Ind. Eng. Chem. Res. 2009, 48, 8121–8133
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Population Balance Modeling of Liquid-Liquid Dispersions in Homogeneous Continuous-Flow Stirred Tank K. K. Singh,† S. M. Mahajani,*,† K. T. Shenoy,‡ and S. K. Ghosh‡ Department of Chemical Engineering, Indian Institute of Technology, Powai, Mumbai, India-400076, and Chemical Engineering DiVision, Bhabha Atomic Research Centre, Trombay, Mumbai, India-400085
This work presents population balance modeling to predict representative drop sizes and drop size distributions in dispersions of dilute phosphoric acid in a mixture of n-paraffin, tributyl phosphate, and di-2-ethyl hexyl phosphoric acid (D2EHPA) created in a continuous-flow stirred tank agitated with a top-shrouded pump-mix impeller having four trapezoidal blades. The stirred tank has been assumed to be homogeneous. Three population balance models, differing in droplet breakage rate models, have been evaluated. A part of the experimental data has been used to obtain the optimum values of the parameters of all the three models. The remaining part of the experimental data has been used to compare the models for their ability to predict the representative drop diameters and the number probability density distributions. On the basis of this comparison, the best model has been identified. 1. Introduction Liquid-liquid dispersions of immiscible phases in continuous-flow stirred tanks are frequently used in hydrometallurgical and chemical process industries. The objective of creating the dispersion is to provide large specific interfacial area between the phases across which a valuable component is preferentially extracted from one phase to the other. The specific interfacial area between the phases should neither be too small nor too large. If the interfacial area is too small the extraction will be poor, if the interfacial area is too large, owing to drop sizes being too fine, the settling vessel required for mechanical phase separation following the mixing will be large. A large settling vessel will lock in large inventory of the expensive solvents thereby tending to make the process uneconomical. The specific interfacial area can be estimated from the knowledge of the Sauter mean diameter evaluated from the drop size distribution and holdup of the dispersed phase using the following formula aj )
6φ a32
(1)
The drop size distributions are affected by several parameters such as impeller type, impeller speed, physical properties of the phases, throughput, holdup of the dispersed phase in the tank, etc. Unlike a batch-stirred tank where holdup is known a priori, holdup in a continuous-flow stirred tank is also affected by these parameters. A quantitative estimate of the effect of these parameters both on drop size distribution and holdup is, therefore, essential for optimum design of a continuous-flow stirred tank. This quantitative estimate can be obtained either from empirical correlations based on the experimental data or from mathematical models. The second approach is likely to give more reliable results than the former approach in which extrapolation beyond the experimental range is always risky. Figure 1 gives an idea of a fully predictive model for the hydrodynamics in a continuous-flow stirred tank. The model is represented by the dotted line in the figure. The information going into the model should be just the geometry, physical * To whom correspondence should be addressed. E-mail: sanjaym@ che.iitb.ac.in. Tel.: 91-22-25767246. † Indian Institute of Technology. ‡ Bhabha Atomic Research Centre.
properties of the phases, dispersed phase flow rate, continuous phase flow rate, and impeller speed. The output of the model should be the power input, holdup and the specific interfacial area. Internally, the model will embed the two-phase computational fluid dynamics (CFD) model and the population balance model. The two have to be solved together. This can be done by starting with a guess value of the representative drop diameter and solving the two-phase CFD model to predict holdup and power input. The predicted holdup and power input then go as the inputs to the population balance model which, in turn, predicts the representative drop diameter. This value is compared with the guess value of the representative drop diameter given earlier as an input to the two-phase CFD model. If the difference between the two is more than a prescribed tolerance, the representative drop diameter predicted by the population balance model goes as the revised estimate of the representative drop diameter into the two-phase CFD model for the next round of iteration. Process continues till the prescribed tolerance on the representative drop diameter is satisfied. Accurate prediction of the holdup and the representative drop diameters will, however, depend on the proper selection of the models for the interphase momentum exchange term in the twophase CFD model and the interaction kernels in the population
Figure 1. A fully predictive model of hydrodynamics in a continuous-flow stirred tank.
10.1021/ie800901b CCC: $40.75 2009 American Chemical Society Published on Web 01/27/2009
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Figure 2. Factors affecting population of droplets of size “a” in a given control volume.
