- Email: [email protected]
The effect of initial bubble size distribution on the interbubble gas diffusion in foam
NOTES The Effect of Initial Bubble Size Distribution on the Interbubble Gas Diffusion in Foam The changes in bubble size produced by the titled effect are computed in detail from theory and mutually compared. The results show a marked effect on the increases in the various averages of bubble size with time, but a much smaller effect on the fractional decrease in the overall surface of the bubbles. readily delineated when the controlling resistance to gas transfer is either at the bubble surface or within the i~terstitial liquid. Equation [1] is solved by conversion to a finite difference equation and conveniently recasting it in dimensionless form as follows:
1. INTRODUCTION The diffusion of gas between bubbles is an important phenomenon by which the distribution of bubble sizes in a liquid foam changes spontaneously. This change leads in time to a significant reduction in the number of bubbles present, and ultimately to the collapse of the foam. Accordingly, Clark and Blackman (1) proposed and Lemlich (2) developed theory for predicting the successive changes in bubble size that result from this diffusional phenomenon acting alone. The foam is presumed to be sufficiently stable to resist any appreciable rupture of iamellae during the time under consideration. The theory begins with the application of the classical law of Laplace and Young to the pressure difference between a bubble and its surrounding liquid. This is then combined with the general rate equation, the geometric properties of a bubble, and the conservation of gas throughout the foam. The combination yields an instantaneous average concentration of gas in the interstitial liquid located midway between bubbles that is equivalent to the equilibrium pressure within a fictitious bubble equal in radius to the instantaneous mean of the radii of the real bubbles taken by the second moment divided by the first moment. The ideal gas law is then incorporated into the aforementioned relationships to yield the following integro-differential equation for the change in bubble radius r with time ~-.
dr - 2J~_.RT---/I~rF(r"c)dr 11 Ic-~- ~ dr Pa ~f~ r2F(r,r)dr
AR
(R2,1
2J3`RTT/Par~,and rlrc.R~a
where the dimensionless time Y is the dimensionless radius R is is the dimensionless mean R taken by the second moment divided by the first moment, and rc is any convenient characteristic radius such as the arithmetic mean radius at z = 0. Thus, starting with any given initial distribution of bubble sizes, Eq. [2] can be employed via digital computation to predict the successive distributions at corresponding successive times. 2. THEORETICAL RESULTS This computation was recently carried out (2) starting initially with the empirical distribution of de Vries (3, 4). The present work shows the results of the analogous computation starting with the Maxwell-Boltz,mann like theoretical distribution attributed to Bayens by Gal-Or and Hoelscher (5). The present communication then presents and compares the variation with time of certain key statistical parameters of bubble size distribution based on these two initial distributions. Each distribution is recast here in generalized dimensionless form by the present authors as follows:
Ill
J is the effective permeability to gas transfer, 3' is the surface tension,/~ is the gas constant, T is the absolute temperature, F is the frequency distribution function, and Pa is the prevailing surrounding pressure (such as atmospheric) which on the absolute scale is taken to be very nearly equal to that within the bubbles. Equation [1] is subject to some given initial condition at r = 0, and to the constraint F(r 0") = 0 for allr -< 0 because every bubble must have a positive nonzero radius. Some special cases are
de Vries: Bayens:
~(R,0) -
2.082R (1 + 0.347R2) 4
d,(R,0) = 32 R2 exp(_4R2&r )
[3] [4]
Throughout the present work, R is based on the initial arithmetic mean radius, rt.0(0) = J'~ In other words, R = Function ~ R , 0 ) is also dimensionless, and equals Correspondingly, in subsequent figures, dimensionless ~ R , Y ) equals
F(r,'r)=F(r,O)
r/r~.o(O). rl.o(O)F(r,O).
rF(r,O)dr.
