Unified Correlations for the Prediction of Drop Size in Liquid−Liquid

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2682

Ind. Eng. Chem. Res. 1996, 35, 2682-2695

Unified Correlations for the Prediction of Drop Size in Liquid-Liquid Extraction Columns Arun Kumar and Stanley Hartland* Department of Chemical Engineering and Industrial Chemistry, Swiss Federal Institute of Technology, CH-8092 Zu¨ rich, Switzerland

Correlation of the average drop size in eight different types of extraction columns, namely, rotating disk, asymmetric rotating disk, Ku¨hni, Wirz-II, pulsed perforated-plate, Karr reciprocating-plate, packed, and spray columns, is presented. A unified correlation for mechanically agitated columns consists of a two-term additive model involving the ratio of interfacial tension to buoyancy forces at low agitation and the theory of isotropic turbulence at high agitation. This model has further been extended to another two-term correlation, wherein the compartment height and gravitational constant have been introduced to adequately allow for the effect of various variables on the drop size. Only in the case of mixer-settler type Wirz-II columns was the drop size found to be affected by the dispersed-phase holdup. The drop size in unagitated packed columns is determined simply by the ratio of interfacial tension to buoyancy forces, the constant of proportionality in the correlation being a function of the physical properties. The treatment accorded to spray columns is similar to that used for agitated columns, except that the drop size at high nozzle velocities (jetting region) is controlled by the ratio of interfacial to kinetic energies. Introduction

lowing equation can be derived (Hinze, 1955):

Knowledge of the drop size is of fundamental importance in the design of liquid-liquid extraction columns. It affects the dispersed-phase holdup, the residence time of the dispersed phase, and the allowable throughputs. Furthermore, together with the holdup, it determines the interfacial area available for mass transfer and affects both the continuous- and dispersed-phase masstransfer coefficients. It is therefore important to be able to predict the drop diameter as a function of the column geometry, agitation conditions, physical properties of the liquid-liquid system, and direction of mass transfer. In the absence of agitation or at low levels of agitation, the breakup of drops is controlled by the ratio of buoyancy to interfacial tension forces. A limiting value of the drop size under such conditions may be predicted from:

d32 ) C1(γ/∆Fg)1/2

(1)

where the constant, C1, is a function of the column geometry and mass transfer and may also depend upon the liquid-liquid system employed. For example, ChangKakoti et al. (1985) find C1 ) 1.3 for n-butyl alcohol drops dispersed in water in a rotating disk column, and Logsdail and Slater (1983) report a value of 0.92 for pulsed perforated-plate columns. For medium levels of agitation, there is no appropriate theory for quantifying the break-up effects, and empirical equations appear to predict acceptable results. Under turbulent conditions (high agitation levels), a drop will break up if the force exerted by motion in the continuous phase exceeds the cohesive forces due to interfacial tension and dispersed-phase viscosity. For low dispersed-phase viscosity, when the interfacial tension provides the dominant cohesive force, the fol* Author to whom correspondence should be addressed.

d32 ) C2-0.4(γ/Fc)0.6

(2)

A number of relationships have been proposed to describe the effect of dispersed-phase viscosity (Hinze, 1955; Sleicher, 1962; Davies, 1985). Calabrese et al. (1986) suggested the following implicit equation for stirred tanks:

d32 ) C2-0.4

()[ () γ Fc

0.6

1/2

Fc Fd

1 + C3

]

µd1/3d321/3 γ

3/5

(3)

which reduces to eq 2 for negligible viscous resistance to breakage. On the basis of arguments given by Hinze (1955), Calabrese et al. (1986) simplified eq 3 to give an explicit correlation:

( ) [ {( )

d32 ) C2-0.4

γ Fc

0.6

1/2

1 + C4

Fc Fd

}]

µd1/3DR1/3 γ

m1 3/5

(4)

The effect of coalescence on drop size is hard to quantify. If the column internals are preferentially wetted by the dispersed phase, coalescence of drops is greatly enhanced, but this situation can be avoided at the design stage. Near to the flooding point, drop to drop coalescence increases in importance as the holdup increases. Coalescence effects may be expressed in terms of the holdup as (see, e.g., Fischer (1973) and Rinco´n-Rubio et al. (1994))

d32 ∝ (1 + Cφφ)

(5)

The presence of a solute tends to lower the interfacial tension between two immiscible liquids. When mass transfer occurs from the continuous to the dispersed phase (c f d), the concentration of the solute in the draining film between two approaching drops will be

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Ind. Eng. Chem. Res., Vol. 35, No. 8, 1996 2683 Table 1. Pertinent Literature Concerning Prediction of Drop Size in Different Types of Columns Rotating Disk and Asymmetric Rotating Disk Columns Moderate agitation: Mı´sˇek (1963), Spa¨the and Weiss (1972), Mumford and Al-Hemiri (1974), Magiera and Z˙ adło (1977), Blazˇej et al. (1978), Marr (1978), Kumar and Hartland (1986b) High agitation: Mı´sˇek (1963), Kagan et al. (1964), Fischer (1971, 1973), Mersmann (1980), Jeffreys et al. (1981), Hartung and Weiss (1982), Kamath and Subba Rau (1985), Zhang et al. (1985), Kumar and Hartland (1986b) Entire range of agitation: Chang-Kakoti et al. (1985), Kumar and Hartland (1994b) Ku¨hni Columns Fischer (1973), Goldmann (1986), Kumar and Hartland (1994b) Wirz-II Columns Rinco´n-Rubio et al. (1994) Pulsed Perforated-Plate and Karr Reciprocating-Plate Columns Moderate agitation: Kagan et al. (1965), Assenov and Penchev (1971), Kubica and Zdunkiewicz (1977), Schmidt and Miller (1982), Reissinger (1985) High agitation: Mı´sˇek (1964), Miyauchi and Oya (1965), Angelino et al. (1967), Pilhofer (1978), Khemangkorn (1976), Maksimenko (1981), Boyadzhiev and Spassov (1982) Entire range of agitation: Ugarcˇic´ (1981), Kumar and Hartland (1986a), Sovova´ (1990), Kumar and Hartland (1994b) Packed Columns Lewis et al. (1951), Gayler and Pratt (1953), Streiff and Jancic (1984), Mac´kowiak and Billet (1986), Seibert and Fair (1988), Moore et al. (1989), Kumar and Hartland (1994a) Spray Columns Single-nozzle relationships: Hayworth and Treybal (1950), Scheele and Meister (1968a), de Chazal and Ryan (1971), Izard (1972), Kagan et al. (1973), Skelland and Huang (1979) Multiple-nozzle relationships: Horvath et al. (1978), Kumar and Hartland (1982), Kumar (1983), Kumar and Hartland (1984), Dalingaros et al. (1986), Kumar and Hartland (1994b)

lower than in the surrounding continuous liquid. For mass transfer in the opposite direction (d f c), the concentration will be correspondingly higher. The resulting gradients of interfacial tension will retard drainage and inhibit coalescence in the former case and accelerate drainage and coalescence in the latter case. Therefore, c f d transfer tends to produce smaller drops and d f c transfer larger drops. The magnitude of the interfacial tension gradients produced and hence their effect on coalescence rate depend upon the degree of mass transfer.

