Estimation of Coalescence and Breakage Rate Constants within a

understanding of coalescence and breakage rate coef- ficients within a Ku¨ hni column. Several authors (Chatzi and Lee, 1987; Laso, 1986) have shown ...
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Ind. Eng. Chem. Res. 1998, 37, 1099-1106

1099

Estimation of Coalescence and Breakage Rate Constants within a Ku 1 hni Column Sandra E. Kentish, Geoff W. Stevens,* and H. R. C. Pratt Department of Chemical Engineering, University of Melbourne, Parkville, Australia, 3052

A simplified population balance model is presented to describe the hydrodynamics of the dispersed phase within a Ku¨hni column. This model is fitted to experimental axial drop size distribution profiles by the optimization of expressions for the breakage and coalescence rate coefficients. Thus, over the experimental range of study, correlations for these rate coefficients are developed respectively as breakage, Brij ) 3.0N(uij/Hs) exp[(-1.8 × 10-4DF(φ))/(N2di5/3)], and coalescence (hλ)ij ) 0.640(uij/Hs)φj-1.3vi0.56. These expressions indicate that the breakage and coalescence rate coefficients are much less dependent upon rotor speed than suggested by correlations in the literature for turbulent stirred tanks. In contrast, as suggested by Gourdon et al. (Chem. Eng. J. 1991, 46, 137), these coefficients are correlated with the drop residence time, which in turn is determined by the individual droplet velocities. Introduction The objective in this paper is to gain a better understanding of coalescence and breakage rate coefficients within a Ku¨hni column. Several authors (Chatzi and Lee, 1987; Laso, 1986) have shown that such rate coefficients cannot be quantified independently from steady-state drop size distributions. Consequently, many authors (Narsimhan et al., 1984; Laso, 1986; Sathyagal et al., 1996; Steiner et al., 1996) used transient drop size profiles within stirred tanks in order to obtain independent estimates of these coefficients. Breakage rate constants have been determined in more sophisticated liquid-liquid contactors through the direct observation of the behavior of single drops. A disadvantage of this technique is that it is limited to situations where a single breakage event occurs during the passage of a drop through the equipment module; thus, multiple breakages cannot be recorded. Further, the effect of increasing holdup upon breakage rates cannot be ascertained. Cabassud et al. (1990) were the first to apply such a technique to a Ku¨hni column. Gourdon et al. (1991) continued this work and obtained an equation for the probability, Pr(d), of a single drop breaking within a stage as a function of the Weber number, We(d), over a range of 3.8 × 103 < ReI < 2.3 × 104, that is, namely,

(

)

(

)

0.112 0.112σ Pr(d) ) exp ) exp - 2/3 5/3 We(d) Fc d

(1)

with the maximum stable drop size prior to breakage given by

We(dmax) ) 0.09 )

Fc2/3d5/3 max σ

(2)

They then expressed the breakage rate through the column, Br(d), in terms of the mean residence time in each stage, Hs/u(d), as follows: * Author to whom correspondence should be addressed. E-mail: [email protected]. Telephone: 61 3 9344 6621. Fax: 61 3 9344 4153.

Br(d) )

Pr(d)u(d) Hs

(3)

Fang et al. (1995) and Godfrey et al. (1996) attempted to improve upon these expressions, based on experiments conducted at impeller Reynolds numbers, ReI < 1.3 × 104 under which conditions the majority of breakage events occurred at the sieve plate. They found that the probability of breakage was well-correlated by an expression involving a modified Weber number. While the work described above has allowed a reasonable understanding of the breakage process within agitated columns, there remains greater uncertainty regarding droplet coalescence. The only direct experimental works conducted with equipment more sophisticated than a stirred tank are those of Hamilton and Pratt (1984), using a packed column, and Garg and Pratt (1984), using a pulsed perforated plate column. These workers passed two streams of size-equilibrated drops, one-half containing dithizone (green) and the other nickel bis(ethyl xanthate) (yellow) into the working section of the column, where the coalescence of green with yellow drops gave red drops, the proportions which were determined photographically. An alternative approach toward an understanding of droplet interactions has been the development of phenomenological models. These models are based upon Kolmogoroff’s theory of local isotropic turbulence, and are thus usually only valid when drop sizes are intermediate between the micro and macro scale of turbulence and when impeller Reynolds Numbers are high (ReI > 5 × 104). Thus, Coulaloglou and Tavlarides (1977) modeled the breakage rate, Br(d), as the combination of a collision rate between a drop and an eddy together with a breakage efficiency for the conversion of this collision into an actual breakage event. This resulted in the following:

