Crystallization of Monohydrate Citric Acid. 2. Modeling through Population Balance Equations Alexandre Caillet,† Nida Sheibat-Othman,† and Gilles Fevotte*,†,‡ LAGEP (Laboratoire d’Automatique et de Ge´ nie des Proce´ de´ s) UMR CNRS 5007, UniVersite´ Lyon 1, Baˆ t. 308G, 43 bld, du 11 noVembre 1918, 69622 Villeurbanne Cedex, France, and Ecole Nationale Supe´ rieure des Mines de Saint Etienne, LPMG, CNRS UMR 5048, 158 cours Fauriel 42023 Saint-EÄ tienne, France
CRYSTAL GROWTH & DESIGN 2007 VOL. 7, NO. 10 2088-2095
ReceiVed September 21, 2006; ReVised Manuscript ReceiVed March 20, 2007
ABSTRACT: Experimental data about both the liquid and the solid phases during seeded batch crystallizations of citric acid (CA) in water were obtained using in situ Raman spectroscopy and image acquisition. These experimental results were reported in a previous paper in this issue (Caillet, A., et al. Cryst. Growth Des. 2007, 7, 2080-2087). The present paper is now focused on the mathematical modeling of the desupersaturation process during the crystallization of monohydrate citric acid (MCA), which is the stable form at 15 °C. Both the crystallization of MCA (monohydrate) and the dissolution of the anhydrous (ACA) form were investigated. The model is based upon population balance equations (PBEs) describing the evolution of the crystal size distribution (CSD) during batch seeded operations. The estimation of the kinetic parameters of MCA nucleation and growth, and of the dissolution of ACA, was performed using nonlinear optimization techniques. For various operating conditions (modifications of the initial supersaturation and of the seed amount), the two PBE models represent satisfactorily the experimental behavior of the process. In particular, activated secondary nucleation is shown to explain particular features of the solute/solvent system that were observed previously. Introduction The design of a calibration strategy for the continuous monitoring of solvent-mediated phase transition using in situ Raman spectroscopy was reported in a previous paper.2 The technique was applied to the monitoring of citric acid crystallization that exhibits anhydrous to monohydrate phase transition below 34 °C. Satisfactory Raman measurements were obtained after estimating in-line the overall solid concentration in suspension and the composition of the solid phase from in situ Raman spectral data. Isothermal seeded batch crystallization of monohydrate citric acid (i.e., the stable form) was performed with varying operating conditions. During these experiments, the monitoring of the concentration of citric acid (in both solid and liquid phase) was ensured using Raman spectroscopy. In addition to Raman spectroscopy, in situ image acquisition was shown to yield valuable information about the time evolution of the crystal size distribution (CSD).1 The two measurement approaches allowed qualitative analysis of the main phenomena involved during the crystallization process. The present work deals with the design of a dynamic model for the generation of monohydrate citric acid (MCA) crystals under isothermal conditions and the dissolution of anhydrous particles when undersaturated conditions are applied: these two typical situations are encountered during the solvent-mediated phase transition of citric acid, from anhydrous to monohydrate solid phases. The modeling is based upon population balance equations (PBE) and is intended to allow further mathematical description of the overall ACA to MCA phase transition process. Phase transitions between polymorphs or between solvates and/or hydrates are very common phenomena in all industries where complex molecules are processed or produced in the solid form, notably, in the pharmaceutical industry. It is obvious that * To whom correspondence should be addressed. E-mail: fevotte@ lagep.cpe.fr. † LAGEP (Laboratoire d’Automatique et de Ge ´ nie des Proce´de´s) UMR CNRS 5007, Universite´ Lyon 1. ‡ Ecole Nationale Supe ´ rieure des Mines de Saint Etienne, LPMG, CNRS UMR 5048.