Literature Correlations A number of empirical and semiempirical equations for predicting drop size in different types of extraction columns are available. Table 1 gives a listing of published literature. In order to save space, only references are cited. Correlations for rotating disk and asymmetric rotating disk columns and pulsed perforatedplate and Karr reciprocating-plate columns have been given under single headings due to similarities within each group of columns. Although for the first group most of the equations have been derived by using data for rotating disk columns, they may be used to predict the drop size in asymmetric rotating disk columns. However, equations for pulsed columns are not always applicable to Karr columns, and vice versa. Correlations for the above two groups of columns have been further classified into categories of moderate agitation, high agitation, and entire agitation levels. Reviews and/ or tabulations of various correlating equations can be found in the papers by Chang-Kakoti et al. (1985) and Kumar and Hartland (1986b, 1994b) (rotating disk and asymmetric rotating disk columns), Kumar and Hartland (1994b) (Ku¨hni columns), Kumar and Hartland (1986a), Pietzsch and Blass (1987), Sovova´ (1990), Baird et al. (1994) and Kumar and Hartland (1994b) (pulsed perforated-plate and Karr reciprocating-plate columns), Kumar and Hartland (1994a) and Stevens (1994) (packed columns), and Kumar (1983) and Kumar and Hartland (1984, 1994b) (spray columns).

Data Bank Table 2 describes the drop-size data base for eight different types of extraction columns. The data for rotating disk and asymmetric rotating disk columns are treated together, since the hydrodynamic behavior of these two types of columns does not appreciably differ. Also listed in this table are the continuous- and dispersed-phase liquids, solutes employed, number of data points corresponding to no mass (or heat) transfer and for transfer in either direction, and ranges of physical properties, column geometry, and operating conditions. Under mass-transfer conditions, the values of the physical properties were calculated as explained elsewhere (Kumar and Hartland, 1995). The errors in the experimental values of drop size depend upon the techniques of observing and measuring and, more importantly, on the number of drops counted. The probable error in the measured values of drop size on the basis of a sample of 300 drops alone is about 10% (Kumar and Hartland, 1994b). Further details on the data for various types of columns are given below. Rotating Disk and Asymmetric Rotating Disk Columns. The data are taken from 24 different groups of investigators. In 30 liquid systems no mass transfer took place, but it occurred in the remaining 22 systems, either from the continuous to the dispersed phase (13 systems) or from the dispersed to the continuous phase (9 systems). Unfortunately, only a single source (Ha¨usler, 1985) furnishes data for asymmetric rotating disk columns. We have not used the drop sizes in both these columns derived by Mı´sˇek (1963) and Stangl and Mı´sˇek (1982), who assumed the terminal velocity of a single drop to be equal to the experimental value of the characteristic velocity. Ku 1 hni Columns. The drop sizes measured by seven different groups of investigators have been considered. Fischer (1973) and Kumar (1985) have determined the variation in the drop size with the stage number, whereas the data of Bailes et al. (1986b) correspond to the measurements taken within a particular stage. Goldmann (1986) performed the measurements by using a capillary suction method, the probe being installed in a calming section attached at the end of the working

