Drop Breakup in Simple Shear Fields of Viscoelastic Fluids Raymond W. Flumerfeltl Chemical Engineering Department, University of Wisconsin, Madison, W i s .
Experimental results are reported on the breakup of Newtonian drops in simple shear fields of viscoelastic fluids. Of particular interest i s the relationship between the drop size, critical shear rate for breakup, and the material properties of the fluids involved. The results indicate the existence of a minimum drop size Dmin (which varies with each fluid system) below which breakup cannot be achieved. The elasticity of the continuous phase tends to increase Dmin as well as the critical shear rate yc required for breakup when D ∈ the viscosity of the continuous phase has the opposite effect. Breakup results under unsteady shear conditions (a step change application of shear) show significantly decreased values of Dmin and from those associated with “steady” shear conditions (a gradual application of shear).
>
T h e breakup of one fluid in another is a process of considerable importance in many commercial mixing and dispersion operations. Although such processes are generally too comples for detailed analyses, it is possible t o gain valuable insights into the underlying phenomena by studying the deformation and breakup of single drops in various well defined flow fields. T o date, most studies of this type have been limited to viscous, Newtonian systems. The first such work was that of Taylor (1934). I n addition to observing the critical conditions for breakup, he compared the deformation characteristics before breakup with a previously given first-order hydrodynamic theory (Taylor, 1932). This work provided the basis for the more extensive studies which followed, and in particular, for the theoretical and experimental studies by Bartok and Yason (1959, 1961), and Rumscheidt and Mason (1961a, b). These investigators were the first to note carefully the droplet configurations and orientations during breakup as well as the effects of surface-active agents and superimposed electrical fields. Xore recently, Karam and Bellinger (1968) accurately established the critical conditions for breakup as well as the droplet configurations before and after breakup. In the present work results are described for the breakup in non-Newtonian systems. Specifically, the breakup of Kewtonian drops in steady and unsteady shear fields of viscoelastic fluids is discussed. Of primary interest are the critical conditions for breakup and the respective roles played by the viscous, elastic, and surface forces. The implications to practical mixing processes are also of concern. *is will be shown, the breakup characteristics in viscoelastic systems can be quite different from those in Newtonian systems. These differences must be carefully noted and understood if efficient mixing and dispersion operations are to be achieved. Critical Conditions for Breakup
Consider an isolated drop suspended in a simple shear field as shown in Figure 1. At low shear rates the surface forces dominate and the drop is nearly spherical in shape. 1 Present address, Chemical Engineering Department, University of Houston, Houston, Tex. 77004.
312 Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
At higher shear rates the shear forces internal and external to the drop become important and the drop becomes elongated. .As the shear rate is increased further, a critical value is reached (the critical shear rate) where the drop undergoes rapid elongation and finally breaks up into a system of smaller drops. I n Xewtonian systems the critical shear rate is given by -ic
= -iClPl
P I , 7,
d , g, D )
where p and p’ are the densities of the continuous and disperse phases, 7 and 7’ are the viscosities, r is the interfacial tension, and D is the initial diameter of the drop. The effects of external body forces are neglected. Dimensional analysis dictates four independent dimensionless groups. These may be put in the form
(y){($),(%),(0)) ~-ieD’
=
G
When inertial effects in both phases are negligible in comparison to the viscous effects, the critical breakup condition can be represented as
These groups arise naturally in Taylor’s small deformation theory (1932) and have been the basis for existing experimental correlations in viscous systems (Karam and Bellinger, 1968). Analogous relations for systems involving rheologically complex fluids are obtainable once the rheological equation of state is specified. The interest here is in the results which arise when a simplified form of the Bird-Carreau viscoelastic model (1968) is employed. This semiempirical model is a nonlinear extension of the “generalized Maxwell model” for linear behavior and was motivated by both experimental observations and results from molecular theory. I n fixed components it takes the form
---..zM
-*I
I
(a)
H
-
(b)
(d) y.0
(C)
t
SPROCKETS ATTACHED TO VARIABLE SPEED DRIVES
y-0
,
v 2 -
+
+
Figure 1. Typical breakup sequence: (a) -+ 0; (b) = ?I; (c) ?; = +Z +I; (d) . i Yo. Shear rate = = (VI
>
>
i FLEXIGLASS ENCLOSURE
+
+
Vz)/B
SUPPORTING PLATES
with the memory function given as
I
PHASE
I
...........................
