aration was obtained using a membrane containing 20% sulfolane. The membrane behavior in the lower sulfolane concentration range probably results from a complicated SO2 solubility effect which changes with pressure and the amount of sulfolane in the membrane in a nonlinear manner. The decreased membrane selectivity and high flux a t sulfolane compositions greater than 12.8% probably is due to over-plasticization which causes pin-hole leaks. An addition level of 8.2% sulfolane was chosen for further parameter studies. Effect of Permeator Temperature Figure 2 shows the effect of cell temperature on membrane selectivity and flux using a membrane containing 8.2% sulfolane and a feed containing 6% SOz. For the runs made at 14" a container of ice was placed in the constant temperature enclosure to reduce the temperature below room temperature. As can be seen, the separation factor decreased rapidly with increasing temperature while there was a nearly linear increase in flux. This response is probably due to a decrease in the solubility of SO2 in the plasticizing medium with increasing temperature coupled with an increase in the diffusion coefficient. Effect of Feed Composition The effect of feed composition on flux. permeate composition, and separation factor is shown in Figure 3. These data were taken at room temperature using membranes containing 8.2% sulfolane. Data were not obtained using the 12.5% SO2 feed material and 500 psig cell pressure because the gas mixture becomes saturated in SO2 below this pressure. As can be seen, the permeation rate and separation factor are highly dependent on feed composition and this probably indicates a nonlinearity in the concentration dependence of the permeability coefficient
Discussion The behavior of the sulfolane plasticized vinylidene fluoride membrane can probably be adequately explained by the high solubility of SO2 in sulfolane. It would be interesting to compare the premeation data with quantitative data on the solubility of's02 and Nz in sulfolane a t various temperatures and pressures but unfortunately these data are not available. The membrane behavior appears to be very similar to the permeability of organic vapors through plastic films which show a complicated dependence on pressure and concentration owing to a strong interaction between the solute and membrane (Li, et al., 1965). This is in contrast to permanent gases which have permeation coefficients independent of pressure. Thus, a high flux and separation factor results at high pressures and high SO2 concentrations where saturation of the gas with SO2 is approached. Since the diluent is a permanent gas the behavior of the system is probably more complicated than vapor permeation, however. The long-term performance of the plasticized membrane has not been determined and probably deterioration with a decrease in selectivity and flux with time could be expected. From this standpoint the lower temperature would probably be better or. it may be possible to use a heavier sulfone as a plasticizer to advantage. Literature Cited Kirk, R. E., Othmer, D. F., Ed., "Encyclopedia of Chemical Technology," Vol. 19, 2nd ed, Interscience, New York, N. Y., 1969, p 252. Li, N. N., Long, R. B..Henley, E. J., lnd. Eng. Chem.. 57, 18 (1965). McCandless, F. P., lnd. Eng. Chem.. Process Des. Develop., 11, 470 (1972). McCandless, F. P., Ind. Eng. Chem., Process Des. Develop.. 12, 354 (1973). Stern, S. A. Gareis, P. J., Sinclair, T. F., Mohr, P. H.. J. Appl. Polym. Sci.. 7, 2035 (1963).
Received for recierc: June 6, 1973 Accepted September 19, 1973
Drop Size Distributions Produced by Turbulent Pipe Flow of Immiscible Fluids through a Static Mixer Stanley Middleman Chemical Engineering Department. University of Massachusetts. Amherst, Massachusetts 01002
Data are presented for six organic liquids of viscosities ranging from 0.6 to 26 CP and interfacial tensions ranging from 5 to 46 dyn/cm dispersed in water as the continuous phase. The effect of mixer pitch and number of mixing elements is illustrated. Some aspects of Kolmogoroff's theory provide a basis for correlation of the data.
