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Ind. Eng. Chem. Res. 2010, 49, 2456–2468
Modeling of Particle Size Distribution for Semibatch Precipitation of Barium Sulfate Using Different Activity Coefficient Models C. Steyer,† M. Mangold,‡ and K. Sundmacher*,†,‡ Process Systems Engineering, Otto-Von-Guericke-UniVersity Magdeburg, UniVersita¨tsplatz 2, D-39106 Magdeburg, Germany, and Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstrasse 1, D-39106 Magdeburg, Germany
Our recent experiments with precipitation of barium sulfate (Steyer et al. In BIWIC 15th International Workshop on Industrial Crystallization; Shaker VerlagAachen, Germany, 2008; pp 73-80. Steyer and Sundmacher J. Cryst. Growth 2009, 311, 2702-2708) showed that an excess of barium or sulfate ions in the crystallizer has a strong impact on the resulting particle size distributions (PSDs). Theoretically, this effect is not well understood, as most kinetic models in the literature are valid only for stoichiometric conditions. The aim of this work was, therefore, to investigate the influence of detailed thermodynamically well formulated activity coefficient models on the PSDs in a one-dimensional population balance model of a semibatch continuous stirred-tank reactor (CSTR). The nonsymmetry of these models with respect to an excess of barium or sulfate ions was studied. It was found that the effect on the supersaturation is too weak to explain the experimental results in terms of supersaturation-dependent kinetics. It was thus concluded that, for the conditions studied here, it is not sufficient to use nucleation kinetics that depend on supersaturation only. Instead, the actual ratio of ions in the solution should be incorporated into the kinetic expression for nucleation. 1. Introduction Fine materials are widely used in industry, for example, for pharmaceuticals, catalysts, dyes, and paints. Precipitation is an important procedure for preparing fine particles. Properties such as shape and size distribution define the product properties. To ensure a well-defined product, during precipitation, the influence of process parameters such as concentrations of reactants, supersaturation, and mixing performance of the reaction device, as well as precipitation mechanisms such as nucleation, growth, and agglomeration, have to be determined and controlled. Barium sulfate is used as a filler and extender in plastics and paints, as well as in pharmaceuticals. Recently, experiments on the precipitation of barium sulfate were carried out to investigate the particle morphology depending on the supersaturation and feeding policy in a semibatch stirred-tank reactor.2 It was observed that crystals precipitated under specific experimental conditions such as supersaturation, ion ratio, and feed policy grow with specific morphologies. Also, depending on the experimental conditions, for semibatch operation, the particle populations had different particle size distributions (PSDs) for the same concentration ratio, depending on whether barium or sulfate ions were present in excess. It is known that excesses of these ions have an influence on the growth and nucleation kinetics. For the system of barium chloride and potassium sulfate, Aoun et al.3 identified that both nucleation and crystal growth change significantly, and also differently, for ion excesses of barium and sulfate. Although, in a later work, Aoun et al. attenuated this statement,4 there are many references in literature that report findings of different influences of barium and sulfate ion excesses on the precipitation * To whom correspondence should be addressed. Address: Professor Dr.-Ing. Kai Sundmacher, Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstrasse 1, D-39106 Magdeburg, Germany. Tel.: +49-391-6110350. Fax: +49-391-6110353. E-mail:
[email protected]. † Otto-von-Guericke-University Magdeburg. ‡ Max Planck Institute for Dynamics of Complex Technical Systems.