Br(d) )

c11/3 2/3

d

(

)

c2σ exp - 2/3 5/3 FD d

(4)

Given that the specific energy dissipation, , is propor-

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1100 Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998

tional to [N/(DF(φ))1/2]3DI2, this becomes

Br(d) )

c3NDI2/3

(

)

c4σDF(φ) exp FD N2d5/3DI4/3 d xDF(φ) 2/3

(5)

where DF(φ) is a turbulence damping factor introduced to account for the effect of two-phase flow on the degree of turbulence. Tsouris and Tavlarides (1994) pointed out that eq 4 has the disadvantage of predicting a breakage probability which goes through a maximum as the drop size increases, and they therefore proposed a more complex expression for this purpose. The coalescence rate coefficient is expressed as the product of a collision frequency, h, and a coalescence efficiency, λ. Most authors use the expression for the collision frequency of drops developed by Coulaloglou and Tavlarides (1977) by assuming an analogy with collisions between gaseous molecules, that is, namely,

h(dq,di) ) ch1/3(dq2 + di2)(dq2/3 + di2/3)1/2

(6)

Conversely, a variety of conflicting expressions have been proposed for the coalescence efficiency. Thus, rigid sphere models have been developed by Tsouris and Tavlarides (1994), Muralidhar et al. (1988), and Das et al. (1987), while Coulaloglou and Tavlarides (1977), Muralidhar and Ramkrishna (1986), and Sovova (1983) all presented differing models for deformable drops. These models all conflict in their relationships between the theoretical coalescence efficiency and parameters such as drop size, interfacial tension, and rotor speed. Many authors comment that the coalescence efficiency is dependent upon the purity of the reagents used (Sovova, 1983; Madden and Damerell, 1962; Laso, 1986; Fitzpatrick et al., 1986; Stevens et al., 1990; Steiner et al., 1996), while the collision rate is a function of the exact equipment configuration (Park and Blair, 1975; Tsouris and Tavlarides, 1994). Thornton and Pratt (1953), Groothuis and Zuiderweg (1964), and Tsouris and Tavlarides (1993) have clearly shown that mass transfer can drastically alter the coalescence process. Consequently, while phenomenological models as described above have their place in determining the structure of the coalescence equation, experimental work will inevitably be required to determine the magnitude of the variables for a particular piece of equipment and a particular liquid system. In the present work, the variation in drop size distribution with axial length in a Ku¨hni column was observed experimentally. This is an equivalent situation to the observation of drop size distribution transients within a stirred tank, as discussed above. These experimental results are compared to a simplified population balance model, and the breakage and coalescence rate coefficients within this model are then varied until an optimal fit to the experimental database is achieved. Population Balance Model The use of population balances to model dispersed phase behavior was pioneered by Valentas et al. (1966), using probability density functions of drop sizes to construct balances for the number of drops of each given size. Such balances have been expressed in terms of both drop diameter and drop volume, although the latter is more appropriate since volume (or equivalently mass)

is a conserved quantity (Ramkrishna et al., 1995; Das, 1996). In addition, with such a basis the small drops formed due to erosive breakage can more readily be considered insignificant (Das, 1996). Finally, such equations can more readily be adapted to include solute balances later. Thus, Casamatta and Vogelpohl (1985) gave an unsteady-state drop population balance incorporating back mixing for a differential contactor as follows:

(

)

dP(z,d) dP(z,d) d d ) (P(z,d) u(z,d)) + ED(d) + dt dz dz dz Πv(z,d) (7) where P(z,d) ) φ(z) f(z,d) is the static volume concentration at height z and represents the static volume density of drops with respect to the total dispersion. The last term, Πv(z,d), represents the net volume density of drops generated per unit time by coalescence and breakage within the vessel. For the present purpose the above equation is discretized for application to a stagewise column model assuming back mixing in the dispersed phase to be negligible. The appropriate steady-state population balance equations can then be written in terms of the static drop volume fraction, Pij, for drops of size i within stage j, as

