Ind. Eng. Chem. Fundam. 1083, 22, 105-109
105
Errors in the Measurement of Bubble Size Distribution in Foam Hslng C. Cheng' and Robert Lemllch" Department of Chemical and Nuclear Engineering, University of Cincinnati, Cincinnati, Ohio 4522 1
The determination of foam bubble size by planimetric measurement of the area of the contact face at the glass wall of a column containing essentially homogeneous new foam yielded excellent agreement with the bubble size as known a priori. Furthermore, the said determination proved to be more reliable than measurements based on either the longest chord or the commonly employed longest median of the contact face. However, for highly heterogeneous new foam, all three measures of foam bubble size underestimated the true mean. Apparently, small bubbles wedge large bubbles away from the wall. The effects are presented quantitatively, along with correlations for their approximate prediction.
Introduction The distribution of bubble sizes is one of the most important characteristics of a foam. It is commonly measured visually or photographically. However, such measurements are subject to several sources of error, as outlined below. 1. Statistical Sampling Bias. The plane (or other surface) that is viewed discriminates against the inclusion of a sufficient number of small bubbles (Clark and Blackman, 1948). They are small targets and so are preferentially missed by a sampling slice through a solid foam. This also applies to the bounding surface of a liquid foam in the absence of counterbalancing local segregation or distortion of bubbles. The error can be corrected theoretically (de Vries, 1957, 1972) by eq 1.
F(r) is the corrected size frequency distribution function of bubble radius r, and f ( r ) is the corresponding uncorrected function. Equation 2 is a useful corollary for the mean radius by the j t h and kth moments rjvk = r > - l , k - l (2) where
without the bias, and
with the bias, that is, sampled by the viewed surface. As a practical matter, the surface need not be planar, provided its radii of curvature are large compared to the bubble radii. Also, for numerical evaluation, integration is discretized to summation. Stricly speaking, eq 1-2 require that the bubbles be either favorably or randomly distributed in space. Of course, polyhedral foam bubbles are not randomly positioned, but the order is only short range. Therefore, by employing an internal sampling plane of sufficient extent, the validity of eq 1-2 is not compromised. However, sampling at a bounding surface is another matter, as will be discussed presently. By applying an approach analogous to that which led to eq 1 and 2, the corrections for statistical bias can be Brookhaven National Laboratory, Upton, NY 11973.
generalized to encompass sampling by a line or a group of random points, as well as the sampling of a plane or line. The generalized result for eq 2 is T ' ? k = ?-)-1,k-1 = rf>-Z,k-2 = r'f>..3,k-3, where the single prime denotes the surface as before, the double prime denotes the line, and the triple prime denotes the points. Details are on file (Cheng, 1980). 2. Bubble Distortion. At the retaining wall of a liquid foam, such as the glass wall of a foam column, the bubbles are distorted. This is depicted in Figure 1for a spherical bubble. By geometry and near-conservation of bubble volume, D, < Dd L D,. This leaves open the important relationship between D, and D,, with D, being the diameter that is most readily seen and usually measured. Jashnani and Lemlich (1973) briefly attacked the problem by attaching a small number of free bubbles to each other on a wet flat glass plate. Each bubble was approximately 0.6 cm in diameter. They found that the span across the contact face at the plate for a bubble that was otherwise completely surrounded by other bubbles was approximately the same as the diameter of the same bubble freely suspended in air. However, it is not certain that this finding would apply to the more common case of a large number of confined smaller bubbles of heterogeneous size. 3. Bubble Segregation. Chang et al. (1956) compared the bubble size distribution measured at the surface with that along a planar slice through frozen foam. They reported no significant difference. However, it is not clear to the present writers whether Chang et al. measured the true diameter of the cut bubbles or just the more accessible diameters of the circles in the cutting plane (Lemlich, 1982). Furthermore, Leonard (1964) reported that in a column of liquid foam large bubbles were hidden from the wall, being wedged away by smaller bubbles. Thus the question of segregation a t a bounding surface is open. 4. Differences in Stability. As a result of interbubble gas diffusion, large bubbles grow a t the expense of small bubbles (de Vries, 1957, 1972). The rate and manner in which this occurs depends on bubble curvature, which in turn depends on bubble distortion and size distribution (Lemlich, 1978; Ranadive and Lemlich, 1979). As indicated above, a t the bounding surface both of the latter may be different from what they are within the bulk foam. Furthermore, gas diffusion through the bubble face on the wall is restricted, Thus, as the foam ages, further error may be expected to develop in bubble size measurements at the surface. Finally, such error may be additionally compounded if, due to differences between surface and bulk, the rate of rupture of the lamellae between bubbles is also different at the surface.
