Optimization of Breakage and Coalescence Model Parameters in a

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Optimization of Breakage and Coalescence Model Parameters in a Steady-State Batch Agitated Dispersion Margarida M. Ribeiro,*,† Pedro F. Regueiras,‡ Margarida M. L. Guimar~aes,‡ Carlos M. N. Madureira,‡ and Jose J. C. Cruz_Pinto§ †

CIETI/Depatamento Engenharia Química, Instituto Superior de Engenharia do Porto, 4400-072 Porto, Portugal CIGAR/Departamento Minas, Faculdade de Engenharia, Universidade do Porto, 4200-465 Porto, Portugal § CICECO/Departamento Química, Universidade de Aveiro, 3810-193 Aveiro, Portugal ‡

ABSTRACT: The highly dynamic behavior of liquid-liquid dispersions in reaction and separation operations still defies accurate and experimentally validated modeling. This behavior is characterized by simultaneous dropsize and operating conditions dependent drop breakage and coalescence, which strongly influence both the hydrodynamics and the reaction yield and selectivity, or separation performance, of such systems. This dynamic character of the behavior is present and of critical importance even at steady state, and not just during the transient evolution toward it or during disturbances in the operating conditions. This work addresses the measurement (by a noninvasive technique) and the optimization of kinetic liquid drop interaction parameters, duly taking into account the full and real complexity of the behavior, which is shown to require the inclusion and quantification of drop coalescence frequencies, no matter how lean and strongly agitated the dispersion may be. The analysis in this paper is limited to batch perfectly agitated vessels with lean dispersions at steady state and uses carefully collected experimental drop size distribution data and a very precise (with 50 logarithmic drop volume classes, to ensure uniform precision of drop size assignment) and fast coupled numerical dynamic simulation and nonlinear optimization algorithm, to quantify the drop breakage and coalescence kinetic parameters of drop interaction models. Significant physical insight has been gained on the interdependence of two (one for breakage and the other for coalescence) of the parameters and on the values of the others, in addition to an excellent agreement of the predicted and experimental drop size distributions at steady state. Further, and as expected, the need to always fully account for interdrop coalescence (in addition to breakage), whatever the operating conditions, by contrast to oversimplified modeling approaches, has also been clearly demonstrated.

1. INTRODUCTION The mass-transfer efficiency and reaction yield and selectivity of liquid-liquid contactors are highly dependent on the hydrodynamics of the dispersed phase, namely, on the drop breakage and coalescence frequencies that result from the turbulence induced by agitation, which is itself indispensable in creating and maintaining a dispersed phase with adequate specific interfacial area. The main practical effects of those drop interactions are, first, a direct influence on the resulting drop size and interfacial area and, second, a redistribution of solute(s)/reactant(s) among drops of different diameters, directly influencing the instantaneous kinetics of the (transfer or reaction) process within each individual drop. Potential application of improved knowledge in this area encompasses wide fields of chemical technology, decisively influencing our control over their steady-state efficiency or yield, dynamics (and, thus, controllability) and economic viability. In this scientific and technological field, experimentation still lags far behind our theoretical understanding of the underlying processes and mechanisms; as a result, the same happens with the practical implementation of dynamic models for design and online control algorithms. The theoretical and computational components of the latter aspect has advanced significantly with the contribution of the present authors,1-5 but such development still needs to be further evaluated, adapted, and complemented r 2011 American Chemical Society

by information and data accessible only by experiment, using typical liquid-liquid systems in specially designed and instrumented liquid-liquid contacting apparatuses.6 During the past decades, several drop or particle-size measurement techniques have been developed. Bae and Tavlarides7 discussed the operating range of each of these techniques; Singh et al.8 summarize some of the studies performed on liquid-liquid dispersions and describe their experimental techniques to obtain drop size distributions. Most of those techniques were employed to measure the drop size distribution (or mean drop size) in lean dispersions under steady-state conditions and with use of intrusive techniques. Our long-term experimental research objectives include obtaining drop size distributions, under both steady-state and dynamic (truly transient) conditions, namely, during very fast changes in the dispersed phase volume and solute concentration distributions inside the vessels after step or pulse changes in the operating conditions. In this case, the most satisfactory analysis methods are video techniques such as those described by Pacek and Nienow.9 As reported by these authors, the video technique allows very quick data collection, accurate measurement of drop Received: February 17, 2010 Accepted: December 20, 2010 Revised: October 27, 2010 Published: January 7, 2011 2182

