1282
Ind. Eng. Chem. Res. 1988,27, 1282-1291
Experimental Determination of Gas-Bubble Breakup in a Constricted Cylindrical Capillary P.A. Gauglitz,+C. M. St. L a u r e n t , t and C. J. Radke* Chemical Engineering Department, University of California, Berkeley, California 94720
T o gain insight into the mechanisms of gas-foam generation in porous media, this work quantifies how smaller bubbles snap off from a single, larger bubble as i t moves through a smooth, cylindrical constriction. Time to snap off, generated bubble size, and bubble velocity are obtained from viewing 16-mm movies taken of snap-off events. The bubble capillary number, CuT, is varied from to 5X We study two narrowly constricted capillaries of different tube radii (neck radius/tube radius of 0.17 and 0.20) and one widely constricted capillary (neck radius/tube radius = 0.40). In ~ , the generated bubble the widely constricted capillary, time to breakup is proportional t o C U ~ -and length is proportional t o CuT-l. In the narrow constrictions, the same results apply above CuT = 5X below this critical CuT, time to breakup is independent of CuT but depends on the Ohnesorge number. Snap-off behavior with surfactant solutions is found t o be similar t o that in surfactant-free solutions. 1. Introduction Gas foams show great promise for decreasing gravity override and achieving mobility control in steam injection processes for enhancing oil recovery (Ploeg and Duerksen, 1985; Dilgren and Deemer, 1982). The understanding of foam flow process has progressed primarily through the investigation of the single pore mechanisms (Falls et al., 1986; Friedmann and Jensen, 1986; Hirasaki and Lawson, 1985; Owete and Brigham, 1984; Mast, 1972). These studies form the basis of the current modeling efforts for foam flow which rely on a quantitative description of the pore level events (Falls et al., 1986). The objective of this work is to quantify the important snap-off mechanism of bubble generation and regeneration in porous media. Fried (1961) is apparently the first investigator to observe visually bubble generation in porous media. He mentions two mechanisms: (1)bubble division which occurs when a new bubble forms as a larger bubble moves through a liquid-filled constriction, and (2) bubble redivision which occurs when long bubbles divide into a series of smaller bubbles as they pass through pore constrictions. The two mechanisms are identical and cqnform with the detailed visual observations reported by Mast (1972) and Ransohoff and Radke (1988) for the snap-off mechanism. The snap-off mechanism occurs when a bubble front moves into a liquid-filled constriction (Goldsmith and Mason, 1963). A local view of a generation site for snap-off is depicted in Figure 1. Wetting liquid collects in a constriction of a generation site due to surface tension forces. In Figure 1,the nonwetting gas generally flows in the larger channels. Periodically, however, bubbles are formed by snap-off at the generation site when gas invades past the neck of the liquid-filled constriction, and a collar of liquid collects a t the pore which snaps off the gas finger. The four key steps in bubble snap-off are depicted in Figure 2. In the first two steps, the nonwetting bubble moves into the constriction, depositing a film of wetting liquid. As the bubble front passes through the constriction, fluid gathers in a collar that grows a t the pore neck as shown in step 3. When sufficient liquid collects in the collar, it blocks the constriction and snaps off a smaller
* To whom
correspondence should be addressed. Currently with Chevron Oil Field Research Company, LaHabra, CA 90631. 3 Currently with Clorox Research Laboratory, Pleasanton, CA 94566.
0888-5885/88/2627-1282$01.50/0
bubble, step 4. A series of several smaller bubbles can originate from each upstream bubble. To quantify the snap-off mechanism, we investigate here bubble snap-off in the simple geometry of constricted, cylindrical capillaries. Section 2 presents simple scaling arguments which guide and correlate the experimental findings. In companion studies, we have reported the results of snap-off in constricted capillaries of square cross section, which probably better mimics the pore spaces of naturally occurring porous materials (Gauglitz et al., 1987; Ransohoff et al., 1987). The principal difference between these two pore geometries is also discussed in this section. Section 3 then describes the experimental apparatus and procedures to observe snap-off visually (Gauglitz, 1986; Gauglitz et al., 1987). Sixteen-millimeter movies record the translation of air bubbles through the constriction and the snap-off kinetics. Experimental results for a series of glycerol-water solutions without surfactants in three constricted, cylindrical capillaries are given in section 4. We first show results for the capillary with the neck radius/tube radius ratio of 0.4. The results for tighter constrictions and the role of gas compressibility in bubble snap-off are considered next. Finally, section 5 presents experimental data for surfactant solutions of 1w t % SDBS (sodium dodecylbenzene sulfonate) and 1wt % Chevron Chaser SDlOOO (a commercial aliphatic disulfonate) in glycerol-water mixtures. We investigate these solutions since surfactants are requisite for generating stable foam, and further, surfactants can create dynamically immobile fluid interfaces which may alter snap-off behavior. 2. Qualitative Theory Effective use of foams in underground porous media requires a quantitative understanding of the bubble snap-off mechanism shown in Figures 1 and 2. Dimensional analysis determines the important dimensionless variables. This also allows our experimental observations in capillary tubing with diameters of -1 mm to be extended to the very small pores (-0.01 mm) of oil-bearing underground porous media. In addition, we gain physical insight into the snap-off mechanism by using simple scaling arguments to determine the dependence of the important dimensionless quantities. There are nine important variables in the snap-off process. Figure 3 portrays most of these. In the experiments, we are interested in the time to breakup, tb. The bubbles are driven through the constriction at a constant volumetric rate, which determines the velocity of the 0 1988 American Chemical Society
Ind. Eng. Chem. Res., Vol. 27, No. 7, 1988 1283
HO
Film remaining from wetting fluid displacement
Figure 4. Bubble movement, film deposition, and snap-off in a constricted capillary.
Figure 1. Local view of bubble generation by snap-off.
Figure 2. Pore-level view of bubble generation by snap-off.
