Ind. E n g . Chem. Res. 1987,26, 77-81
T = temperature, K r = mole fraction z = distance (0) = value of variable at z = 0 Greek Symbols a = defined by eq 19
= viscosity of pure gas, Pa-s p = mean viscosity of a gas mixture, Paes p
Subscripts
i , 1, 2 = species transported through a porous medium Literature Cited Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley International: New York, 1960; pp 22, 511. Bosanquet, C. H.; British, T. A. Report BR-507, 1944.
77
Graham, T. Philos. Mag. 1833,2, 175, 269, 351. Gunn, R. D.; King, C. J. AIChE J. 1969, 15, 507. Jackson, R. Transport in Porous Catalysts; Elsevier: Amsterdam, 1977; p 50. Mason, E. A,; Malinauskas, A. P. Gas Transport in Porous Media: The Dusty Gas Model; Elsevier: Amsterdam, 1983; pp 21, 51,92. McGreavy, C.; Asaeda, M. Presented at the 184th National Meeting of the American Chemical Society, Seattle, 1982, ORGN 473. Olsson, R. G.; McKewan, W. M. Trans. Am. Inst. Min., Metall. Pet. Eng. 1966, 236, 1518. Olsson, R. G.; McKewan, W. M. Metall. Trans. 1970, 1 , 1507. Rowe, P. N.; Claxton, K. T.; Lewis, J. B. Trans. Inst. Chem. Eng. 1965, 43, 14. Turkdogan, E. T.; Olsson, R. G.; Vintners, J. V. Metall. Trans. 1971, 2, 3189. Unal, A. Trans.-Inst. Min. Metall., Sect. C , in press.
Received for review May 7 , 1985 Accepted April 17, 1986
Minimum Agitator Speeds for Complete Liquid-Liquid Dispersion A. H.P. S k e l l a n d * and George G. R a m s a y School of Chemical Engineering, T h e Georgia Institute of Technology, Atlanta, Georgia 30332
Data from three different sources have been pooled to obtain an empirical correlation of the minimum agitator speed needed t o obtain complete liquid-liquid dispersion in baffled vessels. The variables include 5 common types of impellers, 2 generating axial flow and 3 radial flow, in 4 locations, fluid properties in 11systems, tank diameter, liquid height, and volume fraction of the disperse phase. Observations from 481 runs were correlated with an average absolute deviation of 12.7% by the expression (NFJmin = C2(T/D)2*~0.106(NGaN~o)-0.084 where C and a are tabulated according to the impeller type, location, and-in some cases-the ratio H/T . Scale-up relationships are developed from the above expression. Liquid-liquid dispersion in agitated vessels finds extensive application in mixer-settler design in extraction operations and in emulsion polymerization processes. In such work, it is necessary to ensure that the agitator speed is high enough to achieve complete dispersion of one liquid in the other. Skelland and Seksaria (1978) showed that, in some cases, impeller speeds of lo00 rpm are insufficient to ensure complete dispersion. The few previous studies began with Nagata's (1950) work; he used an unbaffled, flat-bottomed vessel, with a centrally mounted, four-bladed flat-blade turbine agitator, with TID of 3 and a blade width of 0.06T. He obtained the empirical expression
where H I T = 1and 4 = 0.5; T did not vary. The constants Co and cyo were functions of the impeller type and location. A related phenomenon is the minimum impeller speed required to suspend solid particles off the bottom of the vessel in an agitated solid-liquid suspension. Zwietering (1958) used turbines, paddles, propellers, and vaned disks to suspend sand and sodium chloride particles in liquids in baffled vessels. He presents g 0 . 4 5 ~ p 0 . 4 5 F c 0 . 1 g , 0 . 2 ( 100R)O.13 Nmin= Cr( (4) g 0 . 8 5 0.55
g)
Pavlushenko et al. (1957) used three-bladed square-pitch propellers for sand and iron suspensions to obtain Nmin= 0.105(
and he reported that Nminis independent of interfacial tension. Van Heuven and Beek (1971) studied dispersion with a six-bladed disk turbine in a baffled vessel. From a combination of theory and experiment, they obtained
where DIT was constant at 0.333. The study by Skelland and Seksaria (1978) included a variety of impellers-propellers, pitched-blade turbines, flabblade turbines, and curved-blade turbines-in a baffled vessel. They found N min. =
C~aO~Lc'/S~d-1/9u0.3Ap0.25
(3)
0888-5885/87/2626-0077$01.50/0
Pc
g)
g 0 , 6 p d 0 . 8 g 0.4 P
g 0 . 6 Pc 0.6F c 0.2
(5)