When a mixture of two immiscible fluids is subjected to pipe flow, a dispersion is created which can be characterized by a "drop-size distribution function," @@). From +(D) it is possible to calculate various average drop sizes. For example, the Sauter mean diameter may be defined as -
D32 =
78
1-P
@(D) d D / J m D 2 W D ) d D
(1)
0
I n d . Eng. C h e m . , P r o c e s s D e s . D e v e l o p . , Vol. 13,
No. 1, 1974
The Sauter mean is a particularly useful average since the interfacial area per unit volume can be obtained directly from
A,
=
WID32
(2)
where 4 is the volume fraction of dispersed phase. The drop-size distributions produced by turbulent pipe flow have been the subject of theoretical and experimental
The energy dissipation rate i is a local quantity that undoubtedly varies spatially across the pipe diameter. Nevertheless, to make progress, we will relate ^t to macroscopic variables by assuming that =
Figure 1. The Kenics static mixer.
studies by Hinze (1955), Paul and Sleicher (1965), Sleicher (1962), Collins and Knudsen (1970), and Hughrnark (1971), the results of which make it possible to predict the behavior of such systems from a knowledge of physical properties and macroscopic flow variables. Our work focusses on the production of dispersions by flow through a pipe containing an in-line motionless mixer, the Kenics static mixer. The Kenics static mixer is shown in Figure 1. It consists of a series of stationary elements, fixed relative to the pipe wall, which divert the flow field and cause the mixing action. The mixing elements are rectangular, and split the pipe cross section into two semicircular sections. Each element is twisted through 180", and alternate left- and right-hand twists are fixed in series down the pipe axis. The dependence of the drop size distribution on mixer geometry has been examined, and the results are considered in the light of a simple hydrodynamic treatment. Theory We begin with the idea that the turbulent flow field may be characterized by an energy spectrum function E ( k ) , defined so that E ( k ) dk is the energy per unit mass of continuous phase associated with turbulent fluctuations of wave number k to k dk. A drop of diameter D is stabilized by surface energy of magnitude 4 ~ 0The ~ drop ~ . is subject to disruptive energy, associated with turbulent fluctuations, of magnitude Y6aD3pc, where
+
Only energy in turbulent eddies of length scales smaller than D (wave numbers k > 1/D) is considered, on the grounds that larger eddies merely transport a drop, but do not disrupt it. We assume that @ ( D )should be some function of the ratio of turbulent energy to surface energy, and write
9(D)
=
9 [ I ; 6 r ~47 ~rD2a l , l E ( kdk I
3
(4)
T o proceed, it will be assumed that the gross features of the turbulent energy spectrum are governed by KolmogorofY's theory of turbulence (see Hinze, 1955, 1959), which gives
a2(QAP/pAL)
(8)
The term QAP is simply the rate of energy input into the fluid, and pAL is the mass of fluid in the pipe of cross sectional area A and length L . In eq 8 we let p be the continuous phase density, since we will only consider fairly low volume fraction dispersions. It is convenient to replace QIA with the average velocity Vin subsequent equations. If eq 8 is substituted into eq 7 one finds. after some algebraic rearrangement, that
@ ( D ) = 9[(D/D")1L',,3 ' f f ' '1
(9)
In eq 9 the pipe diameter DO has been introduced. The Weber number is defined as
ATLve= p V 2 D o / a
(10)
D&P/2pV2L
(11)
The friction factor is
f
=
Since eq 9 indicates only what @ is a function of, all numerical coefficients in front of DIDOhave been dropped. If eq 9 is used in eq 1, it follows immediately that -
D,12/Do
=
CNiv,-J'5f -?
j
(12)
a result which can be subjected to experimental verification. In pipe flows, the friction factor is a weak function of Reynolds number for N R , > 3000, and a reasonable approximation is seen, e . g . , in Bird, e t al. (1960), to be f
- .vKr-;
(13)
Thus we see that -
D,? / Do
= ChT;v,-'
'iVRel 1'
(14)
The point of displaying the Reynolds number explicitly is to indicate the very weak dependence of drop size on Reynolds number a t constant Weber number. This dependence would very likely be masked by experimental error in most cases. If we wish to compare dispersions made in a circular pipe to dispersions made in the same pipe, but one containing a motionless mixer, then eq 1 2 suggests (Q?)>l/(mo
=
(f(,/f>,V'
(15)
a t constant flow rate, and assuming equal physical properties. Friction factors in a Kenics static mixer have been determined over a wide range of Reynolds numbers and mixer geometries, and hence it should be possible to subject eq 15 to experimental test. It is worth noting a t this point, however, that f o / f ~is typically of the order of for N R e > 3000, and so one may expect to reduce the drop size by an order of magnitude through the use of the static mixer. Experimental Methods
in the inertial subrange (to be defined subsequently) where ^t is the energy dissipation rate per unit mass and a1 is some undetermined coefficient. It follows that
i n E ( k ) dk and
=
3/2ai2
(6)
The continuous phase was water in all cases reported here. The dispersed phase fluids are listed in Table I along with the pertinent physical properties. Figure 2 shows the layout of the experimental system. Static mixers varying in pitch ( L e / D )and number of elements in series were placed inside glass pipes. The dispersed phase was introduced through a small glass tube which entered the pipe through a T branch about one diameter upstream of the first element of the mixer. On the outside of the pipe, a t the position of the last element of Ind. Eng. Chem.. Process Des. Develop., Vol. 13, No. 1, 1974
79
dispersed phose inlet
10-IL 1
continuous inlet I
A camera
Figure 2. Schematic of system for production and measurement of drop sizes.