. 6.. .,.Gaps
Bubble
+&.,
l/R1 + 1/R3 so that PL1- PL2 is positive, fluid flows into the collar at the constriction and initiates snap-off. Note that as the collar grows, R2diminishes, and the pressure driving force becomes even more positive. This accelerates the unstable collar growth. Equation 4 also demonstrates the physical significance of the dimensionless time given in eq 1. The driving force is the product of surface tension and the difference in curvature, and the fluid viscosity offers the only resistance to flow along the film. These two parameters combine to yield a characteristic velocity, a l p , for the fluid film. The
1284 Ind. Eng. Chem. Res., Vol. 27, No. 7, 1988
dimensionless time results from combining the length scale, RT, with this velocity. The capillary number is crucial in bubble snap-off since it determines the thickness of the fluid film deposited by the bubbles as they translate through the capillary. Bretherton (1961) and Park and Homsy (1984) have shown that, when Cu 0, the film thickness, Ho,is of the order of the capillary number to the two-thirds power:
Bleed Valve
el
-
Ho/RT = O(Cu2i3)
(5)
Numerical solution of the exact creeping flow equations confirms eq 5 (Shen, 1984). However, experimental data begin to deviate from eq 5 for capillary numbers below lo4 (Bretherton, 1961; Schwartz et al., 1986). Fortunately, the majority of our data is a t capillary numbers above Hammond (1983) has demonstrated that the characteristic response time for flow in thin liquid films is proportional to the inverse cubic power of the film thickness so that Tb 0: (Ho/RT)-~ (6) Combination of eq 5 and 6 specifies that the time to snap off is proportional to the capillary number to the minus two power: 7b
0:
CaT-2
(7)
Hence, on a log-log plot of breakup time as a function of the tube capillary number, the experimental data should lie on a line of slope of minus two. The bubble size generated by snap-off is pertinent to foam texture. The bubble translates through the capillary with a velocity of -UT for a period of time given by the breakup time, tb. The distance the bubble front travels past the constriction when snap-off occurs is Lb UTtb. In dimensionless form, this becomes Lb/RT CUT 7b (8)
-
-
Combining this with eq 7, we find that Lb/RT
0:
CUT-'
(9)
Equation 9 shows that smaller bubbles are generated a t higher tube capillary numbers. This interesting result demonstrates that, although bubbles may move faster through a constriction, they deposit a thicker film which snaps off more quickly, offsetting their higher translation velocity and making smaller bubbles. Shen (1984) proposed different but equivalent forms of eq 7 and 9. In constricted square capillaries, in contrast, liquid flows primarily along channels in the corners of the capillary. The channel thickness depends weakly on the capillary number and, accordingly, the capillary number plays a minor role in the time to breakup. In this case, the time to breakup is essentially independent of the capillary number and the generated bubble length increases proportionally to the capillary number. Square cross-sectional pores have been considered in detail by Gauglitz et al. (1987) and Ransohoff et al. (1987). In the next section, the experimental apparatus designed to observe snap-off is described. We determine the constants of proportionality in eq 7 and 9 and test whether the dimensionless time and capillary number properly scale the breakup mechanism. 3. Experimental Section
Experiments are performed to quantify gas-bubble snap-off over large ranges of CuT. The apparatus facilities visual observations in a variety of constricted cylindrical capillaries. Sixteen millimeter movies record the snap-off events.
u
Syringe Pump +
i
-
2
T Vode Drain
Figure 5. Schematic of experimental apparatus.
3.1. Apparatus. Figure 5 shows a schematic of the experimental apparatus. The key element is the constricted capillary which is placed inside a viewing cell. The capillary is connected on the upstream side to a flow manifold, containing a bleed valve to insert air bubbles, which itself is connected through a graduated pipet and Tygon tubing to a Harvard syringe pump (Model 925). In all experiments, the Harvard pump utilizes a 20-cm3syringe which is calibrated at each pump setting by determining the time required for an air-liquid meniscus to move between two markings on the pipet. This calibration is in agreement with a second calibration performed by measuring the mass efflux from the syringe at each pump setting. On the downstream side, the capillary connects to a constant level bath filled to 2 in. above the level of the capillary. A 500-W lamp, placed 9 in. behind the viewing cell, back-lights the capillary when movies are filmed. Diffuse back-lighting proved superior to front- or back-lighting with a fiber optic light source. When observing snap-off, we reduce optical distortions across the cylindrical glass capillary by filling the viewing cell with a mixture of 40 wt % dibutyl phthalate (Kodak) in mineral oil (Aldrich Chemical Co., nDZo = 1.4673, d = 0.838 g/cm3) which matches the refractive index of the glass capillaries. This makes the Pyrex glass capillary with a refractive index of 1.474 practically invisible (Goldsmith and Mason, 1962). Interestingly, visual observations of the Pyrex glass in the refractive index bath demonstrate that each piece of glass has a slightly different refractive index. Although the refractive index bath can be adjusted for each glass capillary, we employ the 40 w t % mixture. Evaporation from the refractive index bath is slowed by a Plexiglas lid covering the viewing cell. Snap-off is recorded by filming 16-mm movies of the snap-off events. A 16-mm Bolex H16 movie camera films the process a t 50 frames/s with Kodak Black and White Plus-X reversal film. To magnify the small constrictions, 20-mm or 60-mm extension tubes are used with a KernPaillard MACRO-SWITAR 50-mm f11.4 lens. The 60-mm extension tube (with f-stop 11)allows detailed observation of the region near the pore neck (field of view 1 mm x 2 mm), and the 20-mm extension (with f-stop 11) allows observation of bubbles as they pass through the constriction and snap off and is used most often. The camera is mounted on a tripod (Fastax WF 326B, Wollensak Optical Co., Rochester, NY) and carefully leveled before filming a movie. For further details, consult St. Laurent et al. (1986). To observe and quantify the dependence of breakup time on eaT = kuT/(r, experiments are devised to vary the capillary number over a wide range. Fluids covering various viscosities and surface tensions are obtained from mixtures of glycerol (Baker reagent grade), distilled water, and the two surfactants: sodium dodecylbenzene sulfonate (SDBS) and Chevron Chaser SD1000. The viscosities are ascertained with a Brookfeld viscometer (UL adaptor) and
Ind. Eng. Chem. Res., Vol. 27, No. 7, 1988 1285 Table I. Physical Properties of Experimental Solutions (21 OC) surface viscosity, tension, density: solution mPa.s mN/m n/cm3 deionized H 2 0 1.0 72 1.00 1 wt % SDBS in H 2 0 1.0 31 1 wt % active CHASER in 1.1 39 HZO 1 wt % SDBS in 54 wt % 7.6 34 glycerol in H 2 0 1.1wt % active CHASER in 5.3 41 50 wt % glycerol in H 2 0 57.9 wt % glycerol in H 2 0 8.5 68 1.15 70.2 wt % glycerol in HzO 19.9 67 1.18 83.3 w t % glycerol in H 2 0 59.96 67 1.22 a
Perry (1950). Small volume, not UL adapter.
