266
Ind. Eng. Chem. Fundam. lB80, 19, 266-275
4 = dimensionless modulus, defined in eq 9 J. = dimensionless flux ratio, defined in eq 5 and 7 Subscripts Am = amine species i = chemical species i Literature Cited Astarita. G., Marrucci, G., Giola, F., Chem. Eng. Sci., 19, 95 (1964). Clarke, J. K. A., Ind. Eng. Chern. Fundam., 3, 239 (1964). Coldrey, P. W., Harris, 1. J., Can. J. Chem. Eng., 54, 566 (1976). Danckwerts. P. V., Chem. Eng. Sci., 34, 443 (1979). Danckwerts. P. V., McNeil, K. M., Chem. Eng. Sci., 22, 925 (1967). Danckwerts, P. V., Sharma, M. M., 7% Chem. Eng., CE244 (Oct 1966). Dennard, A. E., Williams, R. J. P., J. Chem. Soc. A , 812 (1986). Dixon. D. C., Can. J. Chem. Eng., 55, 487 (1977). Donaidson, T. L., Quinn, J. A., Roc. Mtl. Acad. Sci. USA, 71, 4995 (1974). Donaldson, T. L., Quinn, J. A., Chem. Eng. Sci., 30, 103 (1975). Eberson, L., Chapter 6, "Acidity and Hydrogen Bonding of Carboxyl Groups", in "The Chemistry of Carboxylic Acids and Esters", S. Patai, Ed., Intersclence, New York, 1969. Edsaii, J. T.. "Carbon Dioxide, Carbonic Acid, and Bicarbonate Ion: Physical Propertiesand Kinetics of Interm-", in "CO,: Chemical, Bbchemkxi, and physiological Aspects", R. E. Forster, J. T. Edsall, A. B. Otis, and F. J. W. Roughton,.Ed., NASA SP-188, Washington, D.C., pp 15-27 1969. Emmert, R. E., Pigford, R. L., AIChE J., 8, 171 (1962).
"Handbook of Chemistry and Physics". 44th ed, Chemical Rubber Pubi. Co., Cleveland. 1962. Hikita, H., Asai, S., Ishlkawa, H., Honda, M., Chem. Eng. J., 13. 7 (1977). Ibrahim, S. H. I.. Kuioor, N. R., Chem. Eng. Sci., 17, 1087 (1962). "International crlticel Tables", Vol. 111, First Edition, National Research CouncH, McGraw-HIII, New York, 1928. Jencks, W. P., Chapter 3, "Qeneral AcibBase Catalysis" "Catalysis in Chemistry and Enzymdogy", m a w - H i l i , New York, 1969. Jensen, M. B., Jorgensen. E., Fawhoit, C., Acta Chem. Sand., 8. 1137 (1954). Jorgensen, E., Acta Chem. S a n d . , 10, 747 (1956). Jorgensen, E., Faumdt, C., Acta Chem. Sand., 8, 1141 (1954). Lander, R. J., Quinn, J. A., J . k m . Sci.. 3, 47 (1978). Nguyen, Y. N., PhD Dissertatlon, Unkeristy of Rochester, 1978. Nunge, R. J., Gill, W. N., AIChE J., 9, 469 (1963). Otto, N. C., Quinn, J. A., Chem. Eng. Sci., 28, 949 (1971). Sada, E., Kumazawa, H., Butt, M. A., Can. J. Chem. Eng., 54, 421 (1976a). Sada, E., Kumazawa, H., Butt, M. A.. Hayashi, D., Chem. Eng. Sci., 31, 839 (1978b). Sharma, M. M., Trans. Faraday Soc., 81, 681 (1965). Sharma, M. M., Danckwerts, P. V., Trans. Faraday Soc.,59, 386 (1983). Smlth, D. R., Deparhnent of Chemical and Bbchemicai Engineerlng, University of Pennsykania, private communication, Aug 1978. Smith, D. R., Quinn, J. A., AIChE J., 25, 197 (1979). Union Carbide Corp., "Ethanolamines", physical property data pamphlet.
Received for review June 28, 1979 Accepted April 14, 1980
Postwithdrawal Drainage of a Viscoelastic Separan Solution Kartlc C. Khllar, Charles B. Welnberger, and John A. Tallmadge' Department of Chemical Engineering, Drexel University, Philadelphia, Pennsylvania 79 104
The transient profiles of draining films, formed initially by vertical withdrawal of a flat plate, were studied experimentally for a viscoelastic liquid. In comparison with Newtonian and relatively inelastic power-law liquids, the viscoelastic liquid initially produced a thinner and more slowly draining film. In addition, the viscoelastic coating film was found to have a more uniform thickness during the initial period of drainage. As expected, elastic responses appeared to be significant during withdrawal and negllgible at long drain times. Logarithmic plots of thickness vs. drain time were found to be approximated by two straight lines of different slope. Hence these plots were characterized by three parameters, which were the short-time slope, the long-time slope, and a transition time, at which time the change in slope occurs. The long-time slope agreed with that predicted using a purely viscous, power-law liquid model.
