nal temperature gradients for pellet no. 1, while their data for the pellet of high r value indicate that the total AT is external. Since Kehoe and Butt present observed rate, KO, the mass Biot number and intraphase diffusivity (and therefore k,a) and bulk concentration data for each of their runs, we readily compute our observable qDa (eq 9). As r is also known AT,/ATo can be computed by eq 16. This comparison of theory and experimental data is set forth in Table I. Conclusions
A most conservative estimate of the iniportance of the external AT, is secured solely in terms of observed rate and the ratio of the Biot numbers. Given the fact that for gas-porous solid systems the Biot number ratio is indeed large ( 10-104), the assumption implicitly set forth in Hutchings' work, that external temperature gradients can be far more important than internal (intraphase) gradients, is surely verified for anticipated values of r in most gas-solid systems. We may, then, with some confidence invoke a model of an isothermal catalyst pellet operative a t a temperature dictated by external heat transport in catalytic-reactor simulation (Minhas, 1969; White, 1969). Comparison of the theoretical relation herein derived with experimental data proves to be rather satisfying. Nomenclature a = external surface to volume ratio, 1/L (Bi),,, = mass Biot number (Bi)h = thermal Biot number C = species concentration C,, = heat capacity I1 = intraparticle diffusivity Da = Damkohler number f = reduced concentration, C/Co h = interphase heat transfer coefficient AH = reaction enthalpy change
k = chemical rate coefficient k, = bulk mass transport coefficient L = volume to external surface ratio, l / a Le = Lewis number Sc/Pr P r = Prandtlnumber r = ratio of mass to thermal Biot numbers R = pellet radius Ro = observed, global, reaction rate Sc = Schmidtnumber t = reduced temperature, T / T o T = temperature z = reduced length, x / L
Greek Letters @ = Prater number based on C, = Prater number based on CO p = interphase adiabatic temperature change 4 = overall effectiveness factor q, = internal effectiveness X = pellet thermal conductivity d, = Weisznumber
Subscripts 0 = bulk fluid condition s = surfacecondition Literature Cited Carberry. J. J . , Ind. Eng. Chem., 58, 40 (1966). Cassiere, G., Carberry, J. J . , Chem. Eng. Educ., 7, 22 (1973) Goldstein. W.. Carberry. J. J.. J. Calal.. 28, 33 (1973) Hutchings, J . Carberry. J. J , AlChEJ., 12, 20 (1966). Kehoe. J. P. G . . Butt, J. B . , A I C h E J . . 18. 347 (1972). Lee, J. C. M., Luss, D . . Ind. Eng. Chem., fundam.. 8 , 596 (1969). Minhas, S.. Carberry. J . J., Brit. Chem. Eng., 14, No. 6 (1969). Prater, C. D . , Chem. Eng. S o . , 8 , 284 (1958). Wheeler, A . Adv. Calal.. 3, 249 (1951) White, D., Carberry. J. J., lnd. Eng. Chem., 61 ( 7 ) , 27 (1969).
D e p a r t m e n t of C h e m i c a l Engineering I h i u e r s i t y of Notre D a m e N o t r e D a m e , I n d i a n a 46556
James J . Carberry
ReceiL'ed for reuieu April 5, 1973 .4ccepted January 20, 1975
Coalescence and Conductivity in Dynamic Foam
Prehumidified nitrogen was bubbled through aqueous solutions of anionic, cationic, and nonionic surfactant to form foam that ascended continuously through a vertical glass column, 5.1 c m in diameter and 1 m in height, which was equipped with electrical conductivity cells to measure foam density at various levels in situ. For residence times of several minutes there was generally only a small change in foam density and a moderate shift in bubble size distribution as a result of interbubble gas diffusion, and there was no apparent rupture-induced coalescence, provided an adequate concentration of surfactant was present.
