Polymorph control in MSMPR crystallizers. A case study with

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Polymorph control in MSMPR crystallizers. A case study with paracetamol. Lucrece Nicoud, Filippo Licordari, and Allan S. Myerson Org. Process Res. Dev., Just Accepted Manuscript • DOI: 10.1021/acs.oprd.8b00351 • Publication Date (Web): 13 Mar 2019 Downloaded from http://pubs.acs.org on March 17, 2019

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Organic Process Research & Development

Polymorph control in MSMPR crystallizers. A case study with paracetamol. Lucrèce Nicoud,†,‡ Filippo Licordari,† and Allan S. Myerson∗,† †Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA-02139, USA ‡Current address: Ypso-Facto, 10 Viaduc Kennedy, 54 000 Nancy, FRANCE E-mail: [email protected] Phone: +1-617-452-3790 Abstract Mixed-suspension mixed-product removal (MSMPR) crystallizers can potentially allow the production of metastable polymorphs. A necessary (but not sufficient) condition is to select operating conditions such that the steady state concentration remains above the solubility of the metastable form. In this paper, we provide a framework that helps controlling polymorphism in MSMPR. We consider paracetamol as a model system and target the production of form II, which is metastable and rapidly converts to form I in batch. We combine experimental characterization (inline infrared spectroscopy, Raman spectroscopy, and imaging) with population balance modeling to investigate the possibility to produce the metastable form in a continuous manner. We identify a design space allowing the visualization of the regions where each form is obtained at steady state as a function of the kinetic parameters of both forms. We show that form II cannot be obtained at steady state in pure ethanol. However, the addition of a small amount of metacetamol (1% in mass with respect to paracetamol) in the feed is sufficient to obtain form II at steady state. We interpret the effect of

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metacetamol in light of a kinetic model and quantify its incorporation in the crystals with high performance liquid chromatography.

Key words polymorph control, MSMPR, design space, population balance model, process analytical technology

Introduction Numerous active pharmaceutical ingredients (APIs) are known to crystallize in different structures, referred to as polymorphic forms, which are characterized by a different packing and/or molecular arrangement in the crystal lattice. 1 It is key for the pharmaceutical industry to control crystal polymorphism because different polymorphs may exhibit different properties, for instance in terms of solubility, bioavailability, filterability or compactability. 2,3 It is possible to influence crystal polymorphism by adjusting operational variables, such as the level of supersaturation, the composition of the solvent, as well as the type and concentration of additives. 4–7 In any case, the polymorphic outcome is dictated by the relative nucleation and growth rates of the various forms, and potentially by the rate of solvent-mediated transformation. 6,8 The latter consists in the dissolution of a metastable form followed by the recrystallization of a more stable form. 9–16 Such solvent-mediated transformation starts when the solute concentration drops below the solubility of a metastable polymorph as the system strives to reach thermodynamic equilibrium. Therefore, in a batch crystallizer, a metastable form can convert to a more stable form if left for a sufficient length of time (although the transformation may be kinetically hindered and may not occur in any observable time). In general, it is preferable to produce the most stable polymorph to avoid any transformation issues during production and storage. However, in some particular cases, it may be 2


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advisable to formulate a metastable form, for instance to improve the dissolution rate and the bioavailability. 17 In this regard, continuous crystallizers are attractive because it is always possible to maintain the solute concentration at steady state above the solubility of the target metastable form, thus preventing solvent-mediated transformation. In addition, the success of continuous manufacturing in other industrial sectors has led to an increased interest in the pharmaceutical industry. 18–20 This has resulted in a number of studies on the continuous crystallization of APIs over the last years. 21–30 Potential advantages brought by a transition from batch to continuous crystallizers are reduced turnover times and smaller equipment sizes. There are several types of continuous crystallizers among which the mixed-suspension mixed-product (MSMPR) crystallizer is the most common. 31 Only few studies report experimental results about polymorph control in MSMPR crystallizers. 32,33 An important observation made in these works was that the polymorphic form obtained at steady state was independent of the polymorphic form of the seeds, thus suggesting that seeding is not an efficient strategy to control polymorphism in MSMPR crystallizers. Depending on the system (L-glutamic acid or amino benzoic acid) and the operating conditions, either the metastable or the stable polymorphic form were obtained at steady state. Kinetic models based on population balance equations have proven useful to rationalize these experimental data. 32–34 Nevertheless, polymorph dynamics in MSMPR has not been sufficiently elucidated yet. Systematic studies on additional systems are needed to better understand the kinetic competition between the different polymorphic forms and assess whether MSMPR crystallizers can offer a relatively general approach to crystallize a metastable form (i.e., eliminate the risk of solvent-mediated transformation inherent in batch). In this study, we provide a framework that helps controlling polymorphism in MSMPR crystallizers. We consider paracetamol (acetaminophen) in ethanol as a model system. Paracetamol can crystallize in two main structures: form I (monoclinic) and form II (orthorombic). 35 The target polymorph in this study is form II, which is metastable but has better tableting properties than form I (the commercial form) due to an improved compres-

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sion behavior. 36,37 This has prompted several authors to study the crystallization of form II. 24,38–41 Here, we first build on our previous work performed in batch 38 to identify operating conditions potentially allowing the crystallization of form II at steady state in an MSMPR crystallizer. Then, we rationalize our results by constructing a design space based on a kinetic model. This allows us to quantify “how much faster” should be the crystallization of form II compared to form I so that form II wins the kinetic competition at steady state. Finally, we explore the use of a soluble additive (metacetamol) to control the polymorphism of paracetamol in MSMPR crystallizers.