Differences in the exponents on a given variable in eq 4 and 5 are noteworthy, including the conflicting directional effect of gC indicated by the two expressions. Experimental Apparatus a n d Procedure Fluids Used. Deionized water formed one phase in all runs; the other phase consisted of one of the following presaturated with water in all cases: ethyl acetate, benzaldehyde, chlorobenzene, and carbon tetrachloride. Fluid properties appear in Table I. Interfacial tensions for ethyl acetate and benzaldehyde are from Skelland and Seksaria (1978), chlorobenzene from Moeti (1984), and carbon tetrachloride from the International Critical Tables (1928). 0 1987 American Chemical Society
78 Ind. Eng. Chem. Res. Vol. 26, No. 1, 1987 Table I. Fluid Properties at 23 OC dynamic viscosity, fluid density, kg/m3 N s/m2 ethyl acetate 894 0.000 46 benzaldehyde 1041 0.001 4 chlorobenzene 1106 0.001 0 carbon tetrachloride 1590 0.001 0 997 0.000 9 water
interfacial tension with water, N/m 0.006 27 0.014 5 0.035 2 0.045
++ n
k Shaft n Baffle
-rI 4
L i g h t e r liquid
I H F I II T H H-Ei2
jI1 ’4
Table 11. Apparatus Dimensions internal liquid diameter of height in baffle vessel, m vessel, m width, m 0.019 0.216 0.216 0.019 0.241 0.121 0.019 0.241 0.241 0.019 0.241 0.362 0.025 0.292 0.292
impeller diameters, m 0.102, 0.076, 0.065 0.102, 0.076, 0.065 0.102, 0.076, 0.065 0.102, 0.076, 0.065 0.102, 0.076
Apparatus. The Experimental Agitator Model ELB manufactured by Bench Scale Equipment Co. was used for the liquid-liquid dispersion studies. The unit included a ‘/,-hp drive motor, which supplied an infinitely variable output speed of 0-20 rps. The accuracy of the speed control dial was checked with a stroboscope. Three glass, cylindrical, flat-bottomed vessels were used, providing the ranges 2.13 I T / D I3.83 and 5 HIT I 3/2. Four radial baffles at 90° intervals were employed, with TIB = 12. The continuous phase was identified with the aid of two conductivity electrodes made of copper wire, 0.01 m apart and connected to a conductivity meter. Agitation was supplied by three sizes of a centrally mounted six-flat-bladed turbine on a concentric vertical shaft. The shaft, impellers, and baffles were all made of 316 stainless steel. Dimensions of the equipment are listed in Table 11, and a diagram is shown in Figure 1. Operational Procedure. Before filling with a given pair of liquids, the equipment was washed with detergent and then rinsed with tap water, followed by deionized water. Two immiscible liquids in the desired volume proportions (0.1 IH 2 0 I0.9) were added to the vessel. The impeller was centrally located a t H / 2 in all runs, and its speed gradually increased in small increments, until visual observation showed that complete dispersion was just achieved. This state has been defined by Skelland and Seksaria (1978) as follows: “The minimum mixing speed of the impeller is the rotational speed just sufficient to completely disperse one liquid in the other, so that no clear liquid is observed either at the top or the bottom of the mixing vessel. In some cases, clear liquid pockets of approximately 1 x lo* to 5 x lo4 m3 persisted near the sides of the vessel at the top and the bottom, although the rest of the liquids were mixed. In some instances, small pools of clear liquid adhered to the drive shaft at the top of the vessel. In order to mix these small liquid pockets, speeds had to be increased by 25100% or more. To prevent such anomalous results, a well-mixed or completely dispersed state was defined when only small, relatively nonstationary, liquid pockets remained unmixed in the bulk dispersion. The rotational speed of the impeller corresponding to this state is defined as the minimum mixing speed, Nmi,,and is not necessarily the same as that required for a homogenous dispersion. More nearly homogenous dispersions may have occurred a t speeds higher than the minimum mixing speed, in accordance with the findings of Pavlushenko (1957) and Zwietering (1958)”. In the range 0.4 5 4 I0.6, after the continuous phase had been identified in a given run, phase inversion was
+I)+
Flat botton cylindrical
II
’-
47
Ir
glass j a r Denser l i T J i d
B a f f l e does n o t extend t o v e s s e l botton
S i d e Yiev
@ Impeller
Bottom V i e d
Figure 1. Schematic diagram of the experimental apparatus.