Table 11. Characteristics of Constricted Capillaries and Their Identification constriction, aore iden. radius. cm R,.,L/R, emDirical fit k(5) narrow round 5.08 X 0.2 1 - 0.4[1 + cos (r5/8)] (0.5 mm) wide round (0.5 5.08 X lo-' 0.4 1 - 0.3[1 + cos (?r5/9)] mm) narrow round (1 0.1 0.17 1 - 0.4[1 + COS (*5/6)] mm)
the surface tensions determined with a Wilhelmy plate and a Rosano surface tensiometer (Bipolar Corp.). The fluid properties are shown in Table I for each mixture studied. 3.2. Constricted Capillaries, The constricted capillaries are made from straight lengths of cylindrical capillary tubing, 40-50 cm long. To eliminate gravity effects (Le., Bond number ( p - pg,)gRT2/fl < l ) , we study capillaries with R T I0.1 cm. Three constricted capillaries are studied two from 0.1016 5 X cm i.d. Trubore tubing (Ace Glass, Vineland, NJ), and one from 0.20-cm-i.d. stock capillary tubing. Constrictions are formed by fixing the ends of the capillary in a glass blowing lathe, rotating the capillary, and heating it a t a single location with a narrow blue flame. When the glass melts, surface tension causes the capillary to constrict. After removing the flame, the glass solidifies to form a constriction. No two constrictions are exactly identical. However, the shape of the resulting constriction can be controlled by (1)varying the amount of heating (longer heating makes a constriction tighter), (2) varying the width of the flame (a wider flame makes a longer constriction), and (3) pushing together or pulling apart the ends of the capillary while heating (pushing makes sharper constrictions, while pulling the ends of the capillary apart makes longer constrictions). Snap-off behavior depends strongly on the shape of the pore constriction, which is therefore characterized carefully. Each of the three capillaries is analyzed by projecting a single movie frame onto graph paper and measuring the local radius of the capillary as it compares to the unconstricted tube radius. Open circles in Figure 6 represent the radial position of the pore wall obtained from the projection of the wide round (0.5-mm) pore. Table I1 identifies each pore with its tube radius and neck radius. For further details of the pore shapes, see St. Laurent et al. (1986). The shape of each constricted pore is fit to a cosine function, as shown by the solid line in Figure 6. All constrictions obey the cosine function closely. They are very symmetrical. Several differently constricted capillaries are studied since the neck radius/tube radius is important in determining the snap-off behavior. Referring to Table 11, we
*
1
-12
I
I
-4 0 4 9 = x/RT, Dimensionless Length
8
I
-8
" 12
Figure 6. Pore characterization of the wide round (0.5-mm) pore made from 1.016 X lo-' cm i.d. Trubore capillary tube.
note the following: the wide round (0.5-mm) pore and the narrow round (0.5-mm) pore have the same tube radius, but the neck radius of the former is 2 times the later; the narrow round (1-mm) pore has nearly the same neck radiusltube radius ratio (-0.2) as the narrow round (0.5mm) pore, but its tube radius is twice as large. Additionally, the narrow round (1.0-mm) pore is more sharply constricted than the narrow round (0.5") pore since the constriction length spans 5 = *6 for the narrow round (1-mm) pore compared to 5 = *8 for the narrow round (0.5-mm) pore. We believe this is a minor difference. 3.3. Procedure. Prior to running a set of experiments, the capillaries are soaked in chromic acid and then rinsed with distilled water. The flow system and capillary are connected and flushed with -30 cm3of the fluid mixture. An experimental run begins by inserting a bubble of desired length through the bleed valve into the capillary upstream of the constriction (see St. Laurent et al. (1986) for details). A pump setting is chosen to determine the bubble velocity. After the pump is switched on, and the bubble moves into the camera's field of view, the 500-W lamp behind the viewing cell and the Bolex movie camera are turned on. Although the lamp generates a great deal of heat, it is only on during the short duration of an experiment (1-30 s), and over the course of several experiments the fluid temperature remains nearly at ambient temperature (21 "C). The bubble is filmed as it travels through the constriction, recording the bubble movement and the kinetics of the collar formation. The movies are projected onto a screen to observe snap-off. Time to breakup is measured by counting the number of frames between the instant when the bubble front passes the neck of the constriction until the collar snaps off and blocks the pore. Each frame represents 0.02 s a t the filming speed of 50 frames/s. The breakup time is accurate to within 0.06 s (three frames) due to the sometimes ambiguous specification of the exact frame when the bubble front passes the neck of the constriction. The generated bubble length is measured from the projection and compared to the capillary radius to obtain the dimensionless length of the snapped off bubble. A single bubble of measurable length is needed for each experimental run. Bubbles are drawn into the capillary through the bleed valve in the flow manifold. Since gas compressibility can be important, a range of initial upstream bubble lengths is investigated. Experiments are repeated for identical conditions with bubbles approxicm3 in the 1.016mately 3 cm in length (volume 2 X mm-i.d. capillary), 8 cm in length (volume 6 X cm3in cm3 in the 1.016-mm-i.d. capillary and volume 25 X the 2.0-mm-i.d. capillary), 16 cm in length (volume 1 2 X cm3 in the 1.016-mm-i.d. capillary), and very large lengths (volume 2.03 cm3) achieved by allowing the bubble
1286 Ind. Eng. Chem. Res., Vol. 27, No. 7 , 1988 I
Wlde Round (05")
Wide Round (0.5")
Pore
Pore
\ p z85mPa.s D =68mN/m RT '5 08 (109Cm Run I Bubble Length u 8 c m
0 85 A 19.9
.-
in
68 2 67 I R,: 5 08 (laz) cm ~ u d b l eLength 3 -16 c m I
I IO-^ IO-^ IO-~ Tube Capillary Number, CaT'pUT/'
(6~10%m3) I
I
I
I
Tube Capillary Number, Ca,= p U T / u
Figure 7. Effect of the capillary number on the time to breakup in the wide round (0.5-mm) pore.