392
0021-9797/79/090392-03502.00/0 Copyright © 1979 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal of Colloid and Interface Science, Vol. 70, No. 2, June 15, 1979
NOTES
rl.o(O)F(r,z). It should be noted that F(r,r) and ~b(R,Y) are based on the number of bubbles present initially, that is, a t T = Y = 0 . The computation began with equal class intervals of AR = 0.002. Intervals of AY = 0.001 were employed, along with back-looping at each Ay until successive values of 1/Rz.~ agreed within 0.0005 before proceeding to the next time step. As the computation proceeded in time, the class marks shifted. All class marks that moved to or below zero were of course dropped. Also, at each time step the values of the frequency distribution function were changed in inverse proportion to the corresponding changes in their respective class intervals (so as to preserve continuity in the class limits of R) and each class mark was slightly shifted to the center of its revised class interval. Further details are on file (6). From the vantage of hindsight, it seems likely that looping the quantity (1/R2.~ - l/R) would have promoted even better convergence than looping 1/R2,1. Alternatively, Eq. [2] in differential form can be rearranged and integrated over AY with R2.1 taken as constant over the AY. Then, l/R2.~ can be looped as before. However, use of such an integrated form involves an implicit AR as well as the taking of logarithms, and this additional drain on computer time must be balanced against the more direct use of Eq. [2] with smaller AY. Starting with the aforementioned generalized initial distribution of Bayens, Fig. 1 shows the computed changes that occur in the bubble sizes as time progresses. The progressive reduction in the area under the curves shows (and is directly proportional to) the
1.0 f ~ INITIALLY BAYENS
0.8 /
INITIALLY DE VRIES
0.6 -~- 0.4 o.2 0.0 0.0
1.0 '
L0
3.0 '
4.0 '
R FIo. l. The solid curves show the distributions of dimensionless radii of bubbles at various successive dimensionless times, based on the number of bubbles present at zero time, starting with the distribution of Bayens (5). The broken curve shows the distribution of de Vries (3, 4) for comparison at zero time only.
393 I.O-
Q8
,,,:9
0.6
Q4
0.2
0.0
,
0.0
1.0
,
2.0
5.0
,
4.0
5.0
R Fio. 2. The distributions of dimensionless radii of bubbles at various successive dimensionless times, based on the number of bubbles present at each such time, starting with the distribution of Bayens (5).
progressive reduction in the number of bubbles present. The shift toward larger bubble sizes is also evident. (The generalized initial distribution of de Vries is shown for comparison at zero time.) The shift is even clearer in Fig. 2 in which dimensionless distribution function ~R,Y) is based on the number of bubbles actually present at dimensionless time Y. The interrelationship is ~R,Y) = tk(R,Y)/f~ 6(R,Y)dR. The fractional number of bubbles remaining, which is f~ 4~(R,Y)dR, is shown as a function of dimensionless time in Fig. 3 for the results based on the two initial distributions. For comparison, an analytical solution proposed by de Vries (3, 4), which has been recast by the present authors into dimensionless form, is also shown for comparison. However, this solution of de Vries (which is based on his initial distribution) is highly simplified in that it assumes that bubbles always transfer gas to other bubbles that are by comparison infinitely larger. Unlike the present theory, it makes no allowance for bubbles of intermediate size and does not yield a progression of distributions. The progressive shift toward larger bubbles is also well illustrated in Fig. 4 in terms of representative values of some dimensionless mean radii of interest. These mean radii are defined in accordance with the usual convention of / f~
] 1/(J--k)
According to Eq. 2, R2.1 represents the continually changing division between small [shrinking] bubbles and large [growing] bubbles. Ra.1 is involved in determining the total length of Plateau borders for foam Journal of Colloid and Interface Science, Vol. 70, No. 2, June 15, 1979
394
NOTES 1.0
~C
k 0.2
0.0 0.0
I
I
1.0
J
2.0
5.0
i
4.0
i
5,0
T FIG. 3. The fractional number of bubbles remaining as a function of elapsed dimensionless time. Curve A is computed from theory (2) and is based on the initial distribution of de Vries (3, 4). Curve B is computed from theory and is based on the initial distribution of Bayens (5). Curve C is from the analytical solution (3, 4) to a highly simplified model proposed by de Vries.
drainage and overflow, and R3.2 is employed in determining bubble surface for adsorption as in foam fractionation (7, 8). R 1,0is of course the common arithmetic mean. It is evident from this figure that the variation of these mean radii with time is quite sensitive to the particular initial distribution of bubble sizes.
4.0
1.0 ... R3, 2
3,5
-~
3.(: - \~
///
/'R2 I
//
,// "
.-/'R3'I
/
1
~
0.9
0.8
\\/ ./ .. /~.~/"/ //'"
2.5 . Rj,k
////
¢'R3,2
/.~e"-..<;%
s (T) s (0) 0.7
/
.