Figure 2. Apparatus for drop breakup experiments
and the strain function defined by
(4) where xi and xi' are the Cartesian coordinates of a fluid element a t times t and t', respectively; II(t') is the second invariant of the rate-of-strain tensor. The constants ? I p , A l p , and X z P are related to the zero shear viscosity 70, two time constants XI and b,and two dimensionless parameters al and ( Y Z , by the empiricisms 7 P = VO
~
5
;Alp = hlk
2
A1
p + l
2 p = h 2
h
u2
( ) -
P f l
k = l
(5)
The model contains five parameters which are easily determined from simple viscometric data (Bird, 1969; MacDonald, et al., 1968). Based upon this rheological representation, the conditions for breakup of Sewtonian drops in viscoelastic shear fields are given by
Here again inertial effects and body force effects have been neglected. I n many cases i t is possible to reduce the number of dimensionless groups by two by assuming t h a t the ratios a2/a1 and h1/X2 are constants. Previous data on five different polymer solutions and melts produced a ~ / a 1values of 0.81 f 0.05 and X1/X2 values of 0.8 0.2 (MacDonald, et al., 1968). Of the four polymer solutions used here, three of these showed values for the a z / a l and X1/X2 ratios consistent with those just given. I n such cases, eq 6 can be reduced to
*
(7) Such a representation is convenient since only shear viscosity data are needed for the determination of the rheological parameters involved. Experimental Methods and Materials
The apparatus used was similar to that originally used by Taylor (1934). The simple shear field was produced by the movement of two parallel belts in opposite direction (see Figure 2). The continuous belts were made from regular 35-mm film with the light-sensitive coatings removed. Each
film was tightly stretched over a drive sprocket, roller, and guide plate. Variable speed drives were connected t o the sprockets. The operating speeds of the belts varied from 1 cm/sec to about 30 cm/sec. Operation a t lower speeds was limited by the drive units, and operation a t higher speeds by the rapid wear and subsequent breakage of the film on the sprockets. The guide plates were used to ensure that the belts were parallel in the test section. Since these plates were movable, the gap could be adjusted as desired; normally this was 1 cm. The belts moved along the faces of the plates and were held there by the normal stresses arising in the viscoelastic continuous phase. The drops were placed between the plates with a long capillary syringe. The diameter of each drop mas determined by comparing its image on a photograph with that of a reference of known dimensions. Once the drop was properly positioned in the system the drive units were turned on and the speeds of the belts adjusted until the center line of the shear field coincided with the center of the drop. The translational velocity of the drop was then zero. The speeds were increased in small increments until the critical shear rate was exceeded and the drop estended as in Figure 1 and subsequently broke. The shape of the drop just before breakup and the distribution of drops after breakup were visually noted and recorded. The critical shear rate was determined from the speeds VI and VZ of the belts and the gap B v1
Yc =
~
+ vz B
The belt speeds were measured by timing one or more periods of rotation, with the length of each belt known. Experiments of the type just described will be referred to as "steady" experiments. Additional experiments were run under unsteady conditions. I n particular, the effects of step changes in the shear rate on the breakup process were studied and compared with the steady experiments. These unsteady experiments were run in the same way as the steady experiments, except that after each incremental increase in shear rate, both belts were stopped, the system allowed sufficient time to rest, and then both drives started simultaneously. The shear rate +c,st a t which the drop broke after start-up was of interest as well as the breakup mechanism. With viscoelastic systems+c,stcan be quite different from .ic. A number of fluid systems were studied. These are listed in Table I with the corresponding interfacial tensions (meaInd. Eng. Cham. Fundam., Vol. l l , No. 3, 1972
313
Table 1. Properties of Fluid Systems Continuous phase
XI, sec
XZ, sec
62
5.2
170
348
Disperse phase
170,
5% PS-DBPa 6% PS-DBP 770 PS-DBP 8% PS-DBP 9% PS-DBP 0.75% ET-597 in 20% glyc5% PS-DBP erine and water 6% PS-DBP 8% PS-DBP 9% PS-DBP 11% PS-DBP 12% PS-DBP 1370 PS-DBP 1.0% ET-597 in 20% glycerine 5% PS-DBP and water 7% PS-DBP 9% PS-DBP 11% PS-DBP 13% PS-DBP 1 . 0 % H E C in water 6% PS-DBP 8% PS-DBP 10% PS-DBP 12% PS-DBP Castor oil
0 . 5 % ET-597 in 20% glycerine and water
a
PS-DBP
IO'
I
=
,
I
p
51
I I I
,
I
I
I
,,
,1
,I
I
,
I
I
,,
111
, , ,,
I' 0
- 8-C MODEL FIT
Ld-.--LIO
IO" 10.'
