Ind. Eng. Chem. Fundam., Vol. 17, No. 2, 1978
predictive equation, experimental data on De and k for solid-solid reactions are desirable for a more rigorous test of the model.
Nomenclature C = concentration of the species under consideration, mol/ cm3 CO = concentration of the species at t = 0, x < 0, mol/cm3 C* = concentration of the species at t > 0, x = 0, mol/cm3 C, = concentration of the species at the end of the product zone, mol/cm3 Q = diffusion coefficient, cm2/s D = interdiffusion coefficient, cm2/s D, = diffusion coefficient of the species in the product zone, cm2h D, = diffusion coefficient of the species in the reaction zone, cmz/s k = first-order reaction rate constant, l/s L = length of the pellet, cm t = time, s w = dimensionless concentration x = distance along the length of the pellet, cm x , = width of the product zone, cm x, = total width: product zone + reaction zone, cm z = dimensionless distance x / L z, = dimensionless distance x p / L z , = dimensionless distance x ,/L
89
Greek Letters 0 = dimensionless time as defined by eq 5 4, = dimensionless Thiele modulus as defined by eq 10 and 18
Literature Cited Arrowsmith, R. J., Smith, J. M.. lnd. Eng. Chem. Fundam., 5, 327 (1966). Crank, J., "The Mathematics of Diffusion", 2nd ed, Clarendon Press, Oxford, 1975. Danckwerts, P. V., Trans. Faraday Soc., 46, 701 (1950). Greskovich, C., Stubican, V. S., J. Phys. Chem. Solids, 30, 909 (1969). Kwy, C., "Fifth International Symposium on Reactivity of Solids", Munich, 1964, p 21, Elsevier, Amsterdam, 1964. Krishnamurthy, K. R.,Gopalkrishnan, J., Aravamudan, G., Sastri, M. V. C., J. lnorg. Nucl. Chem., 36,569 (1974). Mantri, V. B., Gokarn, A. N., Doraiswamy, L. K., Chem. Eng. Sci., 31, 779 (1976). Minford, W. J., Stubican, V. S., J. Am. Ceram. SOC.,57, 363 (1974). Schmalzried, H., "Defects and Transport in Oxides", Battelle Institute Materials Science Colioquia, Ohio, 1973, p 83, Plenum Press, New York, N.Y.. 1973. Taplin, J. H., J. Am. Ceram. SOC., 57, 140 (1974). Tudose, R. Z.,Bull. lnst. Polytech. Din /AS/, 16 (20). 241 (1970). Wagner, C., Acta Metall., 17, 99 (1969). Whitney 11, W. P., Stubican, V. S., J. Phys. Chem. Solids, 32,305 (1971). Yamaguchi, G., Tokuda, T., Bull. Chem. SOC.Jpn., 40, 843 (1967).
Received for reuiew December 22,1976 Accepted December 1,1977
Prediction of Changes in Bubble Size Distribution Due to Interbubble Gas Diffusion in Foam Robert Lemlich Department of Chemical and Nuclear Engineering, University of Cincinnati, Cincinnati, Ohio 4522 1
A theory is described for the change in the distribution of bubble sizes that results from the diffusion of gas between the bubbles of liquid foam. The theory involves an earlier concept of gas transfer to and from an effective fictitious intermediate bubble of zero holdup. The result is an integro-differential equation which is converted to a finite difference equation that can be applied via digital computation to any given initial distribution to yield the corresponding successive distributions. Such distributions are shown in generalized dimensionless form for a general unimodal initial distribution.