-
,
IO+
Table I. Properties of Dispersed Phase Liquids a t 25" P,
Anisole Benzene Benzyl alcohol Cyclohexane Oleic acid Toluene
g
ml
p',
CP
1.o
0.99 0.87
0.6 5 .O
1.o
0.76
0.90 0.87
u, dyn
,=,
IO2
I o3
I o4
Nwe
Figure 3. Sauter mean drop diameter for the benzene-water sys-
26 40
tem.
5
0.8 26 .O
46 16
0.6
32
(16)
The number of drops counted, M , was usually of the order of 100 to 200.
Results Figure 3 shows data obtained in the benzene-water system in two different diameter pipes containing static mixers. We find it convenient, and apparently acceptable within experimental accuracy, to plot D 3 2 / D o as a function only of the Weber number. Since the Reynolds number range is only about an order of magnitude for the data shown, the tenth-power dependence predicted by eq 14 is too weak t o make itself felt. This is likely to be the case in any practicable application of' a correlation of the type shown in Figure 3. However, in going from a static mixer to an empty pipe, a t constant Weber number and Reynolds number, there is a large change in friction factor, and eq 15 suggests that a significant effect on drop size should be observed. A limited amount of data were taken in an empty pipe, and these are shown in Figure 3. As expected, the drop size is considerably different in the two cases. At a Weber number of 100, for example, the Reynolds number in the 0.5-in. pipe is 6200, and the ratio of friction factors, f ~ / f o , is observed to be f ~ / f o = 100 from which it follows that & / 80
10
I
cm
the mixer, a small rectangular transparent plastic box was placed. When filled with water the box made it possible to photograph the dispersion flowing beyond the last element with minimal optical distortion due to curvature of the pipe walls. Photographs were taken with a 3 5 m m Pentax camera, using a 135-mm lens on a bellows rack. High-speed flash was provided by an EGG Model 549 microflash of 0.5-psec duration. The negatives were placed in 35-mm slide frames and projected, for measurement, onto sheets of white paper. The volume flow rate was determined by collecting the effluent from the pipe, over a known time interval, in a graduated cylinder. After the dispersion separated into two distinct layers in the cylinder: it was possible to determine the volume fraction of dispersed phase. The drop diameter measurements were classified into size groups of' 10- or 20-p intervals, and the Sauter mean diameter was calculated from these discrete data using
/=I
I
Ind. Eng. Chern., Process Des. Develop., Voi. 13, No. 1, 1974
10-1/
10-2[
D, = 112" I" oleic a c i d ' -' benzylalcohol -
--
C
benzene toluene cyclohexane
10
1
1
I
I anisole 0 1o - 3 L - p - - p J I
-
i
I o2
I o3
I o4
Nwe
Figure 4. Sauter mean drop diameter for six different fluids
= 6.2. Acccrding to eq 15, then, we should observe = 6.2. The observed value is 7.3. %or all-of the empty pipe data shown the f , ' f~o ratio is 100, and the drop size ratio should be constant. This does not appear to be so even over the narrow range displayed, and this may indicate a failure of this aspect of the model. Equation 8 is the most likely source of this discrepancy. Even so, these results are not in bad agreement with the general features of this model. Since our goal was not to study drop size behavior in an empty pipe, we have not pursued this point further. Instead, we turn to a more detailed study of behavior in the Kenics static mixer. Effect of Dispersed Phase Viscosity. Figure 4 shows all of the data obtained in 1 and 0.5-in. static mixers of constant pitch (Le L) = 1.5) and constant number of elements (ne = 21). The properties of the six fluids studied are listed in Table I. We note that the data for dispersed phase fluids of viscosity comparable to that of the continuous phase (water) are grouped together, while data for benzyl alcohol (p' = 5 cP) and oleic acid (p' = 26 cP) are separated from the main body of the data, at higher Weber numbers, and indicate that larger mean drop sizes are produced. Qualitatively, such a result is to be expected, since a high viscosity in the dispersed phase retards disruption of the drop. In writing eq 4 it was assumed. however, that only surf'ace energy stabilizes the drop. Empty pipe data show similar effects, and Hinze (1955) and Collins and Knudsen (1970) suggest methods of' correlating the data t o account for dispersed phase viscosity. The methods do not work well with our data.