10
IO'
4
IO'
IO'
(Sfd')
Figure 3. Viscosity and normal stress data for 1% by wt of HEC in water
,
(Yz
6.8
2.4
1.9
8.8
10.2
2.6
2.1
11.9
13.8
2.8
2.3
2.4
1.9
0.44
0.26
TI, p
dyn/cm
17 l/To
1.92 3.28 5.05 7.06 9.93 1.92 3.28 7.06 9.93 18.9 28.4 40.9 1.92 5.05 9.93 18.9 40.9 3.39 7.06 14.0 28.4 8.18
14.8 14.6 14.2 13.9 12.2 13.1 12.8 12.4 12.0 15.4 15.6 15.2 12.4 12.3 12.0 11.9 14.0 15.6 15.7 16.0 15.8 13.1
0.0310 0.0529 0.0815 0.114 0.160 0.0113 0.0193 0.0415 0.0584 0.112 0.167 0,240 0.00551 0.0145 0.0285 0.0543 0.118 0.0605 0.126 0.251 0.506 0.146
polystyrene in dibutyl phthalate.
I% HEC IN WATER
IO'
Q l (Yl
, , , , ,,,,
, , , /,,,
,
, , ,,,,,, ,
,
, ,,,,,I , , , ,
I@
ET 597- 20% GLY- SOLUTIONS
IO'
v IPOISEI
10
I0 '
sured with DuNouy tensiometer) and the disperse phase viscosities. With one exception, the disperse (or drop) phase solutions consisted of various concentrations of commercial grade, relatively low molecular weight polystyrene (PS) (Union Carbide) in dibutyl phthalate (DBP). These solutions possessed Newtonian properties for the shear rates encountered in this study. The fluids used for the continuous phase were 0.5, 0.75, and 1.0% by weight, Dow ET-597 polymer (a polyacrylamide) in 1 to 4 proportions by weight of glycerin and water, and 1% H E C (100 M,Hercules) in water. A11 solutions were viscoelastic as evidenced by the viscosity and normal stress data given in Figures 3 and 4. These data were obtained with a Weissenberg rheogoniometer. The model parameters qa, XI, Xp, L Y ~ , and a2 for each continuous phase fluid were obtained graphically in the manner outlined by Bird (1969). It should be noted that with the exception of the 1% HEC solution the ratios h / X ? and ( Y ~ / L Y ~ are nearly constant, with values which are in general agreement with those of MacDonald, et d.(1968). The 1% HEC solution shows a significantly higher value for Xl/hz. A few experiments were attempted with a Newtonian continuous phase (corn syrup); however, a number of experimental difficulties arose and a consistent set of data could not be obtained. Specifically, the belts did not ride completely flat on the guide plates, even when stretched very tightly over the sprocket, roller, and plate assembly. This caused an unmeasurable variance in the gap B , and consequently a high uncertainty in the results. I n addition, the lubrication characteristics of the corn syrup were considerably less than those of the viscoelastic solutions. As a result, increased stress was placed on the belts with frequent breakages occurring.
Oi,
IO" IU'
10.'