Introduction The distribution of bubble sizes in a liquid foam can change as a result of at least two different spontaneous phenomena (Lemlich, 1973). One involves the rupture of the lamellae between bubbles. Another involves the transfer of gas between bubbles by diffusion. Some liquid foams are extremely resistant to the first phenomenon, that is, to rupture. However, no liquid foam is immune to the second phenomenon. It is the prediction of the latter phenomenon that forms the subject of the present paper. Interbubble gas diffusion occurs in the following way. The bubble sizes in a foam are never exactly uniform. Indeed, they often vary widely. Consequently, the slightly higher pressure in the smaller bubbles forces gas to diffuse through the lamellae into the larger bubbles. Thus the smaller bubbles shrink and the larger bubbles grow. This changes the distribution of bubble sizes. In fact, the smaller bubbles can shrink to the 0019-7874/78/1017 -0089$01.OO/O
point of disappearance, thus causing a change in the number of bubbles as well. The higher pressure in the smaller bubbles results from their smaller radii of curvature. According to the classical law of Laplace and Young, the pressure difference across a curved surface is given by
.=.(?+$)
(1)
where y is the surface tension and rl and r2 are the principal radii of curvature. Applying eq 1 to the pressure difference between a large submerged spherical bubble of radius rL and a small submerged spherical bubble of radius r s gives = 27
1 1 (6 - --)
(2)
The factor 2 appears in the foregoing equation because for a sphere r l = r2. 0 1978 American Chemical Society
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Ind. Eng. Chem. Fundam., Vol. 17, No. 2, 1978
By combining eq 2 with a form of Fick’s law of diffusion, de Vries (1957) was able to derive an expression for the lifetime of the smaller bubble. From this he was then able to derive an equation for the rate of disappearance of “small” bubbles. However, his results do not show which of the larger bubbles receive the gas from the small bubbles. Thus his theory does not predict the change in the size distribution of the remaining bubbles. In an effort to accomplish the latter, New (1967) assumed that the larger bubbles receive the gas simply in proportion to their surface areas. Such an assumption is of course very approximate. Another limitation of these earlier theoretical approaches is that they are based on the assumption that, insofar as rate is concerned, gas diffusion is always from a small bubble to another bubble that is infinitely larger. In other words, the earlier approaches assume that, in applying eq 2, rL >> rs. This of course is an oversimplification. Not only does it make the application of eq 2 very approximate, but it makes no allowances for bubbles of intermediate size. Such intermediate bubbles receive gas from smaller bubbles and transmit gas to larger bubbles. Indeed, except for the very extremes, all bubbles in a foam should be treated as being intermediate in size! Accordingly, the present writer proposes to employ a hopefully more accurate theory which takes these matters into account.
Theory Instead of viewing the gas as diffusing directly from bubble to bubble, it is more realistic to view the gas as first diffusing into the liquid that exists between the bubbles, or more specifically diffusing to the liquid region midway between the bubbles. The concentration of gas in this liquid can be considered as being equivalent (through Henry’s law) to a gas pressure in the liquid. Then, by virtue of the aforementioned law of Laplace and Young, this equivalent gas pressure can be viewed as that which would exist within a fictitious spherical bubble of radius p. Thus the pressure difference between a bubble of any size and the liquid is AP=27
(;-1;)
1
(3)
Of course, AP in eq 3 may be positive or negative. Now, from the general rate equation
N = -JAM
(4)
where N is the molar rate of gas transfer from a bubble to the liquid, J is the effective permeability to the transfer, and A is the surface area through which the transfer takes place. By conservation of moles, N = -dm/dr where m is the moles of gas in the bubble a t time T . A is approximated by 47rr2.Combining the foregoing with eq 3 and 4 yields dm = 87rJ7(: - r ) (5) ds By assuming conservation of gaseous moles throughout the foam as a whole, Z(dmld7) = 0. The summation is taken over all the bubbles. Then, eq 5 becomes
where n is the number of bubbles of radius r. Equation 6 was first proposed by Clark and Blackman (1948). The present writer will now examine p and carry the development further. By considering an average effective p (that is, independent of r but varying with T ) , eq 6 gives
Znr2
P=Znr
(7)
or Jm
0 =
r2F(r,7) dr
SOrn
rF(r,T) dr
where F(r,s) is the frequency distribution function of r a t T . It should be noted from eq 7 or 8 that p is identical with the particular instantaneous mean radius rZ1 (Lemlich, 1972). This differs from the arithmetic radius rl0. Generally speaking, in absolute terms the pressure in a bubble is only slightly higher than the surrounding pressure Pa (which is typically atmospheric). Thus, from the ideal gas law and the volume of a sphere m = - 47Par3 3RT where R is the ideal gas constant and T is the absolute temperature. Combining eq 5,8, and 9 yields