f0PJ5
(D32)0/(&.2)M
.
1
i
\\
benzene
t
,
I o-2,
IO
I
1 benzene D, :I / 2 "
l o - 3 L lo-?
I 0-
'
I
n,=21
I o3
I 0'
Nwe Figure 7. Effect of number of elements on drop size.
I
iI
8
i
4
I
I
i
I
4L
9 Figure 5. Effect of volume fraction on drop size.
2
______
0 0
4
12
8
20
16
24
28
"e
Figure 8. Effect of number of elements on drop size at a fixed Weber number.
i I
I
10
IO'
V (crn.sec-')
0.1
Figure 6. Etfect of pitch ( L , D ) on drop s u e
Effect of Volume Fraction of Dispersed Phase. The data shown in Figures 3 and 4 were all taken a t low volume fractions, 4, in the range of 0.5 to 1%. It was assumed, in writing eq 4, that drop size is determined only by factors affecting drop break-up, and that once a drop is formed it will either remain a t that size or, possibly, be broken by subsequent interaction with a "high-energy" eddy. This idea ignores the possibility of coalescence of drops as a factor in determining drop size. Coalescence will be promoted by two factors: both a high volume fraction as well as turbulent mixing promote collision between drops. The final drop size distribution, then, represents equilibrium for a dynamic process which balances drop break-up (essentially a Weber number dominated process) against drop coalescence ( a volume fraction and Reynolds number related phenomenon). Figure 5 shows data for the benzene-water system in a 0.5-in. static mixer. ] h o p sizes were obtained as a function of volume fraction 4, with Reynolds number as a parameter. In this case (of constant DO and physical properties) the Reynolds and Weber numbers are related by Nwe = 2.5 X 1 0 - 6 N ~ ~ ~ . For relatively low Reynolds numbers (NRE < 2250) which correspond to very low Weber numbers ( N w e < 12) we see the effect o f coalescence in Figure 5. At high Reynolds numbers. which correspond to relatively high Weber numbers, the increase of drop size with volume fraction is very slight, indicating that drop break-up dominates the process.