Auij Pij - Aui, j+1 Pi, j+1 ) Πvij

(8)

In this expression, the stage order is counted from j ) Nj at the dispersed phase inlet to j ) 1 at the dispersed phase exit. The net volume of drops generated or destroyed per unit time through coalescence and breakage, (Πvij), is defined by Ni

∑ q)i

Πvij ) viΠnij ) viVj i/2

2

viVj

∑

q)1

νqjβq,ijBrqj Pqj vq

(hλ)q,i-q, j Pqj Pi-q, j vqvi-q

- VjBrij Pij + 2

- Vj Pij

Ni-i(hλ) qi, j Pqj

∑

q)1

vq

(9)

where Brqj represents the breakage rate coefficient for drops of size q in stage j, νqj the number of daughter drops formed, and βqij the number of drops of size i resulting from such a breakage. The term (hλ)iqj represents the coalescence rate coefficient for drops of size i and q coalescing in stage j. To simplify the population balance further, the assumptions made by Hamilton and Pratt (1984), Garg and Pratt (1984), and Laso (1986) are now incorporated. First, rather than the normal linear discretization of drop volumes, a geometric discretization is used, that is namely

vi ) 2i-1v1

(10)

This form of discretization reduces the number of drop classes that need to be used without losing resolution at the smaller drop sizes. Second, drops are assumed only to break into two drops of equal volume and coalescence is allowed to occur only between drops of identical size. This assumption is not based on the premise that drops of unequal size do not interact; rather, given the geometrical size discretization discussed above, a breakage

Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 1101

or coalescence event involving unequal-sized drops will tend to result in the majority of the drop volume remaining within one of the original drop classes. Garg and Pratt (1984) and Hamilton and Pratt (1984) also allow coalescence between adjacent drop sizes, but this additional condition is omitted in the present model. Laso (1986) showed that the use of this approach results in a deviation of less than 20% from the exact solution. Steiner et al. (1996) further commented that in more than 200 case studies in which the full scale was compared with the simplified model, the differences were less than 15%. Incorporating these constraints into the present model, the parameter Brij becomes the rate coefficient for breakage of a single drop of size i into two drops of size i - 1 within stage j. Similarly, (hλ)ij represents the rate coefficient for the coalescence of two drops of size i into a single drop of size i + 1. The model equation for drop size i in stage j is written as follows:

Auij Pij - Aui, j+1 Pi, j+1 ) VjBri+1, j Pi+1, j - VjBrij Pij + 2Vj2(hλ)i-1, j Pi-1, j2 vi-1

-

2Vj2(hλ)ij Pij2 vi

(11)

The resulting series of model equations is supplemented by the boundary condition at the inlet to the column, that is namely

gIN i QD ) Aui,Nj+1Pi,Nj+1

(12)

where gIN represents the dynamic volume fraction of i drops of size i in the feed. Further, breakage of the smallest drop size and coalescence of the largest drop size is prohibited so that Br1j ) (hλ)Ni j ) 0. In addition to these constraints, breakage of drops is assumed to occur only above a maximum stable drop size, dmax. The criterion for dmax used by Gourdon et al. (1991) is used in this work (Eq 2). The energy dissipation term, , includes a turbulence damping factor DF(φ), as defined by Tsouris and Tavlarides (1994) to account for increases in holdup. Droplet velocities relative to stationary coordinates, uij, are calculated from the expression presented in Kentish (1996). Given that estimates of Brij and (hλ)ij are available, these model equations can be solved for Pij, as will be described later. Once the model equations had been solved in this manner for Pij, this parameter was converted to the dynamic drop fraction, gij, for comparison with the experimental values. Thus,

gij )

fijuij Ni

fijuij ∑ i)1

)

Auij Pij QD

(13)

This dynamic fraction represents the volume fraction of drops of size i passing through stage j over a period of time, as opposed to the more usual static fraction fij, which measures the volume fraction present in the stage at a given instant of time (see Kentish et al., 1997). The predicted local hold-up values, φ j, and dynamic Sauter mean diameters at each stage can also be calculated.