Introduction Drainage represents a situation where a film of liquid, adhering to a solid surface, drains down the solid and thus thins with time. Here the solid is stationary and liquid flows downward mainly due to gravity. Under many drainage conditions, the behavior of the draining film is affected by the film profile prior to drainage. There are many methods of obtaining a draining film, some of which involve liquid lowering, in which a liquid is moved down along a solid surface (Satterly and Givens, 1933; Deryagin, 1964; problem 2R in Bird et al., 1960). Others involve complete removal of the solid from a liquid bath (White and Tallmadge, 1965) or rotation to vertical of a horizontal surface containing a sessile drop (Denson, 1970). In this work, we form the draining film by partial withdrawal of a flat plate from a liquid bath. This withdrawal is partial because complete removal does not occur. This results in the two step process shown in Figure 1. Drainage of this type has been called postwithdrawal drainage (Lang and Tallmadge, 1971). Specifically the phrase "postwithdrawal drainage" designates the situation in which a finite object, initially immersed in a bath of 0196-4313/80/1019-0266$01.OO/O
wetting fluid, is withdrawn at constant speed from the bath but not completely removed (Figure lA), brought to rest, and the resultant fluid coating then allowed to drain down into the bath (Figure 1B). The solid lifting process is called unsteady withdrawal, during which the initial film profile is established, and the downward flow process occurring along the stationary solid is called postwithdrawal drainage. Film drainage after withdrawal occurs in dip coating processes, in measuring contact angle, and in rinsing electroplated parts and brass mill tubes. Other practical applications of film drainage and the interrelationships among drainage, withdrawal, and removal are discussed in a review article by Tallmadge and Gutfinger (1967). Most of the work on drainage has been done with Newtonian liquids. However, some industrial processes involve viscoelastic liquids, such as dip coating with paints or polymer melts. Postwithdrawal drainage with such fluids involves a complex case of transient flow of nonNewtonian liquids where both shear and elongation effects are important. Thus, for both application and scientific reasons, the authors are interested in studying postwithdrawal drainage of viscoelastic liquids. 0 1980 American Chemical Society
Ind. Eng. POSTWITHDRAWAL DRAINAGE fA TWO STAGE PROCESS1
A WITHDRAWAL STAGE (tetw)
t
U=Uw (CONSTANT)
UNSTEADY WITHDRAWAL
B DRAINAGE STAGE ( t > tW)(tD'O)
TRANSIENT
DIP COATING
I
Chem. Fundarn., Vol. 19, No. 3, 1980 267
confirmed this prediction both theoretically and experimentally by studying the case of the initial profile formed by a large sessile drop on horizontal surface. The surface was subsequently rotated 90° to form a vertical plate and thus drainage. As compared with Jeffreys, Denson showed large deviations at short times and convergence at long times. The early analytical and precise work on postwithdrawal drainage was done by Lang and Tallmadge (1971). They extended the Jeffreys equation to include the effect of initial profile, as suggested by Denson, and verified experimentally their postwithdrawal drainage expression, using a viscous Newtonian oil with a viscosity of 1900 mN s/mz (1900 cP). Obtained by using the Chase and Gutfinger (1967) profile for unsteady withdrawal as the initial profile, the Lang and Tallmadge thickness expression for a postwithdrawal draining thickness is given, in terms of withdrawal time, t,, by c
Figure 1. Stages in postwithdrawal drainage (t, is withdrawal time and t D is drain time). DRAINAGE GEOMETRY
-w+:
F L A T PLATE
(LINE OF CONTACT1
I3iLh
IFLM THICKNESS AT X)
Figure 2. Notation for the drainage geometry.
The main purpose of this work is to investigate, experimentally, the behavior of a viscoelastic liquid during postwithdrawal drainage. Another aspect is to consider how the elastic nature of the liquid affects this drainage phenomenon by comparison with earlier data and models. This paper also describes the drainage experiments that have been carried out over a range of speed, time, and length conditions and compared with new and previous data for Newtonian liquids. The geometry and symbols used for the drainage thickness (h) are shown in Figure 2. The problem in question is, "How does thickness h vary with drain time tD, distance x , fluid properties, and other relevant parameters? " Previous Work Some of the early work on Newtonian liquids was done by Jeffreys (1930), Denson (1970), and Lang and Tallmadge (1971). Jeffreys derived an expression for the thickness of a draining, Newtonian film adhering to a vertical surface by assuming a laminar, quasi-one-dimensional flow, where only viscous and gravitational forces are important. His expression was obtained by integration of the simplified differential equation and use of the usual boundary conditions of no slip at the solid and no shear a t the gas interface. His expression, where x is vertically downward, is h=
(5)"
As mentioned by Jeffreys, this expression is valid for long drain times. Denson (1970) pointed out that the Jeffreys equation can be extended to predict the f i b thickness at short drain times by including the effects of the initial profile. Denson
Comparing eq 2 with Jeffreys eq 1,one can see that eq 2 can also be obtained by replacing the drain time ( t D ) by total time ( t D + t,) in the Jeffreys expression. For non-Newtonian liquids, the works of Pendergrass (1975), Denson (1972), and Fitzjohn (1974) are noted. A recent paper by Pendergrass (1975) includes a general equation by which film thickness might be obtained for most of the inelastic non-Newtonian model fluids. However, the analysis is independent of initial profile, so there is considerable question about the use of these potential expressions at short drain times and for postwithdrawal drainage. Denson (1972) extended his sessile drop, drainage experiments to four different non-Newtonian liquids, two of which were quite elastic. In addition, he derived a theoretical model which describes the data quite well. His model suggests, for his sessile drop case, that shear properties are sufficient to describe drainage; elastic properties were not used in his model. Elongational effects were small, apparently. It thus appears that his experimental work was not oriented to investigate the direct effect of elastic properties (such as relaxation time), and, of course, his work applies best to a different geometry from the one of interest in this paper. The only known non-Newtoniandata on postwithdrawal drainage is unpublished work by Fitzjohn (1974) at our laboratories. Fitzjohn carried out experiments with three liquids, which were described in shear by power law parameters and were relatively inelastic. His data showed a relatively constant slope on a logarithmic plot of total time. To model the drainage, Fitzjohn used the freedrainage, power-law theory of Gutfinger and Tallmadge (1965) and obtained an expression for postwithdrawal drainage of power law liquid His expression is
This expression predicts the slope, m, on a full logarithmic plot of thickness h vs. total time ( t D + t w ) ,given by -m = n / ( n + 1) (4) The slope of eq 3 and 4 agreed with the data of Fitzjohn. However, the magnitude of film thickness predicted by eq 3 did not agree with data. Thus, although not complete or precise, eq 3 and 4 are helpful in predicting time slopes for postwithdrawal drainage of inelastic, power-law liquids.
288
Ind. Eng. Chem. Fundam., Vol. 19, No. 3, 1980 POSTWITHDRAWAL DRAINAGE APPARATUS LANG (19691, LANG-TALLMADGE 119711
SPROCKET WHEEL
MOT
I-----$ I l -DISK
(MOVING)
*MOUNT (FIXED) V , BRASS BEARING
DLR ,I,
ROD
dl
1-to o
GLASS & A T E (EFOAE WITHDRAWAL)
\ 1
Figure 3. Experimental apparatus used for postwithdrawal drainage.
In summary, the previous work on postwithdrawal drainage has been only with Newtonian and inelastic, power-law liquids. That work has established the importance of withdrawal step and indicates that the total time concept is useful for inelastic liquids.