5
~
RI'O
, I,O
,
2,0
I 3,0
I 4.0
5.0
ACKNOWLEDGMENT This material is based upon work supported by the National Science Foundation under Grant No. ENG7709887. REFERENCES 1. Clark, N. O., and Blackman, M., Trans. Faraday Soc. 44, 1 (1948). 2. Lemlich, R.,Ind. Eng. Chem. Fund. 17, 89 (1978). 3. de Vries, A. J., " F o a m Stability." Rubber Stichting, Delft, 1957. 4. de Vries, A. J., in "Adsorptive Bubble Separation Techniques" (R. Lemlich, Ed.), pp. 7-31. Academic Press, New York, 1972. 5. Gal-Or, B., and Hoelscher, H. E.,A.I.Ch.E.J. 12, 499 (1966). 6. Ranadive, A. Y., "Gas-Diffusional Foam Instability." M.S. Thesis, Univ. Cincinnati, 1978. 7. Lemlich, R., in "Adsorptive Bubble Separation Techniques" (R. Lemlich, Ed.), pp. 33-51. Academic Press, New York, 1972. 8. Lemlich, R., in "Chemical Engineers Handbook" (R. H. Perry and C. H. Chilton, Eds.), Section 17, pp. 29-34. McGraw-Hill, New York, 1973. AJIT Y. RANADIVE t
0.6
1.0/~ 0.0
Figure 4 also shows the surface area of the foam bubbles expressed as a fraction, S(Y)/S (0), of the original area. It is worth noting that, starting with either of the two initial distributions, the fractional decrease in surface area is always much less than the fractional decrease in the number of bubbles. Furthermore, the two curves for the decrease in surface area are quite close to each other. Unlike the situation for the mean radii, this implies that the rate of fractional decrease in surface area is comparatively insensitive to the initial distribution. This behavior is of interest with regard to those applications that make use of the high surface area of foam, such as foam separation and other operations that involve surface adsorption.
3.5
T FIG. 4. The progressive increase in various dimensionless mean radii of the bubbles (left ordinate) and the progressive decrease in the relative surface area of the bubbles (right ordinate), all as computed from theory. The solid curves are based on the initial distribution of Bayens (5); the broken curves are based on the initial distribution of de Vries (3, 4). Journal of Colloid and Interface Science, Vol. 70, No. 2, June 15, 1979
ROBERT L E M L I C H 2
Department of Chemical and Nuclear Engineering University of Cincinnati Cincinnati, Ohio 45221 Received September 5, 1978; accepted January 8, 1979
Present address: Westinghouse Research Center, 1310 Beulah Road, Pittsburgh, Pennsylvania 15235. 2 Please address all correspondence to Robert Lemlich.
1. INTRODUCTION The diffusion of gas between bubbles is an important phenomenon by which the distribution of bubble sizes in a liquid foam changes spontaneously. This change leads in time to a significant reduction in the number of bubbles present, and ultimately to the collapse of the foam. Accordingly, Clark and Blackman (1) proposed and Lemlich (2) developed theory for predicting the successive changes in bubble size that result from this diffusional phenomenon acting alone. The foam is presumed to be sufficiently stable to resist any appreciable rupture of iamellae during the time under consideration. The theory begins with the application of the classical law of Laplace and Young to the pressure difference between a bubble and its surrounding liquid. This is then combined with the general rate equation, the geometric properties of a bubble, and the conservation of gas throughout the foam. The combination yields an instantaneous average concentration of gas in the interstitial liquid located midway between bubbles that is equivalent to the equilibrium pressure within a fictitious bubble equal in radius to the instantaneous mean of the radii of the real bubbles taken by the second moment divided by the first moment. The ideal gas law is then incorporated into the aforementioned relationships to yield the following integro-differential equation for the change in bubble radius r with time ~-.