dx.doi.org/10.1021/ie100368t | Ind. Eng. Chem. Res. 2011, 50, 2182–2191

Industrial & Engineering Chemistry Research size (from a few micrometers to a few millimeters), continuous monitoring even for long periods of time (as may be required when an event such as phase inversion occurs at a time difficult to predict), and performance of semiautomatic data treatment. A particularly important aspect of the video technique is that, given an adequately fast, noiseless camera, enough drops can be measured in a short time. According to the same authors, the video technique seems to be more reliable when transients are involved. The major difficulties to overcome in the studies performed up to now, however, involve the choice of a suitable lighting system to avoid the invasive nature of the system used by Pacek and Nienow and to allow accurate measurements in dense dispersions. Due to the fast improvement of image data acquisition and laser techniques, there have been a wide variety of work recently published in this field, such as that of Alopaeus et al.,10 Desnoyer et al.,11 and O’Rourke and MacLoughlin,12 but none of them use noninvasive techniques. Claims that specific invasive techniques yield results similar to those of noninvasive ones may be justified in some cases, but they should not be generalized to the point of doubting the interest and potential of completely noninvasive techniques, particularly when developments in lighting promise to extend their future applicability to much higher dispersed phase volume fractions. Miettien et al.13 have presented a short review of the experimental flow visualization techniques for the characterization of gas-liquid flow in a mixed tank. In what concerns bubble size distribution measurements, only phase doppler anemometry (PDA) and particle image velocimetry (PIV) are indeed noninvasive techniques. Nevertheless, they are also still restricted to low dispersed phase volume fractions and to transparent dispersions. So, for liquid-liquid systems, the implementation of the above and other noninvasive techniques should be further studied and explored. Given the difficulties associated with experimentation on liquid-liquid extraction processes, computer simulation work became increasingly attractive, as both the power and the availability of personal computing increased. Mathematical models for these kinds of systems are currently of such complexity and sophistication that analytical solutions are unavailable and numerical experimentation becomes mandatory for their internal and external validation. Also, computer simulation allows the researcher to extend his knowledge of the behavior of such systems and to test the various theoretical models proposed for their description and find their interdrop breakage-coalescence parameters. Recently Singh et al.14 presents a study of different breakage models. In this paper, we study different coalescence models and consider different options, even an unrealistic one (coalescence null). The behavior of liquid-liquid systems and, more specifically, of their dispersed phases, is the result of a complex combination of processes that occur at the level of a single drop. Individual drops may participate in two kinds of processes. The first kind involves continuous variation (through chemical reaction, mass and/or heat transfer, etc.); the second kind involves sudden variation (mainly of the number and volume of drops) and may result from drop breakage and coalescence, as well as from drop arrival/departure from any given control volume. The dispersed phase is formed by a population of drops which may be suitably described by means of a drop size distribution function f(v,t) dv, which is the number density (at time t) of drops within the infinitesimal volume interval dv, where v stands for the volume of a (generic) drop.

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In the case of a perfectly agitated batch vessel, this drop size distribution function obeys a conservation (or population balance) equation of the form ∂ dv ∂ f ðv, tÞ ¼ Bðv, tÞ - Dðv, tÞ ð1Þ ½f ðv, tÞ þ ∂v dt ∂t where B(v,t) and D(v,t) stand for the drop birth and death terms due to either breakage, coalescence, entry, or exit events. In the absence of Marangoni effects, the same equation would be valid for both equilibrated (with no mass transfer, as in the present work) and continuous to dispersed phase mass-transfer systems, providing the correct, concentration-dependent, physical property values were used in all interaction functions.4,15 The second term of the left-hand side of the conservation equation vanishes when drops are supposed to be incompressible and the righthand side expresses sudden variations that translate into loss of identity of a drop. Although the forms of these terms depend on the particular hydrodynamic system under consideration, they may be written in the general form Z ∂ f ðv, tÞ ¼ 2 f ðv0 , tÞ gðv0 Þ βðv, v0 Þ dv0 0 ∂t v >v Z 1 þ f ðv0 , tÞ f ðv - v0 , tÞ hðv0 , v - v0 Þ λðv0 , v - v0 Þ dv0 2 v0 v) size drops, g(v) is the breakage frequency, and β(v,v0 ) is the daughter drop size distribution; the second term represents the number of drops acquiring the v size by coalescence between v0 (0) sized drops, h(v0 ,ν-v0 ) is the collision frequency and λ(v0 ,v-v0 ) the coalescence efficiency; the third term represents the number of drops losing size v by breakage into any other smaller size; and the fourth term represents the number of drops losing size v by coalescence with drops of any size v0 . The discretized form of these balance equations was used by Laso et al.16 in order to compute the size distribution of drops in a stirred vessel, at both the stationary and transient states. In their implemented discretization, the volume domain was coarsely divided into a comparatively small number of intervals, the ratio of the amplitudes of contiguous intervals being 2. To keep the problem simple, the drops were supposed to break into two equal parts, and coalescence was allowed only between drops of equal size. Unfortunately, such simplifications have no physical support. The resulting system of (simple) ordinary differential equations was solved by means of the Michelson17 algorithm, which is a semiimplicit Runge-Kutta type, third-order, algorithm, using half-steps and full-steps to control the local truncation error. Al Khani et al.18 have employed a similar technique to simulate the behavior of the dispersed phase in an extraction column. The population balance equation for more realistic models, which incorporate a binary breakage mechanism leading to a full distribution of daughter drop volumes but ignore coalescence (which would make the problem a much harder one), has been solved by similar techniques. Their use is easily extended to the case of extraction columns, although at the price of much longer computation time.19 2183

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Industrial & Engineering Chemistry Research Ribeiro et al.3-5 published innovative algorithms for directly (numerically) solving the population balance equations (PBE) in three-dimensional phase space (drop volume, age, and solute concentration), with both breakage and coalescence. These enable the quick and accurate solution of both the transient and steady-state cases for a stirred vessel, under either batch or continuous conditions. The models used were Coulaloglou and Tavlarides’20 for the hydrodynamics of the dispersed phase (cf. Table 1), and both the rigid drop and the oscillating drop models, as proposed by Cruz-Pinto,21,22 for the mass transfer. Other interaction and mass-transfer models may easily be accommodated so that the numerical procedures themselves are of wide applicability. In the present work, and in the absence of mass transfer, we compare the steady-state experimental drop size distributions— obtained with a noninvasive technique (Ribeiro et al.6) for different operating batch system conditions—with those obtained by simulation during sufficiently long times to reach steady state, using the previously developed non-steady-state algorithm, as originally presented by Ribeiro et al.,3,4 for different coalescence models. The present paper describes and discusses in detail the process of ascribing numeric values to the kinetic parameters, by minimization of the deviations between the steady-state experimental and simulated data for a batch system, and derives physically significant conclusions. This optimization algorithm, using a Levenberg-Marquardt approach, obtains and compares the whole experimental and simulated distributions, rather than simply compare Sauter mean diameters (Podgorska23) or the three moments of the distribution (volume to surface mean diameter (d32), volume mean diameter (d30), and mean diameter (d10); Singh et al.14).