product. Barium ions preferably adsorb on the barium sulfate particle surface, forming a positive charge. The resulting electrical potential can lead to repulsive forces between particles, hindering agglomeration and changing the growth and nucleation rates.5 By changing the energy surface, blocking growth sites, or hindering ion diffusion to the surface, adsorbed ions also can change face growth rates and thus habit modifications. In general, the precipitated particles are smaller for excess barium.5-8 For barium sulfate precipitation, several kinetic models have been presented in the literature, of which only a few are mentioned here.3,4,9-12 Aoun et al.4 provided a summary of kinetic expressions from the literature including the experimental conditions, mechanism, and concentration ranges for growth and nucleation rates. Most of the results were derived for batch operations and exhibited no difference in driving force concerning barium or sulfate ion excess. Our own semibatch experiments showed that, for stoichiometric batch precipitation, differences in morphology and PSD in regard to the feeding sequence were indeed small.1 For nonstoichiometric conditions, however, the experimental results are poorly predicted by these models. Vicum et al.11 assumed the reason for the poor fit to lie in the use of simplified thermodynamic models and therefore used the Pitzer model13,14 for the calculation of the activity coefficients to partially improve the fitting of their experimental data. To our knowledge, the only model that includes an empirical approach to consider the nonsymmetrical behavior of the Ba2+-SO42--Cl--K+ electrolyte solution with respect to different ion ratios is from Aoun et al.,3 although it was also derived for and is applicable only to batch experiments. In that model, the different influences of ion excess are considered empirically by fitted parameters to measured growth and nucleation rates for values of the ion ratio of barium to sulfate in the range of 0.1-10. Although the model of Aoun et al. is able to reproduce their batch experiments well, it is not quite satisfactory from a theoretical point of view. In particular, the dependence of their kinetics on the initial conditions in the batch
10.1021/ie901306r 2010 American Chemical Society Published on Web 01/22/2010
Ind. Eng. Chem. Res., Vol. 49, No. 5, 2010
reactor lacks a physical justification. In an adapted form, this kinetic model has already been applied to the semibatch experiments.1 The aim of this article was to investigate how much influence a nonsymmetrical approach for the driving force (activity coefficient model) has if used with batch model kinetics from the literature and whether such a sophisticated thermodynamic precipitation model is sufficient to describe the observed different PSDs in experiments for semibatch precipitation of barium sulfate reported earlier.2 Nucleation and growth kinetics were taken from Vicum et al.11 These kinetics are considered to be quite suitable, as they cover the whole supersaturation range by distinguishing between homogeneous and heterogeneous nucleation and also integration-limited and diffusionlimited growth through a two-step growth model as presented in section 3. For activity coefficients, three approaches were used for the calculation of the supersaturation to investigate the impact of the activity coefficient model on the simulated PSD results. The extended Debye-Hu¨ckel approach as used by Angerho¨fer12 is symmetrical with respect to the barium and sulfate concentrations and is valid for ion strengths up to 0.1 mol/L. Bromley15 proposed a semiempirical extension to the Debye-Hu¨ckel approach with interaction terms for unlike charged ions in solution that is applicable up to an ion strength of 6 mol/L. Pitzer’s proposed approach13,14 takes most shortrange interionic forces such as ion-pair interactions between likecharged ions and ion triplets into account. Both the Bromley and Pitzer models give nonsymmetrical activity coefficients with respect to excesses of barium and sulfate ions. 2. Thermodynamics The driving force for precipitation is the supersaturation. For dissociated Ba2+ and SO42- ions, the supersaturation can be calculated as16 Sa )
