56
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 1, 1978
T(ZX*idmixt,7(ZX*i3)mjxt osmotic pressure of mixed solute aqueous solution corresponding to total mole fraction of all ions in solution, in solution phase 2 and solution phase 3, respectively, atm
Literature Cited Agrawal, J. p., Sourirajan, S., lnd. Eng. Chem. Process Des. Dev., 9, 12 (1970). Glueckauf, E.,Proceedings, First International Symposium on Water Desalination, Washington, D.C., i965. (Published by U S . Department of the Interior, Office of Saline Water, Washington, D.C., Vol. 1, pp 143-156, 1965). Hodgson, T.D., Desalination, 8, 99 (1970). Hoffer, E., Kedem, O., hd. Eng. Chem. Process Des. Dev., 11, 221 (1972). Lonsdale, H. K., Pusch, W., Walch, A., J. Chem. Soc., Faraday Trans. 1, 71,501
(1975). Matsuura, T..Pageau, L.,Sourirajan, S.,J. A w l . PoJym. Sei., 19, 179 (1975). Matsuura, T., Sourirajan, S.,J. Appl. Po/ym. Sci., 17, 1043 (1973). . 16,3165 (1972). Pageau, L., Sourirajan, s., J. ~ p p ~~o ./ y msci., Parsons, R.. “Handbook of ElectrochemicalConstants“, Table 73, Bvtterwofths, 1959. Rangarajan, R., Matsuura, T.,Goodhue, E. C., Sourirajan, S..Ind. Eng. Chem. Process Des. Dev., 15, 529 (1976). Sourirajan, S.,“Reverse Osmosis”, (a) Chapter 1. (b) Chapter 3, (c) Chapter 6, and (d) Appendix, Academic Press, New York, N.Y., 1970.
Received for review January 17,1977 Accepted July 27, 1977
Issued as NRC No. 16333.
Minimum Impeller Speeds for Liquid-Liquid Dispersion in Baffled Vessels A. H. P. Skelland” and R. Seksaria Chemical Engineering Department, University of Kentucky, Lexington, Kentucky 40506
Empirical correlations are developed to predict the minimum impeller speeds required for two-phase liquid-liquid dispersion in baffled vessels. Variables include size, location and form of impeller, and fluid properties in five equal-volume, binary liquid systems. The minimum impeller speeds for complete dispersion were first correlated only in terms of quantities actually varied, as follows: N = C 0 ~ ~ ~ c 1 ’ 9 ~ ~ ~ 1 ’ 9 where u 0 ~ 3Co ~ and p 0 a. ~ 2depend 5, upon the type of impeller and its location. Expressions like this correlated 195 results with an average deviation of 10.62 % . Equations of the form D1’2N/g1’2= Cl( TlD)a1(~cl~d)”9(~plpc)~ o .r2l5D( ~ p , gwere ) ~ , ~next obtained from dimensional analysis and these also correlated the same results, showing an average deviation of 10.17 YO, although T and g were constants in this work. Six distinct types of mixing phenomena were observed in the systems investigated. Consideration is given to impeller selection and location and to some aspects of scale-up.