2. EXPERIMENTAL SECTION The liquid-liquid system used for the experiments consisted of mutually equilibrated toluene, as the dispersed phase, and distilled water, as the continuous phase. A glass vessel, 4.5 L volume, with flat bottom and four baffles was used. Agitation was provided by a standard Rushton turbine of 0.1 m diameter (half of the vessel’s diameter) placed at one-third of the vessel’s height. The toluene-water system was separated in a glass settler. A video-microscope-computer system for measuring the drop size distribution has been selected for a novel, noninvasive, technique. A Sensicam long-exposure camera was mounted on a low-magnification Nikkon SMZ-2T microscope, and these were fixed in a house-made support and placed in the correct position to obtain the pictures of the vessel’s contents. At the opposite side of the vessel, the lightning system was mounted with four lamps (12 V, focused beam, rear heat dissipation support, Reference Masterline Plus, Philips). At the present state of our experimental and data treatment procedures, good quality image data could only be obtained for lean dispersions at steady-state conditions, bearing in mind that our technique is completely noninvasive (with no probes disturbing the dispersions) Further improvements will be necessary and are being planned in the lightning system to extend the same noninvasive technique to dense dispersions in both steady and transient conditions. For the present work, the magnification was 25 and the focus was located up to 3 cm behind the vessel wall, which is much deeper than, for instance, in Pacek’s technique (Pacek and Nienow9). From the video pictures thus acquired, the drops were measured with a routine developed in Scion Image software

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(Release Beta 3b). In this specific routine, a mouse is used to place three points on the perimeter of each drop. The software then inserts a circle showing the size and the location of the drop, which will be stored in an ASCII (txt) file. If the operator misplaces one of the points, which would lead to a wrong size determination, he can visually detect it and repeat the three-point selection. After all drops have been measured for each experiment, the txt file was exported to a Microsoft Excel spreadsheet where the drop size in pixels is converted to centimeters. The drop volumes were then classified into 50 different logarithmic size fractions, from vmin = 10-7 cm3 to vmax = 10-2 cm3, such that HigherBoundVolj = vmin (vmax/vmin)(j/(no. of classes)). The need for discretizations of at least 32 logarithmic size classes followed from recent experimental reproducibility studies. Statistical studies over preliminary experiments have shown that a sample size of 2000 (minimum of 1500) drops was large enough to ensure experimental representativeness. This was obtained from size measurements on samples from the same steady-state dispersions with varying numbers of drops, until stable values of the average and standard deviation of the logarithm of drop volumes were obtained. In the present work, we used experimental drop size distributions obtained for the following four different combinations of nominal values of holdup and agitation speed: 1.85% of holdup with 100 rpm, 1.85% of holdup with 110 rpm, 2.17% of holdup with 100 rpm, and 2.17% holdup with 110 rpm. Each experiment was carefully performed, and a great number of frames (over 1000) were collected in different periods after reaching the steady state (at least 24 h). Although the statistical study mentioned above indicates that representativeness could be assured with 2000 drops, more than 4000 drops were identified and measured to obtain each experimental drop size distribution. Full experimental details and results have been published elsewhere.6,24

3. PROCEDURE The experimental drop size distribution data were compared with those predicted by direct numerical solution of the discretized un-steady-state drop size population balance equations, using exactly the same 50 experimental logarithmic size fractions, which ensured excellent resolution. As is well-known (and good standard practice), the use of logarithmic size fractions is, by definition, the only way of ensuring a uniform, drop size independent, relative discretization error, Δv/vh, in the drop volume distribution, where Δv and hv are the volume interval and average drop volume of any given class. This yields uniform width histogram classes in a log(v) scale (cf. plots). Given the exceptional speed (much faster than the real process, even during disturbances in the operating conditions) and efficiency of the simulation algorithm previously developed by Ribeiro et al.,3,4 the contactor’s hydrodynamic steady state was obtained as the final outcome of the whole, experimentally many hours long, transient start-up period. That steady state, of course, is only dependent on the operating conditions—dispersed-phase fraction (batch operation) or dispersed- and continuous-phase flows (continuous operation), agitation intensity, and dispersedphase feed drop size distribution (only for continuous operation)— and not on the particular vessel’s initial conditions at start-up. The effect of the dispersed-phase feed drop size distribution in continuous mixers (to be considered in future work) expectedly raises additional difficulties, because the agitation intensity also influences the actual, effective, inlet drop size distribution, 2184


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Table 1. Equations for the Interaction Models breakage equation

breakage frequency

daughter drop size distribution

normal distribution

coalescence equations

collision frequency

C2 σð1 þ φÞ2 ε1=3 d - 2=3 exp gðdÞ ¼ C1 FD ε2=3 d5=3 1þφ

!