Materials and Methods Materials Paracetamol form I was purchased from Sigma-Aldrich (Acetaminophen BioXtra, ≥ 99%), metacetamol from TCI America (3’-Hydroxyacetanilide, ≥ 98%) and anhydrous ethanol from Deacon Labs (Koptec 200 proof pure ethanol). HPLC grade water and methanol were purchased from BDH Chemicals.

Preparation of the seeds Seeds of form II were prepared by cooling to 0 ◦ C an ethanolic solution of paracetamol at 300 mg/gEtOH containing 10% of metacetamol in mass with respect to paracetamol. The presence of metacetamol inhibits the crystallization of form I and the transformation of form II in form I. More details about the experimental procedure and the analytical characterization of the seeds are given in our previous work. 38 In particular, no trace of form I could be detected with powder X-ray diffraction (PXRD), Raman spectroscopy, and infrared spectroscopy. The detection limit of form I with PXRD was estimated in the order of 5%. The metacetamol content was assessed in this study by redissolving the seeds in ethanol and analyzing the solution with high performance liquid chromatography (HPLC). It was found 4


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that the seeds contain slightly less than 2% metacetamol.

Solubility measurements The solubility of both forms of paracetamol in pure ethanol was measured as a function of temperature in our previous work. 38 The experimental data were fitted with the following expressions:     c∗I = 0.702 exp(0.019 T )

(1)

   c∗II = 0.593 exp(0.020 T ) where c∗I and c∗II are the solubilities of form I and II, respectively, expressed in mg/gEtOH , while T is the temperature in Kelvin. The experimental data together with the fits are shown in the ESI. In this study, we also measured the solubility of both forms of paracetamol as a function of the concentration of metacetamol at the fixed temperature of −10 ◦ C. For form I, saturated solutions were prepared by magnetically stirring an excess of commercial paracetamol in contact with 1 mL of an ethanolic solution at the selected metacetamol concentration. The vials were placed in a Crystal16 apparatus (Technobis Crystallization Systems) to accurately control the temperature. After a minimum of 4 h, the slurry was filtered with a 0.2 µm syringe filter (VWR, 13 mm diameter, PTFE membrane) and the filtrate was placed in tared vials. The vials were weighted immediately after introducing the filtrate and after solvent evaporation. Ethanol was first evaporated under the hood and the last traces were removed by drying the crystals in a vacuum oven at around 60 ◦ C. Once the mass of the samples was constant over time, the total solute concentration (paracetamol and metacetamol) was computed as the mass of crystals divided by the mass of ethanol that had evaporated. The solubility of paracetamol was then estimated by substracting the metacetamol concentration (in mg/gEtOH ) to the total solute concentration. It was shown with HPLC analysis that metacetamol significantly incorporates in parac-

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etamol crystals and that the seeds of form II contain approximately 2% metacetamol. Therefore, when saturated solutions of form II were prepared by dissolving the seeds of form II in pure ethanol, the resulting metacetamol concentration was above the range of interest. To reduce the amount of metacetamol coming from the dissolution of the seeds and obtain a metacetamol concentration in the range of interest, saturated solutions of form II were obtained by dissolving the seeds of form II in slightly undersaturated solutions (100 mg/gEtOH ), which were previously prepared by dissolving the commercial (metacetamol free) form of paracetamol in ethanol. The rest of the procedure is similar to that employed for form I except that the actual concentration of metacetamol in the filtrate (resulting from solution preparation and form II dissolution) was determined by HPLC. In addition, it was verified with Raman spectroscopy that solvent-mediated transformation does not occur during the time frame of the experiment, i.e., that the crystals are still of form II when saturation is reached. Two measurements were performed at each metacetamol concentration. The points and error bars in the graphs represent the mean and standard deviations of these two measurements.