balance model, as highlighted in Figure 1. There are no universal models for the interphase momentum exchange and droplet interaction kernels as reflected by a large number of empirical or semiempirical models reported in literature. Selection of the models suitable for the liquid-liquid system of interest cannot be done with confidence without the backup of some experimental information. When the experimental data are available for holdup and drop size distribution, the process of identifying suitable interphase momentum exchange model and droplet interaction kernels can be split into two steps. In the first step, with the experimental inputs on holdup, the population balance model can be solved with several different combinations of droplet interaction kernels to zero in on the combination that gives the drop size distributions close to the experimentally measured distributions. In the second step, with experimental inputs on representative drop diameters, the two phase CFD model can be solved with different models of the interphase momentum exchange term to zero in on the model that predicts holdups to be close to experimentally measured holdups. This study focuses on the first step, that is, identification of droplet interaction kernels for the system of dilute phosphoric acid dispersed in a mixture of D2EHPA, TBP, and n-paraffin. This has been done by choosing different combinations of the models for breakage, coalescence, and daughter droplet distributions, as reported in the literature and evaluating the model parameters therein using part of the experimental data followed by validation using unseen experimental data. The combination giving the closest match with the experimental data is identified. Considering a highly inhomogeneous energy dissipation rate in a stirred tank1 and dependence of drop size distributions on energy dissipation rate, it is reasonable to assume that the drop size distributions should exhibit spatial variations which need to be predicted accurately by a population balance model. A few attempts have been reported to model these spatial variations of drop size distributions; however, drop size distributions predicted by the inhomogeneous model are observed to be similar to the ones predicted by the homogeneous model of stirred tank.2 The computational complexities of the inhomogeneous models are obviously more. Considering this, and the experimental observations revealing that for the phase system being investigated in the present paper the representative droplet sizes do not change much with position, a homogeneous model of the stirred tank has been assumed. 2. Population Balance Equations for Continuous-Flow Stirred Tank In general, the drop size distribution in liquid-liquid dispersion in a continuous-flow stirred tank is affected by the size distribution of the droplets in the feed, breakage, and coalescence
of droplets and swelling or shrinking of droplets due to mass transfer. In the hydrodynamic studies where mass transfer is not involved, the last factor can be ignored. Figure 2 illustrates different phenomena affecting the population of a particular drop size in a control volume. The population balance equation for the drops of diameter “a” in a homogeneous control volume can be written as3 d {nA(a)V} ) ninAin(a)Qin + dt
∫
∞
a
∫√
β(a, a′) ν(a′) g(a′) nVA(a′) da′ +
3
a⁄ 2 {( 3 λ a - a′3)1⁄3, a′} 0 3)1⁄3}
a′
h{(a3 - a′3)1⁄3, a′} nA{(a3 -
nA(a′) da′V - nA(a)Qout - g(a) nA(a)V nA(a)
∫
∞
0
λ(a, a′) h(a, a′) nA(a′) da′V (2)
where the term on the left-hand side denotes the rate of accumulation of droplets of diameter “a”, first term on the righthand side denotes the rate at which the droplets of diameter “a” are coming with feed, second term on right-hand side denotes the rate at which droplets of diameter “a” are generated due to breakage of larger drops and the third term on righthand side denotes the rate at which droplets of diameter “a” are generated due to coalescence of smaller drops. The fourth, fifth, and the last term on right-hand side represent the rate at which droplets of diameter “a” are leaving the control volume, rate at which droplets of diameter “a” are disappearing due to breakage, and rate at which droplets of diameter “a” are disappearing due to coalescence with the other drops, respectively. n represents the number of droplets of all sizes per unit volume of the dispersion, V is the volume of the control volume, A(a) is the number probability density of the droplets of diameter “a”. nin represents the number of droplets of all sizes per unit volume of the feed stream, Ain(a) represents the number probability density of the droplets of diameter “a” in the feed stream, and Qin represents the volumetric feed flow rate. g(a′) is the breakage rate of the droplets of diameter a′, ν(a′)is the number of daughter droplets formed due to breakage of droplet of diameter a′, β(a,a′) is the size distribution of daughter droplets formed due to breakage. Usually binary breakage is assumed and hence ν(a′) ) 2. h(a,a′) is the symbol for the collision frequency between drops of diameter a and a′, λ(a,a′) is the symbol for collision efficiency for the interaction between drop of diameter a and a′. The coalescence rate per unit volume is defined as the product of the two. The upper limit of the integral in the third term on right-hand side corresponds to the droplet having volume equal to half of the volume of droplet of diameter “a” and ensures that the coalescence between droplets of diameter (a3 – a′3)1/3 and a′ resulting into generation of droplet
Ind. Eng. Chem. Res., Vol. 48, No. 17, 2009
of diameter “a” are accounted for only once. Qout is the volumetric out flow rate. Note that the control volume has been assumed to be homogeneous, that is, perfectly mixed such that the droplet concentration and the droplet number distribution in the outlet stream is same as that prevailing within the control volume. The above equation for a steady state continuous-flow stirred tank with bulk phases coming at the inlet reduces to
∫
∞
a
β(a, a′) ν(a′) g(a′) nVA(a′) da′ +
{(a - a′ 3
3)1⁄3
, a′}nA{(a - a′ 3
g(a) nA(a)V - nA(a)
∫
∞
0
3)1⁄3}
∫√ 3
a⁄ 2 {( 3 λ a - a′3)1⁄3, a′} 0
h
nA(a′) da′V - nA(a)Q -
λ(a, a′) h(a, a′) nA(a′) da′V ) 0 (3)
Multiplying eq 3 by da, dividing by Q, and defining a new variable such that Y(a) ) nA(a) da
(4)
which represents the number concentration of drops of diameter “a”, eq 3 can be rewritten as Y(a) ) τ
∫
∞
a
β(a, a′) daν(a′) g(a′) Y(a′) - τg(a) Y(a) τY(a)
τ
∫
∫
∞
0
λ(a, a′) h(a, a′) Y(a′) +
a⁄√2 {( 3 λ a - a′3)1⁄3, a′} 0 3
h{(a3 - a′3)1⁄3, a′}Y{(a3 a′3)1⁄3} Y(a′) (5)
where τ is the mean residence time. Equation 5 is the population balance equation for one drop size. Similar equations can be written for other drop sizes. To predict the drop size distribution these coupled nonlinear equations need to be solved. The complex and coupled nature of the equations renders the analytical solution difficult. They can, however, be solved numerically. Equation 5 can be converted to the following discretised form suitable for numerical solution. Nc