Experimental Apparatus and Method The apparatus, Figure 3, was designed to permit continuous measurement of the transient film thickness during postwithdrawal drainage. The draining film was produced by immersing a rectangular piece of a nonreflective glass plate inside a tank full of the experimental liquid; then it was withdrawn upward vertically at constant speed U, to a drainage length L,stopped, and allowed to drain. Part of the plate remained in the liquid tank at all times. The withdrawal speed was varied from 1.3 to 34 mm/s. The glass plate connection device was placed in a drill holder attached to a 0.5411. drill rod; this rod was constrained by a brass bearing to move only vertically. Above the bearing, a disk was mounted on the rod; the disk was attached to a string, chain, and sprocket wheel arrangement developed by White (1965) and used by Rajani (1968). The string length was adjusted so that the disk just touched the brass bearing as the glass plate just touched the bottom of the liquid tank. The glass plate was 300 mm long, 145 mm wide, and 3.0 mm thick, and the transparent Plexiglas tank was 150 X 150 mm and 210 mm deep. All runs were conducted in a constant-temperature room. Measurement of the transient film thickness during drainage was done a t a constant distance x (from the wetting line), by use of a light absorption technique. This apparatus is the same used and described in the thesis of Khilar (1977) and, except for the glass plate, in unpublished reports of Lang (1969) and Fitzjohn (1974). Details follow. Measurement of Thickness and Time The apparatus used for thickness measurement was a single-beam visual-UV spectrophotometer (Perkin-Elmer, Model 139, made by Hitachi). This instrument consists of a light source, an optical system for selecting a given wavelength, a phototube, and a linear-log recorder (Perkin-Elmer Varicord 43); it has a wavelength range of 150 to lo00 nm. The monochromator was turned 90° from the usual horizontal position to a vertical position in order to
provide a light slit with the wide portion of the slit in a horizontal plane. See Figure 3. A small amount of blue dye was dissolved in the colorless test liquid before withdrawal. Then, during each drainage run, a beam of visual, monochromatic light was sent through the film. The transmitted light was measured by a phototube and recorded on strip chart. Because a visually absorbant dye was selected, all runs were made in a closed, darkened room. By means of the recorder, the absorbance was automatically plotted against time on the recorder output chart. Because of the required darkness, the light source compartment was covered with aluminum foil paper to shield out the light coming from it. An intense dye (Pontamine Blue, No. 6BX, duPont Co.) was used. A wavelength of 630 nm was selected and used throughout because it was found, in preliminary tests, that light absorption by this dye reached a maximum at this wavelength. A transparent cell, fabricated by Lang (1969),was used to calibrate the absorbance-thickness relationship and to conduct preliminary dye experiments. The cell was made from a 102 X 114 mm block of 10 mm thick Lucite, into which six cavities were machined. The individual cavities, each about 13 X 76 mm in size, had precisely measured thicknesses of 120, 228 508 1002, 2540, and 5065 pm, respectively (about 0.1 to 5 mm). The cell was used to determine extinction coefficients from the following form of the Lambert-Beers model of linear absorption for dilute solutions
A = h(Ke/lOOO) (5) Here Ke is the modified extinction coefficient; it is a function of the liquid, the dye, and the dye concentration. Absorbance A is defined as the natural log of the ratio of intensity (I,) of the incident beam to the intensity (0 of the transmitted beam. In calibration studies, the four cell cavities used in most tests were the ones in the 0.2 to 2.5 mm range. The cell cover was about 5 mm thick. In preliminary tests, the extinction coefficient was found to be constant up to the highest dye concentration tested, which was 600 mg/L. The dye concentration used for the low viscosity, glycerin-water tests was 600 mg/L and for the viscous, Separan tests was 200 mg/L. For these test liquids, the Ke extinction coefficients were 1.97 and 0.64, respectively, where h is in micrometers (pm). These dye concentrations were chosen so that the measured absorbance (for the double thickness actually measured) was held to a maximum of about 1.0, in order to obtain precise values of absorbance. The observed ranges of single film thicknesses were about 15 to 150 pm for glycerin-water and 200 to 800 pm for Separan solutions. Calibrations were checked for each set of runs and were found to be stable and constant. Drain times were measured on the recorder strip chart. For short runs up to 200 s, continuous measurement was satisfactory. For long runs, which took up to 3000 s or almost 1h, an interrupted chart method was used. In the interrupted method, the chart paper was run for about 300 s, then stopped for a timed interval of about 100 to 200 s, and then run for about 50 s before stopping again. There were about 7 or 8 interruptions in the long time runs. In order to zero the recorder at the beginning of each run, it was found necessary to adjust the slit width of the spectrophotometer. In our work, the slit width varied from run to run into the range of 0.31 to 0.37 mm, with most runs at 0.34 mm. To try to avoid the difficulty of working in a darkened room, a colored light bulb was tested, but the phototube absorbances were reduced considerably. Therefore, this
Ind. Eng. Chem. Fundam., Vol. 19, No. 3, 1980 269 DRAINAGE
up from the bearing, as shown in Figure 4B. The length
DISTANCE ( X )
A MEASUREMENT OF LENGTH
L varied from 140 to 180 mm, depending on the distance
(S)(SLIT TO SURFACE)
x which was desired. Consequently, the withdrawal time t , was fixed primarily by the choice of speed, since t , =
L / UW.
\i /
BEFORE WITHDRAWAL B
MEASUREMENT
The drainage distance x was varied over a sixfold range during the runs, from 8 to 50 mm. It was not difficult to obtain desired x values at long heights near 50 mm, but it was difficult at small heights near 10 mm; therefore, several trials were sometimes necessary to obtain the small heights desired.
OF DRAINAGE
T,vDISK
LENGTH ( L ) (AFTER WITHDRAWAL) (BEFORE WITHDRAWAL)
ZMOUNT (FIXED)
Figure 4. Determination of drainage distance ( x ) .
method was not used. It is suggested, however, that future workers consider phototube filters to avoid the dark room method.