Figure 1. Schematic of the 25-stage Ku¨hni column used for experiments.

drilled with holes for the insertion of sample probes, as shown in Figure 1. The continuous phase was water, and methyl isobutyl ketone (MIBK) acted as the dispersed phase. Full details of the equipment have been given by Kentish (1996). A capillary probe was used to measure the dynamic drop size fractions, gexp,il, at a variety of points axially along the column (see Isselhard, 1993). Experimental conditions covered a 4 × 3 experimental design matrix (i.e., each of four rotor speeds (N ) 100, 120, 140, and 160) were tested at each of three dispersed phase flow rates (QD ) 4.2, 6.6, and 9.2 cm3/s). The continuous phase flow rate was held constant at 4.1 cm3/s in all cases. The dynamic drop size distribution of the dispersed was measured concurrently with the phase feed gIN i axial drop size distribution profiles. The point of measurement was only a few centimeters below the first stage (see Figure 1). As this sampling position is relatively close to the staged column section, there is no need to include allowance for a column entry section in the model. These experimental results are supplemented by measurements of the local holdup recorded under identical experimental conditions. This local holdup was determined directly by the extraction of a 200-mL sample of the dispersion from each measurement point and measurement of the volumes of each phase after separation (Kentish, 1996). A selection of the experimental results is presented in Figures 2-4. Figure 2 shows that for the system under study, equilibrium drop size distributions were achieved in the initial stages of the column. The dynamic Sauter mean diameter reached a steady value after about five stages (i.e., at the second sample point) (stage 21), as shown in Figure 3. Conversely, changes in the shape of the drop size distribution curve persisted through approximately eight stages to the third sample point (stage 18), as shown in Figure 2. In any such optimization it will be critical to achieve a good fit to the data over these initial stages, as only these reflect the transient variation in coalescence and breakage rates.

Experimental Section

Interpretation of Data

A laboratory scale glass Ku¨hni column of 25 stages and 72.45 mm diameter was used; the column was

The objective of this work is now to develop expressions for the breakage rate coefficient Brij and the

1102 Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998

drop size fraction, gil, were obtained which best fitted the 91 experimental values, gexp,il (i.e., 13 drop size fractions at 7 sample points). Specifically, we seek to find the minimum value of N1 Ni

ψ(p) )

Figure 2. Evolution of the dynamic drop size distribution through the column when QD ) 4.2 cm3/s, QC ) 4.1 cm3/s, and N ) 100 rpm.

Figure 3. Changes in the Sauter mean diameter through the column under two sets of experimental conditions.

coalescence rate coefficient, (hλ)ij, which allow the population balance model developed above to fit the experimental data obtained. Initially, empirical expressions are developed which optimize the fit of the model to the data and these are then compared to similar expressions available in the literature. However, it would be more appropriate if relationships could be developed which have some physical meaning. Thus, an attempt has also been made to provide equations which reflect the structure of the theoretical models presented previously by other authors. The fitting of such a suitable rate coefficient equation to the experimental data is a two-stage optimization process. Within each particular experimental run, the rate coefficient is a function only of the drop size or volume, vi, and the local dispersed phase holdup, φj. Thus, in the simplest instance empirical equations are written as

Brij ) p(1)v p(2) i

(14)

p(5) (hλ)ij ) p(3)v p(4) i φj

(15)

These equations contain five adjustable parameters, p(1)-p(5), although p(2) and p(4) are highly correlated, as are p(1) and p(3). The values of these parameters were optimized until model predictions of the dynamic

(gexp, il - gil)2 ∑ ∑ l)1 i)1

(16)

where l ) 1 to Nl represents the sample points at which drop size measurements are available. For this purpose, eqs 11 and 12 were written in the form of a matrix equation in which values of Brij and (hλ)ij on successive stages were obtained from eqs 14 and 15 using assumed values of the parameters p(1)-p(5). This equation involved a tridiagonal matrix which, on inversion, gave values of the Pij for each stage; these in turn were converted to gij values using eq 13. The resulting gij values were compared with the experimental values for the stages for which experimental data were available and were then adjusted repeatedly using the Levenburg-Marquardt optimization method (see Press et al., 1989) until eq 16 was minimized. This first-stage optimization process was repeated for each of the 12 experimental conditions (i.e., the four rotor speeds and three dispersed phase flows), leading to 12 independent estimates of the set p(1)-p(5). In the second stage optimization, each of these five parameters was examined for evidence of any trends with respect to the external experimental variables, rotor speeds, N, and dispersed phase flow rate, QD. The combination of parameters N/xDF(φj) was also considered as an external variable as this should be representative of the average energy dissipation, . As breakage appeared to be the dominant mechanism in the initial stages of the column, the breakage parameters p(1) and p(2) were defined in this manner first. Thus, for example, in the first instance, it was found that p(1) and p(2) were best represented by

p(1) ) 1.05 × 1010

N2 ; DF(φj)

p(2) ) 1.3

(17)