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2684 Ind. Eng. Chem. Res., Vol. 35, No. 8, 1996 Table 2. Summary of Experimental Literature on Drop Size in Different Types of Columnsa Rotating Disk and Asymmetric Rotating Disk Columns data base: Kagan et al. (1964), Olney (1964), Fischer (1971, 1973), Weiss et al. (1974), Husung (1976), Spa¨the et al. (1976), Blazej et al. (1978), Sommeregger (1980), Wolschner (1980), Zhang et al. (1981), Al-Aswad (1982), Hartung and Weiss (1982), Kaplan (1982), Korchinsky et al. (1982, 1983), Zhu et al. (1984), Ha¨usler (1985), Kamath and Subba Rau (1985), Zhang et al. (1985), Bailes et al. (1986a), Korchinsky and Al-Husseini (1986), Korchinsky and Young (1986), Mao and Slater (1994) continuous-phase liquids: water, 20% aqueous glycerine, 40% aqueous glycerine, 60% aqueous glycerine, 40% aqueous ammonium sulfate, 60% aqueous sucrose, isobutyl alcohol, carbon tetrachloride, 1-methyl-2-pyrrolidone (NMP) + ethylene glycol dispersed-phase liquids: toluene, cumene, n-butyl alcohol, isobutyl alcohol, isoamyl alcohol, octanol, cyclohexanol, butyl acetate, diisopropyl ether, trichloroethylene, tetrachloroethylene, gasoline, kerosene, Clairsol 350, paraffin oil, white oil, 68% kerosene + 32% carbon tetrachloride, 37% kerosene + 63% carbon tetrachloride, water solutes: acetone, n-propyl alcohol, phenol, acetic acid, butyric acid, isobutyric acid, succinic acid number of data points: no mass transfer, 414; c f d transfer, 195; d f c transfer, 140 ranges of physical properties: Fc ) 835-1594 kg/m3, Fd ) 725-1614 kg/m3, µc ) 0.74-62.60 mPa s, µd ) 0.33-142.00 mPa s, γ ) 0.8-52.9 mN/m ranges of column dimensions: DC ) 50-450 mm, DR ) 30-225 mm, H ) 14-225 mm, DR/DC ) 0.46-0.80, H/DC ) 0.11-0.50, e or (DS/DC)2 ) 0.21-0.72 ranges of operating conditions: N ) 0-31.6 1/s, ReR ) 0-434 400,  ) 0-3.53 W/kg, φ ) 0.009-0.840 drop diameter: d32 ) 0.30-6.00 mm Ku¨hni Columns data base: Fischer (1973), Kumar (1985), Bailes et al. (1986a,b), Goldmann (1986), Ming and Yao (1986), Dongaonkar et al. (1991) continuous-phase liquids: water, aqueous glycerine dispersed-phase liquids: toluene, cyclohexane, o-xylene, cumene, n-butyl alcohol, methyl isobutyl ketone, butyl acetate, ethyl acetate, kerosene solutes: acetone, acetic acid, isobutyric acid, succinic acid number of data points: no mass transfer, 380; c f d transfer, 223; d f c transfer, 99 ranges of physical properties: Fc ) 986-1114 kg/m3, Fd ) 777-903 kg/m3, µc ) 1.00-4.85 mPa s, µd ) 0.49-3.93 mPa s, γ ) 0.8-41.5 mN/m ranges of column dimensions: DC ) 72-200 mm, DR ) 50-85 mm, H ) 50-90 mm, DR/DC ) 0.43-0.69, H/DC ) 0.45-0.69, e ) 0.10-1.00 ranges of operating conditions: N ) 0-5.8 1/s, ReR ) 0-42 000,  ) 0-0.75 W/kg, φ ) 0.008-0.828 drop diameter: d32 ) 0.33-8.54 mm Wirz-II Columns data base: Ha¨usler (1985), Rinco´n-Rubio (1992) continuous-phase liquid: water dispersed-phase liquids: toluene, n-butyl alcohol, n-octanol, cyclohexanone, butyl acetate, 56% butyl acetate + 44% diethyl carbonate, 32% butyl acetate + 68% diethyl carbonate solute: acetone number of data points: no mass transfer, 195; c f d transfer, 11; d f c transfer, 15 ranges of physical properties: Fc ) 984-998 kg/m3, Fd ) 828-950 kg/m3, µc ) 0.98-1.37 mPa s, µd ) 0.56-9.85 mPa s, γ ) 1.9-32.2 mN/m ranges of column dimensions: DC ) 150-152 mm, DR ) 48 mm, H ) 80 mm, DR/DC ) 0.31-0.32, H/DC ) 0.52-0.53, e ) 0.163 ranges of operating conditions: N ) 1.7-13.3 1/s, ReR ) 2800-30 200,  ) 1.05 × 10-3 -0.56 W/kg, φ ) 0.051-0.320 drop diameter: d32 ) 0.55-3.46 mm Pulsed Perforated-Plate Columns data base: Jones (1963), Kagan et al. (1965), Miyauchi and Oya (1965), Tutaeva and Kagan (1967), Assenov and Penchev (1971), Khemangkorn (1976), Elenkov et al. (1978), Pakdee-Patrakorn (1980), Toller (1981), Ugarcˇic´ (1981), Schmidt and Miller (1982), Zhu et al. (1982), Garg and Pratt (1983), Vassallo (1983), Aufderheide and Vogelpohl (1984), Reissinger (1985), Batey et al. (1986), Lorenz (1990) continuous-phase liquids: water, 2 M nitric acid, 3 M nitric acid dispersed-phase liquids: toluene, o-xylene, n-butyl alcohol, methyl isobutyl ketone, butyl acetate, carbon tetrachloride, kerosene, 10% tributyl phosphate + 90% kerosene, 20% tributyl phosphate + 80% kerosene, 30% tributyl phosphate + 70% kerosene, 20% D2EHPA + 80% kerosene solutes: acetone, o-nitrophenol, iodine number of data points: no mass transfer, 756; c f d transfer, 92; d f c transfer, 48 ranges of physical properties: Fc ) 986-1102 kg/m3, Fd ) 789-1595 kg/m3, µc ) 0.84-1.43 mPa s, µd ) 0.55-3.36 mPa s, γ ) 1.8-45.0 mN/m ranges of column dimensions: DC ) 25-215 mm, H ) 30-100 mm, D0 ) 1.6-6.0 mm, e ) 0.08-0.49 ranges of operating conditions: A ) 0-44.7 mm, f ) 0-4.25 Hz, Af ) 0-80.0 mm/s,  ) 0-5.11 W/kg, φ ) 0.002-0.481 drop diameter: d32 ) 0.23-4.61 mm Karr Reciprocating-Plate Columns data base: Lane (1971), Bensalem (1985), Shen et al. (1985) continuous-phase liquids: water, methyl isobutyl ketone, kerosene dispersed-phase liquids: toluene, kerosene, water solutes: acetone, butyric acid number of data points: no mass transfer, 102; c f d transfer, 20; d f c transfer, 49 ranges of physical properties: Fc ) 805-1000 kg/m3, Fd ) 801-1000 kg/m3, µc ) 0.82-1.07 mPa s, µd ) 0.57-1.72 mPa s, γ ) 8.7-34.3 mN/m ranges of column dimensions: DC ) 51-76 mm, H ) 25-29 mm, D0 ) 12.7-16.0 mm, e ) 0.55-0.58 ranges of operating conditions: A ) 8.4-45.0 mm, f ) 0.33-5.83 Hz, Af ) 8.4-69.4 mm/s,  ) 8.73 × 10-4-0.49 W/kg, φ ) 0.040-0.605 drop diameter: d32 ) 0.99-8.54 mm Packed Columns data base: Lewis et al. (1951), Gayler and Pratt (1953), Billet and Mac´kowiak (1980), Streiff and Jancic (1984), Bailes et al. (1986a), Mac´kowiak and Billet (1986), Seibert (1986), Moore et al. (1989), Nedungadi (1991), Leu (1995) continuous-phase liquids: water, 23% aqueous glycerine, 35% aqueous glycerine, sulfolane dispersed-phase liquids: heptane, isooctane, benzene, toluene, cumene, n-butyl alcohol, isobutyl alcohol, dibutyl carbitol, methyl isobutyl ketone, ethyl acetate, butyl acetate, dichlorodiethyl ether, carbon tetrachloride, dichloroethane, gasoline, kerosene solutes: acetone, isobutyric acid, succinic acid number of data points: no mass transfer, 312; c f d transfer, 52; d f c transfer, 18 ranges of physical properties: Fc ) 985-1258 kg/m3, Fd ) 682-1600 kg/m3, µc ) 0.92-8.35 mPa s, µd ) 0.41-4.70 mPa s, γ ) 0.8-46.5 mN/m ranges of column diameter and packing properties: DC ) 25-152 mm, dp ) 6.3-50.0 mm, ap ) 122-839 m2/m3, e ) 0.640-0.975 ranges of operating conditions: Vc ) 0.0-13.1 mm/s, Vd ) 0.7-20.6 mm/s, φ ) 0.058-0.667 drop diameter: d32 ) 0.89-7.92 mm

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Ind. Eng. Chem. Res., Vol. 35, No. 8, 1996 2685 Table 2 (Continued) Spray Columns data base: Garwin and Smith (1953), Henton (1967), Vedaiyan (1969), Ferrarini (1972), Hupfauf (1973), Horvath (1976), Bu¨hler (1977), Berger (1981), Kumar and Hartland (1982), Dalingaros et al. (1986), Lin and Ford (1988) continuous-phase liquids: water, amyl alcohol, carbon tetrachloride, 60% paraffin oil + 40% toluene dispersed-phase liquids: benzene, toluene, o-xylene, n-butyl alcohol, isoamyl alcohol, methyl isobutyl ketone, butyl acetate, amyl acetate, kerosene, Shell Sol, spindle oil, water solutes: acetone, acetic acid, succinic acid number of data points: no mass or heat transfer, 488; c f d transfer, 222; d f c transfer, 33 ranges of physical properties: Fc ) 817-1583 kg/m3, Fd ) 756-998 kg/m3, µc ) 0.45-4.10 mPa s, µd ) 0.33-3.39 mPa s, γ ) 1.5-44.3 mN/m ranges of column and nozzle diameters: DC ) 38-104 mm, DN ) 0.5-5.0 mm ranges of operating conditions: VN ) 9-472 mm/s, φ ) 0.007-0.739 drop diameter: d32 ) 1.54-9.25 mm a

Note: the ranges are given in italics when the values of a variable are not available for the entire set of data.

section. Ming and Yao (1986) and Dongaonkar et al. (1991) do not specify the measurement locations. The number of liquid systems corresponding to the cases of no mass transfer and c f d and d f c directions of mass transfer are 14, 5, and 2, respectively. The total number of data points is 702, including 580 data for which the variation in d32 with the stage number is known. Kumar and Hartland (1994b) show that the drop size decreases with the stage number (counted from the dispersed-phase inlet), albeit slowly. Wirz-II Columns. Data from only two sources are available. Almost identical stage geometry has been used in both the investigations. Ha¨usler (1985) carried out a limited number of experiments for a toluene (dispersed)-acetone (solute)-water (continuous) system for both directions of mass transfer. Rinco´n-Rubio’s (1992) study involves six liquid systems of low to medium interfacial tension in the absence of mass transfer and covers wide ranges of flow rates and agitation intensity. The total number of data points is 221, including 26 runs for both directions of mass transfer. Pulsed Perforated-Plate Columns. The experimental results for the three stable regions of operation, namely, the mixer-settler, dispersion, and emulsion regions, considered in this work are taken from 18 different sources. In 22 liquid systems, no mass transfer took place, but it occurred in the remaining 8, either from the continuous to the dispersed phase (six systems) or in the opposite direction (two systems). Karr Reciprocating-Plate Columns. Only three data sources could be found in the literature, which furnish data from 171 measurements. The number of liquid systems is seven, including two systems each for either direction of mass transfer. Packed Columns. The data are taken from 10 different experimental studies. It should be noted that all data are for the case when the continuous phase wets the packing. Furthermore, 33 runs corresponding to the random-packing data for which the packing size, dp, is less than the critical packing size, dp,cr, have been excluded. The critical packing size is calculated by using the equation suggested by Gayler et al. (1953). The random packings employed in the available data are Raschig rings, Pall rings, Berl saddles, and Intalox saddles. The ordered packings studied are Sulzer SMV, Montz-Pak B1-350, stacked Bialecki rings, and tube columns. Spray Columns. Results for 743 experimental measurements from 11 different published sources are considered. Only the data corresponding to nozzle velocities up to the critical nozzle velocity, predicted from the correlation of Hughmark (1967), are included. The total number of liquid systems is 25. Heat or mass transfer took place in 12 systems (nine and three