0196-4313/83/1022-0105$01.50/00 1983 American Chemical Society
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Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983
Undistorted
Distorted
Figure 1. Distortion of a spherical bubble by a retaining wall.
To Foam Collector Glass Column Optical Box
\I
Liquid Out-/ \Liquid Gas in
In
Figure 2. Liquid pool and foam column (not to scale).
Accordingly, the present investigation was undertaken to examine some of the foregoing sources of error in detail and, to the extent feasible, to propose methods for estimating their individual and combined effects. Experimental Section General. The approach employed was to generate a foam of a predetermined (and hence definitely known) distribution of bubble sizes in a vertical glass column and then examine it photographically at the glass wall. In this way it was possible to compare mutually the apparent and true bubble size distributions. A stream of nitrogen was passed through a humidifier, an entrainment separator, and a 2-L surge vessel, into one or more carefully prepared vertically mounted bubblers located near the bottom of a 12-L flask as shown in Figure 2. The flowrate through each bubbler was measured with a rotameter and the frequency of bubble formation was measured with a stroboscopic light. The rising bubbles then passed up through a static mixer and formed a foam atop the pool. A steady stream of liquid was withdrawn from the pool and pumped to a constant head tank (not shown) to reenter as feed on top of the foam. This kept the foam wet and so helped prevent bubble breakage. Any overflowing foam was allowed to coalesce in a foam collector (also not shown) and the resulting foamate returned to the bottom of the pool. The entire apparatus was located in an air-conditioned inner room the temperature of which was maintained at 24 f 1 "C. The vertical column was of 5.1 cm i.d. Pyrex pipe and 61 cm tall. I t was fitted internally with three pairs of platinized platinum electrodes of 2.5 cm diameter (Jones and Bollinger, 1935). They were mounted nearly flush at 15.2-cm vertical intervals along the wall. Together with a Beckman RC-18A ac bridge, they served as conductivity cells to monitor stability.
The pool consisted of the nonionic surfactant Triton X-100 in a 0.01 M KCl aqueous solution of 10 vol. % U.S.P. glycerine. The KC1 was incorporated to increase conductivity and the glycerine to increase foam stability. The concentration of the Triton X-100 was commonly 0.001 M, based on a molecular weight of 632.8. Bubble sizes were measured by flash photography through a rectangular optical box which was filled as needed with liquid of pool composition. Except when examining stability, care was taken to photograph foam that was newly formed. This was to avoid any significant changes in bubble size with time. For the examination of stability, foam was also photographed further up the column. For the foam the bubble diameter (or radius) was taken as the diameter (or radius) of the circle having the same area as the wall contact area of the bubble, A , the latter being measured from an enlarged photograph by means of a planimeter with due regard for scale. Thus the radius equals (A/?r)l12. For the pool the projected area of the bubble was employed, and the results were checked against the volumetric mean bubble radius taken as ( 3 V / 4 ~ 2 ) ' / ~ , where V is the volumetric gas flowrate and z is the frequency of bubble emission. Bubblers. Earlier investigations have shown that over certain ranges of gas flowrate, glass capillary tubes can produce bubbles of quite uniform size and frequency of emission (Brunner and Lemlich, 1963; Leonard and Lemlich, 1965). Thus, for example, a single such bubbler can produce a foam of virtually one known bubble size, while two different bubblers employed simultaneously can yield a foam of essentially just two known bubble sizes. However, careful choice, preparation, and use of the bubblers is critical. For the present investigation, four capillary tubes of differing inside diameter were employed, namely, 0.0075, 0.01, 0.035, and 0.05 cm. Each was 0.5 cm long and was ground with fine emery paper to remove any rough edges. Each was also cleaned with chromic acid after every run. When more than one capillary tube was used at a time, the distance between the exits of neighboring capillaries was about 1 cm. None of the bubble streams appeared to interfere with each other. The static mixer was employed to avoid spurious segregation in the foam due to the positioning of the capillaries in the pool. Further details are on file (Cheng, 1980).
