pentagonal film
A Network Model of Static Foam Drainage
Langmuir, Vol. 11, No. 4, 1995 1387
For the hexagonal film
Thus
(26)
Kf,h = 0.84
After substituting eqs 15-21 into eqs 10 and 11 and nondimensionalizing, the following equations were obtained.
(39) Thus in nondimensional form, the initial condition works out to be: a t t = ~
k = 3, 4, 6, 8; n = 1,..., U
q;b,k
= 9; n = 1,..., U
= (qb,d2[p= Af,p’12 Kf,s,the film area constant for quadrilateral defined by Kf,, = AfJ2 I, the length of the Plateau border, cm
Langmuir, Vol. 21, No. 4, 1995 1391 Pk, the number of pentagonal films that make up the kth Plateau border Ppool,the pressure in the liquid pool dynlcm2 P p b , the pressure of any Plateau border in general, dynl cm2 q b & ,the pressure of the kth Plateau border of the nth layer, dynlcm2 Qnb& the volume of liquid flowing out of the kth Plateau borier of the nth layer i&&, the nondimensional volume of liquid flowing out of the kth Plateau border of the nth layer R, the radius of the foam bubble r p b , the radius of curvature of the Plateau border s k , the number of quadrilateral films that make up the kth Plateau border E, time, s t, nondimensional time v;l,h,the volume of hexagonal film, cm3 Ti,the volume of film in general, cm3 qp,the volume of pentagonal film, cm3 qfs, the volume of quadrilateral film, cm3 V, the rate of film thinning cm s-l F h , the nondimensional volume of hexagonal film V& the nondimensional volume of film in general the nondimensional volume of pentagonal film V;ts, the nondimensional volume of quadrilateral film V,i, the volume of the film of type i T b , k , the volume of the kth Plateau border in nth layer, cm3 q b , k , the nondimensional volume of Plateau border
sp,
Greek Symbols
p, the coefficient of drainage of eq 5 the liquid holdup in the foam 6, 61, 6 2 , and 83, the film thickness 8, the angle which the Plateau border makes with the vertical e k , the angle which the kth Plateau border makes with the vertical p , the viscosity of the liquid &, the drop in gravity potential across one length of the kth Plateau border @, the drop in the gravity across a distance of one length of a Plateau border aligned with gravity cr, surface tension, dynlcm z, a time scale for a fluid particle to travel through a Plateau border v, the amount of liquid in a full Plateau border at time E,
t=O
Subscripts f,i, index for films in general f,h, index for hexagonal films f,p, index for pentagonal films f,s, index for quadrilateral films j , k, I, indices for Plateau borders in general pb, indicates Plateau border Superscripts n, index for any layer in general U,index for last layer (layer in the foam-liquid interface) LA940442M