solid-phase transformations involve kinetic features that, for many complex and entangled reasons, are likely to lead to the production of a given desirable form or, on the contrary, to the generation of undesirable solid phases. As far as full comprehension of the dynamics of phase transformation systems is concerned, and despite major reported kinetic studies,3-9 several experimental factors have still to be investigated. At first glance, and as outlined in preliminary general papers dedicated to the phase transformation kinetics, the dynamic behavior of most solvent-mediated phase transformation systems does not seem to require knowledge of kinetic laws that would be specific to the solid transformations in question. In other words, “standard” crystal growth, dissolution, and nucleation kinetic models should be suitable to represent the dynamics of phase transition processes. As usual for consecutive and/or concomitant processes, the overall rate of transformation is mostly determined by the slowest phenomenon involved: the phase transformation can be limited, for example, by the dissolution of the metastable form or by the growth of the stable form. Given the few experimental and modeling reported results, the investigation of various solute/solvent systems remains necessary to confirm and qualify the main assumptions about solvent-mediated phase transition processes. Actually, it is reasonable to state that the rarity of published experimental data was mainly due to the lack of appropriate sensors: until recently, no in-line techniques were available to allow in situ monitoring of liquid and solidphase compositions during solid-phase transition processes. Without appropriate sensors, a significant experimental effort was required to obtain rather poor discrete-time off-line data. Moreover, the reliability of these data was not always satisfactory, due to the difficulty inherent to the withdrawal of samples from the crystallizing medium. The availability of recent sensing in situ technologies, process analytical technologies (PATs) in particular, is major progress allowing undertaking more in depth kinetic studies. Using such in-line measurements, it is the goal of the present paper to report on the design and validation of a PBE model for the crystallization of MCA and the dissolution
10.1021/cg0606343 CCC: $37.00 © 2007 American Chemical Society Published on Web 10/03/2007
Modeling through Population Balance Equations
Crystal Growth & Design, Vol. 7, No. 10, 2007 2089
Figure 1. Picture of the lab-scale crystallization jacketed reactor equipped for the Raman monitoring of phase transitions during solution crystallizations.
of ACA, at 15 °C. This study is intended to provide results and modeling tools required for the proposal of a comprehensive PBE model describing the whole anhydrous to monohydrate isothermal phase transition process of citric acid in water. Experimental Section The measurements strategy developed for this study is briefly recalled. More details about the experimental data obtained using PATs can be found elsewhere.1,2 The crystallization of citric acid in water was selected as a model system. In situ measurements of solute concentration were performed using the ReactRA Raman spectrometer manufactured by Mettler-Toledo, equipped with a 16-mm diameter immersion Hastelloy probe sealed with a sapphire window. As shown in Figure 1 the probe is connected to the spectrometer through fiber optic (200 µm) allowing spectral resolution of 7 cm-1. The calibration of the Raman measurements which was developed for this application was presented previously.1 Using the calibration model, complete information about the solid phase (i.e., partial and overall solid concentration) was obtained in real time. Through the resolution of the solute mass balance, the solid concentration measurements allowed computing estimates of solute concentration, even though this latter concentration was not directly measured. Two definitions of supersaturation are used below:
β(t) )
C(t) C*(T(t))
σ(t) )
)
C(t) C*
C(t) - C* C*
(1)
(2)
where C* is the constant solubility of MCA and C(t) is the estimated solute concentration expressed in [kg anhydrous-basis dissolved CA/ kg of water]. In this paper, as we deal with process engineering issues rather than with precise fundamental crystallization modeling, the two definitions of supersaturation given by eqs 1 and 2 were approximated as the ratio between the actual concentration and the solubility (in other words, the ratio between the current activity and the activity at saturation γ/γ* was assumed close to 1), which is a current practice in industrial crystallization studies.10 Figure 2 displays a schematic representation of the in situ image acquisition sensor developed in our laboratory to monitor the CSD. Because of the difficulty of measuring the size of overlapping particles and/or of out of focus crystals, no automatic processing of the pictures could be applied to compute the CSD histograms. The contour of each crystal was drawn by the operator in a “handicraft” way, using the mouse. Computing the corresponding projected area and sorting the results was then performed using analySIS, an image processing software developed by Soft Imaging System. The sampling period for the acquisition of one given picture was 1 s. The projected area of each particle was determined from the acquired pictures. To obtain a one-dimensional representation of the particles, the diamond-like crystals shown in Figure 3 were compared to spheres. The diameter of
spherical particles with the same projected area as the real crystals was computed and stored as the “measured” particle size. Such equivalent size is referred below to as Dm and Da, for monohydrate and anhydrous particles. As an example, Figure 3 shows both in-line Raman “measurements” of absolute supersaturation computed with respect to the solubility of monohydrate citric acid, and pictures taken during two different seeded experiments. The solubility curve, which is not shown here, was taken from the literature.2,7 It should be noticed that the Raman solubility measurements performed during this study were found to fit the published data. However, to accurately compute the supersaturation of MCA, the monohydrate solubility value at 15 °C was evaluated very cautiously. The following value, C* ) 1.348 kg anhydrous-basis dissolved CA/kg of water, was used in the following section. Figure 4 shows typical examples of weight-CSD resulting from the processing of about 500 particle images. The experimental acquisitions provide complementary data about both the liquid and the solid phase, which has been really useful for modeling purposes. The operating conditions of the seeded batch isothermal experiments will not be recalled here,1 but Table 1 summarizes the main parameters of runs 1-7. The last dissolution experiment (run 7) will be discussed in more detail in the following section. The design and validation of a comprehensive kinetic model describing the nucleation and growth of monohydrate crystals and the dissolution of anhydrous citric acid is now presented.