Measurement of Speed and Distance Two of the most important independent variables studied were the withdrawal speed U, (Figure 1)and the drainage distance x (Figure 2). The range of speeds studied was about 30-fold, from 1.3 to 34 mm/s. The withdrawal speed was found by taking the average of speeds of the withdrawal rod before and after each experimental run. The speed was measured by finding the stopwatch time required for the rod in Figure 3 to move through a fixed distance, which was indicated by two marks on the apparatus. Higher speeds were not used with the current apparatus because vibrations were apparent above 40 mm/s. The approximate desired speed was used to select the potentiometer setting for the reversible Servotek motor. Potentiometer settings of 10 and 100 were used for glycerin-water solutions (resulting in speeds of 3.1 and 28.0 mm/s); settings of 2, 18, and 112 were used in most Separan runs (or 1.25, 6.0, and 33.4 mm/s, respectively). The distance from the wetting line to the light beams defines the drainage distance I,as shown in Figure 2. It was found that the wetting line did not move relative to the plate during either withdrawal or drainage, so that length measurements were simplified. The drainage distance x was calculated as the difference between the drainage length L and the surface length S; thus
x=L-s
(6)
Measurements of S and L are shown in Figures 4A and B. The length S was measured precisely before each run. As noted in Figure 4A, the length S is the distance between the monochromator slit and the initial level of the liquid surface; “initially” is when the plate was immersed to the bottom of the bath. Immersion depths varied from 205 to 209 mm and the length S was fairly constant, ranging from 130 to 140 mm over the runs. The drainage length L was obtained for each run by measuring the distance through which the disk D moved
Procedure First, the motor power unit, monochromator, and recorder were switched on 15 min or more before the run. The zero recorder adjustment was made by changing the slit width after the infinity position was checked with a closed phototube shutter. The slit width and room temperature were recorded. Second, the glass plate was cleaned with a detergent powder solution and dried with hot air. After a potentiometer reading was selected, the corresponding speeds were measured two or three times. Before each set of new runs, or after a week or so, the extinction coefficient was checked by a dye cell. Third, the plate was attached to its holder, dipped into the liquid bath so that it touched the tank bottom, and length S was measured. After turning off the room lights, the phototube shutter was opened and the recorder chart-paper switch turned on. Next, the run was started by simultaneously turning on the motor and the stopwatch. After the desired time of withdrawal, which was generally selected to obtain a specific distance x , the motor and watch were stopped. Thus postwithdrawl drainage was begun. For the initial phase of drainage, the chart paper was allowed to move continuously. Then the interrupted method was used. Finally, after 45 min, the run was ended, the phototube shutter was closed, and length L was measured. Newtonian Drainage Data The Newtonian liquid used in this work was an aqueous solution of 80 mass % glycerin and 20 mass % water. This 80% mixture was selected because it is a good compromise between higher viscosity and lower hygroscopy. A higher percentage of glycerin is much more hygroscopic and tends to be less stable because of changes in composition, and thus viscosity, with time. Five runs were made with this glycerin-water mixture, three at a slower speed of withdrawal (3.1 mm/s and potentiometer reading of 10) and two at a higher speed of withdrawal (28.0 mm/s and potentiometer reading of 100). Drainage heights were varied, at each speed, from about 17 to 42 mm. Consider first the faster speed runs, which were done at 23 “C so the viscosity was 49 mN s/m2 (49 cP). Following the data and theory of Lang and Tallmadge (1971), these Newtonian postwithdrawal drainage data were plotted against total time. See Figure 5A. As expected, a straight line was obtained in each case. The same result is seen in Figure 5B for the lower speed runs. The lower speed runs were done at a slightly different temperature (17 “C) and thus different viscosity (76 mN s/m2),but this difference is taken into account below. The general prediction noted earlier is given specifically for these runs by substituting density (1209 kg/m3 or 1.209 g/cm3) and viscosity into eq 2 and adjusting for units. We
270
Ind. Eng. Chern. Fundam., Vol. 19, No. 3, 1980 NEWTONIAN (80% G L Y / 2 0 WT% WATER)
STEADY SHEAR RHEOLOGY
KHILAR (1977). KHILAR-WEINBERGER-TALLMADGE
a IO6 0 DATA (THIS WORK)
N
I z
A DATA IHUPPLER
E
>-
t
v
O 0
-
L
15% SEPARAN U AQ SOLUTION OF 50 WT% GLY
2 a
T
4
TIME, tw'tD,
SECONDS I
'0;-3 I
7
'
'
"
1
1967)
: )
5w TOTAL
et01
SMOOTHED DATA EXTRAPOLATED
j
10-2
10-1 100 SHEAR RATE, 7, SEC-'
IO1
I02
Figure 6. Apparent viscosity of Separan U.
lo;
IO
30
TOTAL TIME, tw
IO0 +
3 00
I ,600
tD, SECONDS
Figure 5. Effect of total time for Newtonian liquid (aqueous solution of 80 wt % glycerine).
thus obtain the theoretical prediction for each viscosity as follows
Distance x is in millimeters and time is in seconds. At a total time of 100 s, these equations become h4,(pm) = 6 . 4 3 6
(9)
h7&m) = 8 . 0 1 6
(10)
These predictions were compared with the data of Figure 5. For Figure 5A for example, the predictions are 40.7 and 24.1 Wm, which have differences of -11% and +1% from the data. For Figure 5B, the predictions are 56.6,48.1, and 33.1 pm, which have differences of about -10, -8, and -13%. In summary, there is reasonably good agreement within experimental error. The thickness-total time slopes of the Newtonian data of Figure 5 were measured for comparison with theory and for comparison with elastic liquids. Khilar (1977) reported valves of -0.46, -0.43, and -0.45 for the runs shown, with an average slope of -0.45. This slope of data is somewhat below the theoretical Newtonian value of -0.50 (shown by eq 2,7, and 8). The small difference between theory and data was attributed to experimental error, part of which was due to the difficulty of measuring thickness accurately for these low-viscosity,thin films. These films were usually smaller than 100 pm (smaller than 0.1 mm). Khilar (1977) measured the viscosity of this 80% glycerine solution at 24 "C, using a cone-and-plate viscometer, and found the viscosity at this temperature to be 47 mN s/mz (specifically 49, 44, 46, and 48 were measured at 250,400,630, and 1000 s?). For comparison, the
literature values of Lange (1956) for the 80% solution (62, 45.9, and 34.9 at 20, 25, and 30 " C ) were plotted as log p vs. l/T(K) to obtain by interpolation the 24 "C literature value, which was found to be 48 mN s/m2. The solution used in this work, therefore, has a viscosity about 2% below the Lange values. Viscosities reported above for the 17 and 23 "C drainage runs were obtained by subtracting 2% from the 78 and 50 values read from the plot of literature values. In summary, the Newtonian drainage runs generally agreed with theory in magnitude and in the time slope, as expected. The Newtonian runs also give an indication of the size of the experimental errors; this indication is helpful below in considering errors for the elastic liquid. Separan Solution-Rheological Properties The Separan solution used here contained 1.5 wt 5% Separan AP-30 (Dow Chemical Co.) in 50 wt % glycerin in an aqueous solution. This solution, coded fluid U, was chosen because of its high elasticity, good stability, and because it has been used and characterized by others. The method of preparation is given by Khilar (1977). Both steady shearing and oscillatory tests were done in a cone and plate geometry, using a Rheometrics mechanical spectrometer a t 24 "C. The cone radius was 36 mm and the cone angle was 0.04 rad. In the steady shear measurements, apparent viscosities and first normal stress differenceswere obtained at various s-l (sometimes shear rates, which ranged from to called shearing strain rates). The shear viscosities are shown in Figure 6. The apparent viscosity followed the typical "power-law" type of behavior for shear rates above 0.1 s-l; the power-law parameters obtained from our data are given by T
= 192(T)0.30
(11)
Thus the power-law exponent n is 0.30 and the consistency index K is 192 dyn sn/cm2. Also shown in Figure 6 are shear viscosity data from Huppler et al. (1967) for a liquid of the same composition at 25.0 "C. Agreement is very good. Huppler et al. (1967) report a power law exponent n of 0.30 and a consistency index K of 200 for the higher shear rate region, which is also good agreement. Huppler et al. report a zero shear viscosity of 2900 P. Another Figure 6 property of this fluid which should be noted here is the critical shear rate yo which is determined by intersection of the extrapolation of the zero shear and high shear lines; in this case, the critical shear rate y c is 0.02 s-l. See Figure 6. The first normal stress difference was also measured in the steady, viscometric tests. Since this material function
Ind. Eng. Chem. Fundam., Vol. 19, No. 3, 1980 271
is one manifestation of the "elasticity" of the fluid, the dimensionless group R formed by the ratio of first normal stress difference to shear stress, is likewise a measure of the relative importance of elastic behavior. Here the R ratio is defined by 711
- 712
R ( 9 ) = ___
Table I. Drainage Lengths and Withdrawal Times for Separan" ~
~~~
~~
speed,
(12)
721
In this work, the data R values were 6.5,8.6, 10.8, 13.8, 14.5, and 27.3 for shear rates of 2.5, 15.8, 25,40,62.5, and 250 s-l respectively. As shear rate tends to zero, R approaches the Weissenberg number, We, although no such asymptotic behavior can be seen in our data. The observed trend in our data does, however, indicate the increasing importance of elasticity for this Separan solution as shear rate increases. Dynamic shear tests were made by oscillation of the cone-and-plate a t a frequency w. Here we take the response time T to be defined by T = -G'
pot. reading 2 2 2 2 2 18 18 18 18 112 112 112 112 112
mmls
distance, x, mm
run no.