dr - 2J~_.RT---/I~rF(r"c)dr 11 Ic-~- ~ dr Pa ~f~ r2F(r,r)dr
AR
(R2,1
2J3`RTT/Par~,and rlrc.R~a
where the dimensionless time Y is the dimensionless radius R is is the dimensionless mean R taken by the second moment divided by the first moment, and rc is any convenient characteristic radius such as the arithmetic mean radius at z = 0. Thus, starting with any given initial distribution of bubble sizes, Eq. [2] can be employed via digital computation to predict the successive distributions at corresponding successive times. 2. THEORETICAL RESULTS This computation was recently carried out (2) starting initially with the empirical distribution of de Vries (3, 4). The present work shows the results of the analogous computation starting with the Maxwell-Boltz,mann like theoretical distribution attributed to Bayens by Gal-Or and Hoelscher (5). The present communication then presents and compares the variation with time of certain key statistical parameters of bubble size distribution based on these two initial distributions. Each distribution is recast here in generalized dimensionless form by the present authors as follows:
Ill
J is the effective permeability to gas transfer, 3' is the surface tension,/~ is the gas constant, T is the absolute temperature, F is the frequency distribution function, and Pa is the prevailing surrounding pressure (such as atmospheric) which on the absolute scale is taken to be very nearly equal to that within the bubbles. Equation [1] is subject to some given initial condition at r = 0, and to the constraint F(r 0") = 0 for allr -< 0 because every bubble must have a positive nonzero radius. Some special cases are
de Vries: Bayens:
~(R,0) -
2.082R (1 + 0.347R2) 4
d,(R,0) = 32 R2 exp(_4R2&r )
[3] [4]
Throughout the present work, R is based on the initial arithmetic mean radius, rt.0(0) = J'~ In other words, R = Function ~ R , 0 ) is also dimensionless, and equals Correspondingly, in subsequent figures, dimensionless ~ R , Y ) equals
F(r,'r)=F(r,O)
r/r~.o(O). rl.o(O)F(r,O).
rF(r,O)dr.
392
0021-9797/79/090392-03502.00/0 Copyright © 1979 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal of Colloid and Interface Science, Vol. 70, No. 2, June 15, 1979
NOTES
rl.o(O)F(r,z). It should be noted that F(r,r) and ~b(R,Y) are based on the number of bubbles present initially, that is, a t T = Y = 0 . The computation began with equal class intervals of AR = 0.002. Intervals of AY = 0.001 were employed, along with back-looping at each Ay until successive values of 1/Rz.~ agreed within 0.0005 before proceeding to the next time step. As the computation proceeded in time, the class marks shifted. All class marks that moved to or below zero were of course dropped. Also, at each time step the values of the frequency distribution function were changed in inverse proportion to the corresponding changes in their respective class intervals (so as to preserve continuity in the class limits of R) and each class mark was slightly shifted to the center of its revised class interval. Further details are on file (6). From the vantage of hindsight, it seems likely that looping the quantity (1/R2.~ - l/R) would have promoted even better convergence than looping 1/R2,1. Alternatively, Eq. [2] in differential form can be rearranged and integrated over AY with R2.1 taken as constant over the AY. Then, l/R2.~ can be looped as before. However, use of such an integrated form involves an implicit AR as well as the taking of logarithms, and this additional drain on computer time must be balanced against the more direct use of Eq. [2] with smaller AY. Starting with the aforementioned generalized initial distribution of Bayens, Fig. 1 shows the computed changes that occur in the bubble sizes as time progresses. The progressive reduction in the area under the curves shows (and is directly proportional to) the
1.0 f ~ INITIALLY BAYENS
0.8 /
INITIALLY DE VRIES
0.6 -~- 0.4 o.2 0.0 0.0
1.0 '
L0
3.0 '
4.0 '
R FIo. l. The solid curves show the distributions of dimensionless radii of bubbles at various successive dimensionless times, based on the number of bubbles present at zero time, starting with the distribution of Bayens (5). The broken curve shows the distribution of de Vries (3, 4) for comparison at zero time only.
393 I.O-
Q8
,,,:9
0.6
Q4
0.2
0.0
,
0.0
1.0
,
2.0
5.0
,
4.0
5.0
R Fio. 2. The distributions of dimensionless radii of bubbles at various successive dimensionless times, based on the number of bubbles present at each such time, starting with the distribution of Bayens (5).