0 2 1 2d3 - ðd 0 Þ3 C 2:4 B β ðd, d 0 Þ ¼ 0 3 exp@ - 4:5 2 A ðd Þ ðd 0 Þ3

hðd, d 0 Þ ¼ C3

ε1=3 ðd þ d 0 Þ2 ðd2=3 þ ðd 0 Þ2=3 Þ1=2 ð1 þ φÞ "

coalescence efficiency

λðd, d 0 Þ ¼ exp - C4

Coulaloglou/Tavlarides

# μC FC ε dd 0 4 σ 2 ð1 þ φÞ3 d þ d 0

2

3 2 2 0 2 3 0 3 d ð1 þ φÞ σ d þ ðd Þ þ ðd Þ 6 7 λðd, d 0 Þ ¼ exp4 - C5 5 FD ε2=3 d3 ðd 0 Þ3 d2=3 þ ðd 0 Þ2=3

Sovova

" 0

λðd, d Þ ¼ exp - C6

Tsouris

especially when (as in the usual practice) the dispersed-phase feed is introduced near the vessel’s impeller. Actually, the computer algorithm used is prepared to deal with this problem by adjusting the average size and standard deviation of the feed drop size distribution. As mentioned above, the simulation algorithm for the prediction of the drop size distributions in liquid-liquid systems is essentially the numerical solution of the population balance equation (eq 2), where conservation of mass within the dispersed phase must always be (and has in this work been) guaranteed. The breakage and coalescence frequencies used in this equation depend (cf. Table 1) on the physical and chemical properties of the system and on four unknown parameters (two for the breakage and two for the collision-coalescence). Simulation results thus depend not only on the liquid-liquid system and the operating conditions but also on the values assigned to these parameters. From the logic of the model-building process for drop breakage and coalescence, these parameters ought to be universal in character. However, due to both theoretical and experimental limitations, in addition to possible dispersion nonhomogeneity, agreement has not yet been reached in the literature about the values to be assigned to these parameters. Traditionally, each author presents his own set of parameter values (Table 2), based on trial-and-error attempts to reproduce experimental results. Occasionally, other authors (e.g., Pacek et al.25) have proposed methods for their separate determination. The algorithm used in the present work (Regueiras26) circumvents this difficulty by means of the Levenberg-Marquardt method for the nonlinear, multivariate, optimization of the full set of unknown parameters; although nothing more than a sophisticated trial-and-error method, it uses as a systematic and well-defined search strategy and a clearly defined stopping criterion. In the present work, the values taken from the literature for the Coulaloglou and Tavlarides model parameters are used as initial guesses for starting-up Regueira’s algorithm. In the present work, the Sovova27 and the Tsouris28 variations for coalescence frequency are also explored, along with the

#

μC ð1 þ φÞ FC ε1=3 ðd þ d 0 Þ4=3

Table 2. Parameter Values for the Breakage and Coalescence Equations Published by Different Authors (Referred to in Bapat;30 Bapat-1 and Bapat-2, Two Sets of Parameters Founded by This Author) authors

C1

C2

C3 (cm-3)

C4 (cm-2)

Coulaloglou31

0.00487

0.0552

2.17 10-4

2.28 109

32

Ross

0.00487

33

Hsia

0.01031 30

Bapat-1

30

Bapat-2

0.00481 0.00481

0.08 0.06354 0.05510 0.08

2.17 10

-4

-4

4.5 10

2.21 10 1 10

-3

-4

3 108 1.891 109 2 109 2 108

original Coulaloglou and Tavlarides20 model. The equations used in these models are shown in Table 1. The SLLMMARQ program developed by Regueiras,26 which we use for parameter optimization, is a MS Visual Basic personal computer implementation of the Levenberg-Marquardt method, conceived for the analysis of both batch and continuous systems, with a friendly Windows interface for data entry. In this interface, the user may specify the following: operating mode (continuous/batch); coalescence model (CoulaloglouTavlarides/Tsouris/Sovova); daughter drop distribution type (Gaussian/uniform/Euler’s beta); and insert the initial (guess) values for the interaction parameters (C1, C2, C3, C4, or C5 or C6); average volume and variation coefficient (standard deviation to the mean value ratio) of the feed or initial drop size distribution; values of the physical parameters of the liquid-liquid system under consideration (density, viscosity, and interfacial tension); experimental operating conditions (dispersed-phase holdup fraction in the feed stream, agitation power density, and mean residence time, where appropriate); measured steady-state drop size frequencies. The program then allows the user to specify the fitting (to the drop size distributions measured under different conditions, under an appropriate global objective function) of the interaction model parameters and/or mean drop size and variation coefficient 2185


Industrial & Engineering Chemistry Research in the feed stream and/or ratio of the dispersed-phase holdups inside the vessel and in the exit stream for the continuous vessel (the latter being, under steady-state conditions, necessarily equal to the feed stream holdup). The global objective function was originally defined as the sum of the squares of the deviations between the measured and calculated cumulated volumes multiplied by the width of each size fraction (which approximates the integral of the deviations and has the advantage of making the objective function itself much less sensitive to the discretization of the size axis). Under this specification, the optimization performed by Regueiras’ algorithm corresponds to matching the simulated to the measured cumulated distribution, because empirical evidence points toward an intrinsic greater instability of histogram matching, except perhaps when using relatively coarse discretizations of drop size distributions—a low number of linear, rather than logarithmic, size classes. To allow fitting number-size (as opposed to volumesize) distributions, the program includes, as an option, the further weighing of the above squared deviations by the inverse of the average volumes of the corresponding size fractions. To allow the visual assessment of the goodness-of-fit, the program also generates plots of the simulated and measured volume and number distributions, both as histograms and as cumulants. Since it has been established, along our preliminary optimization practice, that a good volume fit always corresponds to a bad number fit, a third option was added to the program, in which the squared deviations are weighed by the inverse of the square roots of the average volume of the corresponding size intervals. This compromise solution between the volume and number objective functions has been shown to yield the best combined fits of the volume- and number-based drop size distributions and has been used in the present work. To compare fits among data sets with different numbers of experimental runs, the objective function is divided by the number of runs within each set. It may be noted that, since the objective function used is defined in terms of cumulants, all the analyses over histograms have the value of mainly visual refinements—not necessarily very significant, given the random errors in the experimental individual drop class frequencies—of the analyses over cumulants. In this paper, we deal with the study of the hydrodynamics of the system under batch, steady-state, conditions. Contrary to the continuous-flow case, the product of the steady-state batch system is independent of the feed (initial) drop size distribution; thus, the elimination of the two feed parameters should make easier the search for physically significant values of the breakage and coalescence parameters. Our plan is to obtain the drop interaction parameter values from the batch experiments and investigate later their possible use for the continuous-flow system. Actually, it was the verification of the possibility of reaching different computational optima that justified the investigation of the influence of the initial model parameter values on the solutions obtained, described in the following paragraphs, and the ensuing clear identification of some excess of model parameters, namely, at least one interdependence between a pair of parameters. This highlights the need for a renewed model reappraisal.