Results Breakage Rate Coefficient. The optimization procedure discussed above resulted in the following empirical equation for the breakage rate coefficient:

Brij ) 1.05 × 1010

N2v1.3 i DF(φj)

(18)

This breakage expression displays a similar dependence upon drop volume, vi, to the expressions presented by Narsimhan et al. (1984), Laso (1986), and Sathyagal et al. (1996). Conversely, the exponent on the rotor speed is considerably lower, at 2.0, than the range reported for stirred tanks (i.e., from 6.4 (Narsimhan et al., 1984) to 8.3 (Laso, 1986)). There are probably several factors contributing to this lower exponent, namely the following: (1) As suggested by Cabassud et al. (1990), Fang et al. (1995), and Godfrey et al. (1996), a significant proportion of the breakage events actually occur at the stator plate dividing each stage, rather than at the turbine. Consequently, expressions based on stirred tank data become irrelevant, and breakage rates would be expected to be much less dependent upon rotor speed.

Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 1103

(2) The present work was conducted at lower impeller Reynolds numbers than those corresponding to the fully turbulent conditions used by other authors, and hence breakage within the region of the turbine is unlikely to be well-described by their equations. (3) As discussed by Gourdon et al. (1991), it would be expected that the number of breakage events occurring per interval of time would be directly related to the number of turbines or stator plates encountered by the drop in such a time period. That is, the breakage rate Brij should be a direct function of the droplet velocity uij. For the low continuous phase velocities used in the present work, the average droplet velocity is in turn approximately a function of N-2 . Thus, such a dependency upon droplet velocity should be reflected in a lower exponent on N in eq 19. The relatively low dependence of breakage rate upon rotor speed indicates that phenomenological expressions developed by other workers for stirred tanks (e.g., eq 4) would not fit the present data. Although the equations of Fang et al. (1995) and Godfrey et al. (1996) may have proved more useful in this respect, their use of a drop size dependent critical rotor speed, with no defining correlation, prevented their application to the present work. The equations proposed by Gourdon et al. (1991) for single-drop breakage in a Ku¨hni column (i.e., eqs 1 and 3) provide a better interpretation of the data. Thus, after expansion of the specific energy term, this gives

Brij ) p(1)

(

)

uij -p(2)DF(φ) exp Hs N2di5/3

(19)

A repetition of the optimization procedure using eq 19 in place of eq 15, resulted in revised values for p(1) and p(2), as follows:

p(1) ) 3.0N p(2) ) 1.8 × 10-4 Expressing eq 19 in the form originally proposed by Gourdon et al. (1991) gives for the present experimental configuration the following:

(

)

Nuij -0.72σ Brij ) 3.0 exp 2/3 5/3 Hs F di

(20)

The exponential term in eq 20 represents the probability of breakage following a collision with either the turbine or the stator plate. The optimized constant in this exponential term is larger at 0.72 than the value of 0.112 given by Gourdon et al. (1991). However, the exponential term is almost identical to the breakage efficiency term developed by Coulaloglou and Tavlarides (1977); see eq 4. Both Sovova (1981) and Hsia and Tavlarides (1980) comment that the value of this constant in this instance is likely to be dependent upon the spatial characteristics of the contactor. Thus, values have been given ranging from c2 ) 0.025 for a stirred tank (Cruz-Pinto and Korchinsky, 1981) to 0.68 for a vibrating plate extractor (Sovova, 1983). In the present case, multiplying 0.72 by the ratio of the dispersed and continuous phase densities gives c2 ) 0.58, which is within this range. The remainder of eq 20 reflects the number of collision events occurring for each drop per second that are capable of causing breakage. By the nature of the