0
1
1 0
2
4
6
8
IO
12
14
16
100 D ( c m ) Figure 9. Drop size distribution curve from a typical run
Effect of Mixer Pitch ( L e / D ) Figure . 6 shows the effect of pitch on drop size for oleic acid-water dispersions. At constant flow rate, the friction factor is observed to decrease with increasing pitch. For example, in the range of flow rates illustrated here, f2.0/fi.0 = 5-7. Equation 15, then, suggests that the ratio of mean drop sizes should be about a factor of 2. The data show some scatter, and the separation of the upper and lower sets is more like a factor of 1.5 than 2. Still, the results are in general agreement with expectations. Effect of Number of Elements (ne). Figure 7 shows data obtained in the benzene-water system. Three mixers, of the same geometry but differing in the number of elements ne, were used. As could be anticipated, longer exposure to the mixing action leads to reduced drop size. Figure 8 shows these data, a t a fixed Weber number, plotted as a function of ne. The case ne = 0 is simply the empty pipe. It would appear that ten elements are sufficient to produce the equilibrium drop size distribution. Drop Size Distribution. While the Sauter mean is a characteristic measure of the drop sizes produced by the mixer, it fails to give any indication of the range of sizes produced. Figure 9 shows a typical distribution curve for the volume fraction fv, defined in such a way that Ind. Eng. Chem., Process Des. Develop., Voi. 13, No. 1, 1974
81
M
f,(D,) dD,
= n D,31xn,D,1
(17)
,=I
is a discrete distribution function, since the drop size measurements were classified into discrete groups. It is the discrete analog of the continuous function @ ( D ) If . the size interval for classification is small compared to the range (which is true in our cases) then it is reasonable to present fv as a continuous distribution, as shown in the figure. The bimodal character in the neighborhood of the mean is statistical noise associated with the fact that only a few hundred drops were counted. It does not appear in most of our f v curves. The small peak a t low drop size may be real. It appears in a high proportion of our distribution curves. It is associated with the fact that when a “mother” drop breaks, it often produces two major (smaller) “daughter” drops, and a much smaller “satellite” drop (Collins and Knudsen, 1970). The satellite drop probably forms from the “neck” which attaches two daughter drops just before they separate from the mother drop. Figure 10 shows the data of Figure 9 replotted as a cumulative distribution function, F , ( D k ) , defined as fv
/
L’
;
i i
0.5c
I
01 0
2
6
4
8
I O 12 14 16
100 D ( c m ) Figure 10. Cumulative distribution curve for data of Figure 9. 5 . 9 99.0 98 0‘
t
95.0L
,=I
F,(D) is simply the total volume fraction associated with drops of diameter less than some size D. From Figure 10 one can determine that 70% of the volume of the dispersion is associated with drops whose diameters lie within &20% of the Sauter mean. In this sense, then, the drop-size distribution is relatively narrow. Figure 11 shows cumulative distribution data for a large portion of our data. When the drop size is normalized to the mean, 0 3 2 , for each individual set of data, all of the data fall on’a single curve. Any f, peaks associated with satellite drops are “lost” in experimental scatter when the data are plotted for several runs of several different dispersed phases. This suggests (as in fact is apparent when all individual f v curves are examined) that the satellite peak does not occur in all data, and that its position is not always at the same fraction of 0 3 2 . The linearity of Figure 11, since-it is plotted on “normal probability” coordinates, suggests that the drop-size distributions produced in the Kenics static mixer are roughly normally distributed about the Sauter mean. A fit of the data gives the distribution as
In interphase transport process a major factor is the manner in which interfacial area is distributed over the drop sizes. A distribution function for area, f * ( D ) , is defined so that f.4 dD is the fraction of interfacial area associated with drops in the size range D to D + dD.f A is related to f\, by
Discussion In the development of the theory for this system several ideas of Kolmogoroff, related to the manner in which energy is distributed over wave numbers in a turbulent flow, were used. It is appropriate a t this point to question the applicability of Kolmogoroff‘s theory to flow in a Kenics static mixer. One restriction of the theory is that the flow field should be both homogeneous and isotropic. Homogeneity relates to the absence of spatial variations of turbulent properties of the flow. Any shear flow with 82
Ind. Eng. Chern., Process Des. Develop., Vol. 13, No. 1, 1974
Ib e n z e n e
7 anisole 1.0
oleic acid
0’5L Y 0. I
0.051 0,011 0
I
I
0.2
0.4
0.6
0.8
1.0
1.2
1.4
J
1.6
DID,, Figure 11. Cumulative distribution data for several fluids and flow rates.