Mixer-settlers are among the most widely used forms of industrial contactor in liquid-liquid extraction systems. Normally, impellers are used to disperse one immiscible liquid in another. At equilibrium, the solute distributes itself between the two liquids in a ratio dependent on the distribution coefficient. In addition to mass transfer coefficients and interfacial area, the minimum mixing speed for complete dispersion of one liquid in the other should be known to enable efficient design of the mixer. The present study shows that impeller speeds up to 1000 rpm are in some cases insufficient to ensure complete dispersion, indicating the need for research on such systems. Only one previous paper considered the minimum impeller speed needed for complete dispersion. Nagata (1950) performed a limited study on liquid-liquid systems in an unbaffled, flat-bottpmed vessel, using a centrally mounted, fourbladed flat-blade turbine agitator with a T/D of 3 and a blade width of 0.06T. He obtained the following empirical correlation
):(
N = 6D-2/3
1/9
pc - pd ( 7) 0.26
Quinn and Sigloh (1963) worked in a related field, dealing with phase inversion in mixing immiscible liquids. In most of their water-organic systems phase inversion occurred a t speeds of two to three times the minimum mixing speed for equal-volume fractions of the two liquids. In the present study no phase inversion was observed when increasing speeds be-
yond the minimum mixing value. However, inversion occurred before complete mixing in some systems. Selker and Sleicher (1965) studied factors affecting the dispersion of two immiscible liquids. The variables investigated were viscosities, densities, speed of mixing, manner of initiating the dispersion, and the materials of construction of the mixing apparatus. They concluded that the range of volume fraction within which either of two immiscible liquids could be continuous depended mainly on the viscosity ratio of the liquids, and the phase that was continuous depended on the mixing procedure. Laity and Treybal(l957) examined the power-input characteristics for two-phase liquid agitation in the absence of an air-liquid interface. Nagata (1950) reported that the minimum mixing speed is independent of interfacial tension, whereas Johnstone and Thring (1957) note that, for low viscosities and low density differences, the power requirement at a given speed is a function of interfacial tension. In this work it has been found that the minimum mixing speed depends on interfacial tension. A different but related phenomenon is the minimum impeller s p e d needed to suspend solid particles off the vessel bottom in an agitated slurry. Zwietering (1958) studied sand and sodium chloride suspensions in baffled vessels. The impellers in his experiments included turbines, paddles, propellers, and vaned disks. He correlated his data by the equation T 0 ’ g 0 . 4 5 (~ ~pc)0.45 f i c 0,lDp0.2(100R)0.13 (2) N = C’ (6) D0.85p,0.55
01978 American Chemical Society 0019-7882/78/1117-0056$01,00/0
Ind. Eng. Chem.
Table 11. Apparatus Dimensions
Table I. Fluid Properties at 25 "C Interfacial Dynamic tension Density, viscosity, with water, kg/m3 Ns/mz N/m
Fluid 5-cSt Dow Corning 200 Fluid 10-cSt Dow Corning 200 Fluid 15-cSt Dow Corning 200 Fluid Benzaldehyde Ethyl acetate Water
920 940
0.0046 0.0094
0.0425 0.0435
948.3
0.0143
0.0437
1041 894
0.0014 0.00046
0.0145 0.00627
1000
0.0010
-
Shaft
Baffle
Side View
@ Impeller
Bottm Vmm
Figure 1. Schematic diagram of experimental apparatus.
Pavlushenko (1957) made a similar study in unbaffled vessels, using three-bladed, square-pitch propellers for sand and iron suspensions, and correlated his results by Pc
Process Des. Dev.,Vol. 17, No. 1, 1978 57
(3)
The conflicting directional effect of pLcindicated by these two expressions is noteworthy.
Experimental Apparatus and Procedure Selection of Fluids. The main objective of this work was t o study the minimum impeller speed for dispersion in twophase liquid systems. T h e fluid properties investigated were viscosity, density, and interfacial tension. T h e liquids selected were water, benzaldehyde, ethyl acetate,' and three Dow Corning 200 silicone fluids of different viscosities. The Dow Corning 200 fluids are clear dimethyl siloxanes with low vapor pressures and relatively flat viscosity-temperature curves. Water was common to all the five systems studied. T h e 15-cSt Dow Corning 200 fluid was made by blending 45% by weight of the 10-cSt and 55% by weight of the 20-cSt Dow Corning 200 fluid. This corresponds to 45.39% and 54.61% by volume of the 10 and 20-cSt fluids, respectively. Interfacial tension was measured using the Fischer Surface Tensiometer Model 20. The force necessary to pull a platinum-iridium du Nuoy ring through the liquid-liquid interface was measured and converted into interfacial tension. All runs were made a t 25 "C and the fluid properties a t this temperature are reported in Table I. Apparatus. Figure 1 shows a sketch of the apparatus used. A 0.01-m3 cylindrical, flat-bottomed glass jar was used to
Internal diameter of vessel Liquid height in vessel Height of vessel Diameter of shaft Baffle length Baffle width Baffle thickness Length of baffle immersed in the liquid from airliquid interface Volume fraction of organic liquid, 4
0.2135 m 0.2135 m 0.2500 m 0.0140 m 0.2300 m 0,0190 m 0.0025 m 0.1930 m 0.50