Figure 8. Effect of the tube capillary number on the generated bubble length in the wide round (0.5-mm)pore.
to spill over into the flow manifold and part of the graduated pipet. Bubbles in each category are roughly the same size, to within f l cm. Bubble lengths are measured with vernier calipers in the straight section of the capillary.
This result contrasts the observations of Strand et al. (1982) for the snap-off of viscous oil drops in constricted glass capillaries. However, in their investigation, a longitudinal groove was cut along the pore wall, and the flow of wetting liquid into the growing collar was dominated by flow in that channel. Since the channel flow depends weakly on the injection rate, the oil front moves farther downstream a t higher injection rates before the collar snaps off. Thus, larger drops are generated at higher injection rates. This result is analogous to snap-off in constricted capillaries of square cross section as discussed in section 2. In the wide round (0.5-mm) pore, fluid flows along wetting films whose thickness (and hence resistance to flow) depends strongly on the tube capillary number. As the bubble travels faster through the pore, it deposits a thicker film which snaps off much faster, producing bubbles that are smaller a t higher injection rates. The generated bubble size is important to applications of foam for enhancing oil recovery. Hirasaki and Lawson (1985) demonstrated that decreasing the bubble size increases the resistance to foam flow through capillary tubes (a model porous medium). A higher resistance to flow aids mobility control and decreases gravity override which leads to improved oil recovery. Figure 8 demonstrates that pores which are not tightly constricted (neck radiusltube radius -0.4) generate very large bubbles at typical reservoir flow rates of C q = 10a-10-3. However, smaller bubbles which can substantially increase the resistance to foam flow throughout porous media can be generated a t high capillary numbers. These flow rates only occur near the injection wells in oil reservoirs; therefore, we expect bubble generation in wide pores only near a well bore. 4.1.1. Bubble Velocity. In the wide round (0.5-mm) pore, the tube capillary number quantifies bubble snap-off because the bubble velocity and the deposited film thickness depend directly on the tube capillary number. To confirm this, the bubble-front velocity, U , was measured from the 16-mm movies during translation through the constriction. The movies are projected onto graph paper and the position of the bubble front is located at a number of different times. Cubic splines are fit through the data; from the derivative of the spline functions, we obtain the bubble velocity (Carnahan et al., 1969; Hornbeck, 1975). A local capillary number, Ca = pU/a, is then determined from the bubble-front velocity, and the
4. Results and Discussion Results for bubble snap-off over a range of capillary numbers, upstream bubble sizes, and pore shapes have been obtained. We present results primarily for the time to breakup, but some data are also reported for the generated bubble size. 4.1. Wide Round (0.5-mm)Pore. Breakup time data are shown in Figure 7 for 8.5 and 19.9 d e s viscosity fluids in the wide round (0.5-mm) pore. Reproducibility of the experimental data is demonstrated by the excellent agreement between two separate experimental runs, run 1and run 2, for the 8.5 mPa.s viscosity fluid. Data lie on a line of slope of minus two, indicating that the scaling arguments presented in section 2 indeed represent the snap-off mechanism. Figure 7 also demonstrates that the fluid viscosity is properly scaled in both the dimensionless time and capillary number since the data for the two different viscosity fluids collapse onto a single line. Also, the capillary number is above lo4, so Bretherton's (1961) result, eq 5, is accurate. As we will show later, the initial upstream bubble size influences bubble breakup in the narrow round (0.5-mm) pore. However, the initial upstream bubble size is unimportant in the wide round (0.5-mm) pore; we have found the time to breakup independent of a 30-fold change in bubble size (Gauglitz, 1986; St. Laurent et al., 1986). Figure 7 shows breakup time data for upstream buPble lengths ranging from 3 to 16 cm. The constant of proportionality in eq 7 is available from the data in Figure 7 . The data agree well with the following relationship: 7 b = 3.0(10-2)CaT-2 (10) Results for the generated bubble size are shown in Figure 8 for the wide round (0.5-mm) pore. As expected from the scaling discussion in section 2, the generated bubble length decreases with increasing capillary number, falling on a line of slope of minus one. The following relationship represents the data well: Lb/RT = 0.12CaT-l (11)
Ind. Eng. Chem. Res., Vol. 27, No. 7, 1988 1287 b
Wide Round (0.5") w = 8.5 m Pa+
3
A(;)
Pore
= 1-0.3[1 +cos(r?/9)]
P 0
1.0
A
19.9 59.9 8.5
m 8.5 h
+
>b IO5
0
\
I
1
I
I
-0
-16
Narrow Round Pore 0.5"
CT
( m P o 4 (mN/m)
I
2 =X/R,,
I
I
0
8
72
68 67 67 68 1.0" Bubble Length 8 cm (6x10e2cm3)
-
I 16
Dimensionless Length
L
Figure 9. Effect of the tube capillary number of the local capillary number for bubbles as they move through the wide round (0.5-mm) pore displacing an 8.5 mP.s viscosity fluid. The solid line indicates the velocity of bubbles moving a t a constant volumetric flow.
Narrow Round(05mm)Pore
P
1
IO-^
A
14
IO-^
10-2
Tube Capillary Number, Co,=pU,/c
Figure 11. Effect of the tube capillary number and the fluid viscosity on the breakup time for bubbles with a length of approxicm3) in both the narrow round (05") pore mately 8 cm (6 X and the narrow round (1-mm) pore.
a
(mPos) (mN/ml -
0 IO 0 85
I o2
72
68 A 199 67 R ~ 5: 0 e ( \ d ) c m
..
Bubble Length 3 CI (2~1Ozcm31
I
I
Norrow Round ( 0 5 m m ) Pore
\
-
H
IO
&
I$-
X
0
:
m
0
10-5
10-4
10-3
.
