FLOW REGIMES OF STABLE FOAMS M I C H E L S. H O F F E R A N D E L I E Z E R R U B I N Department of Chemical Engineering, Technion, Israel Institute of Technology, Haifa, Israel In order of increasing foam velocity the flow regimes observed for stable foams moving in vertical constant cross section glass columns are plug flow, turbulent flow, and bubble column. The prevailing flow regime depends to a large extent on average bubble diameter, D32, and shape, bubble size distribution, foam velocity, G, and volume fraction of liquid in the foam, f. All the experimental data correlate well For normal bubble size b y fDi2 = CGn, where n equals - 1 .O for plug flow and -2.5 for turbulent flow. distribution (obtained with spinnerettes) D32 can be obtained from a correlation of D S ~ / ~ ~Reo, V where S . Do and Re, are spinnerette orifice diameter and Reynolds number, respectively. The transition from plug to turbulent flow can be determined visually and i s characterized b y breaks in the AP/AZ vs. G, f vs. G, and fD& vs. G curves. At the transition f i s much lower than 26y0 (the minimum void fraction of equal spheres) and a substantial fraction of bubbles are polyhedral.
ONTINUOWSLY
floiving stable foams produced by bubbling
C a gas through solutions containing surface active agents are used in the foam separation technique. This mass transfer operation is based on the fact that surfactants concentrate a t the solution-gas interface. A review of the field u p to 1961 has been published (Rubin and Gaden. 1962). For understanding the phenomena involved and for proper design of foam separation equipment, knowledge of foam flow is needed in addition to mass transfer data. Some design and operation problems associated with foam flow have been discussed (Bruner and Lemlich, 1963; Fanlo and Lemlich? 1965; Haas and Johnson, 1965, 1967; Metzner and Brown, 1956; Rubin and Gaden, 1962; LVace and Banfield. 1966; SVeissman and Calvert, 1967). Plug flow prevails a t lo\v foam floiv rates, and the flow becomes turbulent a t higher rates. To date, ho\vever, the published experimental and theoretical investigations on foam flow are limited almost exclusively to the plug flolv regime. The purpose of the present investigation was to study the flow regimes prevailing under different conditions, to find correlations bet\veen the relevant physical quantities in each floiv regime, and to find out what causes a certain floiv regime to prevail under a certain set of conditions, and how the flo\v regime can be predicted from operating condition?. The investigation was carried out experimentally in vertical constant-cross-section columns. The variables studied were column height, geometry of the column head, type of the gas sparger, and gas flow rate. Phenomena observed in a variablecross-section column. in nonvertical column sections, and in columns built of different materials are also reported. literature Survey
The structure of foams has been described and discussed in detail (Bikerman. 1953; Haas and Johnson? 1967; Leonard and Lemlich. 1965; Rubin and Gaden, 1962). Dry foams are composed of polyhedral bubbles. Polyhedral foams contain a lattice of "plateau borders" ivhich function as capillary tubes through which most of the flow of liquid due to drainage takes place. \\.'et foams are composed of round buhhlrs. Plug Flow. At low flo\v rates. foams move in pliig flow (Haas and Johnson. 1965; Matalon, 1953; Metzner and Brown. 1956; Rubin, 1962; IVace and Banfield. 1966) and the bubbles are usually pol?-hrdral. Pluq flow in vertical columns was studied by several investigators (Fanlo and Lemlich. 1965; Haas and Johnson. 1967; Shih and Lemlich, 1967) and several models were developed and compared. In foams moving upward, tivo simultaneous movements
occur: Bubbles move upward a t velocity V, and liquid drains down\vard a t velocity V,, Lvhich depends on the local liquid on the degree of drainage already accomholdup. f,-i.e.. plished. Therefore. after a certain region of varying f z , a point is reached where V d = V,. From there on f i is independent of Z (uniform throughout the column). I n the region of constant f,, Equation 1 has been found valid over a wide range of operating conditions (Haas and Johnson. 1967; Rubin et al.. 1967) : TI
where
K =
(k ~~.