permit visual observation. Four baffles, placed radially a t 90" intervals, were used to prevent vortex formation. Two conductivity electrodes made of soldered wire were placed 0.01 m apart in the liquid and connected to a conductivity monitor for continuous-phase identification. Mixing was effected by an impeller on the vertically centered shaft, rotated by a motor drive. Table I1 gives the apparatus dimensions. The shaft, impellers, and baffles were all made of 316 stainless steel. T h e Experimental Agitator Model ELB manufactured by Bench Scale Equipment Co. was used for mixing the liquids. It was equipped with a %-hp drive motor, and provided an infinitely variable output speed of 0 to 20 rps. The speed control dial was calibrated directly in rps using a stroboscope. T h e top of the vessel was covered with aluminum foil to prevent contamination of the liquid. Four types of impellers, each available in three different sizes, were used. Their characteristics are described below. 1. Propellers. Propellers are high-speed impellers of the axial-flow type. Three-bladed, square-pitched, downwardthrusting propellers with approximately 50% blade area were used, having diameters of 0.1,0.075, and 0.06 m. 2. Pitched-Blade Turbines. These impellers have performance characteristics somewhat similar to those of propellers. The pitched-blade turbines used had six blades at 45" from the vertical, with projected width nominally of the diameter, and were downward-thrusting. Diameters of 0.1, 0.075, and 0.062 m were used. 3. Flat-Blade Turbines. Also called straight-blade turbines, these impellers discharge radially, deriving suction from both top and bottom. Six-bladed turbines with width equal to of the diameter were used. The diameters were 0.106, 0.078, and 0.065 m. 4. Curved-Blade Turbines. Also called "backswept" or "retreating-blade" turbines, the blades of these impellers curve away from the direction of rotation. The curved-blade turbines used were six-bladed, with blade width nominally l/S of the diameter. These impellers were 0.102,0.076, and 0.063 m in diameter. Impellers 1 , 2 , 3 ,and 4 above closely resemble those shown in Figure 10.6a, c, d, and b, respectively, in Treybal (1963, p 406). The three sizes of each impeller type were geometrically similar. Operational Procedure. Before filling with liquids, all equipment was washed with detergent, rinsed with hot water, and air-dried. Two immiscible liquids in equal volumetric proportions were put into the vessel to a total height equal to its diameter. Baffles were mounted as shown in Figure 1. The impeller was placed a t the desired location of H/4, H / 2 , or 3H/4 from the bottom of the vessel and agitation started. Initially, speed was increased slowly in increments of 1.33rps, starting with zero. Some time was allowed after each speed increment to enable the system to attain its new steady-state condition. This normally varied from 100 to 400 s and could be visually determined. As the system approached a completely dispersed state, speed increments were gradually decreased to about 0.167 rps. The minimum mixing speed was
58
Ind. Eng. Chem. Process Des. Dev., Vol. 17,No. 1, 1978
recorded and the liquids were then allowed to separate completely. Runs were repeated until identical results were obtained. This usually required only three or four runs, due to good reproducibility of the data in the systems investigated. Minimum Mixing Speed and Reproducibility. The minimum mixing speed of the impeller was defined as the rotational speed just sufficient to completely disperse one liquid in the other, so that no clear liquid was observed either at the top or the bottom of the mixing vessel. In some cases clear liquid pockets of approximately 1X 10-6 m3 to 5 X 10-6 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 25-100% 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, N , and is not necessarily the same as that required for a homogeneous dispersion. More nearly homogeneous dispersions may have occurred at speeds higher than the minimum mixing speed, in accordance with the findings of Pavlushenko (1957) and Zwietering (1958). The reproducibility of the experimental N values was good, ranging from 1.8%at the highest N to 8.8%at the lowest N . Repeated values of N for a given system did not vary by more than f0.167 rps at low N’s and k0.333 rps at high N’s. This was observed on consecutive runs as well as on runs repeated after a considerable lapse of time. Techniques for Identifying Continuous and Disperse Phases. The techniques used for the identification of continuous and dispersed phases are as described by Quinn and Sigloh (1963). Sparkling droplets were observed in watercontinuous dispersions and relatively dull droplets in organic-continuous dispersions. The continuous phase could also be ascertained by observing the settling of the dispersed liquids. In a water-continuous system droplets move freely in the water phase, whereas they continually coalesce in the organic phase. The reverse phenomenon is observed for organiccontinuous dispersions. This method is explained in detail by Selker and Sleicher (1965). However, difficulty is encountered in the first method in emulsion-like dispersions observed mainly in the water-ethyl acetate systems. The second method tells which phase is continuous only when mixing is stopped. A more reliable method used two electrodes placed 0.01 m apart in the liquid. A conductivity meter continuously monitored the conductivity of the liquid. The organic liquids used here have electrical conductivities close to zero, whereas tap water has a much higher conductivity. These corresponded to a zero and full-scale reading, respectively, on the conductivity monitor (the meter had a full-scale reading of 0.01 mho/m for a cell constant of unity). A combination of all three methods was used for determining the continuous phase and for showing when phase inversion occurred. The phase which proved to be continuous is noted in the detailed tabulations for each run which are given by Seksaria (1976).