aBa2+aSO42-
(1)
Ka,sp
with the activities of the Ba2+ and SO42- ions expressed as ai ) ciγi
(i ) Ba , SO4 ) 2+
2-
(2)
Here, ci is the concentration of Ba2+ and SO42- ions, and γi is the ionic activity coefficient of Ba2+ and SO42- ions. Ka,sp is the activity-based solubility product of barium sulfate, which can be calculated from the concentration-based solubility product of barium sulfate, Ksp, and the saturation ionic activity coefficient of barium sulfate, γsat ( , as Ka,sp )
sat 2 Ksp(γ( )
(3)
γsat (
Because of the low solubility, was assumed to be unity here. The value of the solubility product, Ksp, at 25 °C is17 1 × 10-9.96 kmol2/m6. The ion excess of dissolved barium ions to sulfate ions is quantified by the concentration ratio R, which is defined as R)
cBa2+ cSO42-
(4)
The concentration-based ionic strength of the salt solution can be estimated from the concentration ci and number of charges zi of all present ion species i as18
Ic )
1 2
∑cz
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2
(5)
i i
Application to barium sulfate precipitated by barium chloride and potassium sulfate leads to the expression 1 Ic ) (4cBa2+ + 4cSO42- + cCl- + cK+) 2
(6)
The ionic strength based on molality can be calculated from the concentrations also as Im )
1 + 4cSO42- + cCl- + cK+) (4c 2FH2O Ba2+
(7)
where FH2O is the density of the solvent (water). Barium sulfate behaves as a weak electrolyte, forming ion pairs in solution that can become quite significant for higher concentrations.19 Thus, the ionic strength is less than the theoretical value and defined by the free ions in solution. The free ion concentrations are also taken for the calculation of the supersaturation and ion ratio (eqs 1 and 4). Equilibrium between free ions and ion complexes is described by cBaSO4(aq)FH2OKI ) cBa2+cSO42-γ(2
(8)
with KI being the reaction equilibrium,19 log(KI) ) -2.72. 2.1. Extended Debye-Hu¨ckel Approach. For infinitely dilute solutions, the difference in behavior of real cations and anions from an ideal solution because of Coulomb forces can be described by the Debye-Hu¨ckel limiting law, which was extended for ion strengths up to 0.1 kmol/m3 as described in the equation12 log γ( ) -
A|zBa2+zSO42- | √Ic
(9)
1 + Ba√Ic
with |zBa2+zSO42-| ) 4, Ic calculated by eq 6, and a being the smallest possible distance between ions a ) 4.5 × 10-10 m
(10)
At a temperature of 25 °C, the parameters A and B are defined as A ) 0.5146
( )
B ) 3.2977 × 109
m ( kmol )
m3 kmol
1/2
1/2
) 0.0163
( ) m3 mol
) 1.0428 × 108
1/2
(11) m ( mol )
1/2
(12) The extended Debye-Hu¨ckel approach is the simplest used activity coefficient model. It gives a “symmetrical” result regarding barium or sulfate excess and considers only longrange interionic forces between barium and sulfate ions. For implementation, the extended Debye-Hu¨ckel model is also used with the reduced free ion concentrations due to complex formation. Figure 1 shows the difference in driving force for the total amount of barium and sulfate ions in the solution if the supersaturation is calculated with activity coefficients by the extended Debye-Hu¨ckel approach without and with complex formation of barium sulfate in solution. The influence on the driving force of real effects can be clearly seen. Figure 1 also shows the lines of constant supersaturation if calculated by
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Ind. Eng. Chem. Res., Vol. 49, No. 5, 2010
FSO42- )
∑ B* c
jc,SO42-) c,SO42-(z
2
mc ,
FBa2+ )
∑ B*
jBa2+,a) Ba2+,a(z
2
ma
(15)
a
with 1 jzca ) (|zc | + |za |) 2
(16)
and
B*ca )
(0.06 + 0.6Bca)|zcza | + Bca 1.5Im 1+ |zcza |
(
)
(17)
where c stands for the cations and a stands for the anions in solution. For the ionic solution of barium, sulfate, chloride, and potassium in water, the interaction terms are as follows Figure 1. Supersaturation diagram for BaCl2 and K2SO4 in water at 25 °C: dotted line, supersaturation based on concentrations (eqs 1 and 2 with γBa2+ ) γSO42- ) 1); dash-dotted line, supersaturation, Sa, calculated with extended Debye-Hu¨ckel approach (eq 9); solid line, Sa calculated with extended Debye-Hu¨ckel approach and consideration of Ba2+ SO42- complex formation with equilibrium KI ) 10-2.72 (eq 8).