60
1)
32
-
g
n .-
1.6
k varies bet\veen 1.1 and 2.0. Plug flo\v in horizontal ducts has also been studied (Haas and Johnson, 1967; \.\.'ace and Banfield, 1966; Weissman and Calvert, 1967). I n this case V , = 0 and as V, > 0, drainage occurs during flow. Therefore the local liquid holdup depends on both axial and radial coordinates and appropriate equations were derived (Haas and Johnson, 1967). Turbulent Flow. ,4t higher floiv rates in a vertical column, the flow becomes turbulent (Haas and Johnson, 1965; Metzner and Brown, 1956; Wace and Banfield, 1966). Turbulent foam flow increases the height of a transfer unit in multistage mass-transfer columns (Haas and Johnson, 1965 ; Metzner and Brown, 1956). I t \vas thought (Haas and Johnson, 1965, 1967) that turbulent flow deirelops \Then fi > 0.26 (the minimum voidage of closely packed equal spheres), because the bubbles become free to move relative to each other. Almost no experimental data are available on turbulent foam flo\v. nor has the transition bet\\-een the flow regimes been studied. Bubble Column. I n an attempt to find theoretically the relationship among gas floiv rate: liquid holdup in the liquid pool from \vhich the foam is formed. and liquid holdup of the foam itself, M'ace and Banfield (1966) developld a much simplified model of foam flow. They considered the bubbles as rigid spheres of equal size. Their calculated results deviate considerably, in some caSes by more than an order of magnitude, from available experimental data. HoM-ever, their model predicts qualitatively the phenomena observed in the present \vork at very high gas floxv rates. As the gas flow rate increases: the difference between the liquid holdup of the liquid VOL.
a
NO. 3
AUGUST 1969
483
pool and of the foam decreases. I t disappears above a certain gas flow rate, when foam no longer exists as a separate “phase.” I n fact, a t high gas flow rates, the flow regime is that of gas-inliquid dispersion in highly turbulent motion. Experimental
The apparatus used was a single-stage continuous circulatory type foam system, a slightly modified version of systems described elsewhere (Rubin, 1962; Rubin et al., 1967). The apparatus consisted of a liquid pool with a changeable gas sparger a t the bottom, a vertical column, a centrifugal baskettype foam breaker, and recycle pipes for returning the collapsed foam (foamate) to the liquid pool (Figure 1 ) . The vertical glass column was made of 26-mm. i.d. interchangeable sections which permitted variations in height. Adjacent sections were connected by the device shown in Figure 2, which assured a uniform cross section. The column heights used were 57, 86, and 152 cm. The top and bottom sections of the columns were provided with flat windows (16mm. diameter) to permit photographic determination of bubble size and size distribution. Pressure taps were located a t regular intervals along the columns, to permit measurement of pressure drops with a vertical and an inclined water manometer. Five gas spargers were used: a 27-mm. diameter sintered porous stainless steel plate and four platinum alloy spinnerettes with hole diameters of 0.040, 0.060, 0.080, and 0.100mm. Each spinnerette contained 125 holes distributed uniformly in a circular area 10.2 mm. in diameter, so that the distance between two neighboring holes was about 0.85 mm. The horizontal part leading the foam into the foam breaker was slightly inclined (Figure 1) to prevent liquid which had left the vertical column from returning to it. Most experiments were conducted with an almost 90’ head bend. Some were performed with a rounded head bend (insert, Figure 1). The aqueous surfactant solution used in all the experiments contained 200 mg. per liter of sodium dodecyl benzene sulfonate, 40 mg. per liter of NaOH, and 250 mg. per liter of NaC1. p H was kept in the range 10.7 to 10.9 by occasional addition of N a O H . Solutions containing more than 80 mg. per liter of N a O H Lvere discarded, to avoid appreciable changes in solution and foam properties. The properties of foam produced from freshly prepared solutions exhibited a time dependency which disappeared after several hours of closed-circuit operation of the apparatus. All the experiments described in this work used solutions which had already attained constant properties. During continuous closed-circuit operation, CO1-free and humidified air was continuously bubbled through the sparger into the liquid pool. The foam produced flowed through column and column head into the foam breaker, where it collapsed. The resulting liquid (foamate) was returned to the liquid pool and the air escaped through a vent. Closedcircuit operation prevented concentration changes due to foam fractionation effects. All the experiments were conducted with stable foams (no bubble coalescence). Experiments were conducted in series consisting of 5 to 10 separate runs. Each series of experiments was conducted with constant foam column height, sparger, and column head geometry. Air floiv rate was the variable. Time required to reach steady state \vas 45 to 60 minutes for the first run of a series and 20 to 30 minutes between runs of the same series. The following quantities were measured in each run: air floiv rate, foamate floiv rate at exit of foam breaker (by measuring the time required to collect a given small volume of foamate), linear foam velocity (in cases where plug flow was observed either in the column or in the column head), pressure distribution along the column, and temperature of solution in liquid pool. I n addition, photographs were taken to determine bubble size and size distribution, using a single lens reflex camera equipped with an extension tube. Careful visual observations rvere made of flow regime and other phenomena occurring in different locations of the apparatus. The follorving quantities Lvere calculated from those measured directly: air flo\v rate a t conditions prevailing at foam column exit, liquid volume fraction of foam at foam column 484
I&EC FUNDAMENTALS
Round head bend
Figure 1. Schematical representation of experimental apparatus
Figure 2. Connections between column sections
exit, f (calculations included correction for slight change of liquid level during measurement of foamate flow rate), and pressure drop per unit column height, A P I A Z . Enlarged prints of the photographs and Equations 3 and 4 \yere used for determining mean bubble diameters, 0 3 2 and D I O ,standard deviation of bubble sizes, u, and u / D l o (giving a measure of bubble size nonuniformity). (3)
Minimum sample size for determination of was 50 bubbles.