E lo3-
.-
10-2
c
Tube Capillary Number,COy=pllp/cr
Figure 10. Effect of the tube capillary number and the fluid viscosity on the breakup time for bubbles with a length of approxicm3) in the narrow round (0.5-mm) pore. mately 3 cm (2 X
physical property data are given in Table I. Cubic splines are employed since they fit a smooth curve through the data and yield accurate derivatives. Details of obtaining the velocity data are given in Gauglitz (1986). The bubble front velocity in the wide round (0.5") pore depends on CUT and the pore geometry. The bubbles are driven through the capillary at a constant volumetric flow rate, and according to mass continuity, the local velocity of the bubble front is inversely proportional to the local cross-sectional area of the capillary. Thus, the local capillary number for the bubble front is related to the tube capillary number as follows: Ca = CU,(l/A(3))2
(12)
Experimental measurement of the bubble velocity, shown in Figure 9, confirms eq 12 for bubbles translating through the wide round (0.5-mm) pore. Data are shown for bubbles and 7.3 injected at tube capillary numbers of 2.8 x x These data compare quite well with the solid lines for bubbles moving a t a constant volumetric rate. The capillary number is slightly greater on the exit side of the constriction. This velocity increase, which is negligible on the wide round (0.5") pore, is due to gas-bubble com-
10-5
10-4
IO-^
10-2
Tube Copillary Number, Co,=pU,/cJ
Figure 12. Effect of the initial upstream bubble size on the breakup time for a liquid with an 8.5 mPa-s viscosity.
pression and expansion. However, gas-bubble compression and expansion proves crucial in understanding snap-off in the narrow pores. 4.2. Narrow Round (0.5-mm and 1-mm)Pores. The snap-off mechanism changes in the narrow round (0.5-mm) pore and the narrow round (1-mm) pore because these constrictions have a smaller neck radius/ tube radius ratio by a factor of 'Iz compared to the previous wide pore. Breakup time data are shown in Figures 10-12 for a range of upstream bubble sizes and tube capillary numbers for fluids of 1.0, 8.5, and 19.9 mPa-s viscosity. We observe in Figures 10-12 that, above a critical capillary number of about 5 X lo4, the time to break up data collapse onto a single line of slope of minus two. This is in accord with the results for the wide round (0.5") pore. In particular, Figure 11shows data for -8 cm long bubbles in both the narrow round (1-mm) pore and the narrow round (0.5") pore. These two pores have essentially the same shape as gleaned from the empirical fit
1288 Ind. Eng. Chem. Res., Vol. 27, No. 7 , 1988 I
I
I
[ Norrow Rourd(O.5mm)Pore
b
4
I
,u
~ 8 . 5mP0.s
a . 6 8 mN/m R,=5.08 (10') c m
\
Bubble Leqih-3cm
0
0
(~x10-2cm31
n
3
5 1Ci3
z
E-
"Bubble J u m p "
d
Figure 13. Movement of compressible bubbles in constricted capillaries.
to the pore shape given in Table 11; only the tube radius differs by a factor of 2. Above the critical capillary number, the breakup times for the two pores lie on a single line of slope of minus two, indicating that the tube radius is properly scaled for this behavior. Below the critical capillary number, Figures 10 and 11 show that the time to breakup in the narrow round (0.5mm) pore asymptotes to a constant value that is independent of the capillary number, but which is a function of the fluid viscosity. Over the range studied, the initial upstream bubble size has a minor effect on the breakup time. This is shown by Figure 12 which compares breakup times for different length bubbles with the 8.5 mPa.s viscosity fluid. The critical capillary number in Figure 11for the narrow round (1-mm) pore is essentially identical with that of the narrow round (0.5-mm) pore. However, below the critical capillary number, the time to breakup seems not to asymptote to a constant. Rather, the time to breakup increases with decreasing CaT, but at a much slower rate. The critical capillary number originates in the compressibility of the gas bubble. The Young-Laplace equation demands that, if the curvature of the bubble front is positive (i.e., a nonwetting bubble), the pressure in the gas must increase to squeeze that bubble through the neck of the constriction. For a bubble to move through the constriction neck, it must achieve a pressure that exceeds that of the liquid by an amount which is sufficient to overcome the curvature. The smaller the capillary neck radius, the higher the pressure the bubble must achieve. Thus, the narrow round (0.5-mm) pore is affected most by gas compressibility since it has the smallest neck radius. Gas-bubble compression and expansion behavior occurs for all three constricted capillaries, although it is most apparent in the narrow pores. Figure 13 depicts schematically the compression and expansion of a bubble as observed in the 16-mm movies. The syringe pump injects fluid into the capillary a t a constant volumetric rate, and the bubble moves through the capillary a t a constant velocity upstream from the constriction. Upon approach to the constriction neck, the bubble begins to compress. Once the pressure in the bubble increases such that its front moves slightly beyond the neck of the constriction, an unstable situation develops. The pressure in the bubble no longer needs to increase for the bubble front to move forward since the capillary radius now increases steadily into the straight section of the capillary. Accordingly, the bubble front advances spontaneously from the neck of the constriction driven by the expanding gas in the bubble. This spontaneous motion is a Haines jump (Morrow, 1970). 4.2.1. Bubble Velocity. It is apparent from viewing the 16-mm movies that the breakup time asymptotes in Figures 10-12 result from compressibility and Haines jumps. Since the jumps are crucial, we measured the ve-
z-? L
-0 E
lo4
0 0 0
x
IO5 -16
-8
0
16
8
24
Dimensionless Position Figure 14. Effect of the tube capillary number on the local capillary number for 3 cm long bubbles as they move through the narrow round (0.5-mm) pore displacing an 8.5 mPa-s viscosity fluid. % = x/RT
l6'r
I
I
I
I
I
1
I
Pore
N a r r o w Round (05")
h
I
-=
-8
X
Bubble Length-3cm ,(2x 10-2cm:)
t 0 X/R,,
8
16
24
Dimensionless Length
Figure 15. Effect of the tube capillary number on the local capillary number for 3 cm long bubbles as they move through the narrow round (0.5-mm) pore displacing a 1.0 mPa-s viscosity fluid.