Results a n d Discussion Unless otherwise specified, all results are for newly formed foam. Uniformly Sized Bubbles. First, runs were carried out with only one bubbler at a time. Determinations of rl,o,p, which is the arithmetic mean bubble radius obtained from the planimetric measurements of photographs of the pool through the optical box, agreed within f0.2% with the volumetric mean bubble radius obtained as described above from measurements of gas flowrate and bubble emission frequency. Planimetric measurements of the foam bubble faces in contact with the wall showed them to be of quite uniform size within a run. In terms of the radii, their standard deviation within a run, s $, averaged only 8.7 % . By applying eq 2 to the planimetric measurements, the arithmetic mean bubble radius within the foam, rl,O,f(which of course cannot be seen), is taken as equal to r'+l,f which is the harmonic mean bubble radius in the foam a t the glass wall. For 14 runs with homogeneous foam, r'o,-l,f averaged only a trendless 1.8% less than rl,o,p. This is deemed good agreement. Since segregation cannot be a significant factor here because the foam is quite homoge-
Ind. Eng. Chem. Fundam.. Vol. 22. NO. 1. 1983
107
Table 1. Comparison of Various Means of Bubble Size in Newly Formed Essentially Homogeneous Foam in Terms of Their Ratios to the Rue Bubble Size as Measured
Glass Wall
in the Pool
mean radius
av value of ratio of mean radius to r , , o P
r'o,-l,f ( I ' + t/z)o.-l.r
0.982
r;l.o.f
0.990
+E.l.O.f
1.063 1.028 1.038
'e.0;l.f
1.045 1.053
r:m.O;l.f
m.1.O.f
std dev
of ratio, 3'%
1.2 1.5 1.4 3.3 3.4 3.8 3.1
Figure 3. Stylized new showing Plateau borders between polyhedral bubbles and bounding glass surface.
1 - 49 . 0
7
8
10
11
'1.O.P x
~
9
12
13
14
15
[cml ~ 2
Figure 4. Ratio of unconwttd planimetrically measured apparent bubble size in newly formed essentially homogeneous foam to true bubble size. neous, the foregoing indicates that the effect of bubble distortion is very small. Figure 3 shows a stylized view of polyhedral foam huhbles at the wall in order to depict the Plateau borders (designated as PB). It is clear that the camera sees the surface spanned by a rather than by b. Accordingly, values of the PB thickness, t , were measured from the photographs and the radii were then augmented by f / 2 . As before, eq 2 was incorporated. The resulting ( r ' + t / 2 ) 0 , . l , averaged 4.5% more than rl,o.p.Of course this augmentation includes the film thickness between huhhles, the exclusion of which would decrease the 4.5% error. Since the foam is nearly homogeneous. the correction for statistical planar bias is small. Figure 4 shows a plot of results when the corrections from eq 2 and t / 2 are both omitted. The radius for the foam now averages only 1.0% less than for the pool. Also, there is no trend along the abscissa among the ordinates of the plotted points, which augurs well for extrapolation. Thus, curiously enough. the closest agreement is apparently obtained by ignoring the two corrections. Evidently cancelling errors are at work. In any event, for a homweneous foam. the effect of bubble distortion on planimetric measurement of bubble radius is slight. The results closely approximate the true buhhle radius. Table I summarizes the foregoing, along with the corresponding results for the foam bubble radius taken not as ( A / r ) l hut rather as half the longest chord (vertex to vertex) and half the longest median (vertex to midside), respectively. of the hlunted polygonal bubble face in contact with the wall. The reasonably close approximation to unity for the ratio hased on the longest chord, r'~,~ls/rl,o.p, and for the ratio based on the longest median, r~,o,.ls/rl,o.p, supports the results of Jashnani and Lemlich (1973) mentioned earlier. However, the scatter, as indicated by the tabulated standard deviation among runs, is greater than for r ~ , . l , / r l , o + :hence the reproducibility is less. Also, the difference of the average ratio from unity is g~eater,being 5.3% when h a d on the longest chord and
+Icm
-4
Figure 5. Newly formedfoam of M Triton X-100 in water with 10 voI % glycerine produced by a 0.0075 cm i.d. bubbler at a gas flowrate of 0.116 cm3/jsand a frequency of 73.0 8.' together with a 0.05 cm i.d. bubbler at a gas flowrate of 0.744 em3/s and a frequency of 41.3 8'. The longest chord, c, and the longest median, m, are illustrated for one bubble.