0 3 2
and u/Dlo
The accuracies of measured and calculated quantities are listed in Table I. The existence of plug flow was verified, in addition to visual observation, by a gas balance : Q c c = (1
- f) V i 0,’
A __Q G - Qcc QG
Q
Q
G
~
(5)
(6)
Most runs were performed with glass columns. Several runs were also performed with a Perspex column, of 50-mm. i.d. and 1000 mm. high. Results and Discussion
Three properties differentiate foams from regular gas-liquid dispersions. The volume fraction of the liquid in foams, f,can be very low. Values off of a fraction of 17’ can be easily obtained. A foam can be considered in many respects as a “phase,” since a distinct boundary can be observed between the liquid pool from which the foam is generated and the foam itself. Depending on conditions, the bubbles in a foam are spherical or polyhedral in shape. For liquid-liquid dispersions it has been reported (Letan and Kehat, 1967) that under certain extreme conditions the drops of the discontinuous phase become polyhedral and the volume fraction of the continuous phase is only a few per cent. In addition, a distinct boundary could be observed between the “foam phase” and the liquid pool. Under these conditions the dispersions can be called liquid-liquid foams. However, because of the very low density and the compressibility of the discontinuous phase in gas-liquid foams, they may show marked differences in behavior when compared to liquidliquid foams-for example, it is possible to obtain a turbulent moving gas-liquid foam, whereas a turbulent liquid-liquid dispersion can no longer be called a foam. Qualitative Description of Observed Phenomena. Most present experiments were conducted with glass columns, in which there is slip between the foam and the column wall. A few experiments conducted with Perspex columns showed no slip between the foam and the column wall. At steady state, a layer of stationary bubbles 2 to 3 mm. deep formed a t the wall and prevented visual observation of foam flow. All the results and observations reported in the present work are for glass columns. Typical observations a t steady state with a given spinnerette in a constant-cross-section column (liquid level A , Figure 1) are described below.
Table 1.
Accuracy of Measured or Calculated Quantities Relative Error, % Quantity Remarks G Zko 5-4.0 f f2-8 V f2-3 AP/AZ Difficult to estimate. Large in turbulent flow because of fluctuations Same photograph measured several 0 3 2 , -5 f 2 times DiO Different photographs of same run up to 120 032, (spinnerettes) DID photographs of same run up to 130 Different 0 3 2 , (sintered porous sparger ) DlO
“Velocity profiles” and “stream lines” are those observed near the column wall and refer to the movement of bubbles, not to the interstitial flow of liquid between bubbles. In the case of plug flow it is certain that the flow observed a t the wall prevailed also in the interior of the column, the main proof being that gas balance calculations give values of A Q O i Q G < 0.05. When the flow is turbulent, the identity of flow pattern a t the wall and inside the column is not certain, but it can be reasonably assumed to exist. Validity of this assumption is supported by results of careful visual inspection and by the fact that visually observed changes in flow regime agree well with breaks in the curves f us. G, fDz, us. G: and A P I A 2 us. G. PLUGFLOW. Plug flow was observed a t low foam velocities. I t was typified by relatively dry foam and mostly polyhedral bubbles which did not move relative to each other. Plug flow was time-invariant and always coaxial, even in nonvertical sections. Typical stream lines and velocity profiles are shown schematically in Figure 3. TURBULENT FLOW. As foam velocity rose above a certain value, the plug flow regime in the vertical column began to be disturbed. The transition to turbulent flow was gradual and some kind of intermediate flow seemed to exist: More bubbles became round, could move in relation to each other, and ordered themselves in the space between larger ones. The air velocity at which slight deviations from plug flow appeared was defined as the critical air velocity, Gcr. I n order of increasing flow rates, the following phenomena were observed in the vertical column : nonuniform noncoaxial velocity, all forbvard flow; over-all, time-averaged, forward flow with local temporary stagnation; time-averaged forward flow with local backflow. Backflow led to formation of large eddies with D,/D, increasing with increasing flow rate. The various observed stream lines and velocity profiles are shown schematically in Figure 3. The term “turbulence” used here for classifying the type of flow should not be confused with turbulence of a single con-
Velocity Profiles.