Results and Discussion Types of Mixing. Several different types of mixing phenomena were observed. Henceforth, LID will refer to a lighter phase dispersed in a denser continuous phase and DIL will refer to a denser phase dispersed in a lighter continuous phase. The different types of mixing were broadly categorized as follows. Type 1. This phenomenon was always observed at the impeller location of H/4. A t low rotational speeds, an LID
dispersion formed in the vicinity of the impeller. As speed increased this layer grew upward into the clear lighter liquid until a well mixed LID system was formed. Type 2. Normally, when one impeller was placed midway in each phase a t low speeds, LID and DIL dispersions were observed in the vicinity of the impellers located at HI4 and 3Hl4, respectively. As speed increased three distinct layers were visible, namely (i) an LID dispersion in the lower portion of the vessel, (ii) large chunks of a DIL dispersion in the continuous denser phase in the middle of the vessel, and (iiij a DIL dispersion in the top portion of the vessel. The middle layer was formed by the shearing of the top layer into the lower layer. As mixing time increased, the concentration of droplets of the lighter phase increased in the lower layer and the volume fraction of the denser liquid increased in the middle layer. Eventually the large chunks of DIL in the middle layer inverted to LID and then the lower LID layer grew upward until the liquids mixed completely. This inversion phenomenon, although observed less frequently, also occurred in a more pronounced manner when one impeller was located at HI2 or 3Hl4. In these cases, at rotational speeds close to the minimum mixing speed the bottom quarter of the vessel consisted of clear, denser liquid whereas the rest of the system was a DIL dispersion. Then, droplets of the lighter phase entered this clear denser phase and their concentration continued to increase as mixing time increased. the system then exhibited the three-layer phenomenon described above, inverted, and formed a well mixed LID system. Normally mixing times from 10 to 20 min were required to complete this development. Type 3. A similar type of mixing was also observed in systems where two impellers, one midway in each phase, were used. At speeds close to the minimum mixing speed, the lower and upper halves of the system were LID and DIL dispersions, respectively. Droplets of DIL were sheared into the lower phase but were not large enough to form the distinct third layer observed in Type 2 mixes. As mixing time increased, the concentration of droplets increased in the denser phase and the volume fraction of the denser phase increased in the upper half of the vessel until the upper phase inverted from a DIL to a LID dispersion. Type 4. This type of mixing, which occurred only at impeller locations of HI2 and 3Hl4, started with a DIL dispersion in the vicinity of the impeller at low speeds. This DIL layer grew throughout the liquid mass until a well mixed system was attained at higher speeds of rotation. Type 5. This type of mixing, observed only at impeller locations of Hl2, was analogous to Type 4 but the mixture was of LID form throughout the development. Type 6. This type of mixing was observed only when two impellers, one midway in each phase, were used. The mixing phenomenon was identical with Type 2 except that, instead of the chunks in the middle layer inverting from DIL to LID, the lower layer inverted from LID to DIL. The middle layer also homogenized to DIL, forming a well mixed system. Type 7. A t times, the air-liquid interface was so violently agitated by large eddies near the interface that splashing occurred well before the system could be thoroughly mixed. This happened at impeller locations 3Hl4 and the minimum mixing speed could not be determined in such cases. Some postscripts were used to further qualify each type of mixing. (A) The air-liquid interface was not agitated and no air bubbles entered the system. (B) The air-liquid interface was slightly agitated but no air bubbles entered the system. (C) The air-liquid interface was violently agitated, sometimes to the extent of splashing. In some instances a few small air bubbles also entered the system. They did not, however, alter the type of mixing phenomenon prevalent or the minimum mixing speed substantially.