1.25 1.25 1.28 1.25 1.25 6.1 6.0 6.0 6.0 33.4 33.4 33.4 33.6 33.3
8 15 15 42 46 6 20 20 43 10 12 20 34 43
68U 67U 77U 69U 70U SOU 78U 79U 73U 71U 76U 75U 72U 74U
U,,
withdraw time, tw,s 113 119 126 142 145 26 28 28 30 4.4 4.1 4.9 5.1 5.6
length, L , mm 141 149 162 178 181 156 166 168 183 146 158 164 172 185
Maximum drain time was 2300 s for these runs. USE OF TOTAL TIME IN VISCOELASTK P/W DRAINAGE KHILAR 11977) KWT
, "
q'w"
Here G'and 7' are the in-phase components of the complex modulus and viscosity. Response times T obtained here included 6.51, 5.80, 3.92, 2.36, 1.49,0.21, and 0.011 s at w frequencies of 0.158, 0.250, 0.396, 0.627, 0.993, 10.0, and 99.7 rad/s, respectively. One characteristic of the elasticity is the natural time To,which is T evaluated in the limit as frequency tends to zero (Astarita and Marrucci, 1972). From our data, To would be above 6.5 s. However, there are not enough data here to determine the natural time To precisely, so we looked elsewhere. Another characteristic of elasticity is the relaxation time, X obtained from stress relaxation experiments. Defining X as the time required for the stress to decay l / e of its initial value upon cessation of shear, Weinberger and Goddard (1974) obtained a maximum value of relaxation s-l. That shear rate time as 50 s for a shear rate of is the Same order of magnitude as that in our draining films of Separan. Since another characteristic time can be defined as the inverse of +c, this 50 s relaxation time is also in very good agreement with the 50 s characteristic time calculated using the critical shear rate of * c = 0.02 s-l, as given in Figure 6 and in Huppler et al. (1967). In summary, we conclude that our Separan solution has a relaxation time of about 50 s.
Visual Observations of Separan Drainage Postwithdrawal drainage data of film thickness ( h )at ) taken at three speeds various times of drainage ( t Dwere of withdrawal (U,) for different distances ( x ) . The three speeds of withdrawals were 1.25,6.0, 33.4 mm/s and the distance range was from 6 to 46 mm. A summary of these runs is given in Table I. It is seen that withdrawal times (t,) varied considerably, from about 5 to 150 s, and drainage lengths ( L ) were nearly constant, varying from 141 to 185 mm. In these and in preliminary runs, several visual observations of the film were made during both the unsteady withdrawal and postwithdrawal drainage cycles. It was observed that: (1)The line of contact of the liquid film during both cycles was stationary and constant with time (Le., no up or down movement). This confirms the condition for postwithdrawal drainage. (2) The film surface at the gas-liquid interface was without any ripples and was uniform in the horizontal direction. This also confirms the
AQ
-E 3
SOLN
OF
50 W T % GLY
78 u
3,000
I25
v1
z
I12
68U
B u
Y
-
8
300+
0
004
'
1%
-
220'
'
30
'
" " "
100
,
I
I l l j , l
300
TOTAL TIME, Iw
1,000
1 , 3,000
,
I
,
,
I0)OOO
Io, SEC
Figure 7. Effect of total time on Separan thickness.
desired condition. (3) The thickness of the dynamic meniscus (near the bath level) decayed from a very large size to a much smaller size during the initial period of drainage. This is a useful exploratory result with deserves further study. It would indicate the elastic response of a viscoelastic liquid during a flow situation involving both shearing and stretching. (4) The Separan film thicknesses were much more uniform with distance x than those of Newtonian liquids. This observation is considered to be very important, perhaps the most important result of this study, and is thus quantified below. This result is important in the dip coating of short objects, because one is generally interested in obtaining a coating thickness which is uniform. Results-Effect of Drain Time To show the results quantitatively and following the previous work, the transient thickness data were plotted vs. the total time (tD+ t,) on log-log plots. Figure 7 shows typical results with three runs, which encompass all the withdrawal speeds studied. These runs cover a wide range in withdrawal speed (26x) and in drainage distance (6x1. The first conclusion from Figure 7 is that postwithdrawal drainage for this elastic liquid cannot be described by a single straight line. Therefore this elastic liquid is fundamentally different from the single line behavior of Newtonian liquids, as noted above. It is also fundamentally different from the single-line power law, non-Newtonian solutions, such as those measured by Fitzjohn. The second conclusion is that this drainage can, however, be described by two straight lines with significantly dif-
272
Ind. Eng. Chem. Fundam., Vol. 19, No. 3, 1980 USE OF DRAIN TIME IN VISCOELASTIC P/W DRAINAGE
Table 11. Drain Time SloDes for SeDaran b , c
KHILAR (e771 K W T
-
drainage results
15% SEPARAN U
mm/s
AQ SOLN OF 50 WT% GLY
5
A
rnm
334
50
sec
6
no
variables
74U
speed, U,, mm/s 1.25
6.0 lO0lb
'
30 ' '
'
""'
I
300 1,000 DRAIN TIME, tD, SEC
100
3,000
I
10,000
33.4
Figure 8. Effect of drain time on Separan thickness.