Figure 3. variables
Observed phenomena and typical values of Typical Values of Variables Spinnerette 0.040-mm. hole diameter a
Flow
G,
cc.
sq. cm. min. f,
%
DEDC
b
C
Weakly by turbulent
Plug
Turbulent (stagnation)
d Turbulent (backflaw)
e Highly turbulent
88.5
205
230
380
914
1.6
5.1
6.1
6.1
1.35 0.37 4.8 X 10-2
1.71 0.42 8.9 X 10-2
1.75 0.43 12.4 X 10-2
6.1 1.o 1.62 0.45 25.4 X 10-2
...
...
VOL. 8
NO. 3
. ..
0.5 1.72 0.45 21.5 X 10-2
AUGUST 1969
485
tinuous fluid. It means here time-dependent velocity profiles and stream lines, and nonzero radial velocity components of bubble movement. I n addition, turbulence in foams is typified by much larger amplitudes of velocity fluctuations (approximately 2 cm. per second) and much lober frequencies (0.5 to 2.0 sec.-l) than in a turbulent single fluid. Turbulence was usually more pronounced near the liquid level, where the foam was wetter, and near the column head bend. The latter phenomenon was caused by the fact that as the foam moved through the head bend the vertical velocity component diminished, and eventually became zero. This produced a nonuniform and increased net drainage, with a resultant liquid backflow and formation of an eddy at the inner curvature of the bend. This phenomenon was observed particularly with the rounded head bend, and much less with the 87’ bend. With the rounded head bend some turbulence could be observed near the bend even before any turbulence was visible in the vertical column. I n the almost horizontal part of the column, plug flow was observed even when considerable turbulence prevailed in the vertical column. This was caused by the fact that the vertical velocity component was zero, causing net drainage of liquid and formation of polyhedral bubbles a t most of the upper part of the cross section. However, a t very high turbulence in the vertical column, the flow in the almost horizontal section was characterized by a foam layer moving relatively slowly on top of a faster moving liquid layer embedded with bubbles. BUBBLECOLUMN.At very high air flow rates the foam became so wet in the vertical column and turbulence so strong that the interface between foam and liquid was difficult or even impossible to locate. Occasional very large bubbles (10 to 22 mm. in diameter), originating in the liquid pool, appeared and shot through the foam column a t velocities much above average. The system behaved like a regular gas-liquid dispersion (bubble column). The transition from turbulent flow regime to bubble column regime is much less distinct than that from plug flow to turbulent flow. Quantitative measurements in the bubble column regime could hardly be performed because “foam column height” could not be held constant and very large air bubbles prevented photographic bubble size determination. Bubble column behavior in the vertical column was usually followed by phenomena similar to turbulent flow in the almost horizontal section. VARIABLE CROSSSECTION.A few comparative experiments were performed with the foam moving from a large cross sec-
tion column into a narrower one (Figure 1, liquid level B ) . Even a t air flow rates which would have caused strong turbulence in the constant and narrower cross section column, plug flow and polyhedral bubbles typified the entire variable cross section column. This phenomenon is easily explained by the improved drainage in the enlarged lower section because of low linear velocity and the consequent low f and polyhedral bubble foam entering the narrower upper section. TYPEOE GASSPARGER.Foams produced with the sintered porous sparger exhibited markedly different behavior from those produced \zith spinnerettes because of the smaller bubble size and much less uniform bubble size distribution. Thus, for example, foams produced with the sintered porous sparger showed more pronounced stagnant regions, backflow, and eddies in the turbulent flow regime, and the transition from plug to turbulent flow started at a lomer air flow rate. Quantitative Results
Unless stated differently, all the results reported below are for glass columns of constant cross section and 87’ head bend. Pressure Drop. Plots of static pressure us. foam height were prepared for each experiment. All these plots showed a region of varying slope a t low foam heights followed by a constant slope region at higher foam heights. 4P/AZ for each experiment was calculated from the constant slope region. A typical plot of 4Pl4Z us. G is shown in Figure 4. The curves for foams produced with spinnerettes depend on both D o and H and are remarkable in that they break when the flow regime changes from plug to turbulent. A similar break was reported by Metzner and Brown (1956), but the break in their curve was based on a single experimental point. The curves for foams produced with the sintered porous sparger are erratic and do not show a distinct break between the two flow regimes. Figure 5 shows that the type of head bend affects the pressure drop. Bubble Size. Distribution of bubble size calculated from single photographs of foams obtained with spinnerettes were normal. Typical plots of average bubble diameter and a/Dlo us. G for spinnerettes are shown in Figure 6. 50
40 0
P X h
N
.