locity of the bubble front from the 16-mm movies as various bubbles translate through the narrow round (0.5mm) pore as described in section 4.1.1. The instantaneous or local capillary numbers of 3 cm long bubbles translating through the narrow round (0.5mm) pore are shown for the 8.5 mPa-s fluid in Figure 14 and for the 1.0 mPa-s viscosity fluid in Figure 15. The bubbles move from left to right in these two figures, and the solid lines simply reflect the best eye fits. Results show the bubbles translating a t a constant velocity, represented by the tube capillary number, CUT, as they enter the constriction. The three different sets of velocity data in both Figures 14 and 15 correspond to every other datum starting with the lowest CaT data points in Figure 10. So for each fluid viscosity, two sets of velocity data are below the critical capillary number, and one data set is above. Consider first the data for the two smaller inlet capillary numbers in Figure 14. We observe each bubble traveling a t a constant Ca = CaT into the constriction. Next, due to gas compression, the capillary number decreases as the bubble front approaches the neck of the constriction. Once near the neck (3 = 0), the two bubbles accelerate and exit the downstream side of the constriction at a high velocity. Notice that the capillary number for each bubble is essentially identical on the exit side of the pore. This is the Haines jump. Because the jump velocities are identical, the bubbles deposit wetting films of the same thickness (see eq 5),and accordingly, the breakup times are the same for these two capillary numbers as seen in Figure 10.
These capillary numbers are below the critical capillary number. Above the critical capillary number, which corresponds to the largest inlet capillary number shown in Figure 14, the velocity data do not demonstrate compression and expansion to the same degree. In particular, the capillary number does not decrease as the bubble enters the constriction. Rather, the bubble capillary number and deposited film thickness are determined more by the tube capillary number than the bubble jump. As a result, the time to breakup lies on a line of slope of minus two for this tube capillary number in Figure 10. This behavior is typical of all bubbles traveling at a constant volumetric flow with their velocity determined by continuity and CuT (see eq 12). Similar results are shown in Figure 15 for the 1.0 mPa.s fluid. The capillary number data again correspond to every other snap-off datum in Figure 10 beginning with the lowest injection rate. The velocity data for the two lower injection rates display gas bubble compression and expansion. The data at the highest injection rate show that the bubble does not slow down when entering the constriction. This highest injection rate is above the critical capillary number, and the breakup time for this tube capillary number lies on the line of slope of minus two in Figure 10. The two sets of data below the critical capillary number again show the bubbles exiting the constriction a t identical capillary numbers via the bubble jump. Accordingly, the times to breakup for these tube capillary numbers are the same as seen in Figure 10. For the lowest injection rate, CuT = 2 X the bubble front ocillates slightly about an axial position of x' 16. The oscillation is due to the inertia of the wetting liquid, and this phenomenon is discussed extensively by Gauglitz (1986). Since the bubble jump is due to gas compression, an incompressible nonwetting fluid should not show this effect. To confirm this, we injected a 16 cm long decane drop (p 0.9 mPa.s) into the narrow round (0.5-mm) pore filled with the 1.0 mPa.s fluid at a rate which is below the critical capillary number. The 16-mm movies reveal that the decane drop translates through the pore without a large decrease in velocity or a bubble jump. This experiment confirms that a compressible bubble is a necessary requirement for a bubble jump with bubbles driven at a constant volumetric flow. 4.3. Critical Capillary Number: Haines Jumps. We can further elucidate the low capillary number asymptote for the time to breakup in Figures 10-12, by quantifying the characteristic velocity (i.e., the capillary number) of the Haines jump. When the bubble expands rapidly out of the capillary neck, it must displace the downstream fluid. This requires a force to accelerate the liquid and to overcome its viscous resistance. The driving force is the pressure in the compressed gas bubble, and the resistance to flow is the viscous stress along the capillary wall and bubble front, and the inertia of the fluid in front of the bubble. In section 3, we showed through dimensional analysis that the Ohnesorge number is an important parameter. The Ohnesorge number is the capillary number when inertia characterizes the magnitude of the bubble jump velocity. We realize this by noting that a characteristic velocity based on inertia is the following: Uchm = ( ~ / R T P ) " ' (13) With this velocity substituted into the capillary number, we obtain a characteristic capillary number for an inertially dominated bubble jump which is the Ohnesorge number, Oh = p/(pnRT)l/'. Thus, the Ohnesorge number is important because it indicates the capillary number of the bubble jump, and therefore, the film thickness deposited
-
-
Norrow Round (0.5 mm) Pore
0
c
W
I- IO3!
Bubble Length Volume (cm) (cm3)
o *
A
A -
IO2 IO-)
2)
3
0 16
6 (IO2) 1.2(IO' 1
2.0
Very long
lo-'
10-2
I
0h n e so rg e Number, Oh = p/(pmR Tj'2
Figure 16. Effect of the Ohnesorge number on the time to breakup.