Yi ) τ
∑ β(a , a ) ∆aν(a ) g(a )Y - τg(a )Y i
j
j
j
j
i
i
j)i+1
Mi
τYi
∑ λ(a , a ) h(a , a ) Y + i
j
i
j
j
j)1
Li
τ
∑
λ(ak, aj) h(ak, aj) YkYj (6)
j)1
k:ak)(ai3 - aj3)1⁄3
where, Nc is the total number of drop classes (or the index of the drop class with the largest diameter) and will be a constant irrespective of the drop class for which eq 6 is written. ∆a is the width of each drop class; Mi is the index of the drop class, a drop of which on combining with a drop of class i will give the drop belonging to the drop class with the largest diameter. Mi is not a constant and will vary depending on the class for which population balance equation is written. Li is the index of drop class having characteristic drop volume just less than or equal to half of the characteristic volume of the ith drop class. This will also vary from equation to equation. k is the index of the drop class such that a drop belonging to this class on combining with a drop belonging to drop class with index j gives a drop having volume equal to a drop in class i. Note that before attempting solution, a priori estimate of the maximum possible drop size and minimum possible drop size should be available. While the latter can be safely assumed to be zero, the former can be predicted from one of the correlations
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for maximum stable drop diameter reported in literature or from experimental data, if they are available. Also the functional form of different rate terms of eq 6 and daughter droplet distribution should be known. This information can also be obtained from literature. Several studies on population balance modeling of liquid-liquid dispersions in stirred tanks have been reported in the literature. Some of them are reviewed in the next section and in the process different models for breakage, coalescence, and daughter droplet distributions are also reviewed. 3. Literature Review In their frequently quoted work, Coulaloglou and Tavlarides4 proposed models for drop breakage and coalescence rates. On the basis of the assumptions of locally isotropic flow field, drop sizes being in the inertial subrange of turbulence length spectrum, turbulent energy spectrum having a -5/3 dependence on wavenumber, negligible viscous effects and that an oscillating deformed drop will break if the turbulent kinetic energy transmitted to the drop by the turbulent eddies exceeds the drop surface energy, the breakage rate was shown to be g(ϑ) ) C1
(
ε1⁄3 σ exp -C2 2⁄3 5⁄9 ϑ2⁄9 Fdε ϑ
)
(7)
Binary breakage was assumed and daughter droplet distribution was assumed to be a normal distribution with variance such that >99.6% of daughter droplets formed lie within 0 to ϑ′ which resulted into the following functional form of daughter droplet distribution: β(ϑ, ϑ′) )
(
(2ϑ - ϑ′)2 2.4 exp -4.5 ϑ′ (ϑ′)2
)
(8)
Coalescence was assumed to be a two step process involving the collision between the droplets and some of the collisions resulting into final coalescence. Consequently, coalescence rate was assumed to be a product of collision rate and collision efficiency. The model for collision rate, based on assumption of analogy with the collision between molecules in kinetic theory of gases, was shown to be h(ϑ, ϑ′) ) C3ε1⁄3(ϑ2⁄3 + ϑ′2⁄3)(ϑ2⁄9 + ϑ′2⁄9)1⁄2
(9)
It was argued that for a collision to be fruitful, the contact time between the collided drops must exceed the coalescence time, the time required for the drainage of the intervening film of the continuous phase to a critical thickness after which the spontaneous drainage occurs. Assuming the coalescence time to be constant while the contact time to be a random variable, following expression for the collision efficiency was obtained
(
λ(ϑ, ϑ′) ) exp -C4
µcFcε 2
σ
(
ϑ1⁄3ϑ′1⁄3 ϑ1⁄3 + ϑ′1⁄3
)) 4
(10)
Accounting for dampening of turbulence due to dispersed phase holdup the above equations could be written as g(ϑ) ) C1
(
ε1⁄3 σ(1 + φ)2 exp -C2 2⁄9 (1 + φ)ϑ Fdε2⁄3ϑ5⁄9
h(ϑ, ϑ′) ) C3
)
ε1⁄3 (ϑ2⁄3 + ϑ′2⁄3)(ϑ2⁄9 + ϑ′2⁄9)1⁄2 (1 + φ)
{
λ(ϑ, ϑ′) ) exp -C4
µcFcε
(
ϑ1⁄3ϑ′1⁄3 σ (1 + φ) ϑ1⁄3 + ϑ′1⁄3 2
3
)}
(11)
(12)
4
(13)
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If specific turbulent energy dissipation rate is expressed by the following expression ε ) C5N3D2
(14)
then, eq 11-13 can be rewritten in terms of impeller diameter and speed as g(ϑ) ) C6
(
ND2⁄3 σ(1 + φ)2 exp -C7 2⁄9 (1 + φ)ϑ FdN2D4⁄3ϑ5⁄9
h(ϑ, ϑ′) ) C8
)
ND2⁄3 ( 2⁄3 ϑ + ϑ′2⁄3)(ϑ2⁄9 + ϑ′2⁄9)1⁄2 (1 + φ)
(
µcFcN3D2
(15) (16)
))
(
ϑ1⁄3ϑ′1⁄3 4 (17) σ2(1 + φ)3 ϑ1⁄3 + ϑ′1⁄3 The population balance model employing eq 8 and eq 15-17 was solved for a continuous-flow stirred tank with probability density of droplet size in the feed assumed to be Gaussian. The best values of the model constants were obtained by comparing the predicted distributions with experimental distributions of phase system consisting of dispersion of a mixture of kerosene and dichlorobenzene in water obtained in a 12 L baffled stirred tank agitated by 10 cm diameter turbine impeller. Good match was observed between the predicted and observed Sauter mean diameters. Predicted drop size distributions resembled the observed drop size distributions but no quantitative estimate of error was given. The optimum value of C6 was 0.40, of C7 was 0.08, C8 was 2.8 × 10-6, and C9 was 1.83 × 109 cm-2. Hsia and Tavlarides5 used the Monte Carlo method to predict the drop size distributions of liquid-liquid dispersions in continuous-flow stirred tank. The following diametric form of eq 11 and 13 were used for the breakage rate and the collision efficiency: λ(ϑ, ϑ′) ) exp -C9
g(a) ) C10
(
ε1⁄3 σ(1 + φ)2 exp -C11 2⁄3 (1 + φ)a Fdε2⁄3a5⁄3
(
λ(a, a′) ) exp -C12
aa′ ( ) a σ (1 + φ) + a′ ) µcFcε
2
)
(18)
4
(19)
3
A slightly different diametric form of eq 12 was used for the collision rate: ε1⁄3 (a + a′)2(a2⁄3 + a′2⁄3)1⁄2 (20) (1 + φ) Binary breakage was assumed and for daughter droplet distribution, the following function was used: h(a, a′) ) C13
( )( )
a3 2 a3 2 1 (21) a′3 a′3 Model predictions were compared with experimental data of Coulaloglou and Tavlarides6 and good agreement was observed. The best value of C10 was 4.87 × 10-3, C11 was 5.52 × 10-2, C13 was 2.17 × 10-4, and C12 was 2.28 × 109 cm-2. Sovova7 solved the population balance model for batch agitated liquid-liquid dispersions. They evaluated three different models for collision efficiency, the first model being the one proposed by Coulaloglou and Tavlarides,4 the second model being the one premised on the work of Howarth8 postulating that chances of coalescence of colliding drops depends on the impact of collision rather than on the intervening film drainage. This model expresses the collision efficiency as follows: β(a, a′) ) 30
(
λ(ϑ, ϑ′) ) exp -C14
σ(ϑ2⁄3 + ϑ′2⁄3)(ϑ + ϑ′) FdN2D4⁄3ϑϑ′(ϑ2⁄9 + ϑ′2⁄9)
)
(22)