IO'
I00
Y
Id
Results and Discussion
1SEG"I
Figure 4. Viscosity and normal stress data for ET-597 sohtions in 2070 glycerine and water 314 Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
A. General Observations. The observed deformation and breakup characteristics were similar t o those previously
1.5
I
CONTINUOUS PHASE: 0.5% ET 597 SOLUTION
0 -0
0
0
0.5
10
hi
1.5
20
25
30
Figure 5. Dimensionless representation of breakup data. Continuous phase: 1 % HEC solution
20
“
I
40 ‘
’
60
“
XI%
-
80 ’
100
120
Figure 6. Dimensionless representation of breakup data. Contiuous phase: 0.5% ET-597 solution
observed in Newtonian systems. *it low shear rates the originally spherical drops became prolate ellipsoids with the major axis inclined approximately 45’ from the axis perpendicular to the flow. (See Figure lb). At higher rates of shear this angle increased, and depending upon the fluid properties of the disperse and continuous phases, a stable or unstable configuration resulted. The stable cases were associated with the drops of the relatively high and low viscosity solutions. Specifically, when 20% PS-DBP drops (7’ = 114 P) and pure D B P drops (7’ = 0.13 P) were tested in the ET-597 solutions, the resulting configurations were completely stable for the shear rates obtainable in the apparatus. With the 20% PS-DBP drops the maximum degree of deformation showed ellipsoidal drops inclined significantly from 45’ and nearly aligned with the flow. The pure DBP drops, on the other hand, shon-ed pointed ends in the most extended state and tended not to be inclined significantly from 45’. It was also observed in both cases that the highest shear rates did not necessarily give rise to the largest deformations. I n some experiments the extent of deformation increased and then decreased when the shear rate was increased gradually to the limit of the instrument. An explanation of this phenomenon is given in the next section. I n the unstable cases, where actual breakages were observed, the configurations showed the same phenomena as mentioned above; that is, the drops with relatively loner viscosities gave rise to sigmoidal shaped drops with sharply pointed ends, and the drops of relatively higher viscosities showed sigmoidal shapes n i t h rounded ends (as in Figure IC). Below the critical shear rate for breakup the drops with pointed ends gave off a stream of very small drops a t each end, barely visible with the naked eye. Each breakup process resulted ,in a set of smaller drops with a certain size distribution and pattern. I n most cases a pattern such as s h o m in Figure I d arose, Lvith possibly more or fewer drops being produced. A n odd number of drops was common, with the outside drops being the largest and the inner drops making up EL regular pattern of large and small drops on a n alternating basis. For a given drop phase and continuous phase these patterns could be consistently repeated by successive experiments. Comparison of the patterns resulting with different fluid systems did not show any clearly distinguishable changes in these patterns for signifi-
CONTINUOUS PHASE 0759. ET 597 SOLUTION
I
0
20
40 A*%
60
BO
.
IO0
Figure 7. Dimensionless representation of breakup data. Continuous phase: 0.75% ET-597 solution
cant changes in the absolute and relative magnitudes of the viscous and elastic effects. B. Breakup under Steady Shear Conditions. For experiments in a given continuous phase it follon s from eq 6 that the critical conditions for breakup can be represented in terms of the groups (q’YcD, u ) , (q’, VO), aiid (XyJ. Figure 5 is such a representation for the breakup of PS-DBP and castor oil drops in the 1% HEC continuous phase solution. Figures 6, 7 , and 8 are similar plots for the E T 4 9 7 continuous phase solutions. Each datum point corresponds to one breakup experiment, n ith the corresponding values of the dimensionless groups being determined from experiniental measurements of the critical shear rate YCand the original drop diameter D , coupled with the appropriate rheological and interfacial tension data for the fluids involved. It is interesting to note that the relationship between q’ycD u and XI+, appears to be linear (or very nearly so) over the range of data obtained. This implies that the critical shear rate aiid the initial drop diameter are related by a hyperbolic relation of the form
yJ.0 - b )
c
(8)
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
315
=
4.0
I
,
I
I
CChlTlNUOUS PHASE 1% ET 597 SOLUTION