ferent slopes. This behavior was observed in all the Sepman runs in this work. The third conclusion from Figure 7 is that all the short-time slopes are quite small (0.03 f 0.00, in comparison with the long-time slopes (0.22 f 0.04). For each line, the slopes are significantly smaller than the values (0.45 to 0.50) for Newtonian liquids. These slopes are discussed further a t the end of this section. A fourth conclusion concerns the time at which there occurs the transition from short-time to long-time behavior. Based on all the runs studied, there is a lack of dependence of this transition time on withdrawal speed. This conclusion is discussed in the next section. Since these drainage results cannot be represented with a sihgle line, it appears that, for elastic liquids, the total time concept is not useful. On the other hand, the time of drainage (tD)is more explicit to describe this transient process. For these reasons, it was decided to plot the same thickness data vs. drain time alone. A typical result is shown in Figure 8. As seen from Figure 8, the thickness behavior for drain time is similar to that for total time and all four conclusions still apply. Again the curves consist of two different straight lines of widely different slopes. In this plot, the short-time slope is also quite small (0.04 f 0.02) and the long-time slope is much larger (0.21 f 0.03). Comparison of Figures 7 and 8, therefore, indicates similar results in terms of general qualitative conclusions and also in terms of quantitative values of the slopes. The slight difference in slopes is due to the shift in axis by the withdrawal times, but the difference does not appear to be very large. Because the use of drain time is simpler for elastic liquids and is similar in conclusions to the total time, it was decided to report the rest of the results in terms of drain time. Results for all 14 runs, based on drain times plots similar to Figure 8, are shown in Table 11. Also shown are transition times, which are the times at which the two straight lines intersect. Table 11, which indicates the reproducibility of the slopes for 4 pairs of runs, shows that the repeatability was good. The pair of runs with the closest agreement was runs 78 and 79, and the pair with most difference was runs 71 and 76, but all were considered to be satisfactory. The first result from Table I1 is that all three parameters (two slopes and transition time) are surprisingly constant over the range of speeds and lengths studied. Thus we conclude that the median short-term slope is about 0.04 and the median long-term slope is about 0.21. Fitzjohn (1974) has given a prediction of the drainage slope for a power-law liquid, eq 4, which predicts a slope
median value av dev
distance, x, mm 8 15(1)" 15(1)" 42(2)" 46 (2)" 6 20(3)" 20(3)" 43 lO(4)" 12(4)" 20 34 43
run no. 68U 67U 77U 69U 70U 80U 78U 79U 73U 71U 76U 75U 72U 74U
longtime slope, mT. 0.24 0.20 0.20 0.20 0.22 0.20 0.22 0.22 0.14 0.22 0.23 0.21 0.20 0.18 0.21
shorttime slope, ms 0.05 0.02 0.04 0.00 0.02 0.08 0.02 0.02 0.01 0.08 0.04 0.06 0.06 0.05 0.04
transition time
i0.02
k0.02
+40
tT, s
260 250 270 300 280 200 240 310 500 200 230 300 310 260 270
Repeatability runs, by pairs. See Table I for individual run values of (a) withdrawal speeds, ( b ) withdrawal times, and (c) withdrawal lengths. These results are from plots of drain time, such as Figure 8.
m equal to n / ( n + 1). For Separan with n of 0.30, a slope of 0.30/1.30 or 0.23 is predicted. On this basis, it appears that the predicted power-law slope (0.23) compares favorably with the long-time slope (0.21), but differs substantially from the short-term slope (0.04). Elastic behavior therefore appears to be manifest by a decreased rate of drainage initially, relative to rates expected for inelastic power-law fluids. After a certain period of time, which we call the transition time, drainage rates appear to conform quantitatively with predictions for relatively inelastic fluids. Other manifestations include the effect of the fluid's viscoelasticity on initial film thickness and variation of film thickness with vertical position. These are discussed below, along with comparisons between transition times and fluid relaxation times. Transition Time Transition time (tT)is equal to the drain time at which change in slope occurs. Clearly the transition time depends on both the long-time slope and the short-time slope. It is believed that long-time slope can be related to the stress-strain relationship of the liquid in steady shear and short-time slope depends mostly on the initial thickness, which depends upon elastic responses. It is also believed that the elastic nature of the liquid results in reducing the initial film thickness so we expect transition time to depend on the rheology of the liquid. Values of transition time are shown in Table I1 at various withdrawal speeds and heights. There does not appear to be any systematic influence of height over the range studied. The median transition times for three speeds and times are 270 f 20, 280 f 70, and 260 f 40 for 1.25, 6.0, and 33.4 mm/s speed, respectively, and for withdrawal times of 116, 26, and 5 s, respectively. Thus there is no appreciable influence of either time of withdrawal or speed of withdrawal. Since both viscous and elastic nature of a liquid is involved in determining the film thickness during unsteady withdrawal, it is not possible at this point to predict any explicit relationship between transition time and the nature of liquids. However, it is felt that further work may
Ind. Eng. Chem. Fundam., Vol. 19, No. 3, 1980 273 Table IV. Initial Thickness for Separan Runs
FILM UNIFORMITY KHILAR (1977). KWT
run conditions
$4001
z
$200y-%ppq i
t-
X
3
60
IO
125
IO
withdraw speed, disU,, tance run mm/s x, mm no. 1.25
6.0 DRAIN
HEIGHT X. rnrn
33.4 i
B3
f 20
IL
IOl
2
4 DRAIN
,
,
7 IO 20 HEIGHT, X, rnrn
,
, ,
40
(, I
70 100
Figure 9. Effect of drainage distance on thickness: A, elastic Sepman; B, Newtonian GW. Table 111. Effect of Distance (Short Times) thickness-distance slopes ~
speeds, mm/s 1.25 6.0 33.4 3.1 28.0
68U 67U 69U 80U 79U 73U 71U 75U 74U
ratio of initial pred. to expt. thickness, pm ( A h ~ ~ / h , E ) expt4 theoryb indiv. av 400 500 512 445 550 525 480 620 750
227 980 1255 1136 1488 1720 1944 2200 2680
0.565 1.96 2.45 2.55 2.70 3.27 4.05 3.55 3.57
1.66
2.84
3.73
~ I E experimental , (by extrapolation of data of this work to zero drain time). h ~ p predicted , (by theory, eq 14).
preciable for Newtonian and are not a function of withdrawal speed, as expected and in agreement with theoretical predictions of Lang and Tallmadge (1971). In summary, the film profiles of elastic Separan at low speeds of withdrawal were found to be surprisingly uniform during initial period of drainage.