achieved in the next run by initiating dispersion with an auxilary off-center turbine, temporarily placed in the phase desired to be continuous. The auxiliary turbine was withdrawn after dispersion was established, the centrally mounted turbine being started just prior to this withdrawal, so as to maintain the dispersion. The reproducibility of the minimum mixing speed needed to achieve complete dispersion was good and was quantitatively similar to that found by Skelland and Seksaria (1978). The identification of the continuous phase in a given run was usually possible by the visual methods described in detail by Quinn and Sigloh (1963) and by Selker and Sleicher (1965). However, the most reliable means of identification was provided by the conductivity electrodes in the vessel, as described earlier, and connected to a conductivity meter. The conductivity of the aqueous phase was much greater than that of the organic liquids, making continuous phase identification a simple matter. Data and results for each run are recorded by Ramsay (1984). Correlation of Results To increase the generality of the correlation to be developed, the results from the present 251 runs were combined with those from 35 runs reported by van Heuven and Beek (1971) and from the 195 runs made by Skelland and Seksaria (1978). This gave a total of 481 data points on 5 types of impeller (3 radial flow and 2 axial flow), 4 impeller locations, and 11systems, with 0.01 5 5 0.6, 2.13 I T / D I3.83, and 112 IHIT I312. The impellers featured in the combined work are shown in Figure 2. The data were correlated by using the BMDP Statistical Software, page 264, Program 9R, University of California Press, 1983. In accomodating the densities of the two phases, van Heuven and Beek (1971) were successful in using a mean density defined as PM = 4Pd -k (1 - 4)Pc
(6)
Ind. Eng. Chem. Res. Vol. 26, No. 1, 1987 79 Table 111. Correlations and Average Deviation between Nmia,sxptl and Nmin,ealcdn impeller 95% confidence set HIT location C 01 interval on 01 4.38 square pitch, 0.67 f0.44 1 1 ~ 1 4 3H/4 2.76 downthrusting 2 1 0.95 f0.44 4.33 0.79 f0.44 propeller 3 1 HI 2 H / 4 , 3H/4 1.46 (three blades) 4 1 1.33 f0.44
16.9 14.4 17.7 13.7
301 - 2.87 -0.86 -0.02 -0.50 1.12
% av deviation
5 6 7 8
1 1 1 1
~ 1 4 3H/4 HI2 H/4, 3H/4
1.95 1.96 0.84 0.94
1.44 1.17 1.97 1.27
f0.47 f0.47 f0.47 f0.47
13.9 14.2 18.6 10.0
1.45 0.64 3.04 0.94
9 10 11 12 18 19
1 1 1 1
0.91
*
2.02
f0.45
9.9
3.19
0.95
1.38
f0.13
11.5
112 312
~ 1 4 3H/4 HI2 H/4, 3H/4 HI2 HI2
0.70 1.10
1.24 1.70
f0.28 f0.28
10.2 14.7
0.85 2.23
curved-blade turbine (six blades)
13 14 15 16
1 1 1 1
~ 1 4 3H/4 HI2 H/4, 3H/4
*
1.03
1.86
f0.46
10.2
2.71
1.34 1.20
1.20 0.94
f0.46 f0.46
8.9 10.7
0.73 -0.05
disk turbine (six blades)
17
1
HI2
0.53
1.70
15.8
2.23
downthrusting pitched-blade turbine (six blades) flat-blade turbine (six blades)
*
*
*
*
*
*
*
*
*
*
*
1.27
*
*
Asterisks indicate insufficient data due to splashing.