or more succinctly
c13
r K -d= (11) dT where K = 2J7RT/Pa. Thus a bubble of radius r should grow if r > 1-21 and should shrink if r < r21. In more general terms, eq 10 or 11can be used to predict the changes in bubble size, subject to the initial condition F ( r , r ) = F(r,O) at T = 0, and subject to the constraint F(r,T) = 0 for all r 5 0 because every bubble must have a positive nonzero radius. Also, it is worth noting that in applying eq 10 or 11to a population of bubbles, drld7 can be viewed as representing (brlbT), where q is the frequency Fdr a t time T based on the number of bubbles present initially, that is, a t T = 0. For a unit of surface area, the effective permeability can be viewed as the reciprocal of the sum of two resistances, namely, the surface resistance l l h and the liquid resistance HtAD, where H is Henry’s constant and t is the effective average liquid [lamella] thickness between adjacent bubbles. Thus -1= - -H+t- 1 J 20 h Two special cases follow. If the surface resistance greatly predominates, J = h. On the other hand, if surface resistance is negligible, J = 2D/Ht. For the latter case, if effective average t is taken equal to the volumetric average t , then J = SDIHL where S is the surface area of the bubbles and L is the volume of the liquid. For spherical bubbles, this last expression can also be written as 3D(1- a)) J= Har32
(13)
where 2) is the volumetric fraction of liquid in the foam and r32 is defined along the usual lines (Leonard and Lemlich, 1965; Lemlich, 1972) as J m
r3F(r,7)d r
r32 = .~ r2F(r,r) d r
Ind. Eng. Chem. Fundam., Vol. 17, No. 2, 1978
It should be noted that deviations from bubble sphericity are not very important to the calculations based on bubble geometry. For instance, if one considers the idealized regular dodecahedral shape (Manegold, 1953) the coefficient 3 in'eq 13 changes to only 3.3, where r in eq 14 is for the sphere equal in volume to the dodecahedron (Lemlich, 1968). However, there would be some question as to just how the law of Laplace and Young should be applied in practice to a polyhedral bubble.
0'90
91
1
Theoretical Results Equation 10 or 11is solved by conversion to a finite difference equation. For eq 11this is 00
15
30
45
60
75
90
R
Equation 15 is then applied via digital computation to closely and initially equally spaced class marks of r with small AT and successive back-looping a t each time step so as to obtain a proper average value of l/rzl over each such step before proceeding to the next step. This looping promotes convergence. As the Computation proceeds, calculation of the total gas volume of the bubbles may show some very slight apparent changes due to truncation. Such changes can be apportioned among the bubbles at each time step so as to conserve total gas volume. More importantly, as the computation proceeds, the class marks will shift. Those that go to or below zero at the end of a time step are of course deleted. Also, in general, as a result of the shift some adjacent class intervals [class limits] will overlap each other, while gaps will appear between others. All such overlaps and gaps must be eliminated by contracting or expanding each class interval, asymmetrically as necessary, to preserve the continuity of the class limits. The value of the frequency distribution function a t each class mark is then increased or decreased in inverse proportion to the corresponding change in its class interval. Each class mark can then be adjusted slightly by shifting it to the midpoint of its revised class limits. Of course, in order to begin the computation, the initial distribution of bubble sizes must be given. Experience shows it is generally unimodal, at least for foams produced by shaking, beating, or multi-orifice sparging. Theoretical distributions based on statistical reasoning from particular fundamental assumptions have been proposed (Abe, 1955; Bayens, 1966), as well as a general empirical algebraic fit that was developed from foam produced by a high-speed mixer (deVries, 1957,1972). However pathological distributions are not unknown, bimodal distributions having been obtained by overloading a single-orificed bubbler with gas throughput (Jashnani and Lemlich, 1975). For the sake of generality and utility, eq 15 was recast in dimensionless form as
where dimensionless time interval AT = KAr/rc2 = 2JrRTAr/Parc2,dimensionless radius R = r/rc, dimensionless mean radius R21 = r21/rc, and rc is some convenient characteristic radius such as initial mean, modal, or median radius. de Vries' said algebraic equation was then also recast in dimensionless form and eq 16 was applied to it via the aforementioned technique of digital computation with rc arbitrarily chosen as rlo(0) which is the initial arithmetic mean radius. Intervals of AR = 0.0025 and AT = 0.001 were used. Also, back-looping was employed over each time step until successive values of 1/R21 agreed within 0.0005.