Introduction Purpose. When employing the technique of foam fractionation, an important consideration is the rate a t which the overflowing column of foam delivers liquid. This rate depends in part on coalescence within the rising foam before it overflows. Furthermore, such coalescence furnishes induced internal reflux which further enriches the overflowing foam with the surface active components. Overflow and coalescence are also pertinent to other situations involving foam. Accordingly, the present study was conducted primarily to investigate the nature and extent of coalescence in steadily ascending foam. Also, as an addendum, the relationship obtained between the volumetric density and the electrical conductivity of foam was com-
pared with the corresponding determinations that were employed in earlier studies. Instability. Coalescence stems from instability which is of two general types. The first is the growth of large bubbles a t the expense of smaller bubbles. This is caused by higher pressure in the smaller bubbles which forces gas to diffuse from these smaller bubbles to the larger bubbles. Small bubbles may even shrink to the point of disappearance. By making certain convenient assumptions, de Vries (1957) derived equations for approximately predicting the rate of diffusion-induced coalescence among bubbles the sizes of which are distributed unimodally. The second type of instability is the rupture of the films [lamellae] which separate bubbles. In foams of sufficient stability, such as were employed in the present investigaInd. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
131
tion, this rupture occurs among the bubbles in an apparently irregular manner (Brady and Ross, 1944). Experimental Section Apparatus and Procedure. Most of the apparatus was constructed during an earlier investigation and has been previously described (Shih and Lemlich, 1971). The foam was generated by bubbling prehumidified nitrogen through a suitable solution of surfactant at room temperature of about 25°C. The resulting foam ascended continuously through a vertical glass column of 5.1 cm inside diameter and 1 m height, and overflowed through an inverted U-bend. The foam was then collapsed mechanically and either returned to the top of the vertical column of foam to trickle down through it [total reflux], or recycled to the liquid pool a t the bottom [simple operation], or both [partial reflux]. In still other runs, liquid feed was introduced at the top so as to trickle down through the rising foam [stripping operation]. The column was furnished with appropriate devices to measure externally gas and liquid flow rates. It was also equipped with a vertically spaced series of electrical conductivity cells to measure the foam density at several locations within the rising (or stationary) column of foam. These measurements are discussed later in the Appendix. The bubbles were formed in the pool by one of three different spargers. A coarse fritted glass bubbler with aperatures of roughly 200 p was employed for most of the runs. A metal spinneret with holes of 50-p diameter or a single glass capillary tube of 0.06-cm diameter were used for the remainder. Bubble sizes were measured by photographing the foam as previously described (Jashnani and Lemlich, 1973, 19741, and then corrected for the statistical bias that discriminates against small bubbles (Clark and Blackman, 1948; de Vries, 1957, 1972). Over the various runs, the linear average bubble radius near the bottom of the column of foam ranged from 0.026 to 0.140 cm. The linear velocity of foam ascension ranged from approximately 0.1 to 0.9 cm/sec. Systems. Five different surface active systems were investigated, all of them with distilled water as solvent. They comprised (1) Triton X-100 [Rohm and Haas] a t pool concentrations of 0.69 X to 4.17 X 10-4 M , (2) technical grade hexadecyltrimethylammonium bromide (abbreviated as HDT) plus methyl orange [Matheson Coleman and Bell for both] at pool concentrations of 0.4 x 10-3 to 2.5 x M and to M , respectively, (3) sodium dodecyl sulfate (purified first by foam fractionation at high reflux ratio) at pool concentrations of 0.59 x to 3.32 x 10-3 M , (4)bovine serum albumin [Armour Pharmaceutical] at a pool concentration of 1.19 x 10-5 M (with reference to a molecular weight of 67,000), and ( 5 ) again Triton X-100, at pool concentrations of 1.96 x 10-4 to 9.60 x 1 0 - 4 M , hut with 45 vol % USP glycerine present to increase the viscosity more than fourfold. Systems (l), (41, and (5) included 0.04 M KC1 in order to make the aforementioned measurements of electrical conductivity feasible. Further details have been placed on file (Jashnani, 1971). Results and Discussion Bubble Size Distributions. Most runs yielded unimodal distributions of bubble size. A typical distribution of bubble radii (obtained near the top of the column at total reflux with Triton X-100 and a superficial gas velocity of 0.26 cm/sec) was as follows in terms of class marks [midintervals] of radii: 5% of 0.0078 cm. 63.170 of 0.0234 cm, 24.2% of 0.0390 cm, 5.7% of 0.0547 cm, 1.870 of 0.0703 cm, and 0.2% of 0.0860 cm. However, exceptions to unimodal132
Ind. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