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 1, 1978
Table 111. Correlations and Average Deviation between Ne,,and
Nored'
Eq 11 type correlations
Eq 4 type correlations Set no. 1 2 3 4
0.348148 0.151858 0.293388 0.044722
5 6 7
0.047382 0.063248 0.009150 0.031193
8
9 10 11 12
13 14 15 16
% av
co
Propeller -1.38272 10.71 13.19 -1.65355 11.80 -1.49329 8.53 -2.02317 Pitched-Blade Turbine 10.87 -2.15120 18.21 -1.91877 11.92 -2.69010 9.68 -1.97371 Flat-Blade Turbine -2.72474 6.93
*
Ql
dev
15.3244 9.9687 15.3149 5.2413
0.28272 0.55355 0.39329 0.92317
11.24 11.71 12.28 8.19
6.8231 6.2040 2.9873 3.3545
1.05120 0.81877 1.59010 0.87371
10.52 18.15 12.94 8.55
3.1780
1.62474
6.49
3.9956
0.88099
11.00
*
*
% av
c1
dev
a0
0.009103
*
*
*
0.036654
-1.98099 *
0.013292
Curved-Blade Turbine -2.56244 8.51
*
*
3.6108
1.46244
7.96
0.048231 0.066748
-1.90056 -1.64010
9.54 5.24
4.7152 4.2933
0.80056 0.54010
8.99 4.28
*
*
59
12.88
*
*
*
*
*
*
*
Overall 10.62 10.17 =Asterisksindicate there were insufficient data to correlate results. Sets 1, 5,9, 13: impeller midway in denser phase, Hl4. Sets 2, 6,10,14: impeller midway in lighter phase, 3H/4. Sets 3,7,11,15: impeller at organic-water interface,H12. Sets 4,8,12,16 two impellers, one midway in each phase, H/4,3H/4. Once the minimum mixing speed was determined, further increase in speed did not cause phase inversion. This is contrary to the findings for some systems by Quinn and Sigloh (1963). However, it is possible that all the liquid systems in this study behaved as the water-isobutyl alcohol system they describe. In the latter system, no inversion was observed with impeller speeds up to 500 rpm for a 50% mixture. Phase inversion can occur before the system is completely mixed. This effect was especially pronounced in Type 3 mixes. Detailed measurements and type of mixing for each run are tabulated elsewhere (Seksaria, 1976). Correlation of Experimental Results. Various techniques were used to find an empirical correlation to fit the experimental data, as described by Seksaria (1976). In the equations tried, N was the dependent variable, whereas D , p c , P d , u, p c , Pd, A p , and g were independent variables. There was a total of 16 different data sets, because four types of impeller were used in four different locations. However, complete data for only 13 sets were obtained, due to equipment limitations or splashing. The dependent variable was plotted against each independent variable on log-log coordinates with all the others held constant.. This established the exponential dependency from the slope of the leastsquares fit. Equations of the form
N = ~ , ~ ~ 0 p ~ l / 9 p ~ ~ l / 9 ~ 0 ~ 3 a p 0 . 2 5 (4) were first developed for the 13 data sets, where Co and cy0 are constant for a particular set but vary from one set to another. Table 111gives values of Co, cyo, and also the average deviation between experimental values of N and those predicted from eq 4 for each data set. An expression for the factors that affect the minimum impeller speed for liquid-liquid dispersion in baffled vessels is next assumed to be
N = f(D, T , W , H,B, ~
c Wd, r ~c
AP, 0, g)
(5)
where 4 is fixed and f is an unknown function. Dimensional analysis, followed by the assumption of an exponential form, yields
In this study
W I D = constant = 0
(7)
H I T = constant = i
(8)
BIT = constant = A
(9)
Substituting for W, H, and B from eq 7-9 and using the established exponents on p c , &, Ap and u from eq 4 gives
or
+ +
where CL= C0PirA6 and a 1 = cy y 6. The exponent a 1 was calculated from 00 in eq 4 as a l = -a0 - 1.1and C1 was evaluated as the average of its 15 values in each data set. Table I11 gives values of C1 and a1 in eq 11for each impeller type and location, together with the average deviations between experimental N values and those predicted from this correlation. The listed values of C1 and a1 apply only for HIT = 1,B I T = 0.09,@= 0.5, and for impellers geometrically similar to those used here. Figure 2 shows a plot of Npredicted vs. Nexperlmental for the correlating form given by eq 4. A closely similar plot results when the ordinate values are computed from eq 11 instead of eq 4. Greater deviations are observed in systems which exhibit mixing of Types 2 , 3 , and 6, i.e., the inversion phenomena. In these systems the speed required for inversion usually mixed the system thereafter. However, a much lower speed was then
60
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 1, 1978 30 sat
I
'
svmbol
I ' I ' I I I I I I
I
'
-
Total points * 195
I
I
I
I
I I I I I I I l I i 5
IO
I '
I
.