The third model was a combination of the two. The model given by eq 22 was found to be the best and used in the final runs along with the breakage model given by eq 15 without holdup term, collision rate model given by eq 16 without holdup term, and the daughter droplet distribution model given by eq 8. The population balance model was solved several times to match the predictions with different experimental data reported in the literature. Each time a different combination of the optimum values of the model constants was obtained. The model constants were finally reported as a function of the ratio of impeller blade width to impeller diameter and number of impeller blades. Sauter mean diameters predicted by the population balance model were compared with the experimental values. In most of the cases the match was good; however, in some instances deviations as large as 40% were observed. For determining the best values of the model constants the objective function was to minimize the sum of squares of deviations of calculated and experimental drop size distributions. No quantitative estimate of goodness of fit was mentioned in any of the cases. However, in some case large discrepancies were acknowledged. Laso and co-workers9 presented a simplified way of solving the population balance models by descretizing the equations in such a way that the characteristic volume of drops in any given class was twice the characteristic volume of the previous drop class. They further assumed that breakage was a first order process, coalescence a second order process, breakage resulted into two equally sized drops and coalescence was possible among equally sized drops. They showed that while the mean diameters obtained by solving the population balance equations under the above assumptions were within 14% of the same obtained by solving full population balance equations, the computational time was 2 orders of magnitude less. Chatzi and Kiparissides10 solved the population balance model to simulate bimodal drop size distributions observed by them in their experimental studies on low holdup dispersions of styrene in distilled water laden with 0.1 g/L of polyvinyl alcohol (PVA), agitated by a 6 blade turbine impeller in a tank of 15 cm diameter operated in batch mode. Having observed predominance of breakage over coalescence due to low holdup and presence of PVA in their experiments, a simplified population balance model, without coalescence term, was used. The breakage model of Coulaloglou and Tavlarides6 was found to be inadequate to simulate bimodal distribution, and therfore the breakage model of Narsimhan and co-workers11 was used. This model assumed that the breakage frequency depends on the average number of eddies arriving at the surface of the drop per unit time and on the probability that the arriving eddy will have energy greater than or equal to the minimum increase in the surface energy required to break a drop. The expression for this breakage model is
{
g(a) ) C15 erfc
C16√σ ⁄ Fd
}
(23) a5⁄6ND2⁄3 For daughter droplet distribution, a model of erosive breakage was used. This model assumed that a parent drop of volume ϑ′ breaks into Nda equal volume (ϑ1) daughter drops and Nsa equal volume (ϑ2) satellite drops with a fixed volume ratio x ) ϑ1/ ϑ2, and the daughter droplet are normally distributed as assumed by Coulaloglou and Tavlarides.6 The expression for this daughter droplet distribution was ν(ϑ′)β(ϑ, ϑ′) ) Ndaβ(ϑ, ϑ′) + Nsaβ(xϑ, ϑ′) if ϑ e ϑ′ ⁄ (Ndax + Nsa)
(24a)
Ind. Eng. Chem. Res., Vol. 48, No. 17, 2009
ν(ϑ′)β(ϑ, ϑ′) ) Ndaβ(ϑ, ϑ′) if ϑ′ ⁄ (Ndax + Nsa) < ϑ e ϑ′ ⁄ (Nda + Nsa ⁄ x)
(24b)
ν(ϑ′)β(ϑ, ϑ′) ) 0 if ϑ > ϑ′ ⁄ (Nda + Nsa ⁄ x)
(24c)
The solution based on the assumption of constant Nsa and x did not give good match with experimental results. Consequently, linear dependence of Nsa and x on the volume of the parent drop was assumed such that Nsa ) 1 + int[SNsaϑ′]
(25)
x ) 1 + Sxϑ′
(26)
This altered the breakage model of eq 23 to
[{
g(a) ) C15erfc C16
Ndax2⁄3 + Nsa
}
1⁄2
-1 (Ndax + Nsa)2⁄3
√σ ⁄ Fd 5⁄6
a ND2⁄3
]
(27)
This modification resulted in better match between the predicted and observed distributions. However, the best values of the model constants including the slopes of eq 25 and 26 were not reported. Alopaeous and co-workers3 emphasized on the heterogeneous dissipation of energy in a stirred tank and argued that it should lead to spatial variations of drop size distributions which needed to be predicted by a computational model. To account for spatial variations of a stirred tank, a multiblock model comprising 11 compartments was proposed. These compartments differed in average values of energy dissipation rate and exchanged flow. The information of the average energy dissipation rates and exchange flow rates was obtained from single-phase CFD simulations with suitably averaged physical properties. To demonstrate the concept, transient population balance equations were solved both for single block model and multiblock model of the batch stirred tank. Breakage rate, coalescence rate, and the model parameters of Hsia and Tavlarides5 were used in simulations. The following daughter droplet distribution was used:
( )( )
a2 a3 2 a3 2 1 (28) a′3 a′3 a′3 Comparison of Sauter mean diameter as predicted by the single block model and the multiblock model showed the maximum difference to be within 9% during initial times of the transient simulations and less than 2% for the converged steady state values. Alopaeous and co-workers2 applied the multiblock model to carryout parametric fitting for Exxsol in water dispersions in a 50 L batch stirred tank agitated by a Rushton turbine. Drop size distributions were measured at three different locations in the tank. In the population balance model the following breakage rate model was used: β(a, a′) ) 90
[
g(a) ) C17ε1⁄3erfc
C18
σ Fcε a
2⁄3 5⁄3
+ C19
µd
√FcFdε
1⁄3 4⁄3
a
]
(29)
This model was based on the breakage model of Narsimhan and co-workers11 modified to account for viscous forces within the drop phase as quantified by Calabrese and co-workers.12 Equations 20 and 28 were used for the collision rate and the daughter droplet distribution. Following model of Tsouris and Tavlarides13 was used for collision efficiency:
(
0.26144µd λ(a, a′) ) +1 µc
)
-
C20µc FcNP1⁄3ε1⁄3(a + a′)2⁄3D2⁄3
(30)