unity. Our correlation is as follows, based on dimensional analysis, reformulation of the resulting groups in terms of appropriate force ratios (Boucher and Alves, 1959), and using an assumed exponential form
a) Pitched blade turbine
b) Marine-type propeller
Axial f l o w impellers u s e d .
The correlation coefficient is 0.92 and the 95% confidence intervals on the last two exponents are 0.106 f 0.073 and 0.084 f 0.017. (The presence of 4 in pM and pM may account for the small residual dependence on this variable shown in eq 8.) The experimental ranges covered for the dimensionless groups were 1.75 (lo7) I (NGaNBo) I8.78 and 0.17 INFrI 38.8. The values of C and a,with the attendant 95% confidence intervals on a for the various geometries studied, are given in Table 111. This table also shows the average percent deviation between calculated and experimental Nmin,defined as 100INmin,calcd , Ivmin,erptll / flmin,calcd. The exponent on TID for the disk turbine was estimated to be the-same as that determined by Zwietering (1978) for solid-liquid dispersions; his impeller and vessel dimensions were similar to those of van Heuven and Beek (1971). This was done because the range of TID used by van Heuven and Beek was not sufficient to achieve a reliable exponent (TID equaled 3 for four systems and 3.33 for the other system). The only check for this assumption is the exponent on TID for a propeller located at H/4. Zwietering's exponent was 0.90 compared to the 0.67 of Seksaria. Because the values are reasonably close, it seems safe to assume that the exponent for T I D is the same for the dispersion of liquids as for the suspension of solids, at least until further experimental data are taken. The last dimensionless group on the right-hand side of eq 8 is the product of the reciprocals of the Galileo and the Bond numbers. The former ( N G a ) is proportional to the ratio of inertial times gravitational force to the square of the viscous force in the system. It has previously found application in studies on circulation in baths of viscous liquids (Boucher and Alves, 1959; Kruszewski, 1957). The Bond number is proportional to the ratio of gravitational to interfacial tension forces prevailing in the system. It 1 7
c ) flat-blade turbine
d ) curved-blade turbine
e ) disk turbine
Radial flow impellers used
Figure 2. Five types of impellers for which eq 8-10 have been developed.
This same expression was used by Laity and Treybal (1957) in correlating power consumption in agitated liquid-liquid systems in baffled vessels, together with the Vermeulen et al. (1955) expression for mean viscosity, which is PM =
Pc
(1+-)
(7)
In the present work, Nm, was the dependent variable, whereas H, T, D, Ap, pM, pM, u, and q5 were independent variables; pM and pM are given by eq 6 and 7 , respectively. The pooled 481 runs provided 16 useable sets of data, corresponding to 5 different impellers in 4 different locations (including 2 impellers on a shaft) and with HIT assuming values of 1, and 3/2 for the centrally mounted flat-blade turbine. For all the other impellers H I T was
1 . 1
80 Ind. Eng. Chem. Res. Vol. 26, No. 1, 1987 /’
-I
,
,.$
fi
L
a / /
5
*
L_pp~_p__-2-iL
~
I -
i---
A
*/’
I d
--
1
, I
^ ^
’ k P
1:
s
‘1,w
Figure 3. Application of eq 8 to the published data of Skelland and Seksaria (1978).