Figure 1. Distributions of dimensionless radii of bubbles at various successive dimensionless times, based on the number of bubbles present a t zero time.
0.0
I5
30
45
6.0
7.5
90
R
Figure 2. Distributions of dimensionless radii of bubbles a t various successive dimensionless times, based on the number of bubbles present at each dimensionless time.
The computation yielded the bubble size frequency distribution function 4(R,T) a t successive values of the dimensionless time parameter T = Kdrlo2(0)= 2JyRTr/Par1o2(O). Function +(R,T), which is also dimensionless, equals rcF(r,r). Figure 1shows the results. 4(R,T) in Figure 1 is based on the number of bubbles present initially,that is, at zero time. Thus the decrease in area under the curves as time progresses clearly shows (and is directly proportional to) the corresponding progressive decrease in the number of bubbles present. The progressive increase in average bubble size is also evident. However, the latter effect is better seen in Figure 2 in which the dimensionless frequency distribution function J.(R,T)is based on the number of bubbles present at dimensionless time T.Thus J.(R,T) in Figure 2 is arrived a t from Figure 1by multiplying $(R,T)by the ratio of the area under the curve of 4(R,O),which is unity, to the area under the corresponding curve of
4(R ,T).
The solid curve in Figure 3 shows the fractional number of bubbles remaining, which is Jo"$(R,T) dR, as a function of T. For comparison, the prediction of de Vries, which was mentioned earlier, is converted to dimensionless form (with due regard for the doubling in his diffusional path length and hence resistance) and is shown here as the broken curve.
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Ind. Eng. Chem. Fundam., Vol. 17, No. 2, 1978
in effect a single bubble of infinite radius. Once again, this differs from the general situation that exists within the bulk foam. Finally, in practice one must recognize the possibility of other mechanisms of change in bubble size, such as rupture of lamella (mentioned earlier), release of gas from supersaturated liquid, gross change in external pressure, and chemical reaction.
‘
0O 8 I\
00’
0.0
I O
20
30
40
50
T
Figure 3. The fractional number of bubbles remaining, as a function of elapsed dimensional time. The solid curve shows the present results. The broken curve is based on the results of de Vries (1957,1972).
Thus, starting with a reasonable initial unimodal distribution, Figures 1and 2 present the succession of bubble size distributions with time, and the solid curve of Figure 3 shows the progressive decrease in the relative number of bubbles. Accordingly, these figures should be approximately applicable to a number of situations. Of course, for greater accuracy eq 15 or 16 should be applied directly to whatever initial distribution is actually at hand. Cautions Certain caveats are in order. When foam bubble sizes are measured visually or photographically, the plane of view will discriminate statistically against the inclusion of small bubbles, thus incorporating a statistical sampling bias (Clark and Blackman, 1948). Fortunately, this bias can be corrected with eq 17 (17)
where f (r,T)is the observed frequency distribution function of bubble radii in the plane of view based on the number of bubbles present a t time 7, as contrasted with F(r,7)which is the true frequency distribution function of bubble radii in bulk based similarly on the number of bubbles present at 7 (de Vries, 1957, 1972). Physical segregation among foam bubble sizes a t a surface is reportedly slight (Chang, et al., 1956). However, the distortion of individual bubbles a t a bounding surface presents questions. Some work wherein a small amount of free foam was built up on a flat glass plate, bubble by bubble, showed that the altitude (and also the longest chord) in the glasscontact face of a polyhedral foam bubble surrounded by other bubbles of nearly equal size is approximately equal to the diameter of the same bubble freely suspended as a sphere in air (Jashnani and Lemlich, 