experimental technique used by Gourdon et al. (1991), only one such collision event was permitted per stage. Thus the adjustable constant in their eq 3 is unity, whereas optimization of this constant in the present work predicts a value of 3.0 times the rotor speed, N. There are two reasons for this larger, rotor speed dependent value: (1) More than one collision event may occur per stage. Both Fang et al. (1995) and Cabassud et al. (1990) suggest that collisions can occur at both the turbine level and the stator plate, although again their measurement techniques allowed only single breakage events to be studied. Further, as the rotor speed increases the possibility of multiple passes through the turbine increases, again increasing the number of collision events. The collision rate given by Coulaloglou and Tavlarides (1977) displays such a proportionality with rotor speed (see eq 5). (2) The present model permits breakage only into two drops of identical size. Cabassud et al. (1990), however, suggests that while this may be valid at low impeller speeds, as the impeller speed increases and breakage shifts to the turbine level, a larger number and larger size range of daughter droplets is produced. In the present simplified model such an increase in the extent of breakage would probably be reflected by an increase in the number of breakage events, as several events are required to achieve the same result. Using eq 19 for the breakage coefficient, optimization of the remaining parameters p(3)-p(5) in eq 15 results in the following empirical expression for the coalescence coefficient:

(hλ)ij ) 21.9N-1.9φj-1.3v0.70

(21)

This expression differs in several respects from the limited range of literature correlations. First, most authors (Park and Blair, 1975; Garg and Pratt, 1984; Hamilton and Pratt, 1984; Laso, 1986) found that the coalescence rate coefficient decreases with, or is independent of, drop size, whereas in the present case the coefficient increases with drop size. Similar differences in the drop size dependency have been noted in expressions in the literature proposed by others for the coalescence efficiency, and such discrepancies may explain the present differences. The exponent on rotor speed in eq 21 is significantly lower at -1.9 than literature values, which range from 2.85 (Coulaloglou and Tavlarides, 1976) to -0.02 (Laso, 1986). The reasons for the large negative exponent in the present work are probably similar to those discussed above for the breakage expression. That is, it is likely that in the present case, a significant amount of coalescence occurred directly below, or at the stator plate rather than at the turbine. Local hold-up values were considerably higher in the top half of the stage, and direct observation of the column during operation showed a layer of nearly stationary drops directly below the plate. Coalescence is likely to occur as drops move through this layer of semistagnant drops and also as they are squeezed together during passage through the limited free area of the stator plates. If coalescence does occur in this manner, it would be expected to be less dependent upon rotor speed. As discussed above, if coalescence does occur at the stator plate, it would be expected that the coalescence rate per unit time would also be a function of the

1104 Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998

section of each stage rather than being an inadequate model (see Kentish, 1996). Once the coalescence and breakage rates have stabilized to their equilibrium values, the volume frequencies of coalescence and breakage per unit time should be identical, that is: Ni

ωbv

Figure 4. Changes in the local holdup through the column under two sets of experimental conditions.

number of stator plates passed during this time period. The rate coefficient should therefore again be a function of the droplet velocity, which is approximately related to N-2. Incorporating uij into eq 22 does indeed slightly improve the fit to the experimental data; thus

uij (hλ)ij ) 0.640 φj-1.3vi0.56 Hs

(22)

Expressed in this manner, the coalescence rate coefficient becomes independent of the rotor speed. This tends to confirm that coalescence occurs at the stator plate rather than within the turbine. The exponent on the holdup of -1.3 in both eqs 21 and 22 is somewhat lower than comparable literature values, which range from -0.55 (Coulaloglou and Tavlarides, 1976) to 0.90 (Laso, 1986). The lower value in the present case may indicate the decreasing importance of the layer of drops directly below the stator plate as the overall holdup increases and the stage achieves a greater degree of homogeneity. Model Predictions Using eqs 19 and 22 for the breakage and coalescence rate coefficients, the average difference between predicted (gil) and experimental (gexp,il) values of the dynamic drop size fraction is 0.031. Given a range of gexp,il values from 0 to 0.4, this average error indicates a good fit to the experimental data, as is evident from Figures 2 and 3. In particular, the model can account for changes in the Sauter mean diameter in the lower stages of the column where drop sizes are approaching their steady-state value. The model presented in this work also fits experimental hold-up profiles well at low average holdup values (see Figure 4). However, this simple model is not capable of reproducing the convex holdup curve shapes observed in the experimental results at higher average holdups. The development of such curves requires a more complex model which allows for back mixing in the continuous phase (see Jiricny et al., 1979; Tsouris et al., 1994; Steiner et al., 1993, 1996); such a model is outside the scope of the present analysis. The experimental hold-up results are also generally somewhat higher than the model predictions at the higher rotor speed (Figure 4), but this is a consequence of holdup measurements being made exclusively in the top