concentrations only, that is, neglecting the influence of the electrolyte solution completely. It has been shown that the use of mere concentrations for the calculation of supersaturation is an inappropriate assumption when dealing with precipitation of barium sulfate,20 and the differences in driving force as visualized by Figure 1 give a good indication of why this is true. 2.2. Bromley Approach. Bromley15 developed a multicomponent version of the Debye-Hu¨ckel limiting law as a semiempirical method for the calculation of activity coefficients as a function of ionic strength up to 6 kmol/m3 that gives a nonsymmetric activity coefficient with respect to barium or sulfate excess. It considers long-range interionic forces between barium and sulfate ions and all counterions present in solution. Other ion interactions (between like-charged ions or ion triplets) are not considered 2 log γm i ) -Amzi
√Im 1 + √Im
+ Fi
(i ) Ba2+, SO42-)
(13)
with Am ) 0.5108 (kg/mol)1/2 for water at 25 °C.21 The term Fi represents the interactions between cations and anions. The mean activity coefficient of barium sulfate can be calculated as m log γ( ) -Am |zBa2+zSO42- |
√Im
1 + √Im |zBa2+zSO42- |
FBa2+ ) B*Ba2+SO42-(zjBa2+SO42-)2mSO42- + B*Ba2+Cl-(zjBa2+Cl-)2mCl1 (zj ) B* )2c + FH2O Ba2+SO42- Ba2+SO42- SO42-
[
B*Ba2+Cl-(zjBa2+Cl-)2cCl-
]
(18)
FSO42- ) B*Ba2+SO42-(zjBa2+SO42-)2mBa2+ + B*K+SO42-(zjK+SO42-)2mK+ 1 ) B* )2c + (zj FH2O Ba2+SO42- Ba2+SO42- Ba2+
[
B*K+SO42-(zjK+SO42-)2cK+
]
(19)
For an example of the mean barium sulfate activity coefficient calculated with the Bromley model, please refer to the Supporting Information. 2.3. Pitzer Approach. By expanding the Gibb’s free energy using a virial series, Pitzer developed a widely used morecomponent-electrolyte solution model for activity coefficients13,21 that also gives a nonsymmetric activity coefficient with respect to barium and sulfate excesses. In addition to the long-term Coulomb interactions, the model also includes short-range interionic forces between like-charged ions and ion triplets GE ) f(Im) + jT m ˜ LMR
∑ ∑ λ (I
ij m)mimj
i
j
+
∑ ∑ ∑µ
ijkmimjmk
i
j
k
(20) +
(
FSO42-
FBa2+ + |zBa2+ | + |zSO42- | |zBa2+ | |zSO42- |
)
(14)
The interaction terms Fi for i ) Ba2+ and SO42- are calculated as
The first term represents an expanded Debye-Hu¨ckel term to account for the long-range interaction forces. The other terms are for considering the short-range interactions. Binary pairs of species i and j are taken into account through the parameters λij(Im), interactions between triplets of species i, j, and k are represented by µijk. All contributions are weighted to the mass of the solvent, m ˜ LM.
Ind. Eng. Chem. Res., Vol. 49, No. 5, 2010
The parameters are calculated by
where Φ′ is the ionic strength derivative of Φ and
λij(Im) ) BPij(Im) for zizj * |zizj | (cation-anion) Φij (c-c, a-a, ion-molecule, molecule-molecule) (21)
{
Cij
∑ m |z |
for zizj * |zizj |
k k
k
Ψijk
fγ ) -AΘ 2β(1) ij
B′P ij )
R1 Im 2
The matrices of the interaction parameters BPij and Cij are symmetrical, that is, BPij ) BPji, Cij ) Cji, and BPii ) Cii ) 0. For the functions f(Im) and BPij(Im), Pitzer identified the following correlation to fit best21 f(Im) ) -AΘ(T) 2β(n) ij
q
∑R
BPij(Im) ) β(0) ij +
n)1
2
Im
n
4Im ln(1 + b√Im) b
(23)
[1 - (1 + Rn√Im) exp(-Rn√Im)]
(24)
(i ) Ba2+, SO42-, K+, Cl-)
i i
(31)
i
(22)
otherwise
∑ m |z |
Z)
{
µijk )
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2
[
√Im 1 + b√Im
+
]
2 ln(1 + b√Im) b
(32)
[-1 + (1 + R1√Im + 0.5R12Im) exp(-R1√Im)]
(33) Cij )
CΘij
(34)
2√ |zizj |
The mean activity coefficient is obtained as m ln γ( ) |zBa2+zSO42- |F + (νBa2+ /ν)
∑ m [2B + ZC /ν) ∑ m [2B + P Ba2+a
a
Ba2+a
+
a
2(νBa2+ /νSO42-ΦSO42-a)] + (νSO42-
P cSO42-
c
c
ZCcSO42- + 2(νBa2+ /νSO42-ΦBa2+c)] +
with
∑ ∑m m ν
-1
AΘ ) A m
ln 10 3
(25)
The parameters b, q, and Rn were derived from fitted data to measured mean ion activity coefficients. For a non-2-2 electrolyte, they take the values b ) 1.2,
q ) 1,
R1 ) 2.0
c
c
( )
GE ∂ jT R ln γm i ) ∂ni
∑ ∑ m m (ν c
a
a
c′