"
0
40
20
r
o
I
I
,
I
CATA PT. AT '7'9* 75)
80
60
100
XY
Figure 8. Dimensionless representation of breakup data. Continuous phase: 1% ET-597 solution
1.0 :
-
0.1
4
A
e
0.5Y. ET-597 SOLN 015% ET-597 SOLN 1.0%ET-597 SOLN 1.0%HEC SOLN
e
*
A
;j
7
A 0
A
i
D > - KX1aq'P-l (for breakup) VOP
A
0.01
0002
where K and p would be expected to depend upon the material parameters al, h/h, and C Y Z / C Y , . According to the data in Figure 9, both K and /3 would be positive. U'hen this result is substituted into eq 9 the effects of q' and 70 on Dminbecome apparent
1
r
1
between q f y c D / u and The intercept 4h1like 11.1, is also dependent on q'lqo, QII, X A , and cy2/a1. It is obvious from eq 9 that inherent difficulties would arise in the breakup and dispersion of small droplets in fluids with large time constants XI. For the type of fluids considered here large values of X I correspond to highly elastic fluids. The apparent existence of a lower limit drop size for breakup in these fluids is in contrast with that observed in h'ewtonian systems where within certain limits of viscosity ratio, drops of any size may be broken with the application of sufficiently high shear rates. T o counter the adverse effect of XI in obtaining fine dispersions it would be necessary to have some control over the interfacial tension u and the disperse phase and continuous phase viscosities q' and 70. The use of surface-active agents to reduce u would be of obvious benefit. The correct adjustment of q f and To to decrease Dminis not apparent from eq 9 because of the unknown dependence of +I on q'/qo. For the fluids studied here the dependence of on q'/qo is shown in Figure 9. Each of the data points shown was obtained from the slopes of the curves in Figures 5-8. Although the data are scattered, a n increase in with q ' / q o is certainly indicated. For each continuous phase the dependence on q ' / q o could be approximately represented by
0.01
e
0.1
I.o
in
where b and c are constants which vary with the fluid system involved. Such a simple result is most likely an approximation of a more complex dependence over a wider range of yc than that measured here. .Is will be discussed subsequently, significant deviations could be expected in the limit of both small + 0) and large --t a ) values of the Deborah group. Setting these aside for the moment we note that eq 8 implies that there is a minimum drop size below which breakup cannot be achieved, regardless of the shear rate. In particular, the diameter D must be larger than the constant b for breakup to occur. I n terms of the fluid properties involved, this condition takes the form (for breakup) where $1 = $ l ( q f / q o , all XI/X~, in the linear relation
a 2 / a 1 corresponds )
316 Ind. Eng. Chern. Fundam., Vol. 1 1 , No. 3, 1972
*
*
?i?.
Figure 9 Experimental data showing dependence of eq 10 on d/m
I n particular, a n increase in qo decreases Dmin. The role of q f depends upon whether @ is less than or greater than 1. The data for the ET-597 solutions, with a 2 / a 1E 0.80 0.02 and X 1 / h E 0.81 0.05, show a p slightly less than 1. It then follom that p - 1 would be small and the effect of 7' on Dmin would be slight. The data for the HEC solution, with a ~ / a = l 0.79 and X1/X2 = 1.7, indicate that /3 is somewhat greater than 1 and the effect of 7' on Dmin is significant. It would appear that higher values of Xl/hz increase the effect of q'. Although the diameter of a given drop may be greater than Dminlthis does not ensure that breakup can be achieved with reasonably obtainable values of shear rate. From eq 10 the critical shear rate is given by
(9) to the slope
*2
(13)
The dependence of #, on q ' / q was discussed above. The dependence of li.2 on q f / q 0 is shown in Figure 10. I n all cases fiZ increases with q'/qo. I n addition, the results for the ET-597 --t 0) of solutions seem to fit the Newtonian results Karam-Bellinger. It then follows from eq 13 and the results of Figures 9 and 10 that the level of shear rate required t o disperse a given Sewtonian phase in a viscoelastic continuous phase is decreased by increases in the continuous phase viscosity and decreases in the continuous phase elasticity and the interfacial tension. The significant deviations of the HEC results from the Newtonian results in Figure 10 indicate that eq 8 or alter-
A
+
CURVE RELATING CRITICAL
05% ET-597 SOLN 0.75% ET-597 SOLN 1.0%ET-597 SOLN 1.0%HEC SOLN
BREAKUP CONDITION
I I I
7
LNEAR REGION
0.8 -
.