)

fijBrij ) ∑ i)1

Ni

ωcv

) 2Vjφj

∑ i)1

(hλ)ij fij2 vj

(23)

This relationship holds with the present results in the center of the column. The range of volume frequencies, (20-72%/s) is of the same order of magnitude as the coalescence frequencies of 4-65%/s. measured by Park and Blair (1975) with an MIBK-water system in a turbulent stirred tank. Conclusions (1) A simplified stagewise population balance model has been developed for the Ku¨hni column used in this work. (2) The results suggest that coalescence and breakage rate coefficients are much less dependent upon rotor speed than suggested by correlations in the literature for turbulent stirred tanks. In contrast, these coefficients are correlated with the drop residence time, which in turn is determined by the individual droplet velocities. The use of eqs 19 and 22 to predict these rate coefficients is recommended for future work. Acknowledgment Thanks are accorded to the Advanced Mineral Products Special Research Centre and the Australian Research Council for their support of this work. Notation A ) cross-sectional area, m2 Br(d) ) breakage rate coefficient or breakage frequency of drops of diameter d, s-1 Brij ) breakage rate coefficient for drops of size i in stage j, s-1 c1-c4 ) dimensionless constants ch ) constant in eq 6, m-3 d ) drop diameter, m dmax ) maximum stable drop size prior to breakage, m DI ) impeller diameter, m DF(φ) ) turbulence damping factor, as defined by Tsouris and Tavlarides (1994) ED ) dispersed phase back mixing coefficient, m2 s-1 f(d) ) static volume density of drop of size d, m-1 fij ) static volume fraction of drop of size i in stage j gij ) dynamic volume fraction of drop of size i from stage j giIN ) dynamic volume fraction of drop of size i in the dispersed phase feed h(dqdi) ) collision frequency of drops of diameter dq and di (hλ)ij ) rate coefficient for coalescence of two drops of size i in stage j, s-1 (hλ)iqj ) rate coefficient for coalescence of drops of size i and size q in stage j, s-1 Hs ) height of a stage, m N ) rotor speed, s-1 Ni ) number of drop size classes Nj ) number of stages

Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 1105 Nl ) number of sample points p(k) ) parameter to be optimized (k ) 1-5) P(d) ) static volume concentration of drops within the dispersion, m-1 Pij ) discretized static volume concentration of drops Pr(d) ) probability of a given drop breaking within a single stage QD ) volumetric flow rate of the dispersed phase, m3/s ReI ) impeller Reynolds number (FCNDI2/µc) t ) time, s u ) drop velocity relative to stationary coordinates, m/s u(z,d) ) velocity of drop size d at column height z relative to stationary coordinates, m/s v ) drop volume, m3 Vj ) single stage volume, m3 We(d) ) Weber number (FC2/3d5/3/σ) z ) height coordinate (increasing in the direction of continuous phase flow), m Greek Symbols βqij ) number fraction of drops of size i resulting from breakage of size q in stage j  ) specific energy dissipation, m2/s3 Πv(z,d) ) volume density of drops of size d formed per unit time at height z by coalescence and breakage, m-1 s-1 v Πij ) net volume of drops formed per unit time through coalescence and breakage, m3/s FC ) continuous phase density, kg/m3 FD ) dispersed phase density, kg/m3 σ ) interfacial tension, kg/s2 νqj ) number of drops formed per breakage of drop size q in stage j φ ) column average holdup φj ) local holdup at stage j ψ ) goodness of fit parameter (eq 17) ωcv ) coalescence volume frequency, s-1 ωbv ) breakage volume frequency, s-1 Subscripts exp ) experimental i ) drop size fraction number i-q ) drop size fraction number defined by vi-q ) vi - vq j ) stage number k ) optimizing parameter number l ) sample point number Ni ) number of the largest drop size class Nj ) number of the bottom stage (i.e., dispersed-phase inlet end) q ) alternate drop size fraction number

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Received for review May 12, 1997 Revised manuscript received December 5, 1997 Accepted December 8, 1997 IE970336Q