4. SIMULATED VS EXPERIMENTAL BATCH SYSTEM RESULTS 4.1. Influence of the Initial Values of the Interaction Parameters. In this first stage of the use of the fitting algorithm,

it was necessary, before anything else, to investigate the way in

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Figure 1. Effect of an 11% change in the agitation rate on the steadystate drop volume distribution at constant (very low) dispersed phase holdup fraction.

which the optimized values of the parameters of the hydrodynamic drop interaction parameters (C1, C2, C3, and C4—cf. the equations in Table 1) may be affected by their initial guesses. To investigate this, and for the experimental batch, the algorithm was run starting with the best fit values suggested by different authors (cf. Table 2), in order to obtain the best statistical compensation of the measurement errors. The small drop diameters (,1 mm) expected in batch conditions limited the explored operating range, but the effect of the above variation in agitation speed, corresponding to more than 44% in agitation power input by a rotor of half of the vessel’s diameter, nevertheless significantly changed the steady-state distribution, as may be seen in Figure 1, and so is a useful test of the model and algorithm. Table 3 shows the results obtained from those optimization runs, which testify to a uniformly acceptable fit (the average objective function per experimental run of the order of 3 10-6) for all initial guesses, as the plots for drop volumes (Figures 2 and 3) and drop numbers (Figure 4) also show in greater detail. Volume distribution histograms show dispersed-phase volumes (or percentages), and not volume densities, as functions of log (volume), which is the correct, physically meaningful, representation when using logarithmic size fractions. The relevant finding is that, although the final goodness-of-fit is similar for the results from different initial guesses, the fitted values of the kinetic parameters are not the same. This may be a widely familiar result of the fortuitous fall of the solution in different local optima but may of course also uncover some correlation(s) between the parameters, if their number might be excessive. The value of C2 is invariable and, although the optimized values for C1 and C3 are slightly different for the different initial guesses, they nevertheless strongly suggest a fixed ratio between them. The values in Table 4 support this, where one sees that the same final C1/C3 ratio (4.664) results for all initial guesses. As discussed below, this seemingly physically genuine relationship between C1 and C3 means that their values may be defined only up to a multiplying constant, so that the value of one of these parameters should be made constant in order to ensure a more stable optimization process.fractions. The strictly constant C1/C3 ratio, together with the wide variability of the individual values of C1 and C3, may be physically interpreted as resulting from the fact that steady-state batch conditions imply a virtually infinite residence time, which means that it is impossible to set a definite time scale. In fact, the results 2186



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Table 3. Results of Fitting the Model to Batch Experiments, Starting from Different Initial Parameter Values optimized parameters authors (for initial guesses see Table 2) Coulaloglou Ross Hsia Bapat-1 Bapat-2 a

C2

C1

Obj-Fa

C3 (cm-3)

C4 (cm-2)

3.342 10-6

2.122 10-3

3.175 10-2

4.550 10-4

6.414 10þ2

3.342 10

-6

1.687 10

-3

-2

-4

3.707 10þ1

3.342 10

-6

3.833 10

-3

-4

8.059 10þ1

3.342 10

-6

3.017 10

-3

-4

8.147 10þ1

3.342 10

-6

4.299 10

-3

-4

1.257 10þ1

3.175 10

-2

3.175 10

-2

3.175 10

-2

3.175 10

3.617 10 8.219 10 6.469 10 9.218 10

Obj-F = objective function.

Figure 2. Experimental and simulated results of drop volume distributions for the Coulaloglou and Tavlarides’ model, in a batch system, for N = 100 rpm and φ = 1.85% (very similar agreement is obtained between experimental and simulated results for the other experimental conditions).

Figure 3. Experimental and simulated (Coulaloglou and Tavlarides’ model) results of cumulative drop volume distributions, in a batch vessel, for 111 rpm and 1.85% holdup (very similar agreement is obtained between experimental and simulated results for the other experimental conditions).