20
E
h(
52
Y h
lJ
25
%
20
N
10
EV
8
.
6
v
N
Q
?I
. 2
4
a.
I
2
a 3 LO
20
60 80 100
200
4 0 0 600
G ,. cm3/crn2 min Figure 4.
Pressure gradient vs. gas flow rate
H = 570 mm.; spargers, spinnerettes 0 Do = 0.040 mm. V D o = 0.100 mm.
486
4
l&EC FUNDAMENTALS
10
1000
20
I
I
40
I 1 1 1 1
60 80 100
G, cm3/crn* min Figure rate
5. Pressure gradient
vs. gas flow
H = 860 mm.; sintered porous sparger bend V 87” head bend
0 Round head
E N n P
T h e bubble size distribution of foams obtained with the sintered porous sparger were not normal and a correlation as in Figure 7 could not be obtained. Plots of D32 and u/Dl0 us. G for the sintered porous sparger (Figure 8) show, in addition to a considerable scatter of points, different trends than the corresponding curves for spinnerettes (Figure 6). These are caused by the nonuniform orifice sizes. At low gas flow rates, most of the gas passes through the larger orifices and bubbles are relatively large. As G increases, part of the gas passes through the smaller pores and D32 decreases. A further increase in G brings the flow into the constant frequency regime and 0 3 2 increases. I t is expected that a t still higher values of G jet regime will be reached and 0 3 2 will decrease. u/Dl0 in Figure 8 (lower), is almost independent of G, probably because of a combined action of the factors mentioned above. Foam Ratio, f. Typical families of curves obtained by plotting f us. G are shown in Figures 9 and 10. T h e curves break a t a value of G corresponding closely to the transition from plug to turbulent flow as observed visually. For spinnerettes as for the sintered porous sparger (Figure lo), curves were sometimes dependent on foam heights, though not in all cases. Previous investigators (Rubin et al., 1967) found that in plug flow f was independent of H. The exceptions in the present work could not be explained.
2.0 1.5 1.0 0.8
0.6 0.6
5! 0 . 4
0
s
0
0.2
1
I
I
I
1
I
I O
I
4
I
I
, I , ,
3.0
E 2.5 E, 2.0 el N
c3 1.5
1.0
20
40
60 80 100
400
200
600 800 1009
G , crn3/crn2 rnin Figure 6.
E
and cr/DI0 vs. gas flow rate
D32
N
Sparger, spinnerettes
Do = 0.040 mm. D o = 0 . 1 0 0 mm. = 5 7 0 mm.
a, b. c, d.
VH 0H 0
n
= 8 6 0 mm. 1 5 2 0 mm.
0.8 Y
0.6 I
I
I
I
I
I
I
1
1
1
1
I
I
I
I
I
I
I
I
I
0.8 0.6
H =
2
Assuming that in the case of uniform orifice size of spinnerettes the relation between bubble diameter and G is similar to that obtained with a single orifice, the rising parts of the curves in Figure 6, a and c, correspond to the constant bubble frequency regime, slope E 1 / 3 , whereas the falling part corresponds to the jet regime. For spinnerettes, the average bubble diameter 0 3 2 can be correlated well by a plot of D 3 2 / % 5 us. G/Do or, equivalently, us. Do Uo P ~ / I c ( ~ .This correlation (Figure 7) is general for all Do and independent of foam height. T h e coordinates for this correlation were chosen according to theoretical and empirical considerations in the formation of bubbles from single submerged orifices (Leibson et al., 1956). I
I
I
I 1 I / /
G
0.4
I
0.2 1
I
60 80 100
LO
20
10
200
400
G , cm3/cm2min Figure 8.
D32
and a/Dlo vs. gas flow rate
Sintered porous sparger V H = 5 7 0 mm., 8 7 ' head bend X H = 860 mm., round head 0 H = 8 6 0 mm., 8 7 ' head bend 0 H = 1 5 2 0 mm., 8 7 ' head bend
I
1x1
I
I
I l l 1
-6 0
P 2 X
\ N
o P
'
4
2
v
1 10
I
2
8 1 0
6 1
I
4
Figure 7.