~~~
drain time, t D , s ~
10 30 100 A. Elastic Separan 0.11 0.15 0.10 0.18 0.23 0.18 0.21 0.23 0.27
8 15 42 6 20 43 10 20 43
results of thicknesses
av 0.12 0.20 0.23
B. Newtonian G/W 0.41 (I 0,42(I
a These Newtonian slopes are slightly below the 0.50 of theory, partly because they were based on plots of drain time instead of plots of total time (the theory was based on total time).
lead to a relationship between transition time and some characterizing parameter of elasticity, such as the relaxation time. In this case X is 50 s and t T is 270 s; they are not equal, but they are within one order of magnitude. Uniformity with Distance (Short Times) In dip coating of short objects, it is very useful to know how uniform the coating thickness will be. To achieve some information regarding the short-time uniformity, crossplots of thickness vs. height were made at constant drain times (less than the transition time) and at constant withdrawal speeds. Figure 9A shows three of these plots, each at a different speed of withdrawal, but all for a 10-s drain time. Some of the runs were part of the runs 28 to 66 which were taken for drain times up to 200 s. Figure 9A shows that the effect of height (i.e., slope of these plots) depends on withdrawal speed. Slopes were 0.27, 0.18, and 0.10, for speeds of 33, 6, and 1 mm/s, respectively. Thus there is a smaller effect at the lower speed of withdrawal. In other words, these coating films are most uniform at the lowest speed of withdrawal. Slopes at other drain times have a similar behavior. Table I11 shows that the influence at 30 s and 100 s is similar; namely, the slowest withdrawal speed produces the most uniform f i i . This uniformity at short drain times is not found with Newtonian liquids. Figure 9B shows that slopes are ap-
Initial Thickness (After Unsteady Withdrawal) It is clear that the withdrawal step is very important here. Because no direct measurements were taken, an indirect analysis was made. We obtain some information regarding the effect of elasticity during transient withdrawal by comparing our data for initial thickness (h,)with that derived from equations of Chase and Gutfinger (1967) for a power-law theory for unsteady withdrawal of short objects. By combining eq 7 and 13 of Chase and Gutfinger (1967) and then substituting withdrawal time (t,) for time ( t ) ,we predict that, as the onset of drainage
Here hI is the predicted film thickness at the onset of drainage and L is the drainage length. Table IV, which compares experimental initial thicknesses with those predicted using eq 14, shows that (a) the predicted values are generally 2 to 4 times larger than the experimentalthicknesses and that (b) the ratio of predicted to experimental is higher for higher withdrawal speeds. This behavior can be rationalized in terms of the liquid's elastic properties. These properties tend to keep the fluid in the undeformed state, that is, in the bath rather than on the plate, thereby leading to thinner films than predicted by theory. As withdrawal speed increases, the Deborah number likewise increases, and the fluid should behave as a relatively more elastic fluid; this is consistent with the increasing difference between experimentally observed thicknesses and those predicted using an inelastic power-law fluid model for increasing withdrawal speed. This behavior is also reminiscent of the wire-coating findings of Middleman (1978), who observed an asymptotic coating thickness as wire speed increases for viscoelastic polyacrylamide solutions.
Other Long-Term Behavior As noted above, the effect of elasticity tends to be negligible at long times-for the case of postwithdrawal
274
Ind. Eng.
Chem. Fundam., Vol. 19, No. 3,
1980
COMPARISON OF DRAINAGE WITH RHEOLOGY KHILAR (1977), K W T L
v, vi
MAX ( 1 )
AND MIN(2) THICKNESSES /RUN
Y
Y
*$
300
74U
RUN 68U
100
1,000
10,000
DRAIN TIME, tD, SEC
ZY, B [SEP U 104 10-3
SHEAR
\
I 10-2 RATE, y
lo-'
100
, SEC-'
Figure 10. Shear rates in the draining films: A, drainage data; B, rheological data.
drainage of a viscoelastic liquid. It was seen that the long-time slope for Separan was the same as that predicted by power law theory. Now let us consider the relationship between free drainage and the rheological data of Figure 6. In drainage studies with an inverted sessile drop, Denson (1972) commented on free drainage behavior. Denson (1972) has stated that the "free drainage behavior of a non-Newtonian liquid depends to a large extent on whether the shear stresses in the film are greater or less than the shear stress at which viscosity of the given fluid exhibits the dependency on shear stress or shear rate". This seems quite reasonable and may be applicable here. To apply the above concept in our case, we have chosen the two runs (74U and 68U) which yielded the thickest and thinnest draining film, respectively. For each, the maximum shear stress (7,) occurring in the film was calculated using the following expression for stress at the plate, T, = pgh. The largest thickness for thick run 74 was about 750 pm (point 1in Figure 10A) and the smallest thickness for thin run 68 was about 240 pm (point 2 in Figure 10A). The resulting maximum shear stresses were found to be 82 and 24 dyn/cm2, respectively. To find the corresponding shear rate (i.) a t these stresses, trial and error calculations were done by using Figure 6 and the expression for apparent viscosity (7) given by T, = 7+. The calculation path used was to assume a shear rate, determine viscosity from Figure 6, calculate maximum stress, and then compare with the desired value of stress until convergence. The resultant values of shear rates, 0.009 and 0.07 s&, respectively, are shown in Figure 10B. As seen in Figure 10B, point 2 indicates that the thinnest films appear to lie within the zero shear region (of constant shear viscosity). However, point 1 suggests that part of most thick films lie within the nonlinear regions of power-law and transitional viscosities, so that agreement with the power law behavior is not unexpected, especially at the long times of this work. Now consider the effect of height on thickness at long times. Using the same cross-plot technique used to obtain the data of Figure 9 and Table 111, slopes of heights for Separan were calculated at 300,600, and 2000 s drain time
and all three speeds. Slopes at increasing speeds were 0.30, 0.29, and 0.27 (average of 0.29) for 300 s; they were 0.31, 0.30, and 0.27 (average of 0.29) for 600 s; and they were 0.25,0.28, and 0.27 (average of 0.27) for 2000 s. Thus the effect of height a t long times was independent of drain time and withdrawal speed, as expected. The median value was a log-log slope of 0.28. Summary From the above data, results, and discussions, it is believed that, in postwithdrawal drainage of a viscoelastic liquid, the effect of elasticity is significant during unsteady withdrawal and considerable during the initial period of drainage. The effect of elasticity tends to be negligible at long drain times. It is believed that this paper presents the first postwithdrawal drainage data for an elastic liquid. It presents the details for a precise experimental method to measure drainage. It also shows that elastic liquids provide for a surprisingly uniform film, using dip coating. Acknowledgment One of the authors (K.K.) acknowledges with thanks the financial support of Drexel University for this work. All rheological measurements were done by personnel at the Rheometrics Co., Union, N.J. We thank them all for their help, suggestions, time, and discussions. The ink figures were drawn and lettered by David Fithian of Drexel University. Nomenclature A = absorbance, In (Io/O g = acceleration of gravity, m/s2 h = film thickness, mm or Irm hI = initial film thickness, mm or pm G' = complex modulus, in-phase component, eq 13 I = intensity of light, transmitted Io = intensity of light, initially K = power-law consistency index, eq 3 and 11 Ke = extinction coefficient, eq 5 L = drainage length, mm, Figure 4 m = slope of h vs. t on log-log plot m L = slope at long times ms = slope at short times n = power law exponent, eq 3 and 11 R = ratio of normal stress difference to shear stress, eq 12 S = length to surface, mm, Figure 4 t = time, s t D = time of drainage, s tT = transition time, s t , = time of withdrawal, s T = response time, eq 13, s To = natural time, s U, = speed of withdrawal, mm/s n = distance downward from top of film, mm, eq 6, Figure 2 Greek Letters 4 = shear rate, s-l yc = critical shear rate, s-l 7' = complex viscosity, in-phase component, eq 13 X = relaxation time, s p = viscosity, mN s/m2 (same as cP) p = density, kg/m3 T = shear stress, dyn/cm T , = maximum shear stress w = frequency of oscillation, Hz Literature Cited Astarlta, G., Marrucci, G. "Principles of Non-Newtonian Fluld Mechanics", Chapter 7, pp 248-249, McGraw-Hill Book Co., New York, 1972. Bird, R. B.. Stewart, W. E.. Lahtfoot, E. N., "TransDort Phenomena", Prob. 2R, Wiley, New York, 1960. Chase, C. A., Gutfinger, C., AIChEJ., 13, 393 (1967). Denson, C. D., Ind. Eng. Chem. Fundam., 9, 443 (1970).