progressive reduction in the number of bubbles present. The shift toward larger bubble sizes is also evident. (The generalized initial distribution of de Vries is shown for comparison at zero time.) The shift is even clearer in Fig. 2 in which dimensionless distribution function ~R,Y) is based on the number of bubbles actually present at dimensionless time Y. The interrelationship is ~R,Y) = tk(R,Y)/f~ 6(R,Y)dR. The fractional number of bubbles remaining, which is f~ 4~(R,Y)dR, is shown as a function of dimensionless time in Fig. 3 for the results based on the two initial distributions. For comparison, an analytical solution proposed by de Vries (3, 4), which has been recast by the present authors into dimensionless form, is also shown for comparison. However, this solution of de Vries (which is based on his initial distribution) is highly simplified in that it assumes that bubbles always transfer gas to other bubbles that are by comparison infinitely larger. Unlike the present theory, it makes no allowance for bubbles of intermediate size and does not yield a progression of distributions. The progressive shift toward larger bubbles is also well illustrated in Fig. 4 in terms of representative values of some dimensionless mean radii of interest. These mean radii are defined in accordance with the usual convention of / f~
] 1/(J--k)
According to Eq. 2, R2.1 represents the continually changing division between small [shrinking] bubbles and large [growing] bubbles. Ra.1 is involved in determining the total length of Plateau borders for foam Journal of Colloid and Interface Science, Vol. 70, No. 2, June 15, 1979
394
NOTES 1.0
~C
k 0.2
0.0 0.0
I
I
1.0
J
2.0
5.0
i
4.0
i
5,0
T FIG. 3. The fractional number of bubbles remaining as a function of elapsed dimensionless time. Curve A is computed from theory (2) and is based on the initial distribution of de Vries (3, 4). Curve B is computed from theory and is based on the initial distribution of Bayens (5). Curve C is from the analytical solution (3, 4) to a highly simplified model proposed by de Vries.
drainage and overflow, and R3.2 is employed in determining bubble surface for adsorption as in foam fractionation (7, 8). R 1,0is of course the common arithmetic mean. It is evident from this figure that the variation of these mean radii with time is quite sensitive to the particular initial distribution of bubble sizes.
4.0
1.0 ... R3, 2
3,5
-~
3.(: - \~
///
/'R2 I
//
,// "
.-/'R3'I
/
1
~
0.9
0.8
\\/ ./ .. /~.~/"/ //'"
2.5 . Rj,k
////
¢'R3,2
/.~e"-..<;%
s (T) s (0) 0.7
/
.
5
~
RI'O
, I,O
,
2,0
I 3,0
I 4.0
5.0
ACKNOWLEDGMENT This material is based upon work supported by the National Science Foundation under Grant No. ENG7709887. REFERENCES 1. Clark, N. O., and Blackman, M., Trans. Faraday Soc. 44, 1 (1948). 2. Lemlich, R.,Ind. Eng. Chem. Fund. 17, 89 (1978). 3. de Vries, A. J., " F o a m Stability." Rubber Stichting, Delft, 1957. 4. de Vries, A. J., in "Adsorptive Bubble Separation Techniques" (R. Lemlich, Ed.), pp. 7-31. Academic Press, New York, 1972. 5. Gal-Or, B., and Hoelscher, H. E.,A.I.Ch.E.J. 12, 499 (1966). 6. Ranadive, A. Y., "Gas-Diffusional Foam Instability." M.S. Thesis, Univ. Cincinnati, 1978. 7. Lemlich, R., in "Adsorptive Bubble Separation Techniques" (R. Lemlich, Ed.), pp. 33-51. Academic Press, New York, 1972. 8. Lemlich, R., in "Chemical Engineers Handbook" (R. H. Perry and C. H. Chilton, Eds.), Section 17, pp. 29-34. McGraw-Hill, New York, 1973. AJIT Y. RANADIVE t
0.6
1.0/~ 0.0
Figure 4 also shows the surface area of the foam bubbles expressed as a fraction, S(Y)/S (0), of the original area. It is worth noting that, starting with either of the two initial distributions, the fractional decrease in surface area is always much less than the fractional decrease in the number of bubbles. Furthermore, the two curves for the decrease in surface area are quite close to each other. Unlike the situation for the mean radii, this implies that the rate of fractional decrease in surface area is comparatively insensitive to the initial distribution. This behavior is of interest with regard to those applications that make use of the high surface area of foam, such as foam separation and other operations that involve surface adsorption.
3.5
T FIG. 4. The progressive increase in various dimensionless mean radii of the bubbles (left ordinate) and the progressive decrease in the relative surface area of the bubbles (right ordinate), all as computed from theory. The solid curves are based on the initial distribution of Bayens (5); the broken curves are based on the initial distribution of de Vries (3, 4). Journal of Colloid and Interface Science, Vol. 70, No. 2, June 15, 1979
ROBERT L E M L I C H 2
Department of Chemical and Nuclear Engineering University of Cincinnati Cincinnati, Ohio 45221 Received September 5, 1978; accepted January 8, 1979
Present address: Westinghouse Research Center, 1310 Beulah Road, Pittsburgh, Pennsylvania 15235. 2 Please address all correspondence to Robert Lemlich.