systems corresponding to c f d and d f c directions of transfer, respectively), but no transfer occurred in the remaining 13 systems. Correlation Mechanically Agitated Columns. As in a previous paper (Kumar and Hartland, 1995), the power dissipation per unit mass of the phases, , is selected as a common and most appropriate predictor variable. The methods for the estimation of  in various columns are summarized below. In rotary-agitated columns, the power dissipation per unit mass is given in terms of the power input per agitator, P, by

 ) 4P/(πDC2HFc)

(6)

in which P is obtainable from a relationship between the power number, NP, and the agitator Reynolds number, ReR. For rotating disk and asymmetric rotating disk columns, Kumar and Hartland (1995) gave the following equation:

NP )

[

]

1000 + 1.2ReRm2 C5 + C6 ReR 1000 + 3.2ReRm2

m3

(7)

The values of the parameters in this equation are:

Rotating disk columns: C5 ) 109.36, C6 ) 0.74, m2 ) 0.72, m3 ) 3.30 Asymmetric rotating disk columns: C5 ) 90.00, C6 ) 0.62, m2 ) 0.73, m3 ) 3.17 Fischer’s experimental data on power required to operate Ku¨hni columns can be depicted by the correlation recently proposed by Kumar and Hartland (1995):

NP ) 1.08 +

10.94 257.37 + ReR0.5 ReR1.5

(8)

For Wirz-II columns, NP becomes constant at 1.3 if ReR > 1000 (Kumar and Hartland, 1995). Table 2 shows that ReR varies between 2800 and 30 200 for the available data, so the power requirements may be calculated from

NP ) 1.3

(9)

The equation proposed by Hafez and Baird (1978) will be used for the calculation of  for both pulsed perforated-

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2686 Ind. Eng. Chem. Res., Vol. 35, No. 8, 1996 Table 3. Values of Parameters in Equation 11 and Error in Drop Size, d32, for Mechanically Agitated Columnsa CΨ

a

column type

no. of data

rotating disk/asymmetric rotating disk Ku¨hni Wirz-II pulsed/Karr all data (unified fit)

749 702 221 1067 2739

cfd

dfc

CΩ

CΠ

n

AARE, %

1

1.33

1.70

0.61

0.67

23.0

1 1 0.95 1

3.05 1 1.48 1.67

2.49 1.22 1.30 1.25

0.52 0.35 0.67 0.49

0.57 0b 0.50 0.38

26.8 16.6 19.7 26.1

For no mass transfer, CΨ ) 1. b Fixed value of e.

plate and Karr reciprocating-plate columns, which follows:

)

2π2(1 - e2) 3HC02e2

(Af)3

(10)

The value of the discharge coefficient, C0, is assumed to be 0.6. Kumar and Hartland (1994b) show that the effect of holdup on drop size in various agitated columns is negligible. Only in Wirz-II columns, in which each stage behaves as a mixer-settler, does the drop size depend upon holdup in accordance with eq 5 (Rinco´n-Rubio et al., 1994). If the effect of holdup for this type of column is also neglected, then a simple correlation, which is applicable over the entire range of agitation, can be inferred from eqs 1 and 2 as follows:

d32 ) CΨ en

/[[ (

1 γ CΩ ∆Fg

)] [ 0.5 2

+

1

()]

CΠ-0.4

γ Fc

0.6 2

]

1/2

(11)

where the parameter CΨ allows for the effect of mass transfer on drop size. In the equation above, the squares of the two resistances have been added, since this was found to give a better fit than when the linear resistances were simply added. It should be noted that the effect of µd on drop size has been neglected, because only for seven data from measurements carried out by Husung (1976) by using a paraffin oil (dispersed)-water (continuous) system in a rotating disk column the value of µd exceeds 10 mPa s beyond which µd becomes a relevant correlating variable (Arai et al., 1977). Furthermore, in the case of Ku¨hni columns, the effect of stage number is neglected, since the information on stage number is not available for all the data. The parameters in eq 11 were determined by using Marquardt’s (1963) algorithm. It is to be noted that the data for pulsed and Karr columns are treated together due to their similar hydrodynamic behavior. The value of CΨ was taken as unity for the case of no mass transfer for all types of columns. Its value for c f d and d f c directions of mass transfer together with those of CΩ and CΠ was obtained from the experimental data summarized in Table 2. For rotating disk, asymmetric rotating disk, and Ku¨hni columns, the data corresponding to c f d direction of mass transfer were found not to be appreciably different from those without mass transfer, so these two cases were treated together. In the case of Wirz-II columns, the data corresponding to mass-transfer runs are scanty. Therefore, optimization of CΨ was considered to be statistically unreliable. Table 3 lists the values of CΩ, CΠ, CΨ, and n in eq 11 for different columns. Since e was held constant in the case of Wirz-II columns, this was omitted from eq 11, so the index, n, is irrelevant.

The average absolute value of the relative error, AARE, is used to compare the predicted results with the experimental data. This is defined as follows:

AARE ) 1

NDP

∑ NDP i)1

|predicted value - experimental value| experimental value (12)

in which NDP is the number of data points. The values of AARE in d32 for the different extraction columns, listed in Table 3, show that eq 11 is somewhat unsatisfactory for Ku¨hni columns. The value of AARE for rotating disk and asymmetric rotating disk columns is also more than 20%, but the experimental data appear to be burdened with errors as explained by Kumar and Hartland (1994b). The drop size data for six types of columns have also been brought into a unified correlation, wherein single values of the parameters in eq 11 were determined on the basis of all data from 2739 measurements. Using this single set of values, shown in Table 3, eq 11 reproduces the data with an average error of 26.1%. This compares with an error of 22.2% for all data when the parameters in this equation were individually optimized for different columns. A convenient form of the unified correlation is

d32 ) CΨ e2/5

/[[ (

1

1 4 γ 3 ∆Fg

)] [ ()] 0.5 2

+

1 -0.4 γ  2 Fc

]

1/2

(13)

0.6 2

with CΨ ) 1 for c f d and no mass transfer and CΨ ) 1.64 for the d f c direction of mass transfer. For Ku¨hni columns, the variation in the drop size with the stage number is known for 580 runs. By use of data from these measurements, the following equation has been developed:

d32 ) CΨ e0.64S-0.10

/[[

1 γ 3.42 ∆Fg

( )] [ ()] 0.5 2

+ 1

0.65-0.4

γ Fc

]

0.6 2

1/2

(14)