30
Nexpclrimentrrl"P'
Figure 2. Predicted N from eq 4 vs. Nexperimental.
adequate to maintain the well mixed state of the inverted system. This did not occur in systems with mixing of Types 1 , 4 , and 5. Choice of Impeller and Its Location. For a given impeller form and size, the N values at Hl4, Hl2, and 3Hl4 were determined for each liquid system. The highest and lowest of these were designated N h i g h and Nl,,,respectively. Repetition for each impeller size and liquid system gave additional corfor a total of 15 pairs of responding values of N h i g h and Nlow, such values for each form of impeller. Table IV shows the frequency with which these highest and lowest values of the minimum impeller speed occurred at a particular location for each impeller type. The table also shows the percentage of occasions in which the denser phase was continuous in the well mixed state. The highest values for minimum mixing speed for the propeller, pitched-blade turbine, flat-blade turbine, and curved-blade turbine usually occurred when the impeller was located at Hl2, Hl4, H / 4 , and H / 4 , respectively. Conversely, the lowest values of minimum mixing speed for these impellers usually occurred at 3H/4, H / 2 , H / 2 , and H/2, respectively (see Table IV). The impeller which provided the overall lowest minimum mixing speed for a complete set of data is the flat-blade turbine located at Hl2. Considering overall performance, the impellers exhibited increasing N in the order flat- curved-, and pitched-blade turbine, and propeller. The same sequence, allowing for some incompleteness in the data, prevailed when two impellers were
used. In general, radial flow impellers dispersed the liquids completely at a lower rotary speed compared to axial flow impellers and also produced smaller droplets. Table IY shows that, although the denser phase was always continuous when the impeller was located at Hl4, the lighter phase was continuous on only 51.63% of the occasions when the impeller was located at 3Hl4. (The latter figure was calculated from all completed runs at 3Hl4). This conflicts somewhat with the statement by Treybal (1963) that the liquid in which the impeller is located at rest will normally be the continuous phase. Scale-up. A long-standing industrial criterion for duplicating effects on two scales of mixing is to ensure that the power input per unit volume is the same in the large and small vessels. The considerations below examine the validity of this specification, first, when full geometric similarity prevails, and then when the ratio T I D may be varied at constant WID, HIT, and BIT. (a) Constant T / D in Scaled-up Operation. The power number is independent of Re in the turbulent region when mixing Newtonian systems in baffled vessels (Skelland, 1967). In the systems studied, ReM generally ranged between lo4and lo5,implying that the system was turbulent when complete dispersion was achieved. Under these conditions the power number, P / N 3 D 5 p ~is, constant for a given system and, since V is proportional to D3
P P N3D5 a-a-
(12) D3 0 3 Also, for constant physical properties, eq 11shows that N = (constant) D-I.'. Substituting
V
P a 0-1.3 -
(13)
V
This shows that power consumption per unit volume decreases as the apparatus size increases to achieve the minimum impeller speed for complete dispersion. Evidently the "rule" of equal power input per unit volume, although incorrect here, is on the safe side. When physical properties are constant and full geometric similarity prevails between two different-sized vessels, denoted by subscripts l and 2, eq l l shows that
(b) Variable T / D in Scaled-up Operation. To see how the power consumption per unit volume a t minimum mixing speed varies with the geometric ratio TID, eq 11is written as follows for constant physical properties:
Table IV. Impeller Behavior at Various Locations (All Data Are in Percentages) Type of impeller Propeller Pitched-blade turbine Flat-bladen turbine Curved-bladea turbine
Characteristic Highest speed Lowest speed Denser phase continuous Highest speed Lowest speed Denser phase continuous Highest speed Lowest speed Denser phase continuous Highest speed Lowest speed Denser phase continuous
"Based on incomplete data.