during the jump. The time to breakup given by eq 7 with Oh as the characteristic capillary number now becomes Tb
0:
Oh-'
(14)
When fast inertial spouting characterizes the Haines jump, time to breakup versus Ohnesorge number should lie on lines of slope of minus two on a log-log plot. However, when viscous forces overshadow inertial forces, the bubble jump is better characterized as a slow viscous oozing. Here, the jump velocity and the thickness of the deposited liquid film are independent of inertia; accordingly, the time to breakup should asymptote to a constant value at high Ohnesorge numbers. To test these hypotheses, we compare the breakup time with the Ohnesorge number when bubble jumps are important. Figure 16 plots the time to breakup for each of the low capillary number asymptotes in Figures 10-12 corresponding to each bubble size and fluid viscosity as a function of the Ohnesorge number as the characteristic capillary number. The results demonstrate that, a t low Oh, a thin film is deposited which evolves slowly, giving a long breakup time. At higher Oh, a thicker film is deposited and the breakup time is less. A t the lowest Oh value, the slope of the curve in Figure 16 is nearly minus two in accord with eq 14. At the highest Oh values, the time to breakup asymptotes to a constant value, indicating that inertia is not important in the bubble jump. The data agree with the curve drawn in Figure 16. However, we have insufficient data a t low Oh to confirm that the time to breakup is proportional to Oh-'. Larger bubbles have a smaller breakup time since they store more energy during compression and subsequently give a slightly faster bubble jump. Thus, the larger bubbles deposit thicker liquid films which snap off faster. Although we have not studied experimentally the effects of the dimensionless groups PLRT/cror L/RT on snap-off (see eq l),in a theoretical study of bubble jumps, we determined that decreasing either P&/a or L/RT increases the jump velocity of the bubble (Gauglitz, 1986). 4.4. Surfactants. Current applications of foam displacement for enhanced oil recovery processes consist of injecting dilute aqueous surfactants with gases, typically steam or COP. For bubble snap-off, surfactants are important since they lower the surface tension, but in addition, they can create dynamically immobile interfaces
1290 Ind. Eng. Chem. Res., Vol. 27, No. 7, 1988
(to at least some degree) (Sadhal and Johnson, 1983; Levich, 1962). This occurs from the relative hardening of the interface which then retards the interface motion. In the case of an inviscid gas bubble, the gas-liquid interface should not support a stress. However, a gas-liquid interface can support a mechanical stress in the presence of surfactants, and this retards the bubble movement. We expect similar effects in bubble snap-off with surfactants, and we investigate two surfactants for comparison: a relatively well-characterized surfactant, sodium dodecylbenzene sulfonate (SDBS); and a commercial foaming agent, Chevron Chaser SDlOOO (a poorly characterized mixture of surfactants). In cylindrical capillaries, the boundary effect of completely immobile interfaces on the time to breakup can easily be determined from an extension of the analysis given in section 2. With a completely immobile surface, the non-slip boundary condition is applied a t the gasliquid interface rather than the no-stress condition. Following Hammond's (1983) analysis, the no-slip condition decreases the flow rate in the liquid film by a factor of 4 compared to the flow with a no-stress boundary condition. Accordingly, the characteristic time to breakup with surfactants increases due to the dynamically immobile interface as follows: 7surfactants = 47b (15) Conversely, the film thickness deposited by a bubble with an immobile interface, determined by Bretherton (1961), is 22/3times larger than with a zero-stress interface, viz., Ho/RT = 22131.337CaT2/3
(16)
where the deposited film thickness with a zero-stress interface is Ho/RT = 1 . 3 3 7 C ~ ~To ~ /determine ~. the capillary number dependence of the breakup time with dynamically immobile interfaces, we combine eq 10 with eq 15 and 16 above to yield the following: TS,,fadnts
= 3.0(10-2)Ca~-2
(17)
Thus, eq 17 and 10 are identical, and the dimensionless time to breakup is predicted to be the same with or without surfactants. Of course, the absolute value of the breakup time with surfactants will be larger, since the equilibrium surface tension for the surfactant solutions is less (see Table I). The same dimensionless groups scale snap-off with surfactants: the capillary number and the dimensionless time. In the narrow round (0.5-mm) pore, the Ohnesorge number and the bubble volume are important due to inertial bubble jumps. A compilation of all breakup time data with surfactants (closed symbols) is shown in Figure 17 for the wide round (0.5-mm) pore. Surfactant-free data (open symbols) are also shown for comparison. With surfactants, the time to breakup is generally larger than without surfactants. For SDBS solutions, the difference is fairly small in accordance with eq 17. However, the breakup time is 3-4 times longer for the 1.1and 5.3 mPa-s viscosity solutions with Chaser SD1000. Chaser shows the most dramatic effect of the surfactants, and it is noticeably different from SDBS. Our simple analysis does not distinguish between surfactants and, therefore, does not completely predict the behavior of the surfactant solutions. However, the data show that surfactants change the breakup time by a relatively small amount compared to the major effect surfactants play in stabilizing foam lamellae. Bubble jumps are observed in the narrow round (0.5mm) pore with surfactants; the behavior is similar to the surfactant-free solutions (Gauglitz, 1986; St. Laurent et al., 1986). Thus, surfactants play a similar role when
Surfactonis in Wide Round (0.5mm) Pore
c 0 85
A 199
67
1.0
31
A
IO-^
I
SDBS
e
10)
Tube Capillary Number, Ca:,
Kr2
pUT/o-
Figure 17. Role of surfactants and the tube capillary number on the time to breakup in the wide round (0.5-mm) pore.
Haines jumps are important. 5. Summary and Conclusions Above a critical capillary number of about 5 X the snap-off behavior of gas bubbles in symmetrically constricted circular capillaries closely follows the expected results of the breakup time being proportional to the tube capillary number to the -2 power, and the generated bubble length being inversely proportional to the tube capillary number. In this regime, experimental results for fluid viscosities ranging from 1to 20 mPa.s and for three different capillaries indicate that the upstream bubble size is unimportant and that the fluid viscosity is properly scaled with a dimensionless time and the capillary number. Below the critical capillary of 5 X inertial Haines jumps due to gas compressibility dominate snap-off behavior and the time to breakup asymptotes to a constant value independent of the capillary number. The low capillary number limit of the breakup time correlates with a characteristic inertial capillary number for the bubble jump, the Ohnesorge number, to the -2 power for O h < W3,but asymptotes to a constant value independent of the Ohnesorge number when Oh > The results for smaller Oh values indicate inertial jumps, and the results at higher Oh values demonstrate a viscous jump. Time to breakup also depends weakly on the initial upstream bubble size for bubble lengths ranging from 3 cm to larger than 16 cm in capillaries with an unconstricted radius of 5.08 X cm. Surfactants lower the surface tension and can create dynamically immobile interfaces. Nevertheless, in the pores investigated, due to offsetting tendencies, bubble breakup behavior is essentially identical without surfactants or with 1wt % solutions of SDBS or Chevron Chaser SDlOOO in glycerol-water mixtures.
Acknowledgment This research was supported by the U.S. Department of Energy under Contract DE-AC03-76SF00098 to the Lawrence Berkeley Laboratory of the University of California. P.A.G. gratefully acknowledges financial support from the Shell Development Company. We express our thanks to Tom Lawhead for his skill and patience in constricting the capillaries.