The values of CΨ are unity for both c f d and no mass transfer and 3.20 for d f c mass transfer. The average error of prediction is 21.6%, so there is a definite improvement in the agreement between experimental and predicted values. As mentioned above, Rinco´n-Rubio et al. (1994) show that the drop size in Wirz-II columns does depend upon holdup. The fit of the data for this column has therefore been improved by modifying eq 11, wherein the holdup function given by eq 5 is included. The final shape

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Ind. Eng. Chem. Res., Vol. 35, No. 8, 1996 2687 Table 4. Values of Parameters in Equation 16 and Error in Drop Size, d32, for Mechanically Agitated Columnsa CΨ column type

no. of data

rotating disk/asymmetric rotating disk Ku¨hni Wirz pulsed/Karr all data (unified fit)

749

a

702 221 1067 2739

cfd

dfc

CΩ

n

n1

n2

AARE, %

1

1.29

2.54

0.97

CΠ

0.64

-0.45

-1.12

22.4

1 1 0.90 1

3.04 1 1.66 1.65

1.60 2.24 1.19 1.43

3.40 × 10-2 0.87 1.12 0.45

0.45 0b 0.41 0.45

-0.63 -0.47 -0.62 -0.57

-0.38 -1.24 -1.20 -0.96

22.4 16.1 18.1 25.5

For no mass transfer, CΨ ) 1. b Fixed value of e.

ous-phase turbulence be determined by the density difference, ∆F, rather than the density, Fc, i.e., Fc be replaced by ∆F in eq 16. Using this assumption, the equation for pulsed and Karr columns becomes

of the correlation is

/[[

d32 ) (1 + 1.78φ)

1 γ 0.98 ∆Fg

( )] [ ()] 0.5 2

+

1

0.27-0.4

γ Fc

]

1/2

0.6 2

(15)

CΨ e0.32

which agrees with the experimental data with an average error of 14.9%. Although eq 11 generally provides reasonable estimates of the drop size, it is possible to improve the agreement between experimental and predicted values if the effects of various predictor variables (particularly  and γ) on the drop size are properly allowed for. Introducing the compartment height, H, and gravitational constant, g, a dimensionless correlation for the drop size may be written as

d32 ) H

CΨ en 1 γ CΩ ∆FgH2

(

)

CΠ

[( )( ) ] [ ( ) ]  Fc g gγ

1/4 n1

Fc g H γ

1/2 n2

(16)

which contains two adjustable indexes, n1 and n2. Best estimates of the parameters in the above equation for different columns as well as for the unified fit, obtained by performing regression analyses, are presented in Table 4. The average error for all columns, when the parameters are individually optimized, is 20.2%. The corresponding figure for the unified fit is 25.5%. Equation 16 thus provides better estimates of the drop size as compared to eq 11, particularly for the Ku¨hni columns. A useful form of the unified fit is

d32 ) H

(

)

1/2

1

+

[( )( ) ] [ ( ) ]

2  Fc 5 g gγ

1/4 -3/5

(

1

)

1/2

+

[( )( ) ] [ ( ) ]

 ∆F 0.42 g gγ

1/4 -0.35

H

∆Fg γ

1/2 -1.15

(18)

The values of CΨ are 1, 0.92, and 1.67 for no mass transfer and c f d and d f c directions of mass transfer, respectively. Furthermore, the value of the average error now reduces to 16.1%. In the case of Ku¨hni columns, eq 16 may be modified to include the effect of stage number on drop size. The resulting correlation by using the relevant data is

H

) CΨ e0.56S-0.10

1 γ 2.13 ∆FgH2

(

1

)

1/2

+ 7.08×10

[( )( ) ] [ ( ) ]

-2

 Fc g gγ

1/4 -0.63

H

Fc g γ

1/2 -0.50

(19) with CΨ ) 1 for c f d and no transfer and CΨ ) 3.16 for d f c transfer. This equation reproduces the data from 580 measurements with an average error of 18.4%. As shown earlier, the data for Wirz-II columns can be better described if the effect of holdup is taken into account. The following extension of eq 16 is then valid:

d32 ) H

CΨ e2/5 1 3 γ 2 ∆FgH2

1 γ 1.55 ∆FgH2

d32

1

+ 1/2

d32 ) H

Fc g H γ

(1 + 2φ)

1/2 -1

(17)

the values of CΨ being unity for c f d and no mass transfer and 1.67 for the d f c case of mass transfer. The values of the average errors for different columns, listed in Table 4, are close to those obtained from the best available correlations suggested by Kumar and Hartland (1994b). However, it is possible to improve the correlation for pulsed perforated-plate and Karr reciprocating-plate columns. In these columns, the theory of isotropic turbulence (and its modification used above) is not quite suitable for describing the breakup of drops (Kumar and Hartland, 1986a, 1994b). However, a simple correction to this theory may be made by considering that the disruptive energy due to continu-

1 γ 2.22 ∆FgH2

(

)

1/2

1

+

[( )( ) ] [ ( ) ]

 Fc 0.88 g gγ

1/4 -0.47

H

Fc g γ

1/2 -1.32

(20) which reproduces the experimental data with an average error of 14.0%. Graphical comparisons between the experimental and predicted data are given below, wherein specific values of the parameters in eqs 11 and 16 for different columns are used. Figure 1 compares the experimental variation in d32 with N measured by Hartung and Weiss (1982) during transfer of n-propyl alcohol and acetic acid from trichloroethylene drops to the aqueous phase in a rotating disk column of 200-mm diameter with that predicted by eqs 11 and 16. In Figure 2, the variation

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2688 Ind. Eng. Chem. Res., Vol. 35, No. 8, 1996

Figure 1. Experimental variation in d32 with N for a trichloroethylene (dispersed)-water (continuous) system when n-propyl alcohol and acetic acid are transferred from the dispersed to the continuous phase measured by Hartung and Weiss (1982) in a 200mm diameter rotating disk column (O) compared with that predicted by various correlations. The curve numbers refer to the correlations given in Table 5.

Figure 2. Experimental variation in d32 with S for an o-xylene (dispersed)-water (continuous) system measured by Kumar (1985) in a 150-mm diameter Ku¨hni column compared with that predicted by various correlations. Data are as follows: N (1/s), symbol; 1.67, O; 2.33, b; 3.00, × ; 4.33, Q, 5.00, +. The curve numbers refer to the correlations given in Table 6. The letters a, b, c, d, and e refer to the data sets O, b, ×, Q, and +, respectively.

in d32 with S measured by Kumar (1985) for an o-xylene (dispersed)-water (continuous) system in a 150-mm diameter Ku¨hni column for five different values of the agitation speed, N, is shown. Since eqs 11 and 16 do not allow for the variation in the drop size with the stage number, the predicted data are represented by using eqs 14 and 19. Figure 3 demonstrates the application of eqs 11, 15, 16, and 20 to the data of Rinco´n-Rubio (1992) for n-butyl alcohol (dispersed)-water (continuous) and n-octanol (dispersed)-water (continuous) systems obtained in a Wirz-II column of 152-mm diameter. The plots from eqs 15 and 20 were developed by using the average value of the holdup for each data set. The experimental variation in d32 with Af for a pulsed

Figure 3. Experimental variation in d32 with N for n-butyl alcohol (dispersed)-water (continuous) (O) and n-octanol (dispersed)water (continuous) (b) systems measured by Rinco´n-Rubio (1992) in a 152-mm diameter Wirz-II column compared with that predicted by various correlations. The curve numbers refer to the correlations given in Table 7.