One impeller located at HI4 HI2 3Hl4 6.67%
13.33 100.00 100.00 0.00 100.00 100.00 0.00 100.00 86.67 0.00 100.00
93.33% 0.00 100.00 0.00 60.00 66.67 0.00 100.00 80.00 0.00
roo.00 100.00
0.00% 86.67 60.00 0.00 40.00
33.33 0.00 0.00 40.00
13.33 0.00 60.00
Two impellers; one midway in each phase
73.33 -
-
66.67 60.00
-
60.00
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 1, 1978 61
Table V. Values of 3 a l - 1.7 in Eq 1 6 O Set no.
3Ql - 1.7
Propeller
1 -0.85184 2 -0.03935 3 -0.52013 4 1.06951 Pitched-blade 5 1.45360 turbine 6 0.75631 7 3.07030 8 0.92113 Flat-blade 9 3.17422 turbine 10 * 11 0.94297 * 12 Curved-blade 13 2.68732 turbine 14 * 15 0.70168 16 -0.07970 Asterisks indicate insufficient data. Set numbers are as in Table 111.
Subscripts c = continuous d = disperse exp = experimental M = mean, using PM and p~ as in Treybal(l963, p 415) pred = predicted
so that
(16) The values of 3al
v.
C, C', CO,C1, Cz = constants D = impeller diameter, m D, = particle diameter, m g = acceleration due to gravity, m/s2 H = height of liquid in the vessel, m N = minimum rotational speed of impeller for complete liquid-liquid (or solid-liquid) dispersion in agitated vessels rev/s P = power input to the system, W R = weight fraction of solids, dimensionless Re = Reynolds number, D2Np/p, dimensionless T = tank diameter, m V = volume of total liquid, m3 W = width of impeller blade, m a , a', ao, P, ., E, 6,r , h = constants p = dynamic viscosity, Ns/m2 p = density, kg/m3 Ap = positive density difference between continuous and disperse phases, kg/m3 u = interfacial tension, N/m 4 = volume fraction of organic liquid, dimensionless
- 1.7 for each data set are given in Table
Consider scale-up to a fixed, large-vessel diameter, T . For single propellers and double curved-blade turbines 3Ul- 1.7 is negative; Le., power input per unit volume decreases with decreasing D. The reverse is true for the pitched-blade turbines, flat-blade turbines, and single curved-blade turbines. Evidently when using single propellers or double curved-blade turbines, it is more economical to achieve full dispersion with small impellers rotating at high speeds. The opposite conclusions hold for the other impellers studied here. Nomenclature a l = constant B = widthofbaffle,m
L i t e r a t u r e Cited Johnstone, R. E., Thring, M. W., "Pilot Plants, Models, and Scale-up Methods in Chemical Engineering," pp 84-85, McGraw-Hill, New York, N.Y., 1957. Laity, D. S., Treybal, R. E., A./.Ch.€. J., 3, 176-180 (1957). Nagata, S., Trans. Soc. Chem. Eng. (Jpn.),8, 43-58 (1950). Pavlushenko, I. S.. Kostin, N. M., Matveev, S. F.. Zh. Prikl. Khim., 30, 1160 (1957). Quinn, J. A., Sigloh, D.E.,Can. J. Chern. Eng., 41, 15-18 (1963). Seksaria, R., M.S. Thesis in Chemical Engineering, University of Kentucky, 1976. Selker, A. H., Sleicher, C. A., Jr., Can. J. Chem. Eng., 43, 298-301 (1965). Skelland, A. H. P., "Non-Newtonian Flow and Heat Transfer," pp 31 1-312, Wiley, New York N.Y., 1967. Treybal, R. E., "Liquid Extraction," 2nd ed, p 410, McGraw-Hill New York, N.Y., 1963. Zwietering, T. N., Chem. Eng. Sci., 8, 244-253 (1958).
Received for review January 25,1977 Accepted August 8,1977
This work was partially supported by National Science Foundation Grant No. ENG74-17286.