I I I I I I
6
8 lo2
1
I
2
Correlation among
1
1
4 D32,
1
1
6
l
1
1
8 lo3
Reo
1.5
Do, and G
Spargers, rpinnerettes 0 Do = 0.040 mm. 0 Do = 0 . 0 6 0 mm. X Do = 0.080 mm. V D, = 0.100 mm.
VOL. 8
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487
The curves off us. G for spinnerettes and the corresponding curves for the sintered porous sparger look markedly different, probably because of the large difference in bubble size and size distribution discussed above. The value off a t G,,, the flow rate a t which the transition from plug flow to turbulent flow occurs, is between 2 and I%, This is much less than the hitherto assumed value of 26%, the minimum void fraction of closely packed equal spheres, and much less than 20'%, the approximate minimum void fraction
10
8 6
4
s rc
.
2
/
1 0.8
15 20
40
60 80 100 150 200
I
400 500
G , cm3/cm2min Figure 9. Liquid volume fraction of exit foam vs. gas flow rate
H = 860 mm., 87' head bend, sparger-spinnerettes 0 Do = 0.040 mm. Do = 0.060 mm. X Do = 0.080 mm. V Do = 0.100 mm.
20 15
10 8 6
s
I
rc
2
1.0 0.8,
0.6
of closely packed unequal spheres of a size distribution resembling that of the discussed foams. This means that turbulence appears when a large fraction of the bubbles are still polyhedral. This finding is confirmed also by determining the approximate average fraction of round bubbles in the foam in different flow regimes (Table 11). Details of this determination are on file (Hoffer, 1967). General Correlation. A generalized correlation for spinnerettes, independent over wide ranges of Do and H , is obtained by plottingfD;, us. G (Figure 11). The curve breaks a t G = 160 cc./sq. cm. min., which is in good agreement with the value of G where the transition from plug to turbulent flow was visually observed. For plug flow /D:2 = 7.1 x 10-8 For turbulent flow /D:2 = 5.5 x G2.35. Both equations were obtained by least squares method. At very high values of G, /D,", becomes almost independent of G for Do = 0.040 mm. (broken line in Figure l l ) , and it may be that this third region exists for other values of Do a t still higher values of G. I n the case of sintered porous sparger, a plot of /D:, us. G (Figure 11) does not provide such a good correlation. The scatter of data is large, the curve does not break sharply with change of flow regime, but for G = 150 cc. per sq. cm. per minute the points are definitely on the spinnerette curve for turbulent flow. Figure 11 also compares the experimental curves, 1 and 2, with the theoretical lines of Lernlich et al. (Leonard and Lemlich, 1965; Shih and Lemlich, 1967) and Haas and Johnson (1967), 3 and 4, respectively. Both theoretical lines deviate considerably from the experimental points for turbulent foam flow. This deviation is expected, since both theoretical models were developed only for plug flow. For the plug flow region the Haas and Johnson theoretical line seems to be in better agreement with theexperimental curve for spinnerettes than the Lemlich et al. line. However, the Haas and Johnson model predicts a value of 1.1 to 2.0 for the constant k in Equation 2. The value of k used for calculating line 4 in Figure 11 is 1.5. Other values of k, within the predicted range, have also been reported (Rubin et al., 1967). In addition, since no surface viscosity data are reported in the literature for the surfactant used in the present work, a value of ps = dyne sec. per cm. was assumed for calculating the theoretical line of Lemlich et al. Critical Air Flow Rate. T h e visual observations and quantitative experimental data indicate that plug flow in vertical-constant-cross-section columns is characterized by slip a t the column wall and a large fraction of polyhedral bubbles. T h e transition from plug to turbulent flow depends primarily on the ability and drive of the bubbles to move relative to each other, This ability and drive depend on a complex relation among f , G, bubble size, bubble size distribution, and bubble shape. Other factors are the uniformity of drainage, which depends on column geometry, and the ratio D32/Dc(wall effects), The complex relationship between these
0,5 15 20
40
G
60 80 100 150 200
400
, cm3/cm2min
Figure 10. Liquid volume fraction of exit foam vs. gas flow rate Sintered porous sparger
V H
= 560 mm., 87' head bend
H = 860 mm., 87' head bend 0 H = 860 mm., round head bend 0 H = 1520 mm., 87' head bend 0
488
l&EC FUNDAMENTALS
Table II. Approximate Fraction of Round Bubbles in Different Flow Regimes Sintered Flow Regime
Plug flow Intermediate Turbulent Highly turbulent
Porous ,Yparger 0.59 0.93 0.96 1 .oo
Sjinnerettes 0.15 0.26 0.50 0.86
Figure 1 1 .