of the optimization of the two competing kinetics (breakage vs coalescence) can only be determined up to a common multiplying factor, as C1 and C3 are multipliers of kinetic constants (i.e., reciprocal time scales). Thus, the identity of the final steady states simulated with different (proportional) parameter values is to be expected, as the result of the kinetics of both formation and destruction of drops in the various size classes being affected in exactly the same way. An entirely analogous situation would arise in the study of a chemical equilibrium between two chemical reactions; such a system would be characterized by one single rate constant and the equilibrium constant. The particular value of the C1/C3 ratio detected by the Levenberg-Marquardt algorithm cannot, in fact, be seen as a pure matter of convenience in the fitting of the model to the experimental data; otherwise a variation, however small, of this value would be expected, as happens with the other parameters. Further and most important, if coalescence were physically absent within the system, the algorithm would not possibly converge to any fixed finite value of C1/C3 but would approach infinite. Otherwise, one would have to completely abandon the possibility of knowing drop interaction models depicting the reality to within any degree of certainty. This strongly supports the suggestion that coalescence must generally be present in systems of this nature, as discussed further below. Another inference from the results in Table 3 is that the C4 optimized values are always much lower and more variable than suggested by other authors. On the one hand, studies by Pacek29 suggest that the origin of this fact may be due to different drainage mechanisms of

the liquid film separating drops on their way to coalescence. On the other hand, simulations performed by the present authors, but not reported here, have shown that even large intentional variations of this parameter (C4) have smaller effects on the dispersion’s properties/structure than variations of the other parameters, as well as the optional use of alternative coalescence models (Sovova, Tsouris), as analyzed in section 4.3. 4.2. Study of the Dependence between C1 and C3 and Choice of the Value of the Parameter To Be Preset. Several studies were carried out to investigate the correlation between C1 and C3 and to “freeze” or preset one of these parameters. C3 was preset to values between 1 10-4 and 1 10þ2, and various fitting runs were performed on the other parameters (C1, C2, and C4). Table 5 shows the results of these runs, and they support the above explanation of the correlation between C1 and C3 but do not suggest any particular value for C3. Since, according to our plan, C3 must nevertheless be fixed, we selected C3 = 1 10-3 cm-3 (the same value found by Bapat,30 as Bapat-2; cf. Table 2). 4.3. Influence of the Coalescence Model on Parameter Optimization. Table 6 and Figures 2, 5, and 6 show the results of the optimization of the three coalescence models considered, setting C3 = 1 10-3 cm-3 . From these results, we may conclude that all three models lead to the same goodness-of-fit and that the common parameters converge to substantially the same values. Since the noncommon parameters have different physical meanings, it is useless to try to compare their optimized values. We will further elaborate only on Coulaloglou and Tavlarides’ model. 2187



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4.4. Results of Oversimplified Modeling (No Coalescence and Oversimplified Breakage). There has been much discus-

Figure 4. Drop number cumulated curves from the fit of the model (Coulaloglou and Tavlarides’ model) to the batch system, for the same run of Figure 3 (very similar agreement is obtained between experimental and simulated results for the other experimental conditions).

Table 4. Ratio C1/C3 Obtained When Starting from Different Initial Parameter Values initial guesses optimized parameters author

C1/C3 (cm3)

C1/C3 (cm3)

Coulaloglou

22.44

4.664

Ross

22.44

4.664

Hsia

22.91

4.664

Bapat 1

21.76

4.664

Bapat 2

4.81

4.664

Figure 2 also illustrates the very good agreement typically achieved between the predicted and experimental drop size distributions for batch operation of the vessel. The best drop interaction parameter values obtained were C1 = 4.664 10-3, C2 = 3.175 10-2, C3 = 10-3 cm-3, and C4 = 5.446 10þ2 cm-2, which do not greatly differ from the values previously proposed by Bapat30 (C1 = 4.81 10-3, C2 = 8 10-2, C3 = 1 10-3 cm-3, and C4 = 2 10þ8 cm-2), with the obvious exception of C4 (which has a small effect when the overall coalescence rate is already significantly low, as in the lean dispersions used here). As mentioned above, the predicted results are truly independent of the assumed dispersion’s initial state, so that for most of the calculations initial values vav = 0.0005 cm3 and σ/vav = 0.125 have been arbitrarily assumed. Figure 3 shows the same predicted and experimental results as cumulative distributions. As explicitly mentioned in the following section, a very significant feature of the above parameter values (and, of course, of the underlying model) is the fact that they apply to different operating conditions (dispersed-phase fractions and agitation intensities), which lends important support to the idea (so far limited to mere hope) that the parameters may be (relatively) universal, as they indeed should, if the physical soundness of the model (this or any other alternative one) is to remain unquestioned. Nevertheless, we explicitly recognize that a much wider range of operating conditions and different liquid-liquid systems will have to be studied in order to fully resolve the issue.