Figure 4. Experimental variation in d32 with Af for a butyl acetate (dispersed)-water (continuous) system measured by Aufderheide and Vogelpohl (1984) in a 80-mm diameter pulsed perforatedplate column with e ) 0.22 and D0 ) 2 mm (O) compared with that predicted by various correlations. The curve numbers refer to the correlations given in Table 8.

perforated-plate and a Karr reciprocating-plate column is shown in Figures 4 and 5, respectively. These figures also provide the variation predicted by eqs 11, 16, and 18, the last equation comparing most favorably with the experimental data. Nonmechanically Agitated Columns. The concept of power dissipation per unit mass is not relevant for these columns, since the values of , although calculable using suitable relationships, are relatively small. This was confirmed by using Ergun’s (1952) equation for packed columns, wherein the pressure drop was calculated only on the basis of continuous-phase flow. The values of  for 341 data points, for which continuousphase flow rates are known, vary between 0 and 7.38 × 10-3 W/kg. Similarly, power dissipation due to flow

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Ind. Eng. Chem. Res., Vol. 35, No. 8, 1996 2689

Figure 5. Experimental variation in d32 with Af for a kerosene (dispersed)-butyric acid (solute)-water (continuous) system when butyric acid is transferred from the continuous to the dispersed phase measured by Shen et al. (1985) in a 51-mm diameter Karr reciprocating-plate column (O) compared with that predicted by various correlations. The curve numbers refer to the correlations given in Table 8.

of the phases in spray columns may be neglected. Special analyses required for these columns are given below. The drop size in packed columns can be correlated by a correlation of the type presented in eq 1 (Mac´kowiak and Billet, 1986; Seibert and Fair, 1988; Kumar and Hartland, 1994a). The preconstant in this equation appears to be a function of the physical properties of the liquid systems (Mac´kowiak and Billet, 1986; Kumar and Hartland, 1994a). Moreover, holdup is not an important factor unless the column is operated near the flooding point (Mac´kowiak and Billet, 1986). In the present work, a correlation similar to that given by Kumar and Hartland (1994a) has been derived:

d32 ) 0.74CΨ

( ) ( ) ∆FFdγ Fw2γw

-0.12

γ ∆Fg

Figure 6. Experimental variation in d32 with Vd for an isooctane (dispersed)-water (continuous) system with four different ceramic rings at Vc ) 0.8 mm/s measured by Lewis et al. (1951) in a 51mm diameter packed column compared with that predicted by various correlations. Data are as follows: packing size and type, symbol; 9.5 mm Raschig rings, O; 12.7 mm Raschig rings, b; 19 mm Raschig rings, +; 12.7 mm Berl saddles, ×. The curve numbers refer to the correlations given in Table 9. The letters a, b, c, and d refer to the data sets O, b, +, and ×, respectively.

Harkins and Brown (1919) for single nozzles, an equation for the drop size is as follows:

d32 ) CΩ

6DNγ ∆Fg

1/3

(22)

At high nozzle velocities (jetting region), we balance the kinetic energy of liquid flowing from the nozzle with the interfacial energy of the drops formed on breakup of the jet:

d32 ) 12γ/(FdVN2)

1/2

(21)

in which Fw ) 998 kg/m3 is the density of water and γw ) 0.0728 N/m the surface tension of water at 20 °C. The values of CΨ are 1, 0.84, and 1.23 for no mass transfer and c f d and d f c directions of mass transfer, respectively. Equation 21 reproduces the drop size data with an average error of 15.7%. Figure 6 displays the experimental variation in d32 with Vd for an isooctane (dispersed)-water (continuous) system for four different packings measured by Lewis et al. (1951). The data are fairly scattered, and it is hard to identify the effect of Vd on d32. Equation 21, which does not include the effects of phase flow rates, is shown as a horizontal line. Equation 1 is not suitable for the estimation of limiting value of the drop size in spray columns, in which the drops are formed at multiple-nozzle or perforated-plate distributors. At extremely slow supply of the dispersed phase, the drop size is determined by the process of detachment at a nozzle tip (or at a perforation). For this kind of formation, described by

( )

(23)

A correlation covering the entire range of nozzle velocities can be derived from eqs 22 and 23 as follows:

d32 )

CΨ 1 6DNγ CΩ ∆Fg

( )

+ 1/3

1 12γ Cκ FdVN2

( )

(24)

On the basis of data from 743 measurements, the values of the parameters CΩ and Cκ, determined by using Marquardt’s (1969) algorithm, are 1 and 2.04, respectively. The data for the c f d direction of heat or mass transfer were found not to be appreciably different from those without transfer, so CΨ ) 1 for these two cases. For d f c transfer, the optimized value of CΨ is 1.06. The average absolute value of the relative error in the predicted values of d32 from the experimental points is 13.0%.

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2690 Ind. Eng. Chem. Res., Vol. 35, No. 8, 1996 Table 5. Comparison of Correlations for the Prediction of d32 in Rotating Disk and Asymmetric Rotating Disk Columns in Terms of the Average Absolute Value of the Relative Error (Total Number of Data ) 749) curve in Figure 1 1 2 3 4 5 6 7 a

correlation

no. of data tested

error, %

Relationships Valid for N g 0 present work (eq 11) 749 present work (eq 16) 749 Kumar and Hartland (1994b) 749 Mı´sˇek (1963) 749

23.0 22.4 25.6 55.8

Relationships Valid for N > 0 Kumar and Hartland (1986b) 746 Chang-Kakoti et al. (1985) 746 Mersmann (1980) 746 Marr (1978)a 517

26.3 29.9 103.4 80.7

(6000 < ReR < 70 000).

Table 6. Comparison of Correlations for the Prediction of d32 in Ku 1 hni Columns in Terms of the Average Absolute Value of the Relative Error (Total Number of Data ) 702) Figure 7. Experimental variation in d32 with VN for a water (dispersed)-amyl alcohol (continuous) system measured by Kumar and Hartland (1982) in a 100-mm diameter spray column (O) compared with that predicted by various correlations. The curve numbers refer to the correlations given in Table 10.

curve in Figure 2 1 2 3 4

As with mechanically agitated columns, eq 24 may be modified to:

d32 ) DN

(

)

1/3

+

(

4.15

) [ ( )]

12γ FdVN2DN

DN

Fd g γ

Relationships Independent of Stage Number present work (eq 11) 702 present work (eq 16) 702 Kumar and Hartland (1994b) 702 b Goldmann (1986) 696

26.8 22.4 20.1 96.4

1/2 0.62

(25) with CΨ ) 1 for c f d and no transfer and CΨ ) 1.05 for d f c transfer. The equation above predicts the drop size with a reduced average error of 10.7%. In Figure 7, eqs 24 and 25 are compared with the data of Kumar and Hartland (1982) for a water (dispersed)-amyl alcohol (continuous) system in a spray column of 100mm diameter. Comparison with Published Equations A fair comparison of present correlations with previously published equations is not always possible, since many of them only apply over limited ranges of variables. Furthermore, some of the earlier equations are implicit, and a solution for d32 is not always possible. Data selection criteria for published correlations, wherever appropriate and possible, have been formulated. These are given in Tables 5-10, which compare the values of d32 predicted by pertinent published equations with those by present correlations in terms of the average absolute value of the relative error for different columns. Rotating Disk and Asymmetric Rotating Disk Columns. Literature correlating equations selected for the comparison with the experimental data are those of Mı´sˇek (1963), Marr (1978), Mersmann (1980), ChangKakoti et al. (1985), and Kumar and Hartland (1986b, 1994b). Unfortunately, it was not possible to consider those general correlations which allow for the variation in the drop size with stage number, since the measurement locations are not always known. Table 5 shows the average absolute values of the relative error. The

a

error, % 21.6 18.4 16.9 57.9

1

0.73

no. of data tested

Relationships Including Effect of Stage Number present work (eq 14) 580 present work (eq 19) 580 Kumar and Hartland (1994b) 580 Fischer (1973)a,b 494

CΨ 1 6γ ∆FgDN2

correlation

Includes the effect of holdup. b Only valid for N > 0.