Ind. Eng. Chem. Res. 1988,27, 1291-1296
Nomenclature Ca = p U / u , bubble capillary number CaT = pUT/u, bubble capillary number in straight section of the capillary d = density of mineral oil, kg/m3 F = denotes general function g = acceleration of gravity, m/sz Ho= film thickness deposited by bubble, m i.d. = internal diameter of glass capillaries, m L = initial upstream bubble length, m Lb = bubble length generated by snap-off, m nDZo= refractive index Oh = p / ( p ~ R ~ Ohnesorge ) ~ / ~ , number PL= liquid pressure, Pa Pc = gas pressure, Pa PL1= liquid pressure at point 1, Pa PL2= liquid pressure at point 2, Pa r = radial position, m F = r / R T , dimensionless radial position Rneck= radius of pore neck, m RT = radius of unconstricted capillary, m R1,2,3 = radii of curvature, m t = time, s t b = time to breakup, s U = instantaneous velocity of bubble front, m/s Uchar= characteristic velocity of bubble jump, m/s UT = velocity of bubble front in straight section of the capillary, m/s x = axial position relative to constriction neck, m 2 = x/RT, dimensionless axial position Greek Symbols X = rPmwd/RT,dimensionless radial position of the pore wall p = viscosity of wetting liquid, mPa.s r = 3.141 59 p = density of wetting liquid, kg/m3 pgas = density of gas in the bubble, kg/m3 u = surface tension, mN/m 7b
= t b / ( 3 p R T / u ) dimensionless , time to breakup = t b / ( 3 p R ~ / u dimensionless ), time to breakup for
Tsurfactanta
surfactant solution
Literature Cited Adamson, A. M. Physical Chemistry of Surfaces; Wiley: New York, 1976. Bretherton, F. P. J. Fluid Mech. 1961, 10, 166.
1291
Carnahan, B.; Luther, H. A.; Wilkes, J. 0. Applied Numerical Methods; Wiley: New York, 1969; p 63. Dilgren, R. E.; Deemer, A. R. Presented a t the SPE California Regional Meeting, San Francisco, March 24-26, 1982; SPE 10774. Falls, A. H.; Gauglitz, P. A.; Hirasaki, G. J.; Miller, D. D.; Patzek, T. W.; Ratulowski, J. Presented at the SPE/DOE 5th Symposium on EOR, Tulsa OK, April 20-23, 1986; SPE 14961. Fried, A. N. “The Foam-Drive Process for Increasing the Recovery of Oil”; U. S. Department of the Interior, Bureau of Mines R. I. 5866, 1961. Friedmann, F.; Jensen, J. A. Presented a t the SPE California Regional Meeting, Oakland, CA, April 2-4, 1986; SPE 15087. Gauglitz, P. A. Ph.D. Thesis, University of California, Berkeley, 1986. Gauglitz, P. A.; St. Laurent, C. M.; Radke, C. J. J . Pet. Technol. 1987, 39(9), 1137. Goldsmith, H. L.; Mason, S. G. J . Fluid Mech. 1962, 14, 42. Goldsmith, H. L.; Mason, S. G. J. Colloid Sci. 1963, 18, 237. Hammond, P. S. J. Fluid Mech. 1983, 137, 363. Hirasaki, G. L.; Lawson, J. B. Soc. Pet. Eng. J . April 1985, 176. Hornbeck, R. W. Numerical Methods; Quantum: New York, 1975. Levich, V. G. Physicochemical Hydrodynamics: Prentice-Hall: Englewood Cliffs, NJ, 1962. Mast. R. F. Presented at the 47th Annual Fall Meetine of SPE. San Antonio, TX, 1972; SPE 3997. Morrow, N. R. Ind. Eng. Chem. 1970, 62, 32. Owete, 0. S.; Brigham, W. E. “Flow of Foam Through Porous Media”. SUPRI TR-37, July 1984; Stanford University Petroleum Research Institute, Stanford, CA. Park, C. W.; Homsy, G. M. J . Fluid Mech. 1984, 139, 291. Perry, J. H. Chemical Engineers’ Handbook, 3rd ed.; McGraw-Hill: New York, 1950. Ploeg, J. F.; Duerksen, J. H. Presented at the California Regional SPE Meeting, Bakersfield, March 27-29, 1985; SPE 13609. Ransohoff, T. C.; Radke, C. J. SPE Reservoir Eng. 1988, in press. Ransohoff, T. C.; Gauglitz, P. A.; Radke, C. J. AIChE J . 1987,33(5), 753. Roof, J. G. Soc. Pet. Eng. J. 1970, 10, 85. Sadhal, S. S.; Johnson, R. E. J. Fluid Mech. 1983, 126, 237. Schwartz, L. W.; Princen, H. M.; Kiss, A. D. J. Fluid Mech. 1986, 172, 259. Shen, E. I.-C. Ph.D. Thesis, University of California, Berkeley, 1984. St. Laurent, C. M.; Gauglitz, P. A.; Radke, C. J. “An Experimental Study of Snap-off in Constricted Glass Capillaries of Circular and Square Cross Section”. Undergraduate Research Report, 1986; Lawrence Berkeley Laboratory, University of California, Berkeley, LBID-1165. Strand, S. R.; Hurmence, C. J.; Davis, H. T.; Scriven, L. E.; Mohanty, K. K. “Choke-off of Non-wetting Fluids in Porous Media”. Research Report, 1982; University of Minnesota, Minneapolis.
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Received for review September 22, 1986 Accepted February 10, 1988
Synthesis of 4A Zeolite from Calcined Kaolins for Use in Detergents Enrique Costa,* Antonio de Lucas, M. Angeles Uguina, and Juan Carlos Ruiz Department of Chemical Engineering, Universidad Complutense, 28040 Madrid, Spain
The synthesis of 4A zeolite from calcined kaolins has been investigated. T h e process variables of the different synthesis steps have been optimized in order t o produce 4A zeolite at a lower price with the established specifications for use in detergents. The recovery of the mother liquors required for the economical viability of this process has been verified. An economical evaluation of this process has been carried out with a n estimated price for the zeolite of 5 4 peseta ($0.43)/kg. The necessity to reduce the contamination of rivers and lakes caused by phosphates has led to the development of a phosphate substitute as a builder in detergents (Layman, 1984). The builder content in detergents can reach 30% by weight (Burzio and Pasetti, 1983). A suitable builder as a substitute for phosphates should meet the following requirements: no degradation of detergent qualities,
toxicologically and ecologically safe, and economically feasible with the possibility of large-scale production from easily available raw materials (Ettlinger and Ferch, 1978). The search for phosphate substitutes was initially concentrated on water-soluble inorganic substances such as soda and water glass, and organic complexing agents such as citrate and NTA and EDTA sodium salts (Berth et al.,
0888-5885/88/2627-1291$01.50/0 0 1988 American Chemical Society