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Parameters were estimated for both the single block model and 11 blocks model of the stirred tank. Comparison of experimental Sauter mean diameters and predicted Sauter mean diameters by the single block model was shown to be good; however, some deviations in drop size distributions were admitted. Comparison of experimental and predicted Sauter mean diameter was shown for the multiblock model but no comparison of the drop size distributions was shown. However, it was mentioned that the drop size distributions obtained by multiblock model were quite similar to single block situation. A recent study14 attempted to model drop-size distributions for the dispersion of toluene in water in a batch stirred tank. Combinations of breakage rate, coalescence rate, and daughter droplet distributions, as used by Coulaloglou and Tavlarides4 and Alopaeus and co-workers2 were tried. Free parameters of the models were obtained by fitting transient drop-size distributions observed for a particular impeller speed (550 rpm). The models with optimized parameters could predict the Sauter mean diameters for low holdups but failed to predict the Sauter mean diameters for high holdups. Even for small holdups, significant deviations (around 30%) for small Weber numbers were observed. Podgorska15 modified droplet interaction kernels to account for intermittency of turbulence and used them to predict sizes and distributions of high viscosity silicone oil droplets for low holdup (φ ) 0.0038). Though the population balance model could predict the Sauter mean diameters, the predicted drop size distributions were significantly different from the experimental distributions. From the review of the above studies, it can be observed that the processes affecting the drop size distributions of liquid-liquid dispersions in a stirred tank are very complex. There is no universal model applicable to all phase systems as evident by a large number of models for breakage rate, coalescence rate, and daughter droplet distribution reported in literature. Even if a given combination of droplet interaction models is able to predict the drop-size distributions for two different phase systems, the model constants are bound to be different. In view of this, identification of a phase specific combination of droplet interaction kernels and optimization of parameters therein is necessary. In fact, creation of such a database for industrially important systems is essential. Our study on the system of dilute phosphoric acid dispersed in a mixture of D2EHPA, TBP, and n-paraffin, an industrially important system used in recovery of rare earths from phosphoric acid,16 is a new contribution to the existing database scattered in the form of several research papers in literature. 4. Experimental Details The experimental setup, shown schematically in Figure 3, consisted of a cylindrical tank of 240 mm diameter and 240 mm height with four 10% baffles, each of 220 mm height. The inlet of the phases was from the bottom of the mixer and the outlet of the dispersion was from a central opening at the top. A top shrouded pump-mix turbine having four trapezoidal blades (D ) 14.8 cm, B ) 3.1 cm, L ) 3.7 cm, C/T ) 0.5, Np ) 3.0) was used to create the dispersion. The impeller is shown in Figure 4. A settler vessel was provided after the mixer to separate and recycle the phases to have the setup operated in a closed loop. The mixer had two ports for sample withdrawal with rubber-clip arrangement, one in the plane of the impeller disk and the other in a plane halfway between the impeller disk and bottom of the mixer. The organic phase used in the experiments was a mixture of n-paraffin, D2EHPA, and TBP. The aqueous phase was 30% phosphoric acid. This phase system was selected because of its industrial importance in recovery of rare earths from phosphoric acid.16 All the experiments were
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Figure 3. Schematic diagram of the experimental setup.
Figure 5. Typical image of the stabilized dispersion.
5. Parameter Estimation As mentioned earlier, the first task is to predict the best values of the model constants such that the drop-size distributions predicted by solution of population balance equations are close to the experimentally observed drop-size distributions. Since it is difficult to quantitatively compare the predicted and experimental distributions completely, it is better to compare some characteristics of the distributions. The obvious choice for this purpose can be some moment of the distribution. Since two widely different distributions can have the same given moment, it is better to compare several moments instead of comparing just one. In the present study, instead of moments of distributions, functions of ratios of moments of distributions, each corresponding to a representative droplet diameter, have been used to quantify closeness of the predicted distributions to the experimentally observed distributions. This function of ratio of the moments can in general be defined as
Figure 4. The pump-mix impeller used in the mixer. Table 1. Physical Properties of the Phases (at 25 °C) phase organic aqueous
F (kg/m3) 859 1328
µ (cP) 4.01 4.28
σs (N/m) 0.0267 0.0194
performed with aqueous phase as the dispersed phase. The physical properties of the phases are given in Table 1. The drop-size distributions were measured by image analysis of the dispersions stabilized by withdrawal from sample ports into a Petri dish containing continuous phase laden with a coalescence inhibiting surfactant (see Supporting Information video). The video shows that in the absence of surfactant, drops undergo coalescence at a rapid rate. In the presence of surfactant, coalescence reduces drastically with only one coalescence event observed in the left lower corner of the field of view. A typical image of the stabilized dispersion is shown in Figure 5. The representative drop sizes at the two withdrawal ports were found to be more or less similar, with a variation of (10%. Considering this and a small value of interfacial tension, the dispersion can be assumed to be homogeneous for all practical purposes. The complete details of the experiments, setup, and methodology are reported elsewhere.17 From the experimental data, following correlation for the Sauter mean diameter was obtained: a32 ) 1.849 × 10-3 N-1.7025(1 + 0.392φf + 3.2435φf2) × D exp(0.4302τ) (31)
amn )
[
∫ ∫
∞
0
∞
0
amA(a) da n
a A(a) da
]
[ ] Nc
1 m-n
)
∑
ami Yi
∑
ani Yi
i)1 Nc
1 m-n
(32)
i)1
When m ) 3, n ) 2, eq 32 gives volume surface mean diameter (a32) alternatively known as the Sauter mean diameter which is directly related to the specific interfacial area. When m ) 3, n ) 0, eq 32 gives the volume mean diameter (a30) and when m ) 1, n ) 0, eq 32 gives the mean diameter (a10). These diameters can be computed from the drop-size distributions predicted from the solution of population balance equations as well as from experimentally measured drop-size distributions and can be used for quantitative comparison of the predicted and observed distributions. Another reason for choosing these functions of ratios is that very often these, specially a32, are used in design rather than the whole drop-size distribution and hence a prerequisite for a population balance model is to be able to accurately predict these diameters. Out of the experimental data set 14 points were used for finding the optimum values of the model constants. Since for each distribution 3 representative diameters were computed, a total of 42 points have been used to optimize the model constants.