I
b
0.6
-
\
\
I I
1.0
0
‘A
0.4
-
0.2
-
’r
0 0.001
0.01
171.
0.1
I .o
Figure 10. Experimental data showing dependence of in eq IO on q ’ / q o
x, p $2
nately eq 10 is not strictly valid in the limit as AI.);, -+ 0. From an intuitive point of view this is to be expected. I n particular, as A1.i gets smaller and smaller a point is reached where the viscosity of the viscoelastic phase becomes constant (equal to qo) and the viscous effects dominate the elastic effects. Below this value of Xly, the behavior of the fluid is Newtonian and the breakup group q’”r;D/u assumes a specified value for each value of q’/qo. Such behavior gives rise to a “zero shear rate region” on a q‘^jcD/uus. AI.);^ plot as suggested in Figure 11. It is clear t h a t eq 10 represents a n adequate approximation in the zero shear rate region only if the slope and the range of the zero shear rate region (over X1+,) are sufficiently small. I n the limit as 4 1 -+ 0, eq 10 becomes strictly valid over the entire range and corresponds directly to the behavior exhibited by Newtonian systems (see eq 1). The relatively high values of 4, observed with the HEC solution could account for much of the deviation between $2 and the Kewtonian results shown in Figure 10. Turning now to another point, the postulated relationship between q’.);D/u and AlYc a t the lower end of the AI.);^ scale in Figure 11 would seem to explain the observation described in the previous section that in some experiments the extent of drop formation first increased and then decreased with the increasing shear rate. If the increase in shear rate is considered to correspond to the lower dotted line in Figure 11, it is seen t h a t this line a t first approaches, then moves away from, the critical break-up curve. If the degree of deformation is assumed to be related to the closeness of these curves, then the observations would be explained. I n the limit of very large values of A,).,; just as in the limit of small values of A,.);,, the material functions of the continuous phase would be expected to become constant and an “infinite shear rate” value of q’ycD/u should be reached as suggested in Figure 11. Whether the intermediate region between the zero shear rate and infinite shear rate regions is truly linear as given by eq 10 is speculative a t this point. It is not unreasonable to expect a nonlinear dependence in light of the complex phenomena involved. -4s a final point in this section, it should be noted that eq 13 implies that breakup of a given drop would occur a t lower shear rates in Newtonian fluids (Al = 0) than in viscoelastic fluids, all other parameters being equal. This does not imply, however, that the shear stress requirements are less for dispersing drops in Newtonian fluids than in viscoelastic fluids
Figure 1 1. Illustration of breakup relation for a given fluid system
CONTINUOUS WAS€ (NEWT): ?d62 POISE
\-9% PS-DBP
0
0
I
IO
U
20
W
30
A,uf $0
Figure 12. Comparison of shear stresses required for breakup in viscoelastic (VE) and Newtonian (NEWT) systems under similar conditions
under similar conditions. I n fact, the contrary seems to be true (at least over the ranges of data taken here). I n Figure 12 the shear stresses required to break PS-DBP drops in a 0.501, ET-597 solution are compared with shear stresses required in a Kewtonian fluid, with q’, TO, u, and D being assumed equal in both cases. The curves shown were calculated from the results embodied in Figures 4 and 6 and the Kewtonian results of Karam and Bellinger. It is clear that for the ranges shown the stress requirements to break drops in a viscoelastic fluid can be significantly less than those necessary in a Newtonian fluid under similar conditions. C. Breakup under Transient Conditions. As discussed in the Experimental Methods and Materials section, a series of breakup experiments were conducted under start-up conditions. The interest here was in obtaining significantly lower shear rates for breakup than required under steady conditions. Dimensional reasoning suggests t h a t