Table 7. Comparison of Correlations for the Prediction of d32 in Wirz-II Columns in Terms of the Average Absolute Value of the Relative Error (Total Number of Data ) 221) curve in Figure 3

correlation

no. of data tested

error, %

1 2

Relationships Independent of Holdup present work (eq 11) 221 present work (eq 16) 221

16.6 16.1

3 4 5

Relationships Including Effect of Holdup present work (eq 15) 221 present work (eq 20) 221 Rinco´n-Rubio et al. (1994) 221

14.9 14.0 15.8

equation of Mı´sˇek (1963) grossly underpredicts, whereas those of Marr (1978) and Mersmann (1980) generally overpredict the values of drop size. Published correlations listed in Table 5 have also been compared graphically with the data of Hartung and Weiss (1982) in Figure 1. It should be noted that Marr’s (1978) equation is not shown, since its range of application lies outside this set of data (for which 1.24 × 105< ReR < 2.33 × 105). Ku 1 hni Columns. Average absolute values of the relative error in the predicted values of the drop size from the experimental points as given by various correlations are listed in Table 6. While evaluating Fischer’s (1973) correlation, which gives drop size in terms of the stage number and holdup, the following data were excluded: experiments by Goldmann (1986), Ming and Yao (1986), and Dongaonkar et al. (1991) for which measurement locations are not specified; runs for which hold-up values are not available; and measurements with N ) 0. His equation only applies to his own data for which it was obtained. Similarly, Goldmann’s

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Ind. Eng. Chem. Res., Vol. 35, No. 8, 1996 2691 Table 8. Comparison of Correlations for the Prediction of d32 in Pulsed Perforated-Plate and Karr Reciprocating-Plate Columns in Terms of the Average Absolute Value of the Relative Error (Total Number of Data ) 1067) curves in Figures 4 and 5

correlation

no. of data tested

Table 10. Comparison of Correlations for the Prediction of d32 in Spray Columns in Terms of the Average Absolute Value of the Relative Error (Total Number of Data ) 743) curve in Figure 7

error, %

9 10 11 12 13

1 2 3 4

correlation

no. of data tested

error, %

1 2 3 4

Random and Ordered Packingsa present work (eq 21) 382 Kumar and Hartland (1994a) 382 Seibert and Fair (1988) 382 Mac´kowiak and Billet (1986) 382

15.7 15.6 24.9 17.0

5 6

Random Packingsb Lewis et al. (1951) Gayler and Pratt (1953)

37.8 17.5c

220 220c

a Independent of phase flow rates. b Includes the effect of phase flow rates. c Solution was only possible for 184 data points.

(1986) formula well describes his own data, but it is inadequate when applied to data from other sources. Of the two correlations suggested by Kumar and Hartland (1994b), the one incorporating the effect of stage number delivers the best estimates of drop size. Correlations due to Fischer (1973) and Kumar and Hartland (1994b), which incorporate the effect of stage number on drop size, are plotted in Figure 2. In using Fischer’s (1973) equation, an average value of the holdup corresponding to each data set was used. His method overpredicts the drop size for all the data sets shown in Figure 2. Wirz-II Columns. Only Rinco´n-Rubio et al. (1994) have attempted the correlation of drop size in this column. They have proposed a model and two dimensional empirical equations. Their model, which allows for the effect of holdup on drop size, is compared with present correlations in Table 7. Drop sizes predicted by using this model are shown in Figure 3, the average holdup corresponding to each data set being used in the calculations. Pulsed Perforated-Plate and Karr Reciprocating-Plate Columns. Various published correlations of general applicability for predicting drop size in these columns are compared in Table 8. It may be noted that most of these equations have not necessarily been endorsed by the original authors for both pulsed and Karr columns. The methods due to Mı´sˇek (1964), Baird and Lane (1973), Pilhofer (1978), Maksimenko (1981),

Nonjetting and Jetting Regions present work (eq 24) 743 present work (eq 25) 743 Kumar and Hartland (1994b) 743 Kumar and Hartland (1984) 743

Jetting Region (VN > Vj ) Kumar (1983) 312 Horvath et al. (1978) 312

13.0 10.7 10.6 11.5

15.3 21.0

Single-Nozzle Relationships

a Includes separate values of A and f. b Solution was not possible for 90 data points. c (Af > 0, 0.2 < d32VTFc/µc< 800). d Solution was not possible for three data points.

curve in Figure 6

error, %

Separate Equations for Nonjetting and Jetting Regions Dalingaros et al. (1986) 743 13.1 Kumar and Hartland (1982) 743 13.1

28.1 94.0 31.8 47.6 31.8

Table 9. Comparison of Correlations for the Prediction of d32 in Packed Columns in Terms of the Average Absolute Value of the Relative Error (Total Number of Data ) 382)

no. of data tested

Multiple-Nozzle Relationships

Valid for Mixer-Settler, Dispersion and Emulsion Regions 1 present work (eq 11) 1067 19.7 2 present work (eq 16) 1067 18.1 3 present work (eq 18) 1067 16.1 4 Kumar and Hartland (1994b) 1067 15.7 5 Sovova´ (1990)a 1005 28.2 a 6 Kumar and Hartland (1986a) 1005 21.8 7 Pietzsch and Pilhofer (1984) 1067b 27.9b 8 Logsdail and Slater (1983)c 902d 150.8d Only Valid for the Emulsion Region Boyadzhiev and Spassov (1982) 479 Maksimenko (1981)a 438 Pilhofer (1978) 479 Baird and Lane (1973) 479 Mı´sˇek (1964) 479

correlation

a

Nonjetting Region (VN < Vj ) Kagan et al. (1973) 431a Izard (1972) 431 de Chazal and Ryan (1971) 431 Scheele and Meister (1968a) 431 Hayworth and Treybal (1950) 431

10.0a 21.6 11.5 22.1 11.8

Jetting Region (VN > Vj ) Skelland and Huang (1979) 312

31.5

Solution was not possible for three data points.

and Boyadzhiev and Spassov (1982) were applied only in the emulsion region, the criterion suggested by Boyadzhiev and Spassov (1982) being adopted for excluding the data points in the mixer-settler and dispersion regions. The value of Vc required in the correlation of Mı´sˇek (1964), when not quoted by the authors of literature data, was taken as 2 mm/s. While using the correlation by Baird and Lane (1973), the values of the preconstant in their equation correspondinng to c f d and d f c cases of mass transfer reported by Kumar and Hartland (1988) were used. The hold up term in the equation suggested by Maksimenko (1981) was ignored while computing the results for 11 data sources due to lack of data. His equation together with that of Logsdail and Slater (1983) does not apply at all and grossly overpredicts the drop size. Figures 4 and 5 display the variation in d32 with Af obtained using published formulas compared in Table 8. The range of application of many of the correlations is restricted to the emulsion region (curves 9-13). Curves 1-6 predict finite drop sizes in the mixer-settler and dispersion regions. Curves 1-3 corresponding to eqs 11, 16, and 18 correctly predict the limiting drop size at low zero agitation, since these equations then reduce to eq 1. This is not true for the correlation of Logsdail and Slater (1983), which predicts an infinite drop size at zero agitation in Figure 4 for pulsed columns and lies completely outside the range of Figure 5 for Karr columns. Packed Columns. The predictive ability of various previously published equations (Table 1) is compared with that of the present correlation in Table 9 in terms of the average error. Note that the correlations proposed by Streiff and Jancic (1984) and Moore et al. (1989) could not be evaluated for the reasons given elsewhere (Kumar and Hartland, 1994a). Correlations by Lewis et al. (1951) and Gayler and Pratt (1953) were only applied to data for random packings for which they were obtained. Unfortunately, 15 data points could not be treated due to lack of information on phase flow rates.