fDi2 vs. gas flow rate
Experimental points. All calumn heights, oll head bends. Sintered porous sparger; spinneretter; 0 Do = 0.040 mm. 3 Do = 0.060 mm. A Do = 0.080 mm. V Do = 0.100 mm. 1 . Experimental for spinnerettes 2. Experimental for sintered porous sparger 3. Theoretical for p = 0.01 poise, p8 = 1 0-4 dyne sec./ cm. (Leonard and Lemlich, 1965; Shih and Lemlich, 1967) 4. Theoretical (Haar and Johnson, 1 9 6 7 )
+
G
,cn?/
em2 rnin
factors is evident on comparing the experimental data obtained with the two types of spargers. Thus, Figures 9 and 10 indicate that G,, for the sintered porous sparger is considerably lower than for the spinnerettes. Comparing Figures 6 and 8 a t the corresponding G,, indicates that the foams produced with the two types of spargers differ in average bubble diameter, and particularly in bubble size distribution. Since G,, for spinnerette-produced foams is the higher, it is probable that for foams having equal diameter bubbles it will be higher still. On the other hand, Figures 9 and 10 and Table I1 show that f and the fraction of spherical bubbles a t G,, are much larger for sintered porous sparger than for spinnerette-produced foams. This indicates that the value of G, and consequently the shear forces, play a n important role in the transition from plug to turbulent flow. There probably exists a minimum G below which plug flow will prevail regardless of bubble shape and f. Finally, the fact that turbulence commences even when a substantial fraction of the bubbles are polyhedral indicates that a t sufficiently high G the shear forces will be large enough to cause turbulence even if the fraction of round bubbles is truly small. If all bubbles were polyhedral, the high shear rates resulting from high G would probably cause bubble rupture and coalescence. Conclusions
The present work shows the existence of three flow regimes for stable foams moving in vertical constant cross section glass columns: in order of increasing velocity, plug flow, turbulent flow, and bubble column regime (where foam as such no longer exists). Because of the multiple and complex factors involved, the
present work did not lead to a unique expression, equivalent to a critical Reynolds number, from which the transition from plug to turbulent flow can be accurately predicted in all cases. I t does provide, however, important general correlations and insight into the mechanisms and behavior of dynamic foams. The transition from one flow regime to the other, in particular from plug to turbulent, can be easily determined visually and is characterized by breaks in the AP/'AZ L'S. G, f us. G, and f D i 2 us. G curves. The prevailing flow regime depends on a large number of factors, of which the following have been investigated a t least qualitatively: bubble size and shape, bubble size distribution, foam velocity, foam ratio, foam height, and uniformity of liquid drainage-i.e., column geometry. Foams having normal bubble size distribution (produced with spinnerettes) are characterized by the following empirical equation, independent of foam height, f D : , = CG"
(7)
and D32 can be obtained from a correlation of D 3 2 / d & us. Re,. Plug flow is characterized by n E 1 and a large fraction 2 . 5 , and the of polyhedral bubbles, turbulent flow by n transition by a distinct break in the curve. For foams having nonnormal bubble size distribution (produced with sintered porous sparger), the break in the curve offDi, us. G is less distinct, though values of n seem to be the same. The correlation of D82/1/% us. Re, does not holc!, however. The value off a t G,, is much lower than 26%. G,, may be increased by more uniform bubbles and more uniform drainage (87" head as compared with the rounded head). A few qualitative experiments indicated that important parameters in foam flow phenomena are uniformity of column cross section and material of construction of the column. VOL. 8
NO, 3
AUGUST 1969
489
Nomenclature =
constant, defined by Equation 7
SUBSCRIPT G = gas
= column diameter, cm.