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Figure 6. Effect of number of drop classes on droplet number probability density distribution.
Figure 7. Effect of number of drop classes on representative diameters.
Figure 8. Effect of number of drop classes on satisfaction of material balance. Table 2. Constituents of Different Models Compared in the Present Study model-1 breakage rate collision rate coalescence rate daughter droplet distribution fitted parameters
eq eq eq eq
18 20 19 28
C10C11C12C13
model-2 eq eq eq eq
model-3
38 20 19 28
eq eq eq eq
C21C22C12C13
29 20 19 28
C17C18C19C12C13
Any optimization algorithm will be iterative because of nonlinear dependence of drop-size distribution on the model constants and would therefore require repeated evaluation of drop-size distributions for a large number of combinations of values of the model constants. Each evaluation of drop size distribution would itself require a large number of iterative calculations as the equations being solved are coupled and nonlinear. This will make the optimization
Table 3. Optimum Values of the Model Constants for Model-1, Model-2, and Model-3 model-1 model-2 model-3
C10 ) 0.17 C21 ) 1.17 C17 ) 7.70 m-2/3
C11 ) 0.016 C22 ) 0.120 C18 ) 0.015
C12 ) 1 × 1013 m-2 C12 ) 1 × 1012 m-2 C19 ) 0.010
C13 ) 0.0970 C13 ) 0.001 C12 ) 2 × 1012 m-2
C13 ) 0.007
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Table 4. Results of Validation of Population Balance Models 1-3 with Experimental Data model-1
a32 a30 a10 overall
model-2
model-3
average % error
beyond (20%
average % error
beyond (20%
average % error
beyond (20%
11.68 13.28 16.21 13.72
2/10 0/10 2/10 4/30
17.93 15.27 16.70 16.63
5/10 3/10 3/10 11/33
17.92 13.82 11.37 14.37
3/10 2/10 1/10 6/30
routine very slow. If the initial guess is far from the final solution the process can be prohibitively long. The solution to this problem is to carry out quick parametric studies by solving population balance equations. The results from the parametric studies can be then used to obtain correlations between the predicted representative drop diameters and the varied parameters. These correlations can then be used in the optimization algorithm to have a quick and good estimate of the model parameters. This estimate can then go as the initial guess into the final optimization problem embedding the complete population balance model in the optimization algorithm. With this, the final optimization algorithm converges in a few iterations. The coupled nonlinear population balance equations were solved using Newton-Raphson algorithm. Population balance equations were solved for drop classes 1 to Nc-1 along with the
Figure 9. Comparison of measured and predicted Sauter mean diameters.
following equation to satisfy the holdup constraint instead of the population balance equation for the last drop class: φ)
∫
∞
0
Nc
∑
π 3 π a nA(a) da ) a3Y 6 6 i)1 i i
(33)
While for a batch system the holdup value is constant and known, for a continuous-flow stirred tank holdup changes with process parameters. For the system investigated in the present study, the following is the dependence of holdup on process parameters:17 φ ) 2.1214φf0.4012N-1.097τ-0.0631 ) 2.1214
(
)
0.4012 Qd N-1.097 Qd + Q c -0.0631 V (34) Qd + Q c
(
)
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Figure 10. Comparison of measured and predicted volume mean diameters.
Figure 11. Comparison of measured and predicted mean diameters.
Value of specific power input to be used in breakage and coalescence models, can be evaluated from the known value of the impeller power number and the impeller diameter (Np ) 3, D ) 0.148 m) as follows: ε)
NpN3D5 V
(35)
where V is the volume of the stirred tank, Qd and Qc are the dispersed phase and continuous phase flow rates, N is the impeller speed, φf is the dispersed phase holdup in the feed.
In the simulations drop diameters were assumed to range between 0 and 1000 µm, with the upper limit being well above the maximum droplet diameter observed in experiments. To begin with, the effect of discretization on dropsize distribution was studied. For this, eq 18, 19, 20, and 28 were used for breakage rate, collision rate, collision efficiency, collision rate, and daughter droplet distribution, respectively, with the model parameters of Hsia and Tavlarides.5 The results are shown in Figure 6 and Figure 7. As can be seen, the distribution for Nc ) 10 substantially differs
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Figure 12. Comparison of measured and predicted drop size distributions.
from the distributions for higher Nc values. For a larger number of drop classes, though the distributions do not differ significantly, the peak values do, with the latter increasing with the number of drop classes. However, for Nc ) 40 and Nc ) 50, the peak values are practically the same. For Nc )
50, the different representative diameters also approach their asymptotic limits, as shown in Figure 7. While solving the population balance equations it is equally important to cross check the material balance which can be done, for example, by comparing the total volume of droplets being born
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Figure 13. Overall comparison of three population balance models.
of breakage per unit time per unit volume of dispersion and total volume of droplets dying in breakage per unit time per unit volume of dispersion. These values can be computed as Nc-1 Nc-1
Vb )
∑ ∑
i)1 j)1+1
( π6 a ) β(a , a )∆aν(a ) g(a ) Y 3 i
i
j
j
j
j
(36)
for volume being born of breakage per unit time per unit volume and
model parameters. As a rule of thumb, the number of data used in optimization should be at least equal to the square of number of parameters. Though 42 data points have been used in optimization of 4 model parameters, these 42 data points actually belong to 14 distinct drop-size distributions. This almost satisfies the rule of thumb mentioned above. For all the models the best values of the model constants were evaluated using the method described earlier. The results are given in Table 3.