Ind. Eng. Chem. Fundam. 1980, 79, 275-281 Denson, C. D., Trans. Rheol., 16(4), 697 (1972). Deryagin, B. V., Levi, S. M., "Film Coating Theory", Focal Press, New York, 1964. FltzJohn, J. L., UnpubilshedRepat, Department of Chemical Engineering, Drexel University, Philadelphia, Pa., 1974. Gutfinger, C., Tallmadge, J. A., AIChE J., 11, 403 (1965). Huppler. J. D., Ashare, E., Holms, L. A,, Trans. Soc. Rheol., 11(2), 159 (1967). Jeffreys, H. R o c . Camb. Phil. Soc., 26, 204 (1930). Khilar, K. C., M.S. Thesis, Department of Chemical Engineering, Drexel University, Philadelphia, Pa., 1977. Lang, K. C., Unpublished Report, Drexel University, Philadelphia, Pa., 1969. Lang, K. C., Tallmadge, J. A., Ind. Eng. Chem. Fundam., 10, 648 (1971). Lange, N. A., Ed., "Handbook of Chemistry", 9th ed, p 1669, Handbook Publishing CO.,Inc., Sandusky, Ohio, 1956; taken from Sheeley, M. L., Ind. Eng.
275
Chem., 24, 1060 (1932). Middleman, S.,Polym. Eng. Sci., 18, 355 (1978). Pendergrass, J., AIChE J . , 21, 487 (1975). Rajanl, M.S. Thesis, Department of Chemlcal Engineering, Drexel University, Philadelphia, Pa., 1968. Satterly, J., Givens, G., Trans. Roy. Soc. Canada III, 27, 145 (1933). Tallmadge, J. A., Gutfinger, C., Ind. Eng. Chern., 59(11), 19 (1967). Weinberger, C. B., Goddard, J. D., Int. J . Muniphase Flow, 1, 456, (1974). White, D. A., Ph.D. Dissertation, Yale Unlversity, New Haven, Conn., 1965. White, D. A., Talimadge, J. A., Chem. Eng. Sci., 20, 33 (1965).
Received for review July 16, 1979 Accepted May 23, 1980
Droplet Breakup and Distribution in Stirred Immiscible Two-Liquid Systems Menso Molag, Geert
E. H. Joosten, and Adelbert A. H. Drlnkenburg"
Laboratory for Chemical Engineering, Rijksuniversiteit Groningen, The Netherlsnds
In a stirred vessel the equilibrium droplet size distribution for a liquid/liquid system was investigated: benzenehrbon tetrachloride was the dispersed phase and water was the continuous phase. Experiments were performed with different types of agitators in baffled and unbaffled vessels. The droplet break-up mechanisms was followed in a special experimental setup, in which uniform small droplets were passed through the agitator. A model is set up that describes the droplet size distribution in an agitated system in the absence of coalescence.
Introduction Droplet size distribution in a two-phase liquid system is important in chemically reacting systems and in extraction processes, in the former because of the distribution and redistribution of chemical reagents, and in the latter since the efficiency of the process depends on the ratio of droplet surface to volume and on the rate of elutration, which also depends on the droplet size. Also the compounding of products is often connected to the droplet size, e.g., in foodstuffs. Many investigators have measured the average droplet size as a function of agitator speed, vessel dimensions, and physical parameters of the system like surface tension, density, viscosity, and fraction dispersed phase. Vermeulen et al. (1955) experimentally found a dependency on the Weber number defined as pn2D3 We = Q
Results showed that
d
- = e(@).We4.6
D
in which d = mean droplet diameter and e(@)is a function of the fraction dispersed phase only, accounting for coalescense effects. Rodger et al. (1956) found, at equal volume fractLons continuous and dispersed phase, a dependency of d / D t o Weber number to the exponent -0.3 and dependency upon the ratio of the diameter of the stirrer to the tank diameter to an exponent which was also a function of the Weber number. The same dependency on the Weber number as given by Vermeulen was found by Calderbank (19581, and Ro-
driquez et al. (1961), although the function c($) is quite different for some of the results, sometimes caused by different stirrer and vessel geometries but also by differences in analyzing methods. All authors mentioned used some type of light-absorption technique to measure the average specific surface and from there the mean droplet diameter. Mlynek and Resnick (1972a,b) and Van Heuven et al. (1977) analyzed the droplet distribution by using a polymerization reaction in order to coat the droplets instantaneously with a nylon jacket, after which the dispersed phase could be separated and photographed. These authors also measured a dependency of the average droplet size on the Weber number to the exponent -0.6. Droplet size distributions were measured by Chen and Middleman (1958);the standard deviation (SD)found was 0.23; by Sprow (19671, SD = 0.5; Olney (1964), SD = 0.45; Huttig and Sadler (1957) SD = 0.35. Collins and Knudsen (1970) measured the droplet size in a turbulent pipe flow and Middleman (1974) in a static mixer (Kenics type). Hinze (1955) applied the Kolmogoroff theory to droplet breakup. He showed that droplets oscillate around an equilibrium: kinetic energy which is brought into the droplet by viscous shear or by turbulent inertial effects is counterbalanced by the extra surface energy when the droplet is removed from its equilibrium sphere shape. This counterbalance is represented by pd3U2
We'= d2u
the ratio between kinetic energy due to oscillations and the surface energy. Above a critical Weber number the droplet becomes unstable and splits. For larger droplets, that is for diam-
0196-4313/80/1019-0275$01.00/0@ 1980 American Chemical Society