7 i
c
is-
Figure 4. Application of eq 8 to the published data of van Heuven and Beek (1971).
has previously been used in work on the atomization of liquids into droplets (Boucher and Alves, 1959; Richardson, 1953). The group on the left-hand side of eq 8 is the Froude number, proportional to the ratio of inertial to gravitational forces. These considerations enable eq 8 to be expressed in the abbreviated form
Expansion of eq 8 yields the expression LI
g0.42~p0.42pM0.08~0.04~0.05
(10) D0’71PM0’54
Comparison between eq 10 and Zwietering’srelationship for solid-liquid suspensions in eq 4 is consistent with the statement by van Heuven and Beek (1971) to the effect that “there are no good reasons to suppose that a great difference exists between the mechanism of “complete dispersion” (liquid-liquid) and “complete suspension” (solid-liquid) ”. Equation 10 also compares closely in many respects with eq 2 for liquid-liquid dispersion using six-bladed disk turbine impellers, as given by van Heuven and Beek (1971). In their work, TID was constant at 3.33 and was therefore a part of their coefficient 3.28. Figures 3, 4,and 5 show plots of Nmin,calcd vs. Nmin,sxptl for the data sets provided by Skelland and Seksaria, van Heuven and Beek, and the present work, respectively. It is perhaps cautionary to note that the equation obtained by application of the standard statistical techniques used here to the 481 data points depends on the form assumed for correlation. Thus, if we assume “in
= CpDbApCpL,djldecTf@pgpMh
(11)
application of the BMDP program 9R gives Nmin= ~ ~ D b ~ p 0 . 4 3 ~ c 0 . 0 8 p d 0 . 0 7 ( r 0 . 0 1 ~ 0 . 1 0 p M - 0 . 5(12) 6 However, if pc and p d in eq 11are replaced by the single quantity pM,the result from program 9R becomes
Nmin=
C~DbAp0.41pM0.05(r0.06~0.09pM-0.68
(13)
The correlation coefficients for eq 12 and 13 are 0.952 and 0.949, respectively. (These values exceed that for eq 8 simply because more variables are present in eq 12 and 13 than the five independent group variables in eq 8.) One
.,.
1 \ILXP (s-
30
)
Figure 5. Application of eq 8 to the data obtained in the present work.
may note the changes in exponent on and on pM in eq 12 and 13 resulting from replacement of pc and & by pM. Scale-up. The criterion of equal power input per unit volume on large and small scales of operation has long been advocated for duplicating effects on the two scales of mixing. This proposition is examined below for the cases of full geometric similarity and of variable TID at constant W I D , H I T , BIT, and 4, respectively. (a) Full Geometric Similarity on Two Scales of Operation. The present work was in the region of NRe > lo4,denoting turbulence throughout. In this case, with full baffling, N p is not a function of NRe(Skelland, 1967). Then P/I@D5pMis constant for a given system, and since V is proportional to D3, P WD5 D3 D3 For constant physical properties, TID, and 4, eq 10 shows that Nmin= ( c ~ n s t a n t ) D ~ , ’Substituting, ~. P - 0-0.13 V Evidently the power consumption per unit volume decreases with increasing size of the apparatus to obtain the minimum impeller speed for complete dispersion. Clearly
P -
v
=-a-
Ind. Eng. Chem. Res. Vol. 26, No. 1, 1987 81 the "rule" of equal power input per unit volume is in error in the safe direction. For constant physical properties, full geometric similarity, and constant 4, eq 10 gives the following for scales 1 and 2: 0.71
(b) Variable T / D on Two Scales of Operation. Equation 10 may be written in the following form for constant physical properties and 4:
Then
Values of 3a - 2.87 for each set of data appear in Table 111. Suppose that scale-up is to be accomplished in a large vessel of fixed diameter T. The negative nature of 301 2.87 in the case of single propellers and double curvedblade turbines means that power input per unit volume decreases with decreasing D for these impellers. The opposite conclusion holds for the pitched-blade turbines, flat-blade turbines, single-curved-blade turbines, and the disk turbine. I t is therefore concluded that, when using pitched-blade turbines, flat-blade turbines, singlecurved-blade turbines, or disk turbines, it is more economical to obtain complete dispersion with large impellers rotating a t low speeds. The reverse is true for the other impellers featured in this investigation.