1973). However, these bubbles were all of approximately 0.3 cm radius. There is some uncertainty as to how well such results might apply to smaller bubbles, bubbles of widely varying size, or wetter foam. Distortion at a bounding surface also affects the degree of curvature of the distorted bubbles, and so affects hp and the rate of interbubble gas diffusion at the very location of measurement, that is, at the optical sampling plane. The proposed theory is for gas diffusion between bubbles within the bulk of the foam and not for locally distorted bubbles a t a boundary. Of course, if the foam presents a free surface, say to the atmosphere, then there is less local distortion. However, such free-surface bubbles are involved with gas exchange partly with adjacent bubbles and partly to the atmosphere which is
Comparison with Experiment Both Clark and Blackman (1948) and de Vries (1957) have observed the growth of large bubbles as well as the shrinkage and disappearance of small bubbles. Unfortunately, neither has published the successive distributions of bubble size that are necessary to test the present theoretical results quantitatively, and the present writer knows of no other published data that can be thus employed. However, de Vries (1957, 1972) claims reasonable agreement between his theoretically predicted counts of surviving bubbles and his experimental counts. So, to the extent that the present work may be deemed from Figure 3 to agree with that of de Vries, the present work is also in agreement with experiment. Conclusions Starting with any given initial distribution of foam bubble sizes, the successive distributions that result from interbubble gas diffusion acting alone can be predicted by digital computation with eq 15 or 16, subject to the natural constraint that no bubble can take on a negative size. Based on a reasonable initial unimodal distribution, the successive distributions are shown in generalized dimensionless form in Figures 1and 2. The vanishing of small bubbles is also shown by these figures and by Figure 3. The main theoretical development and results are for bubbles within the bulk foam. Caution must be exercised when considering the bubbles at either a free or bounding surface. Nomenclature A = surface area for gas transfer D = diffusivity of gas through liquid = volumetric fraction of liquid in foam F = frequency distribution function of bubble radii f = apparent frequency distribution function of bubble radii in plane of view H = constant from Henry’s law h = reciprocal of surface resistance J = permeability to transfer of gas K = parameter 2JyRT/P, L = volume of interstitial liquid m = moles of gas within a bubble N = rate of transfer of moles of gas n = number of bubbles of radius r P = pressure Pa = pressure of surroundings (atmosphere) q = frequency of bubble radius R = dimensionless radius, r / r c Rzl = dimensionless mean radius of bubbles by second moment over first moment, r21/rc R = ideal gas constant r = bubbleradius rc = characteristic bubble radius rL = radius of larger bubble r s = radius of smaller bubble r l = principal radius of curvature rz = other principal radius of curvature r10 = arithmetic mean radius of bubbles rlo(0) = initial arithmetic mean radius of bubbles r Z 1 = mean radius of bubbles by second moment over first moment
Ind. Eng. Chem. Fundam., Vol. 17, No. 2, 1978
r32 = mean radius of bubbles by third moment over second moment S = surface area of bubbles T = absolute temperature t = effective average thickness of lamella Greek Letters y = surface tension A = differencein = dimensionless frequency distribution function of bubble radii based on the number of bubbles present initially p = radius of fictitious bubble of zero holdup T = time T = dimensionless time, K 7 / r C 2or K r / r l o 2 ( 0 ) $ = dimensionless frequency distribution function of bubble radii based on the number of bubbles actually present
Literature Cited Abe, T., Pap. Meterol. Geophys., 5 (3,4), 243 (1955). Bayens, C. A., as reported by Gal-Or, B.. Hoelscher, H. E., AlChEJ., 12, 499
(1966).