sion in the literature (e.g., Gerstlauer et al.19 and Steiner et al.34) about the possible usefulness of simplified interaction models, namely, by neglecting coalescence (which is the most difficult process to adequately describe, model, and compute) or by assuming that breakage occurs mostly into two equally sized daughter droplets and coalescence is only between equally sized drops. The effect of some of these assumptions was assessed for batch conditions (simultaneously using data from all four different experimental runs with different operating conditions), taking two possible initial states of the dispersion-feed A, coarse (and wider distribution), vs feed B, fine (and narrower distribution) dispersed-phase drops. The results (where the interaction parameter values again collectively apply to all four runs) are summarized in Table 7. From both theory and known experimental behavior of liquid-liquid dispersions, one undoubtedly expects that the final steady state, after the start-up transient, must be independent of the assumed initial state of the dispersion. Therefore, identical results must be obtained for any initial size distribution of the dispersed-phase drops, and that is what the model and algorithm predict, when (and only when—see Table 7) both breakage and coalescence are duly taken into account, for any set of operating (agitation intensity and dispersed-phase fraction) conditions. Actually, no real steady state (not merely a metastable one) can in fact be physically expected, and actually be reached, without simultaneous drop breakage and coalescence, even for very low levels of turbulence; if anything, such low levels of turbulence will only yield an inhomogeneous dispersion, without changing the physical nature of the underlying hydrodynamic, breakage/ coalescence, approach to equilibrium. Therefore, no model should predict a truly steady state if drop coalescence is neglected, so its elimination from the calculations (as discussed below) can be and is here used just to illustrate the consequences of such assumption. Of course, when coalescence is ignored, the only possible way the algorithm may closely reproduce the experimental results is by reducing the breakage frequencies until the resulting drop size distribution changes become lower than the algorithm’s stopping criterion—in this instance, a calculated drop entry/exit imbalance below 10-4% within any of the size classes—leading to a metastable rather than a truly steady state. The tabulated results show that extremely poor agreement between the predicted and experimental size distributions (cf. Figure 7) is obtained. We therefore strongly suggest that allowing for and modeling interdrop coalescence is mandatory for physically sound and accurate predictions of liquid-liquid systems hydrodynamic behavior. Experimental work in both agitated tank6,24 and column35-37 contactors (particularly in the latter, where wider and often bimodal distribution generally prevail) unequivocally documented the occurrence of interdrop coalescence, even in the leanest liquid-liquid dispersions (dispersed-phase holdup fractions lower than 5% in batch and continuous flow mixers, and 1-6% in continuous agitated columns). The tabulated results also show that the assumption of binary breakage into two equally sized drops, with the corresponding coalescence counterpart, does however not significantly change the predicted results. Nevertheless, we note that this small difference does not mean that the true model may be considered equivalent. The less realistic nature of the simplifying modeling would perhaps only come out clearly in a dynamic situation, 2188



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Table 5. Results of Fittings to the Batch System When Freezing the Value of C3 C3

C1

Obj-F

C2

C4 (cm-2)

C1/C3 (cm3)

1 10-4

3.342 10-6

4.664 10-4

3.175 10-2

2.566 10þ1

4.664

-3

-6

-3

-2

5.446 10þ2

4.664

3.342 10

4.664 10

3.175 10

1 10-2

3.342 10-6

4.664 10-2

3.175 10-2

3.781 10þ2

4.664

1 10-1

3.342 10-6

4.664 10-1

3.175 10-2

8.885 10þ0

4.664

1

3.342 10-6

4.664

3.175 10-2

1.430 10þ2

4.664

1 10þ1

3.342 10-6

4.664 10þ1

3.175 10-2

3.727 10þ2

4.664

1 10þ2

3.342 10-6

4.664 10þ2

3.175 10-2

1.264 10þ2

4.664

1 10

Table 6. Results of Fitting the Model to Batch Experiments, for Different Coalescence Models coalescence model

Obj-F

C1

C2

C3 (cm-3)

C4 or C5 or C6

Coulaloglou/Tavlarides Sovova

3.342 10-6 3.341 10-6

4.664 10-3 4.663 10-3

3.175 10-2 3.178 10-2

1 10-3 1 10-3

C4 = 5.446 10þ2 cm-2 C5 = 7.022 10-6

Tsouris

3.341 10-6

4.664 10-3

3.175 10-2

1 10-3

C6 = 1.469 10-4

Figure 5. Experimental and simulated results of drop volume distributions for the Sovova’ model, in a batch system, for N = 100 rpm and φ = 1,85% (very similar agreement is obtained between experimental and simulated results for the other experimental conditions).

Figure 6. Experimental and simulated results of drop volume distributions for the Tsouris’ model, in a batch system, for N = 100 rpm and φ = 1,85% (very similar agreement is obtained between experimental and simulated results for the other experimental conditions).

e.g. in a short time transient simulation.3 Our algorithm will however have to be subject to a greater modification to test such combined strictly binary breakage and coalescence behavior. 4.5. Unresolved Issues on the Drop Interaction Model. Despite the two, previously commented on, formal deficiencies of the interaction model used, there is nevertheless some support for its general basic soundness. However, the fact is that such a model, and really any of the physically realistic alternatives proposed so far, shows evidence of some excess of parameters. As shown, in several computer runs where C3 was varied by orders of magnitude (from 10-4 to 104 cm-3), an optimum C1 value could always be found, which varied in exact proportionality, thus unveiling a theoretically predictable physical dependence of (at least) these two parameters. This question is not yet resolved (and has not even been fully appreciated previously by many, if not all, other authors), but its physical basis may be briefly analyzed here. As a matter of fact, the constant C1 appears in the expression of the volume of the turbulent eddies that collide with a given drop in a given time interval (a fraction of which, given by the multiplying negative exponential, will lead to effective breakage), and C3 appears in the collision frequency of other drops with the same given drop

(a fraction of which, given by the coalescence efficiency, will lead to effective coalescence). Bearing in mind that the colliding drops are brought together by the very same turbulent eddies responsible for breakage, one may clearly recognize that both (collision) frequencies should be closely related (proportional). It is perhaps worth mentioning that Delichatsios and Probstein38,39 provided an alternative but similar formulation of the breakage frequency that yields an equivalent definite numerical value for C1, which could prove extremely useful for modeling and numerical prediction purposes. This should be fully explored in future work. Another outstanding issue, at least when the coalescence frequency is low (but not close to zero), as with most lean dispersions and systems similar to those investigated here, is the relative insensitivity of the coalescence frequency to C4, when this parameter is higher than, for instance, 8 cm-2. This may eventually be resolved by combining further, much required, coalescence theoretical model analysis with experiments with denser, highly coalescing dispersions, which cannot yet be easily probed by our noninvasive image capture and analysis systems. 2189



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Table 7. Results of Simplified Modeling C3 (cm-3)