Conclusions The present study is believed to be an advance on the earlier work by Skelland and Seksaria (1978) in the following ways: A new correlating form is used, involving the composite properties pM and pM. Results from three different sources, van Heuven and Beek (1971),Skelland and Seksaria (1978), and the present work, are combined into an overall correlation. This has resulted in a 2.47-fold increase in the number of data points, from 195 to 481. Six new liquid-liquid systems are introduced-four from van Heuven and Beek and two from this study-to give a total of 11 systems. This broader base has permitted a better distribution of dependencies upon physical properties than in the previous study. Three new variables-H, T , and @-have been included in the experimental data. Observations for a fifth type of impeller-the six-bladed disk turbine used by van Heuven and Beek-are now included in the data collection. Improved statistical correlation has resulted from the use of the BMDP Statistical Software program. Nevertheless, the qualitative scale-up conclusions remain essentially the same, as do the observations regarding choice of impeller and its location, the types of mixing
phenomena that occur, and which phase becomes dispersed.
Acknowledgment We are grateful to L. Forney for helpful discussion. Portions of this work were supported by National Science Foundation Grants CPE 80-19617 and CPE-8203872.
Nomenclature E = baffle width, m C, Co, C' = constants-"shape factors" D = impeller diameter, m D, = particle diameter, m g = acceleration due to gravity, m/sz H = height of liquid in the vessel, m NBo= Bond number, D2gAp/u NFr = Froude number, D p p M / g A p N,, = Galileo number, D 3 p ~ A p / h M 2 Nmin= minimum rotational speed of impeller for complete liquid-liquid dispersion in agitated, baffled vessels without regard to uniformity, rev/s P = power input to the system, W R = weight fraction of solids 5" = tank diameter, m V = volume of total liquid, m3 W = width of impeller blade, m Greek Symbols C Y , a0,CY' = constants K ~& , = viscosities of continuous and disperse phases, N s/m2 pM = defined by eq 7, N s/m2 pc, Pd = densities of continuous and disperse phases, kg/m3 PM = 4Pd + - 4)Pp kg/m3 AP = b c - Pdl, kg/m u = interfacial tension, N/m 4 = volume fraction of disperse phase Literature Cited Boucher, D. F.; Alves, G. E. Chem. Eng. Prog. 1959,55 (9), 55-64. "International Critical Tables", 1st ed.; McGraw Hill: New York, 1928; Vol 4. Kruszewski, S. J. SOC.Glass Technol. 1957, 41, 259. Laity, D. S.; Treybal, R. E. AZChE J . 1957, 3, 176-180. Moeti, L. M. M.S. Thesis, Georgia Institute of Technology, Atlanta, GA, 1984. Nagata, S. Trans. SOC.Chem. Eng. Jpn. 1960, 8, 43. Pavlushenko, I. S.; Kostin, N. M.; Matveev, S.F. Zh.Prikl. Khim. (Leningrad) 1957, 30, 1160. Quinn, J. A.; Sigloh, D. B. Can. J . Chem. Eng. 1963, 41, 15-18. Ramsay, G. G. M.S. Thesis, Georgia Institute of Technology, Atlanta, GA, 1984. Richardson, E. G. "Flow Properties of Disperse Systems"; Hermans, J. J., Ed.; Interscience: New York, 1953; Chapter VI. Selker, A. H.; Sleicher, C. A., Jr. Can. J . Chem. Eng. 1965,43, 298. Skelland, A. H. P. "Non-Newtonian Flow and Heat Transfer"; Wiley: New York, 1967; pp 311-312. Skelland, A. H. P.; Seksaria, R. Ind. Eng. Chem. Process Des. Deu. 1978, 17, 56. van Heuven, J. W.; Beek, W. J. Solvent Extr., Proc. Int. Solvent Extr. Conf., 1971, 1971 Paper 51, 70. Vermeulen, T.; Williams, G. M.; Langlois, G. E. Chem. Eng. Prog. 1955,51, 85F-94F. Zwietering, T. N. Chem. Eng. Sci. 1958, 8, 244.
Received for review January 6 , 1984 Revised manuscript received January 14, 1985 Accepted March 13, 1985