ity were found with the albumin system wherein tiny satellite bubbles appeared. These are attributed to the use of the single capillary tube as the sparger. While a t sufficiently low gas rates a single capillary tube gives more uniform bubble sizes than other spargers, a t higher gas rates satellite bubbles are formed. The presence of these tiny bubbles produced a bimodal distribution of bubble sizes. Coalescence. With all systems, the foam ascended essentially in plug flow. The conductometric measurements showed that the foam density was relatively uniform along (up) the column when the foam was stable but decreased sharply up the column when rupture coalescence was present. However, no rupture coalescence was observed in the vertical column of foam except when operating with Triton X-100 at the comparatively low pool concentration of 0.69 x 10-4 M , which was well below the critical micelle concentration (Shih and Lemlich, 1971; Leonard and Lemlich, 1965b). Furthermore, the mass density of the foam was of the order of 0.1 g/cm3. Thus, with a column height of 1 m, the resulting hydrostatic pressure was only about 0.01 atm, and therefore of negligible effect on the sizes of the rising bubbles. The frictional pressure drop of the moving foam, which was measured with a pressure probe, was even smaller. All this leaves interbubble gas diffusion as the sole remaining mechanism to account for changes in the distribution of bubble sizes up the column. Figure 1 illustrates results for the simple mode of operation. In view of the presence of appreciable interbubble gas diffusion, the aforementioned theory of de Vries was tested quantitatively as follows. For each run, the downward rate of interstitial flow relative to the ascending gas was calculated from D and the drainage theory (Leonard and Lemlich, 1965a; Fanlo and Lemlich, 1965), and an estimate of the film thickness was found first from Figure 5 of Leonard and Lemlich (1965b) and again (for comparison) from the semidimensionless relationship on page 28 of the same reference. (While the said figure and relationship were developed from experiments that involved only Triton X-100 in water, to the present writers’ knowledge they are the only correlations of their type available.) Values of the film thickness so obtained were then substituted into de Vries’ theory along with appropriate parameters based on bubble sizes or obtained otherwise. The fractional decrease in the number of bubbles present was then calculated and compared with experiment over the lower half of the column, the upper half of the column, and the entire column. (Albumin was not examined for this purpose due to its aforementioned bimodal distributions of bubble size to which de Vries’ general equation for unimodal distributions does not apply.) Results of the comparison were poor, even for Triton X-100. The theoretical predictions averaged several times larger than the experimental results, and the comparison scattered considerably. The precise reason for this lack of agreement is not known. Apart from possible experimental error, it could be due to the approximations inherent in de Vries’ theory and/or due to inapplicability of the correlations for film thickness since they were developed from experiments conducted in a different way with much larger bubbles. Additional Evidence. Coalescence was also evidenced by an increase in the spread of the bubble size distribution as the foam ascended through the column. Of course, rupture coalescence can only increase bubble size. However, interbubble gas diffusion can reduce the size of a small bubble still further. Thus, some photographs revealed that the smallest bubble at the top of the column was smaller than the smallest at the bottom.
1 ' 0 7 7
0 20 /
,
/
0.9
, /
/
016-
0.6 -
0.50.4-
0.3 -
1,2
3
4
5
6
7
8 2 0
RESIDENCE TIME (minutes 1
Figure I . Coalescence: some results for the fractional reduction in the number of bubbles in a n aqueous foam rising continuously through a vertical column 1 m tall. (Rupture coalescence was visible during the runs of curve A ) .
Symbo1 A 0 A 0 V
0
Curve
A B C D
E F G
0.04
0.06
CONDUCTIVITY
0.10
0.02
0
0.2 -
Concn of surfactant in pool x 104, M
Solute(s)
0.69 1.96 3.40 4.0 12.5 5.9 34.1
Triton X-100 & 0.04 M KC1 Triton X-100 & 0 04 .I4KCI Triton X-100 & 0.04 .I4KC1 H D T & 1 0 - 4 M methyl orange H D T & l O - 5 M m e t h l l orange Sodiuni dodecyl sulfate Sodium dodecyl sulfate
It can be shown (Jashnani, 1971) that r31 and r32 increase with either rupture or interbubble gas diffusion. However, r10 decreases slightly with gas diffusion until bubbles shrink to the point of disappearance. For example, the diffusion of a differential volume of gas, dV, from a bubble of unit volume to a similar bubble produces a 1/2 (1 fractional change in r10 of 1/2 (1 + dV)l 3 dV)1 - 1 = -1/9 ( d w 2 . Thus a transfer of say 10% of a bubble volume changes r10 by -0.11%. With two dissimilar bubbles where the radius of one is twice the radius of the other. a transfer of gas between the bubbles that shrinks the smaller bubble to half its initial radius (and of course enlarges the larger bubble) changes r10 by -14.3%. A decrease in i-10 u p the column with a simultaneous increase in r31 and r32 was in fact observed in a few runs. When the effect of bubble disappearance begins to predominate, r10 increases because the bubble count decreases. Rupture coalescence also causes rlo t o increase. Films from all systems except albumin were mobile to the eye. Of these, sodium dodecyl sulfate revealed high mobility. while the presence of glycerine in the Triton X-100 gave the lowest mobility (except for albumin). The film thickness in most of the runs was between 0.1 and 1 j L , as determined from the aforementioned correlations between film thickness and flow through a Plateau border. Thus! the absence of rupture coalescence over most of the experimental range accords with the report (Kitchener, 1964) that spontaneous rupture seems to occur only when the film thickness is less than 0.1 g .