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2692 Ind. Eng. Chem. Res., Vol. 35, No. 8, 1996

As suggested by Gayler and Pratt (1953), the values of the holdup required in their correlation were estimated by using the method of Gayler et al. (1953). However, the hold-up values for 36 data were found to be greater than unity, thus invalidating the result. Predictions of previously published correlations listed in Table 9 are displayed in Figure 6. Except for Gayler and Pratt (1953), the other authors assume the drop size to be independent of dispersed-phase velocity, Vd. The correlation of Gayler and Pratt (1953), which depicts an increase in d32 with Vd, overpredicts the drop size for the data sets shown in Figure 6. It should be noted that the values of holdup needed in their correlation were obtained from the equation of Kumar and Hartland (1995), since the originally recommended procedure due to Gayler et al. (1953) was found to be inadequate as mentioned above. Spray Columns. A number of selected single-nozzle as well as multiple-nozzle relationships have been applied to the present set of data, the average errors of prediction being listed in Table 10. The jetting velocity, Vj, required to restrict the application of various equations in their recommended regions, was calculated from the correlation of Scheele and Meister (1968b). The values of the Harkins and Brown (1919) correction factor in the correlations due to Hayworth and Treybal (1950), Scheele and Meister (1968a), de Chazal and Ryan (1971), and Kagan et al. (1973) were calculated by using an equation suggested by Horvath et al. (1978). Out of six correlations for single nozzles, the formulas by Scheele and Meister (1968a), Izard (1972), and Skelland and Huang (1979) are unsatisfactory. Similarly, the multiple-nozzle relationship by Horvath et al. (1978) is not acceptable in terms of the accuracy. Figure 7 shows the plots developed from multiple-nozzle relationships that are applicable over the entire range of nozzle velocities. Correlations due to Kumar and Hartland (1982) and Dalingaros et al. (1986), which consist of separate equations for nonjetting and jetting regions, are omitted. Conclusions 1. Published experimental data on drop size for eight different types of differential and stagewise columns are considered. In the absence of agitation, the drop size is determined by the ratio of interfacial tension to buoyancy forces. The effect of external agitation is accounted for in terms of the power dissipation per unit mass. In spray columns, the value of the drop size at high nozzle velocities is obtained from a balance of interfacial and kinetic energies. 2. For agitated columns, a simple two-term correlation is presented which reproduces the data with an average of 22.2%. An extended version of this correlation gives a better representation of data and predicts the drop size with a reduced average error of 20.2%. Similar two-term correlations for spray columns represent the measured drop size with average errors of 13.0 and 10.7%. 3. Except for Wirz-II columns, the effect of dispersedphase holdup on drop size is insignificant. 4. The data for Ku¨hni columns show a definite but weak dependence of the drop size on the stage number. Nomenclature A ) stroke (twice the wave amplitude), m

ap ) packing surface area per unit volume of column, m2/ m3 C0 ) discharge coefficient for flow through holes in the perforated plate, dimensionless C1, C2, ... ) parameters of empirical equations, dimensionless CΠ, CΩ ) parameters in eqs 11, 16, and 22, dimensionless CΨ ) parameter allowing for the effect of mass transfer on the drop size, dimensionless Cφ ) parameter in eq 5, dimensionless Cκ ) parameter in eq 24, dimensionless DC ) column diameter, m DN ) nozzle diameter, m DR ) rotor diameter, m DS ) stator opening diameter, m D0 ) perforation diameter, m dp ) packing size, m d32 ) Sauter mean drop diameter, m e ) fractional free cross-sectional area ((DS/DC)2 for rotating disk columns), dimensionless f ) frequency, Hz g ) acceleration due to gravity, m/s2 H ) compartment height, m m1, m2, ... ) indexes, dimensionless N ) rotor speed, revolutions/s NP ) power number ) P/(N3DR5Fc), dimensionless n, n1, n2 ) indexes, dimensionless P ) power input per compartment, W ReR ) rotor Reynolds number ) NDR2Fc/µc, dimensionless S ) stage number, dimensionless V ) superficial velocity, m/s VN ) nozzle velocity, m/s VT ) terminal of a single drop in an infinite continuous phase, m/s Greek Symbols γ ) interfacial tension, N/m ∆F ) density difference between phases, kg/m3  ) mechanical power dissipation per unit mass, W/kg µ ) viscosity, Pa s π ) 3.1416 F ) density, kg/m3 φ ) volume fraction holdup of the dispersed phase, dimensionless Subscripts c ) continuous phase cr ) critical d ) dispersed phase

Literature Cited Al-Aswad, K. K. M. Liquid-Liquid Extraction in a Pilot Scale Rotating Disc Contactor. Ph.D. Thesis, University of Aston in Birmingham, Birmingham, U.K., 1982. Angelino, H.; Alran, C.; Boyadzhiev, L.; Mukherjee, S. P. Efficiency of a Pulsed Extraction Column with Rotary Agitators. Br. Chem. Eng. 1967, 12, 1893-1895. Arai, K.; Konno, M.; Matunaga, Y.; Saito, S. Effect of DispersedPhase Viscosity on the Maximum Stable Drop Size for Breakup in Turbulent Flow. J. Chem. Eng. Jpn. 1977, 10, 325-330. Assenov, A.; Penchev, I. Effect of Pulsing Intensity upon Droplet Size in a Plate-Pulsed Extraction Column. Dokl. Bolg. Akad. Nauk 1971, 24, 1381-1384. Aufderheide, E.; Vogelpohl, A. Prediction of Flooding Point in Pulsed Sieve-Plate Extraction Columns Using Experimental Drop-Size Data. Chem.-Ing.-Tech. 1984, 56, MS 1272/84. Bailes, P. J.; Gledhill, J.; Godfrey, J. C.; Slater, M. J. Hydrodynamic Behaviour of Packed, Rotating Disc and Ku¨hni Extraction Columns. Chem. Eng. Res. Des. 1986a, 64, 43-55. Bailes, P. J.; Gledhill, J.; Godfrey, J. C.; Slater, M. J. Investigation of Hydrodynamic Behaviour of a Ku¨hni Extraction Column. Chem.-Ing.-Tech. 1986b, 58, 807-809.

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Received for review November 6, 1995 Revised manuscript received May 2, 1996 Accepted May 3, 1996X IE950674W

X Abstract published in Advance ACS Abstracts, June 15, 1996.