boundaries, as in the case here, will he inhomogeneous. In fact, there is strong evidence, described by Hughmark (1971), that most of the drop break-up in an empty pipe occurs in the wall region. Isotropy relates to the question of whether parameters such as the fluctuating components of the flow depend on the coordinate directions. Again, in a pipe flow, it is unlikely that isotropic turbulence is achieved except far from solid surfaces. Nevertheless, Kolmogoroff‘s ideas provide the only simple vehicle for producing a semiquantitative theory. The equations developed above are, a t least, testable through examination of experimental data. We conclude that several features of our results are consistent, in a general way, with the predictions of the theory. In cross section, a circular pipe containing a static mixer has a boundary which can be considered to be two semicircles sharing a common diameter. The hydraulic radius of such a boundary, when defined as the ratio of cross sectional area to wetted perimeter, is just Rh = I/& (1 + 2 , ’ ~ ) -One ~ . might be tempted to use a hydraulic radius either as a more appropriate length scale than D Oitself, or in defining a Reynolds number for consideration of some aspects of the dynamics of the flow field. Neither approach sheds much light on the results presented here. If one examines the treatment of frictional effects for flow in a noncircular cross section, along the lines given for example in Bird, et ai. (1960), it is apparent that the concept of hydraulic radius is of value only when most of the friction producing flow is axial. If the elements of the
static mixer were straight, instead of helical, the mean flow would indeed be axial. The helical nature of the device, however, sets u p very strong secondary flows which are radially directed, and which contribute significantly t o the dynamics of the flow field. We find, then, that consideration of the hydraulic radius is no aid in elucidating these results. The conditions of applicability of eq 5 require consideration of eddies of a size that is very small with respect to macroscopic dimensions, such as pipe diameter, but large in comparison to the "dissipation scale," 7 , which is given by Hinze (1959) as 77 =
(u3/;)1/4
(n)
where v is the kinematic viscosity of the continuous phase. This restriction is equivalent to the requirement that
which is true for most of the data presented here. Under conditions of extremely high-energy input it is possible to produce drops of a size comparable to or smaller than 7. Such drops would be in a dynamic regime known as the viscous subrange, for which eq 5 must be replaced by
E ( h ) = a3vz13k-1
(23)
If the analysis is carried through as in the case for the Inertial subrange, one finds eventually that
(24) where p V / u is a dimensionless group easily seen to be the same as NwelNRe. Equation 24, by comparison t o eq 12, suggests a much stronger dependence of 0 3 2 on flow rate, and on friction factor. The magnitude of the dissipation scale 7 can be estimated using eq 8 in eq 21, and taking a2 = 1. The result is
q/D, =
NRe-34f-14
(25)
At a Reynolds number of lo4, in a Kenics static mixer, we estimate DO = 10-3. From Figure 3 we can see that a t the highest Weber numbers, which correspond to Reyn-
olds numbers in the neighborhood of lo4, the drop size is comparable to 7 . Nomenclature A = cross sectional area of pipe A , = area per unit volume, cm-l D = diameter of a drop, cm DO = pipe diameter, cm D3, = Sauter mean diameter, cm E ( k ) = energy spectrum function, cm3/sec2 1 = friction factor f*(D) dD = area fraction of drops of size D to D dD fv(D) dD = volume of fraction of drops of size D to D
+
dD
+
Fv(D) = cumulative distribution function, cm-I k = wavenumber, cm-I Le = axial length of mixer element, cm N R e = p V D o / p , Reynolds number Nwe = p V D o / u , Weber number ne = number of mixer elements A P / L = pressure drop per unit length of pipe, dyn/cm3 Q = volume flow rate in pipe, cm3/sec Greek Letters t = turbulent energy per unit mass, cm2/sec2 ^t = local energy dissipation rate per unit mass, cm2/sec3 7 = dissipation microscale, cm p(p') = continuous (dispersed) phase viscosity, g/cm sec u = kinematic viscosity of continuous phase, cm2/sec p = density of continuous phase, gm/cm3 (r = interfacial tension, g/sec2 4 = volume fraction of dispersed phase Subscripts 0 = emptypipe M = pipewithmixer Literature Cited Bird, R . B.. Stewart, W. E., Lightfoot, E. N., "Transport Phenomena," Wiley. New York, N. Y., 1960, pp 183-188. Collins, S. B., Knudsen, J. G., AIChEJ., 16, 1072 (1970). Hinze, J. O., "Turbulence," McGraw-Hill, New York, N. Y., 1959, pp 183-1 86. Hinze, J. O., A I C h E J . , 1, 289 (1955). Hughmark, G. A , , A I C h E J . , 17, 1000 (1971). Paul, H. I.. Sleicher, C.A. Jr , Chem. Eng. Sci., 20, 57 (1965). Sleicher, C. A . Jr., A I C h E J . , 8, 471 (1962)
Received for rei'ieu: June 19, 1973 A c c e p t e d September 26, 1973
Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 1 , 1974
83