+
Principal Conclusions U'ith an adequate concentration of surfactant in the liquid pool, the extent of coalescence is generally moderate in foam that is continuously ascending through a vertical
0.08
0.10
0.12
0.14
RATIO
Figure 2. Volumetric fraction of liquid in foam as a function of electrical conductivity of foam divided by electrical conductivity of liquid. A, Clark (1948), using five different systems; B. Fanlo and Lemlich (1965). using Triton X-100 with NaCl in water: C. Shih and Lemlich ( 1 9 i l ) , using Albumin, Saponin, and Triton X-100, in turn. with KCI in water; D. present work with stationary foam, using Triton X-100 with KCI in water; E. present Lvork with ascending foam, using Triton X-100 with KCI in water.
column about ,1 m in length with a residence time of several minutes. Furthermore, under these conditions the coalescence is caused principally by interbubble gas diffusion rather than by the rupture of films. Appendix The conductivity cells were calibrated from auxiliary measurements of foam density that were obtained from the volume of liquid released by the complete collapse of foam after shutdown. However, systems 12) and (3) were omitted from calibration and subsequent electrical measurements because the selective adsorption of an ionic surfactant could have distorted the results through the presence of surface conductivity, The foam density for runs with these two systems was measured solely by foam collapse. Figure 2 shows the relationship obtained between iu and the ratio of the electrical conductivity of the foam to that of the liquid. The results of earlier studies are also presented (Clark, 1948; Fanlo and Lemlich, 1963; Shih and Lemlich, 1971). For the present results, a slight difference was found between stationary foam and ascending foam at steady state. However. a sudden change in foam velocity was found to have no discernable effect on the conductivity, This was shown by introducing a step input in the foam velocity ( c i a a sudden change in gas rate) and simultaneously measuring the conductivity. Thus. the modest differences among the several curves in Figure 2 ma?; well be due largely to differences in foam morphology, which is a subject yet to be fully explored. Nomenclature 33 = fraction of foam volume occupied by liquid .'VO = number of bubbles per unit volume near the bottom of the vertical column of foam n, = number of bubbles of radius ri LVt = number of bubbles per unit volume near the top of the vertical column of foam ri = radius of individual bubble. cm r10 = simple [linear] average bubble radius. ( X n l r , ) / L n , , cm r31 = edge average bubble radius, [ ( X n i r i 3 ) / ( L n ! r ! ) I21, cm r32 = surface average bubble radius, ( L n ! r 1 3 ) / ( L n i r i 2cm ), V = gas volume, expressed as a fraction of a bubble volume Ind. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
133
Literature Cited Brady. A. P., Ross, S., J. Amer. Chem. SOC.,52, 173 (1944). Clark. N . 0 . . Trans. Faraday SOC.,44, 13 (1948) Clark, N. 0 . .Biackman, M . , Trans. FaradaySoc., 44, 1 (1948). de Vries. A. J . , "Foam Stability," Rubber Stitching. Delft, 1957. de Vries, A. J . , in "Adsorptive Bubble Separation Techniques," R. LemIich. Ed.. pp 7-31, Academic Press, New York. N Y . . 1972 Fanlo, S . , Lemlich. R . , A.I.Ch.E.-/. Chem Eng. Symp. Ser.. 9 , 75, 85 (1965). Jashnani, I . L , Ph.D. dissertation, University of Cincinnati, 1971 Jashnani, I . L., Lemlich. R . . ind. Eng. Chem.. Process Des. Develop.. 12, 312 (1973). Jashnani, I . L.. Lemlich, R . . J . Colloid Interface Sci.. 46, 13 (1974). Kitchener, J. A . , in "Recent Developments in Surface Science," Vol. 1, J. F. Danielli, K. G A Pankhurst, and A. C. Riddiford, Ed., pp 51-93, Academic Press, New Y o r k , N. Y . , 1964. Leonard, R A., Lemlich. R.,A.I.Ch.E. J . , 1 1 , 18 (1965a)
Leonard, R . A , , Lemlich, R . , A./.Ch.E. J , , 11, 25 (196513) Shih. F. S . , Lemlich, R.. Ind. Eng. Chem., Fundarn.. 10, 254 (1971)
D e p a r t m e n t of Chemical and Nuclear Engineering Uniuersity of Cincinnati Cincinnati, Ohio 45221
Indrakumar L. Jashnani Robert Lemlich*
Received for review August 15, 1974 A c c e p t e d December 5, 1974 T h i s w o r k was p a r t l y s u p p o r t e d b y F.W.P.C.A. Research G r a n t W-00161 from t h e U. S . D e p a r t m e n t of t h e I n t e r i o r a n d by a n ind i r e c t g r a n t f r o m t h e Procter a n d G a m b l e C o m p a n y . T h e authors t h a n k J o h n P e r r y for h i s assistance with some of t h e calculations.