Nc
Vd )
∑ ( π6 a )g(a ) Y 3 i
i
(37)
i
i)2
for volume dying in breakage per unit time per unit volume. Figure 8 shows the effect of number of drop classes on Vb and Vd. As can be seen on increasing the number of drop classes the two volumes come close to each other and are equal for Nc ) 50. Considering the observations made from Figures 6-8, for subsequent simulations Nc ) 50 was used which corresponded to a drop class interval equal to 20 µm. Having frozen the value of Nc, different combinations of breakage rate, collision rate, collision efficiency, and daughter droplet distributions were tried. Apart from eq 19 for collision efficiency, eq 20 for collision rate, and eq 28 for daughter droplet distributions, other models tended to give distributions widely different from experimental ones, whereas, several models of breakage rate tended to give distributions closer to the experimental one. On the basis of this observation three population balance models were formulated. The constituent breakage rate, collision rate, collision efficiency, and daughter droplet distributions of these models are listed in Table 2. In the second model the following breakage rate model of Narsimhan and coworkers,11 but with the modification to account for damping by dispersed phase holdup, was used:
{
g(a) ) C21erfc
C22(1 + φ)√σ ⁄ Fc
}
(38) a5⁄6ε1⁄3 Since the goodness of fit will depend on the number of adjustable parameters, for model-3 C19 was assumed to be constant so as to compare the models strictly with identical number of adjustable parameters. C19 was chosen to be 0.01 as with this value of C19 drop-size distribution predicted by population balance model for a exploratory case matched qualitatively with the experimental dropsize distribution. In any optimization problem a large number of experimental data should be used to obtain statistically accurate values of
6. Validations Population balance models 1-3 with the optimized values of model constants were used to predict the representative diameters and drop-size distributions for the conditions of 10 experimental runs. The results are summarized in Table 4. Figure 9 shows the graphical comparison of predicted and experimental values of a32 with (20% error bands. Along with that of model 1-3, predictions of the model of Alopaeus and coworkers2 with the original model parameters and reoptimized model parameters and empirical correlation are also shown. While predictions of model-1 and model-3 are clearly better than the model-2, the predictions of model-1 are slightly better than model3. This is in line with the deviations listed in Table 4. There seems to be a trend in the deviations of model-3 with the smaller Sauter mean diameters overpredicted and the larger Sauter mean diameters under predicted. The dispersed phase holdup values corresponding to the larger Sauter mean diameters are larger than the ones corresponding to the smaller Sauter mean diameters. The breakage model used in model-3 does not account for the dampening effect of the dispersed phase holdup as this was found to be better than its counterpart accounting for the dampening effect when all the experimental data used for optimization were analyzed together. The trend in the deviation, however, suggests that the predictions can be improved by treating the cases of the larger and the smaller holdups separately with the breakage model accounting for damping to be used for the higher holdup cases and breakage model without dampening for the lower holdup cases. This will ensure more breakage for smaller holdup cases and less breakage for high holdup cases, thereby improving the predictions of model-3. Using two different breakage models and finding the optimum model parameters therein is, however, not being attempted here in view of the limited number of the experimental data. The processes of redispersion and coalescence dictating the drop size distributions are very complex. Any deterministic model of these complex processes is expected to have some
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limitations as evident by some recent studies.14,15 In view of this and the range of process conditions varied during the experimental runs, predictions of model-1 and model-3, though not very accurate, are satisfactory. Considering that even the empirical correlation in almost 25% cases predicts the Sauter mean diameter to be beyond (15% of the experimental values,17 the predictions of model-1 and model-3 can be considered good. The model of Alopaeus and co-workers2 with original model parameters exhibits large deviations. The same model with reoptimized parameters shows reduced but still significant deviation. This observation once again highlights the need of identifying the best combination of droplet interaction kernels for the system of interest. Figure 10 shows the graphical comparison of predicted and experimental values of a30 with (20% error bands. As observed in case of the Sauter mean diameters, the volume mean diameters are predicted better by model-1 and model-3 than model-2. The model of Aplopaeus et al. (2002) gives poor prediction both with original and reoptimized parameters. The similar observations hold for the mean diameter a10, predictions and measurements of which are compared in Figure 11. Evidently, results of model-1 and model-3 are comparable and better than model-2. To further decide on the best model for the system investigated in this study, a comparison of complete drop-size distributions is necessary. Defining the following error function will facilitate this comparison: M
∑ |A(a )
i experimental - A(ai)model
E)
i)1
M
∑ A(a )
| × 100
(39)
i experimental
i)1