C2

coalescence

feeda

Obj-F

gauss

yes

A

3.342 10-6

4.664 10-3

3.175 10-2

0.001 b

-6

4.670 10

-3

-2

b

-4

B gauss binary a

C1

daughter droplet distrib

null yes

3.341 10

-5

3.176 10

-1

0.001

C4 (cm-2) 5.446 10þ2 1.336 10þ2

-11 b

A

4.624 10

4.569 10

1.430 10

1.000 10

B

1.619 10-4

2.852 10-3

1.568

1.000 10-11 b

A

3.455 10-6

3.281 10-3

2.861 10-2

0.001 b

7.972 10þ2

B

3.302 10-6

3.281 10-3

2.861 10-2

0.001 b

1.702 10þ3

Feed A (coarse): vav = 0.0005 cm3 and σ/vav = 0.125. Feed B (fine): vav = 0.00001 cm3 and σ/vav = 1. b Fixed value as data.

Figure 7. Experimental and simulated (Coulaloglou and Tavlarides’ model) results of drop volume distributions, in a batch vessel, for 100 rpm and 1.85% holdup. The simulated results were obtained assuming no coalescence (gauss, null, B in Table 7).

5. CONCLUSIONS (1) A powerful optimization tool for drop interaction theoretical model parameter calculation was developed and tested against precise experimental drop size distributions in liquid-liquid dispersions. (2) An attempt at exploring the adequacy of different coalescence models has shown that, at least in this particular case, it is virtually indifferent the use of Tsouris, Coulaloglou/Tavlarides, or Sovova models. Good agreement has been obtained in all three cases. (3) Simulations where modeling of the coalescence processes is neglected, even for a very lean dispersion in an intensely agitated mixer, cannot predict the dispersion’s experimental steady state, therefore yielding physically unrealistic and inaccurate results, as physically expected and shown by the poorer agreement between the predicted and experimental drop size distributions. (4) The results have also shown that the ratio of the two preexponential parameters (C1, C3) in the expressions of the breakage and coalescence frequencies assumes a constant finite value, which supports a physically interpretable interdependence between the kinetics of these two processes. So, further work in the physical and mathematical development of the drop interaction models independent in just the right number seems to be still required. ’ AUTHOR INFORMATION Corresponding Author

*Tel.: 00-351-228340500. E-mail: [email protected].

0b 0b

’ NOMENCLATURE ε = energy dissipation (L2 3 T-3) Fc = density of the continuous phase (M 3 L-3) Fd = density of the dispersed phase (M 3 L-3) σ = interfacial tension (M 3 T-2) φ = volume fraction of the dispersed phase μc = viscosity of the continuous phase (M 3 L-1 3 T-1) d = drop size diameter (L) N = impeller speed (T-1) C1 = constant factor in the drop breakage frequency C2 = constant exponent in the drop breakage frequency C3 = constant factor in the drop coalescence frequency (L-3) C4 = constant exponent in the drop coalescence frequency (Coulaloglou; L-2) C5 = constant exponent in the drop coalescence frequency (Sovova) C6 = constant exponent in the drop coalescence frequency (Tsouris) Obj-F = object function (predicted vs experimental drop size distributions) vav = initial average drop volume (L-3) σ/vav = initial drop size distribution standard deviation/initial average drop volume ’ REFERENCES (1) Guimar~aes, M. M. L.; Cruz Pinto, J. J. C. Mass Transfer and Dispersed Phase Mixing in Liquid-Liquid Systems—I. Comput. Chem. Eng. 1988, 12 (11), 1075. (2) Guimar~aes, M. M. L.; Regueiras, P. F. R.; Cruz Pinto, J. J. C. Mass Transfer and Dispersed Phase Mixing in Liquid-Liquid Systems—II. Comput. Chem. Eng. 1990, 14 (2), 139. (3) Ribeiro, L. M.; Regueiras, P. F. R.; Guimar~aes, M. M. L.; Madureira, C. M. N.; Cruz Pinto, J. J. C. The Dynamic Behaviour of Liquid-Liquid Agitated Dispersions— I. The Hydrodynamics. Comput. Chem. Eng. 1995, 19 (3), 333. (4) Ribeiro, L. M.; Regueiras, P. F. R.; Guimar~aes, M. M. L.; Madureira, C. M. N.; Cruz Pinto, J. J. C. The Dynamic Behaviour of Liquid-Liquid Agitated Dispersions—II. Coupled Hydrodynamics and Mass Transfer. Comput. Chem. Eng. 1997, 21 (5), 543. (5) Ribeiro, L. M.; Regueiras, P. F. R.; Guimar~aes, M. M. L.; Cruz Pinto, J. J. C. Efficient Algorithms for the Dynamic Simulation of Agitated Liquid-Liquid Contactors. Adv. Eng. Software 2000, 31, 985. (6) Ribeiro, M. M. M.; Guimar~aes, M. M. L.; Madureira, C. M. N.; Cruz Pinto, J. J. C. Non-invasive System and Procedures for the Characterization of Liquid-Liquid Dispersions. Chem. Eng. J. 2004, 97, 173. (7) Bae, J. H.; Tavlarides, L. L. Laser Capillary Spectrophotometry for Drop-Size Concentration Measurements. AIChE J. 1989, 35 (7), 1073. (8) Singh, K. K.; Mahajani, S. M.; Shenoy, K,T.; Ghosh, S. K. Representative Drop Sizes and Drop Size Distributions in A/O Dispersions in Continuous Flow Stirred Tank. Hydrometallury 2008, 90, 121. 2190


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Optimization of Breakage and Coalescence Model Parameters in a

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