2.8% when based on the longest median. (If the radii are augmented by t / 2 , the differences become approximately 1 2 % and 9%, respectively). Thus these two quick measures of bubble size, the latter of which is in common use, do not exhibit the reliability of that based on the more tedious planimetric procedure. Accordingly, the planimetric procedure was used as the principal measure of foam bubble radius for the remainder of the present investigation. Nonuniformly Sized Bubbles. Runs were conducted with combinations of bubblers of the various sizes. Photographic examination of the upper portion of the pool revealed no significant segregation of bubbles by size therein. No coalesence was observed either. Figure 5 shows the foam for a run with just the two bubblers of smallest and largest inside diameter. They produced essentially just two bubble sizes and the most extreme combination thereof. Planimetric measurement of bubbles in the central portion of the figure reveals that the largest bubble has an apparent radius about 10% smaller than that of bubbles produced in the pool by the large bubbler. Even when increased by t / 2 , the said radius is still about 6% smaller than in the pool. The large bubbles as a group have an apparent arithmetic mean radius that is about 19% smaller than in the pool. When augmented by t / 2 it is about 15% smaller. Accordingly,
108
Ind. Eng. Chem. Fundam., Vol. 22, NO. 1, 1983 1.2
Number 01 Bubblers
SP b,O,p
Figure 6. Effect of inhomogeneity in bubble size on the planimetrically measured apparent bubble size in newly formed foam corrected for statistical planar sampling bias. 1.2
Figure 8. Effect of inhomogeneity in bubble size on the uncorrected planimetrically measured apparent bubble size in newly formed foam.
Number of Bubblers
0.8 I
0.7 1-7.0 0.0
0.1 0.2 0.3 0.4 0.5
0.0
0.1 0.2 0.3 0.4
rd,-1.f
r1 ,O,P
the small bubbles do seem to wedge the large bubbles away from the wall, thus causing the large bubbles to have smaller contact faces at the wall. As for the small bubbles, their apparent arithmetic mean radius is only about 2% smaller than that of the bubbles produced in the pool by the small bubbler. When increased by t/2 the said radius is about 6% larger than in the pool. These percentages for the small bubbles are quite comparable with the results presented above for homogeneous foam. This indicates that the large bubbles do not significantly distort the size measurements of the small bubbles. Again for the run of Figure 5 , the frequency of large bubbles in the pool is 39.3%. According to statistical planar sampling bias, the frequency of large bubbles at the glass wall should be 58.2%. However, a bubble count of the central region of Figure 5 reveals that only about 21 YO of the bubbles a t the wall appear to be large. This apparent segregation by size is further evidence of small bubbles wedging large bubbles away from the wall. The aforementioned qualitative observation of Leonard (1964) is thus supported. Also for this run, the arithmetic mean radius in the foam, r’l,o,f,even when increased by t/2, is 26% smaller than what it would be were it based on radii and frequencies measured in the pool with statistical planar sampling bias included. By comparison, a hypothetical based on radii from the pool but apparent frequencies in the foam a t the wall is 25% smaller. This leaves a difference of just 1% for evidential distortion overall. Thus the apparent segregation is by far the more important effect on the apparent average bubble size in the foam. For the various runs, the degree of inhomogeneity produced is given by sp/rl,o, , where sp is the standard deviation of bubble radii in tke pool within a run. Values for spranged from 0.0069 to 0.0391 cm, and rl,o,pranged from
5
si -
sp Figure 7. Effect of inhomogeneity in bubble size on the planimetrically measured apparent bubble size in newly formed foam corrected for statistical planar sampling bias and Plateau border thickness.
1
Figure 9. Relationship among statistical parameters observed in newly formed foam at the wall and the true bubble size.
::I1, ,
0.0
0.0
0.1 0.2
0.3 0.4
si r0,-1 ,f
Figure 10. Relationship among statistical parameters observed in newly formed foam at the wall and the true inhomogeneity of bubble size.
0.0784 to 0.1195 cm. Figures 6,7, and 8 (which correspond respectively to the first three lines in the body of Table I) show that the bubble size measured in the foam decreases appreciably as the inhomogeniety increases. Similar plots of results based on the quick but less precise methods of longest chord and longest median (omitted here for the sake of brevity) show a similar trend of decrease with inhomogeneity. Thus, Figures 6-8 correlate pertinent statistical parameters of bubble size distribution that are hidden within the foam with readily observable parameters a t the glass boundary. The matter is carried further in Figures 9 and 10. Only one hidden parameter is included in each, namely, the arithmetic mean radius and the standard deviation, respectively. Thus Figures 9 and 10 can be employed to estimate approximately the true values of these two important statistical parameters. Of course expression in terms of relative effects necessarily involves some autocorrelation, and predictions should not be ex-
Ind. Eng. Chem.