93
Chang, R. C., Schoen, H. M., Grove, C. S . , Jr., lnd. Eng. Chem., 48, 2035
(1956). Clark, N. O., Blackman, M., Trans. Faraday Soc., 44, l(1948). de Vries, A. J., "Foam Stability", Rubber-Stichting, Delft, 1957. de Vries, A. J., in "Adsorptive Bubble Separation Techniques", pp 7-31, R. Lemlich, Ed., Academic Press, New York, N.Y., 1972. Jashnani, I. L., Lemiich, R., lnd. Eng. Chem. Process Des. Dev. 12, 312
(1973). Jashnani, i. L., Lemlich, R., lnd. Eng. Chem. fundam., 14, 131 (1975). Lemlich, R., Ind. Eng. Chem., 60(10), 16 (1968). Lemlich, R., J. Cosmet. Chem., 23,299 (1972). Lemlich, R., in "Chemical Engineers Handbook", Vol. 17, 29-34,R . H. Perry and C. H. Chiiton, Ed., McGraw-Hill, New York, N.Y., 1973. Leonard, R. A., Lemlich, R., AlChEJ., 11, 18(1965). Manegold, E.,"Schaum", Strassenbau, Chemie und Technik Verlag, Heidelberg,
1953. New, G. E., Proc. lnt. Congr. Surface Active Substances 7964, 2, 1167
(1967).
Received for review March 15,1977 Accepted February 6,1978
The author thanks Ajit Ranadive for writing and running the computer program, and the Herman Schneider Laboratory for financial support for this assistance.
The Latent Heat of Vaporization Prediction for Binary Mixtures Dragoljub Santrach and Janis Llelmers' Chemical Engineering Department, The University o f British Columbia, Vancouver, British Columbia, Canada
Using the dimensionless mixture coordinates SM"(SM* = (A~/M/TM)/(A~/BM/TBM)), where Am/M = heat of vaporization of the binary mixture at the given mixture temperature, TM (K), while Am/BM = heat of vaporization at the normal boiling point temperature of mixture, TBM (K), and TM* ( TM* = ( TcM/ TM I)/( TcM/TBM 1) where TCM= the critical mixture temperature (K), an empirical equation, SM* = ( TM* -k TM* P)/(1 4- TM* g), has been established to calculate the latent heats of vaporization for binary mixtures along the mixture liquid-vapor saturation curve over the entire liquid range from the normal melting to the critical point for the given mixture. The proposed method (this work) compares very well in overall accuracy with other prediction methods over the complete range of investigation (15 binary systems, 135 data point pairs).
-
Introduction and the Proposed Method T o describe the coexistence behavior for binary mixture along its liquid-vapor saturation curve, a modified form of the differential Clausius-Clapeyron equation may be used
(2) S,X
M V M - TM(VGM - VLM)
(1)
where M V M is the heat of vaporization of the binary mixture at the given mixture temperature TM(K), and the term ( VGM - VLM) is the associated molar volume change between gaseous (G) and liquid (L) phases along the binary mixture liquid-vapor saturation curve. The subscripts S and X refer to this liquid-vapor pressure equilibrium state and the mixture concentration X of component (l),respectively. For the given binary mixture the latent heat of vaporization may be calculated from eq 1 as the product between the differential mixture pressure-temperature ratio and the temperature times the corresponding mixture volume change. However, eq 1 is difficult to apply directly to calculate the latent heat of vaporization for binary mixtures. There is an inherent inaccuracy in obtaining the slopes of the experimental PM-TM curves since accurate binary mixture saturated vapor pressure data are available only a t a few temperatures for the given 0019-7874/78/1017-0093$01.00/0
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concentration X of the binary system. This inaccessibility of needed experimental data has led to other considerations in calculation of latent heat of vaporization for binary mixtures (compare with Reid and Sherwood, 1966). Noting the overall accuracy of the recently introduced Lielmezs-Fish latent heat temperature functions for calculation of the vaporization heat of pure substances (Fish and Lielmezs, 1975), it was felt that these functions could be extended for latent heat prediction of binary systems. To do this, the previously reduced dimensionless coordinates, S*and T* (valid for pure substances (Fish and Lielmezs, 1976)) are rewritten to describe the binary mixtures as s ~and *TM*.
where AHVM = heat of vaporization of the binary mixture at the given mixture temperature (cal/mol), MVBM = mixture heat of vaporization at the normal boiling point temperature of this binary mixture (cal/mol), and TM,TCM,and TBMare the mixture, critical mixture, and normal boiling point of mixture temperatures, respectively (K). Then for this coor0 1978 American Chemical Society