eddy diameter, cm. diameter of sparger orifice, cm. mean bubble diameter, defined by Equation 3 liquid volume fraction of foam a t foam column exit, ml. liquid/ml. foam = liquid volume fraction of foam a t height Z above liquid level, ml. liquid/ml. foam = acceleration of gravity, cm./sec.* = gas flow rate per unit cross section (empty column), cc./sq. cm. min. = critical gas flow rate, cc./sq. cm. min. = height of foam column = constant, defined by Equation 2 = constant, defined by Equation 1 = exponent, in Equation 7 = pressure gradient, cm. HaO/cm. foam height = calculated gas flow rate, cc./min. = measured gas flow rate, cc./min. = Reynolds number of gas through orifice, D,U,p,/~c = mean velocity of gas through orifice = foam velocity, cm./min. = maximum drainage velocity, cm./min. = Z component of foam velocity V , cm./min. = vertical coordinate = viscosity of surfactant solution, poises = surface viscosity, dyne sec./cm. = density of surfactant solution, g./cc. = standard deviation of bubble size distribution, defined by Equation 4 = = = =
J
g
G
G,, H k K n
AP/AZ Qcc Qcm
Re,
U, V
Vd V, Z
literature Cited Bikerman, J. J., “Foams, Theory and Industrial Applications,” Reinhold, New York, 1953. Bruner, C. A,, Lemlich, R., IND.ENG.CHEM.FUNDAMENTALS 2, 297 (1963’1. Fanlo, S., Lemlich, R., A.Z.Ch.E. J . Symp. Ser. 9, 75 (1965). Haas, P. A . , Johnson, H . F . , A.Z.Ch.E. J . 11 (No. 2), 319 (1965). Haas, P. A.. Johnson, H. F., IND.ENG.CHEM.FUNDAMENTALS 6, 225 (1967). Hoffer, M. S., M.Sc. thesis, Technion, Haifa, Israel, 1967 (in Hebrew, with English synopsis). Leibson, I., et al., A.Z.Ch.E. J . 2 (No. 3), 296 (1956). Leonard, R. A , , Lemlich, R., A.Z.Ch.E. J . 11 (No. l ) , 18 (1965). Letan, R., Kehat, E., A.Z.Ch.E. J . 13 (No. 3 ) , 443 (1967). Matalon, R., “Foams,” in “Flow Properties of Disperse Systems,” J. J. Herman, ed., pp. 323-44, North Holland Publishing Co., Amsterdam, 1953. Metzner, A. B., Brown, L. F., Znd. Eng. Chem. 48,2040 (1956). Rubin, E., Ph.D. dissertation, Columbia University, New York, 1962. Rubin, E., Gaden, E. L., “Foam Separation,” in “New Chemical Engineering Separation Techniques,” H. M. Schoen, ed., pp. 319-85. Interscience. New York. 1962. Rubin, E., LaMantia,’ C. R., Gaden, E. L., Chem. Eng. Sci. 22 (No. 8), 1117 (1967). Shih, F. S., Lemlich, R., A.Z.Ch.E. J . 13, 751 (1967). Wace. P. F.. Banfield, D. L., Chem. Process Ene. 47 (No. lo), 70 (1966). ’ Weissman, E. Y . ,Calvert, S., A.Z.Ch.E. J . 13, 788 (1967).
RECEIVED for review March 29, 1968 ACCEPTED November 18, 1968
PARTICLE D Y NAMlCS I N SOLIDS-GAS FLOW I N A VERTICAL PIPE K. V . S . R E D D Y New Jersey Zinc Co., Palmerton, Pa. 18071 D . C . 1. P E I University of Waterloo, Waterloo, Ontario, Canada
An extensive investigation was made of the particle dynamics of narrow spherical glass beads transported vertically by a turbulent air stream in a 10-cm. i.d. pipe. The particle axial velocity profile in the pipe core region was found to be similar to that of the gas and to follow a power-law type relationship. Within the range investigated the slip velocity at the center of the pipe is a function of the loading ratio and the particle terminal velocity, the particle concentration is uniform across the pipe cross section, and particle velocity fluctuations follow the pattern of the free stream turbulence of the gas phase.
TWO-PHASE solid-gas systems have been encountered in
engineering for many years. T h e concentration of solids in the dispersed solid-gas flows may be dense, as in gas catalyst systems, or very dilute, as in transport reactors. Literature on solid-gas systems is abundant. T h e fundamental aspects of the solid-gas systems have been reviewed by Torobin and Gauvin (1959, 1960). Books by Zen2 and Othmer (1960), Leva (1959), Dalla Valle (1948), Orr (1966), So0 (1967), and Brodkey (1967) review the fields of motion of a single particle, fluidization, and pneumatic transportation studies. Although much has been done theoretically and semiempirically t o predict gas velocity profiles in fully de490
I&EC
FUNDAMENTALS
veloped single-phase turbulent pipe flows (Hinze, 1959; Knudsen and Katz, 1958; Schlichting, 1960) and the motion of single particles in an infinite fluid medium, little theoretical work has been done t o predict the particle and gas velocity profiles in solid-gas flows. Progress in experimental investigations has been slow because of difficulties in developing accurate methods of measurement. This work was undertaken t o study the effects of particle diameter and concentration and the gas mean velocity on the particle velocity, the gas-phase velocity, the slip velocity between the phases, and the axial pressure gradient along the pipe.