tended unduly to conditions beyond those of the present investigation. Stability. The investigation was limited to conditions under which no lamellar rupture was observed. However, as the foam aged, changes in bubble size due to interbubble gas diffusion were readily seen. Small bubbles shrank, large bubbles grew, and intermediately sized bubbles first grew and then shrank, all in qualitative accord with theory (Lemlich, 1978; Ranadive and Lemlich, 1979; Cheng and Lemlich, 1980). However, as shown above, even with new foam the bubble size distribution at the wall differs from the distribution in the interior. Moreover, as pointed out in the last part of the Introduction, the distribution at the wall undergoes a change with time that is not representative of the change in the interior. Therefore, the validity of the said referenced theory for interbubble gas diffusion cannot be properly tested in a quantitative manner by observations that are limited to changes in bubble size at a boundary wall. Conclusions 1. Although tedious, planimetric measurement is the most reliable way of optically measuring foam bubble size a t a boundary wall. 2. Alternatively, the longest chord or preferably the longest median may be used when accuracy can be sacrificed for convenience. 3. For a homogeneous foam, the unadjusted measurement of bubble size at a wetted boundary wall is a reliable way of determining true bubble size. 4. For a heterogeneous foam, which is the more usual situation, significant error may occur in interpreting the results of bubble size measurements at a wetted boundary wall. While bubble distortion may not seriously affect the mean, physical segregation at the wall which discriminates against large bubbles may more than override statistical planar sampling bias which discriminates against small bubbles. Figures 6-8 illustrate the effects of error and Figures 9 and 10 can serve as approximate predictors of the same, provided conditions are not extended too far beyond those of the present study including wettability at the wall. Nomenclature A = wall contact area of the foam bubble or projected area of the pool bubble, cm a = span between Plateau borders, cm b = span across bubble, cm D = diameter of bubble, cm F ( r ) = frequency distribution function of bubble radii with the correction for statistical planar sampling bias incorporated, cm-'
Fundam., Vol. 22, No.
1, 1983
109
f ( r ) = frequency distribution function of bubble radii un-
corrected for statistical planar sampling bias, cm-' j = order of moment k = order of moment r = radius of bubble, cm s = standard deviation of bubble radii within a run, cm t = thickness of Plateau border contact with the wall, cm
V = volumetric gas flowrate, cm3/s z = time frequency of bubble emission, s-' Subscripts c = longest chord in the contact face
d = distorted bubble f = in the foam j = mean bubble radius partly by the jth moment j-1 = mean bubble radius partly by the 0'- 1)th moment (-2 = mean bubble radius partly by the 0'- 2)th moment 5-3 = mean bubble radius partly by the 0'- 3)th moment k = mean bubble radius partly by the kth moment k-1 = mean bubble radius partly by the (k - 1)th moment k-2 = mean bubble radius partly by the (k - 2)th moment 12-3 = mean bubble radius partly by the (k - 31th moment m = longest median in the contact face p = in the pool s = surface of contact between bubble and wall u = undistorted bubble 0 = mean bubble radius partly by the zeroth moment 1 = mean bubble radius partly by the first moment -1 = mean bubble radius partly by the negative first moment Superscripts = in the surface or measured at the glass wall If 'If
= on the line
= selected by the points
Literature Cited Brunner, C. A.; Lemiich, R. Ind. Eng. Chem. Fundam. 1063, 2 , 297. Chang, R. C.; Schoen, H. M.; Grove, C. S . , Jr. Ind. Eng. Chem. 1958, 4 8 , 2035. Cheng, H. C. Ph.D. Dissertation, University of Cincinnati, 1980. Cheng, H. C.; Lemlich, R. Ind. Eng. Chem. Fundam. 1080, 19, 133. Clark, N. 0.; Blackman, M. Trans. Faraday SOC. 1048, 4 4 , 1. de Vries, A. J.; "Foam Stability", Rubber-Stichting: Delft, 1957. de Vries. A. J. I n "Adsorptive Bubble Separation Techniques", Lemlich, R., Ed.; Academic Press: New York, 1972; Chapter 2. Jashnani, I . L.; Lemiich, R. Ind. Eng. Chem. Process Des. Dev. 1073, 12, 312. Jones, G.; Boiiinger, D. M. J. Am. Chem. Soc., 1035, 5 7 , 280. Leonard, R. A. Ph.D. Dissertation, University of Cincinnati, 1964, p 83. Leonard, R. A.; Cemiich, R. AIChEJ. 1965, 11, 18, 25. Lemlich, R. Ind. Eng. Chem. Fundam. 1978, 17, 89. Lemlich, R. Chem. f n g . Commun. 1982, 16, 153. Ranadive, A. Y.; Lemlich, R . J. ColloM Interface Sci. 1070, 7 0 , 392.
Receiued for review September 10, 1981 Revised manuscript received October 18, 1982 Accepted October 21, 1982
This material is based upon work supported by the National Science Foundation under Grant No. ENG77-09887.