Where, M denotes the index of the highest class for which some drop was observed in the experiments. Physically, eq 39 represents the ratio of the difference of the heights of experimental and model drop-size distributions averaged over all drop classes and height of experimental drop-size distribution averaged over all drop classes. Graphical comparison of predicted and measured drop-size distributions is shown in Figure 12. Figure 13 shows the overall comparison of the models. As can be seen from Figure 12, in most of the cases model-3 gives the better match with the experimental data. This is more clearly evident from Figure 13 showing that in 7 out of 10 cases model-3 give the best match. On the other hand, the model-1 which gives the best predictions of the Sauter mean diameters, gives the worst match in 7 out of 10 cases. A similar observation was observed in a recent study wherein the prescribed population balance model, though predicted Sauter mean diameters very well, gave large deviations in drop-size distributions.15 Therefore, among the compared models, the model-3 is the best population balance model to predict the representative drop sizes and drop-size distributions for the phase system investigated in this study. Having obtained the best population balance model for the phase system of our interest, it would be worth checking whether the recommended population balance model justifies our assumption of homogeneous dispersion. This assumption was premised on the experimental observation that the difference between the representative drop diameters at two different locations in the tank was within (10%17. In their study on batch system, Coulaloglou and Tavlarides6 found the Sauter mean diameters away from the impeller to be 7% more than in the impeller region. With this difference in Sauter
mean diameters, they too considered the dispersion to be homogeneous and justified the assumption by showing that the circulation frequency was much higher than the coalescence frequency. The coalescence frequency was defined as volume fraction of dispersed phase coalescing per unit time. In our model the volume being generated due to coalescence per unit volume of the dispersion per unit time can be expressed as Li
Nc
Vbc )
∑
∑
i)2
( π6 a ) λ(a , a ) h(a , a ) Y Y 3 i
j)1
k
j
k
j
k j
(40)
k:ak)(ai3 - aj3)1⁄3
Using the definition of Coulalglou and Tavlarides,6 the coalescence frequency can be expressed as N
Vbc 1 c ) wc ) φ φ i)2
∑
Li
∑
j)1
( π6 a ) λ(a , a ) h(a , a ) Y Y 3 i
k
j
k
j
k j
k:ak)(ai3 - aj3)1⁄3
(41) The circulation frequency can be computed from the volume of the tank and impeller flow number. If NQ is the flow number of the impeller, the circulation frequency can be computed to be NQND3 (42) V An ongoing CFD study of the system reveals that the flow number of the impeller used in this study is 0.56. The ratio wc/fc can be used as a measure of the degree of homogeneity of the dispersion. When the models of λ and h of model-3 are used, this ratio can be shown to be small. For example, for run no. 1, this ratio can be computed to be 0.093 which means the circulation frequency is much higher than the coalescence frequency, justifying our assumption of neglecting the spatial variations of drop-size distributions. fc )
7. Conclusions Population balance framework has been applied to model the drop-size distributions in dispersion of dilute phosphoric acid dispersed in a mixture of n-paraffin, TBP, and D2EHPA in a continuous-flow stirred tank agitated by a four-bladed top shrouded pump-mix turbine with trapezoidal blades. Three population balance models differing in the breakage rate models have been compared. A part of the experimental data has been used to find out the optimum values of the model constants. Remaining experimental data have been used for validation of models. Model-3 defined by Table 2 and Table 3, is observed to be the best population balance model for the system investigated in this study. The study also highlights the fact that the best combination of the droplet interaction kernels to be embedded in the population balance model is highly system specific and use of the same combination to predict the dropsize distributions for other systems may lead to error. Supporting Information Available: Video showing rapid coalescence of the droplets in the absence of surfactant and almost no coalescence of the droplets in the presence of surfactant. This material is available free of charge via the Internet at http://pubs.acs.org. Appendix Nomenclature a ) droplet diameter [m] aj ) specific interfacial area [m2/m3]
Ind. Eng. Chem. Res., Vol. 48, No. 17, 2009 amn ) function of ratio of mth and nth moments of drop size distribution [m] a10 ) number mean droplet diameter [m] a32 ) Sauter mean diameter [m] a30 ) volume mean diameter [m] A(a) ) number probability density distribution [1/m] Ain(a) ) number probability density distribution in inlet stream [1/m] B ) impeller blade width [m] C ) impeller off-bottom clearance [m] Ci ) model constant [some dimensionless some with dimensions] D ) impeller diameter [m] E ) error function defined by eq 39, unitless fc ) circulation frequency [1/s] g(a) ) breakage frequency of droplet of diameter a [1/s] h(ai,aj) ) collision frequency between droplets of diameter ai and aj per unit volume [m3/s] L ) impeller blade length [m] Li ) index of drop class a drop of which has the volume just less than or equal to half of the volume of drop of class i, unitless Mi ) index of drop class a drop of which on combining with a drop of class i gives a drop belonging to the highest drop class, unitless n ) number concentration of droplets [1/m3] nin) number concentration of droplets in inlet stream [1/m3] N ) impeller speed [1/s] Nc ) number of drop classes in population balance model, unitless Np) power number of the impeller, unitless NQ ) flow number of the impeller, unitless P ) power input [kg m2/s3] Q ) volumetric flow rate [m3/s] Qin) inlet volumetric flow rate [m3/s] Qout ) outlet volumetric flow rate [m3/s] ϑ ) volume of the drop [m3] V ) volume of the stirred tank [m3] Vb) volume of drops born of breakage per unit volume per unit time [1/s] Vbc ) volume of drops born of coalescence per unit volume per unit time [1/s] Vd ) volume of drops dying in breakage per unit volume per unit time [1/s] wc ) coalescence frequency [1/s] Y(a) ) number concentration of droplet of diameter a [1/m3] Greek Letters β(a,a′) ) daughter droplet probability distribution [1/m] ε ) specific energy dissipation rate [m2/s3] λ(ai,aj) ) efficiency for collision between droplets of diameter ai and aj, unitless µc ) viscosity of continuous phase [kg/m/s] ν(a) ) number of droplets formed on breakage of droplet of diameter a, unitless φ ) dispersed phase holdup, unitless φf ) dispersed phase feed holdup, unitless Fc ) density of continuous phase [kg/m3]
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Fd ) density of dispersed phase [kg/m ] σ ) interfacial tension [kg/s2] σs ) surface tension [kg/s2] τ ) mean residence time [s] AbbreViations CFD ) computational fluid dynamics NPD ) number probability density [1/m] 3
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ReceiVed for reView June 8, 2008 ReVised manuscript receiVed November 14, 2008 Accepted November 24, 2008 IE800901B