Population Balance Modeling (PBE) of the Isothermal Crystallization of Monohydrate Citric Acid PBEs for Monohydrate Crystals Nucleation and Growth. The goal is now to obtain a detailed kinetic scheme and a PBE model describing the crystallization of the stable monohydrate form. The separate characterization (the anhydrous form being absent in the slurry) of this latter crystallization is expected to yield kinetic parameter estimates that could be used to further describe the solvent-mediated phase transition of citric acid from the anhydrous to the monohydrate form. As far as the dispersed size of monohydrate crystals is concerned, the following PBE describes the time evolutions of the characteristic particle size Dm. Let Ψm(Dm,t) be the number density function of monohydrate crystals of equivalent size Dm at time t, then the PBE can be written as follows:11
∂Ψm(Dm,t) ∂ + (G (D ,t)Ψm(Dm,t)) ) 0 ∂t ∂Dm m m
(3)
with Ψm(Dm,t) in Nb‚m-4. Equation 3 assumes that neither agglomeration nor breakage of the particles takes place during the experiments. The following boundary condition (eq 4) describes the generation of new particles in the suspension through secondary nucleation, assumed to occur significantly when MCA particles are already present in the crystallizer.1 It is well-known that new particles should exhibit a size exceeding the critical size D/m. However, in the sequel the critical size will be neglected, which from a purely numerical point of view is a reasonable assumption (i.e., setting D/m ) 0 in the simulation program has no effect on the numerical results). The following expression was therefore set as a left boundary condition:
Ψm(D/m,t) ≈ Ψm(0,t) )
RNm(t) Gm(D/m,t)
≈
RNm(t) Gm(0,t)
(4)
where RNm is the overall nucleation rate of monohydrate particles and Gm is the growth rate, which was assumed to be size independent. The right-boundary condition is given by the following expression where the “infinite size” (Lmax in practice) represents
2090 Crystal Growth & Design, Vol. 7, No. 10, 2007
Caillet et al.
Figure 2. Schematic of the in situ image acquisition system used for monitoring the dispersed phase and acquiring pictures for further off-line CSD analysis.
Figure 3. In-line measurements of supersaturation during batch isothermal crystallizations of monohydrate citric acid performed with two different seed weights, and discrete time pictures of the crystallizing slurry acquired at times pointed out on the supersaturation trajectory.
the smaller estimated size that cannot be reached by the growing crystals:
Gm(∞,t)Ψm(∞,t) ) Gm(Lmax,t)Ψm(Lmax ,t) ) 0
(5)
The population density function used as the initial CSD condition is given by the distribution of monohydrate sieved seed CSD, ψm,seed, which was also measured using image analysis, prior the introduction of the particles in the crystallizer:
Ψm(Dm,0) ) Ψm,seed
(6)
PBEs for Anhydrous Crystals Dissolution. Equation 3 allows describing the time variations of the anhydrous CSD, even though, due to undersaturated conditions, no crystal growth can now take place. From a mathematical viewpoint, the main difference between the two PBEs relating the crystallization and the dissolution processes lies in the set of boundary conditions:
for dissolution experiments the metastable phase is not subject to nucleation and, due to the undersaturated conditions, even small particles detached from the parent seed crystals should dissolve. Consequently, one can write that crystals reaching the limit size 0 vanish through dissolution. Finally, the left-boundary condition was simply left undefined, and negative growth rate was considered (i.e., dissolution) in the second PBE describing the population of metastable particles:
∂ ∂ (D (D ,t)Ψa(Da,t)) ) 0 [Ψ (D ,t)] + ∂t a a ∂Da a a
(7)
where Ψa(Da,t) is the weight density function of anhydrous particles and Da is the dissolution rate of the anhydrous crystals:
Da(t) )
dDa