X o attempt was made to determine the exact nature of this correlation; however, a number of isolated experiments were conducted and the results are shown in Table 11. It is clear t h a t + c , s t can be considerably reduced from ?;, depending upon existing conditions. illso, for a given disperse phaseInd. Eng. Chem. Pundam., Vol. 1 1 , No. 3, 1972
317
Table 11. Results from Start-up Experiments Continuous phase
0 . 5 % ET-597
0.75% ET-597
1 . 0 % ET-597
Disperse phase
tl ’/TO
Xi~/tloD
+c
9% PS-DBP 8% 7% 6% 55% 13% PS-DBP 11% 8% 6% 5% 11% 9% 7% 5%
0.16 0.114 0.0815 0.0529 0.031 0.24 0.112 0.0415 0.0193 0.0113 0.0543 0.0285 0.0145 0.00551
14.4 21.5 34.4 49.0 82.5 7.9 19.6 40.2 79.0 132 23.4 49.4 88.6 186
0.47 0.39 0.45 0.70 0.22 1.1 0.81 0.76 0.87 0.58 1.0 0.82 0.66 0.66
+a.st/
continuous phase system, this effect seems to be exaggerated as 7’ decreases. I n light of these data, dispersion in viscoelastic systems might best be accomplished under transient conditions. Concluding Remarks
It should be emphasized that the above discussion is limited to systems where inertial and body force effects are negligible. Care must be taken in using the results to explain breakup behavior in highly agitated, low-viscosity systems where viscous effects are not necessarily dominant. I n addition] it was suggested that in many cases a simplified form of the Bird-Carreau model for viscoelastic behavior can be used which involves three material parameters-a characteristic viscosity 70, a time constant Ai, and a slope parameter ( ~ 1 I. n these cases, the two remaining parameters, a second time constant XZ, and a second slope parameter (YZ, are taken as constant multiples of A 1 and all respectively. The results based upon such a procedure are not expected t o be generally valid and should only hold for fluids with similar XI/Xz and a1/cy~ values. Acknowledgment
The author wishes t o thank Professor R. B. Bird for suggesting this area of research and for guiding his research
318
Ind. Eng. Chem. Fundam., Vol. 11, No. 3, 1972
activities while a postdoctoral fellow at the University of Wisconsin. The assistance of W. K. Lee and T. Tavgac in various calculations is also acknowledged. Nomenclature
b B
K VI, V2
= = = = = =
parameter, eq 8 distance between plates, Figure 1 parameter, eq 8 diameter of drop under zero deformation condition parameter, eq 11 velocities of upper and lower plates, Figure 1
GREEKLETTERS slope parameters in B-C Model parameter, eq 11 shear rate shear rate a t which breakup occurs, steady conditions = shear rate a t breakup, start-up conditions = viscosity of continuous phase = viscosity of continuous phase in the limit as A1+ + 0 = viscosity of disperse phase = time constants in B-C Model = density of continuous phase = density of disperse (or drop) phase = interfacial tension = slope parameter, eq 10 = intercept parameter] eq 10 = shear stress, viscoelastic continuous phase = shear stress, Newtonian continuous phase =
= = =
literature Cited
Bartok, W., Mason, S. G., J . Coolloid Sci. 14, 13 (1959). Bartok, W., Mason, S. G., J . Colloid Sci. 6, 354 (1961). Bird, R. B., “Lectures in Transport Phenomena,” AIChE Continuing Education Series, Chapter 1, 1969. Bird, R. B., Carreau, P. J., Chem. Eny. Sci. 23, 427 (1968). Karam, H. J., Bellinger, J. C., IND.ENG.CHEM.,FUND.4M. 7, 576 (1968).
MacDonald, I., Carreau, P. J., Bird, R. B., Chem. Eng. Sci. 23, 901 (i968).
Rumscheidt, F. D., Mason, S. G., J. Colloid Sei. 16, 210 (1961a). Rumscheidt, F. D., Mason, S. G., J. Coolloid Sci. 16, 238 (1961b). Taylor, G. I., Proe. Roy. SOC.,Ser. A 138,41 (1932). Taylor, G. I., Proc. Roy. SOC.,Ser. A 146,501 (1934). The National Science Foundation (Grant GK-1275), the Petroleum Research Fund, (Grant 1788-6), and the University of Wisconsin Experimental Station are thanked for their financial support during the period of investigation. RECEIVED for review April 8, 1971 ACCEPTEDApril 25, 1972