Particle Transport across a Plane Turbulent Jet
Particle transport across a plane turbulent jet has been analyzed and self-similar solutions obtained numerically using the Bradbury eddy viscosity profile. The concentration profiles are compared with predictions based on the Goertler and Reichardt theories of turbulent transport, and significant differences are noted. The significance of these results in evaluating experimental determinations of particle Schmidt numbers i n turbulent jets is discussed.
Recently, Lilly ( 1973) performed experimental studies of particle transport across plane turbulent jets in order to determine the effect of particle size on particle eddy diffu~ ) resivity. Turbulent Schmidt numbers (Sc = t ~ / t were ported for a carbon monoxide tracer and two sizes of magnesium oxide particles of mass median diameters 0.96 and 9.0 p, respectively. The values of Sc were found to be 0.34 for the molecule and small particle and 0.17 for the large particle. The implication is that the larger particles were transported more effectively by turbulent eddies than were the small particles or molecules. a rather surprising result. The experimental procedure involved measuring concentration profiles across the jet and comparing these with theoretical predictions of Niitsu and Katoh (1963), whose theory is based on Reichardt's analysis of transport in a jet (1941, 1942). By matching profiles the appropriate value of Sc was deduced. Since Lilly's result of an apparent increase in turbulent transport with particle size contradicts simple intuition, and since no satisfactory physical explanation has been advanced, the result is bound to be the subject of some controversy, and a closer look at Lilly's work is warranted. The objective of this communication is to show that the absolute values of Sc quoted by Lilly could be significantly in error due to the model of turbulent transport used in the data reduction procedure. The Reichardt theory is described as an "inductive" theory by Schlichting (1960). Iteichardt observed that the velocity profile in a self-similar turbulent jet was well approximated by the Gauss function and postulated an empirical law of momentum transport which would yield such a velocity profile. In this manner he avoided solving the governing differential conservation equations. Goertler (1942) did solve the differential momentum conservation equation for a self-similar jet but used the unrealistic assumption of a constant eddy viscosity across the jet; nevertheless Goertler's velocity profile also agrees well with experimental data. Recently Bradbury (1965) made a detailed experimental study of 134
Ind. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
the structure of the self-similar jet and deduced an appropriate eddy viscosity profile. We present here a solution for particle transport across a self-similar turbulent jet by numerical solution of the governing differential conservation equations with the Bradbury eddy viscosity profile. Our solution will be compared with those based on the Reichardt and Goertler theories. The coordinate system is such that x is measured along the jet and y perpendicular to the jet; the corresponding velocity components are u and u At a scale distance x = s the centerline velocity is U,, and the jet half-width is b. On one side of the jet the particle concentration is maintained at N e , and on the other it is zero. For constant properties the governing conservation equations are
a/(
mass: -
ax
particles:
+
a@
- = 0
ay
a s + ax
?t-
which must be solved subject to the boundary conditions
-
In order to obtain self-similar solutions we try the transformation x , ) x , ? where 7 = a(y/x). In addition we write C7 = U , ( s / x ) l 2; b = b , ( n / s ) in recognition of the known characteristics of the centerline velocity C and half-width b of the self-preserving turbulent plane jet. A dimensionless stream-function F and particle concentration C$J are defined by IL = o - ~ U $ 2 F ( a ) and CI# = N / N e , where u is a free constant, yet to be chosen. Bradbury's (1965) eddy viscosity profile is presented as a graph of 1/Rel us y / b where Re1 E c T b / e M , with ( l / R e l ) , = o =