MSMPR experiments MSMPR experiments were performed in an EasyMax 102 Advanced Synthesis Workstation (Mettler Toledo) using a 100 mL two-piece glass reactor equipped with a PTFE cover and an overhead pitch-blade impeller (Alloy-C22, 38 mm diameter, downwards). Peristaltic pumps from Thermo Scientific (Masterflex P/S pumps with Easy-Load II pump heads) were employed to feed the solution and withdraw the slurry. Masterflex Tygon Chemical Tubing from Cole-Parmer was used for both the inlet and outlet streams. The feed solution was kept under stirring at 50 ◦ C by means of a water bath. A small internal diameter was selected for the inlet tubing (L/S 13 – 0.8 mm) so as to limit the time that the feed solution spends in the feed line, and thus reduce the risk of crystal formation. To further reduce that risk, a heating tape was placed on the inlet tubing. The end of that tube was placed into the 6


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slurry to avoid the drying of the feed solution around the tubing otherwise exposed to air. An intermittent withdrawal scheme at high flow rate (170 mL/min) every one tenth of the residence time was implemented so as to avoid size classification at the crystallizer outlet. A large internal diameter was selected for the outlet tubing (L/S 15 – 4.8 mm) to allow such high flow rate. It was ensured that one tenth of the reactor volume was pulled out during withdrawal by adjusting the position of the outlet tubing in the slurry. The start-up phase was as follows: (i) equilibration of 100 mL of feed solution in the crystallizer at 50 ◦ C, (ii) cooling to the set temperature under low stirring (75 rpm) as fast as possible (around 4 ◦ C/min), (iii) increase of the stirring speed to the set value when the temperature reaches 1 ◦ C above the set point, (iv) addition of 500 mg of seeds of form II, (v) start of the inlet and outlet pumps right after seeding. The outlet pump was started in off mode to maximize the time that the seeds spend in the vessel before the first withdrawal. No crystals could be detected with inline microscopy during the cooling phase.

Analytical characterization ATR-FTIR spectroscopy The concentration of dissolved paracetamol was monitored by inline attenuated total reflection Fourier transform infrared (ATR-FTIR) spectroscopy using a ReactIR 15 system from Mettler Toledo. The system was equipped with a 6.3 mm AgX DiComp immersion probe and a diamond as ATR crystal. Thanks to this ATR crystal, the penetration depth of the IR beam in the slurry is in the order of few microns, which allows to probe the liquid composition independently of the presence of solids. The unit was regularly filled with liquid nitrogen (approximately every 6 h) to cool the detector and avoid drifting of the signal. At

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each time point, 256 spectra were recorded. The following calibration was used:   ◦   T = −10 C    T = 0 ◦C

A1514 = 2.00 × 10−3 cliq + 0.205 (2) −3 liq

A1514 = 1.98 × 10 c

+ 0.197

where A1514 denotes the absorbance at 1514 cm−1 after baseline substraction and cliq is the solute concentration in mg/gEtOH . The comparison between the experimental data and the fits is shown in the ESI. More details about the calibration procedure are given in our previous work. 38 The probe had to be cleaned regularly during the experiment (roughly every hour) with ethanol due to fouling. It was verified that no peak appeared nor disappeared in the ATR-FTIR spectra during the experiments, ruling out any potential degradation mechanism of paracetamol.

Microscopy The crystals were visualized by high resolution inline imaging using a Blaze900 probe from BlazeMetrics. The probe uses a 532 nm laser, has a field of view of 900 µm and allows detecting crystals larger than approximately 2 µm. The image plane was set to 120 µm.

Raman spectroscopy The content of form II in dried solid samples was estimated by offline Raman spectroscopy using a Raman WorkStation from Kaiser Optical Systems. The mass fraction of form II f II in % was estimated from the area under the peaks at 454 cm−1 (A454 ) and at 465 cm−1 (A465 ) according to: A454 = 8.0 × 10−3 f II + 0.12 A454 + A465

(3)

The comparison between the experimental data and the fit is shown in the ESI. More details about the experimental procedure and the calibration with powder mixtures are given elsewhere. 38 8


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HPLC HPLC measurements were performed to quantify the metacetamol content in the solid phase. To do so, the crystals were dissolved and diluted to reach a solute concentration in the order of 0.2 mg/gEtOH . The samples were analyzed by reversed phase chromatography using a Symmetry C18 column (Waters, 4.6 × 150 mm, 3.5 µm) assembled on an Agilent 1100 HPLC unit. Elution was performed with a water–methanol mixture at 0.5 mL/min. First, the samples were eluted with a linear gradient from 10% to 80% methanol over 15 min. Then, the column was washed during 10 min at 100% methanol. Finally, the column was re-equilibrated during 10 min at 10% methanol. The solutes were detected with a diode array detector at 243 nm and 280 nm. Paracetamol was eluted first (at around 10.1 min) and then metacetamol (at around 11.8 min). By injecting an equimolar sample of metacetamol and paracetamol, it was found that the ratio of the absorbance coefficients metacetamol/paracetamol is equal to 0.94 at 243 nm and 1.44 at 280 nm. Illustrative chromatograms are shown in the ESI.

Kinetic model The kinetic model used in this work has been developed in our previous study performed in batch. 38 It accounts for the primary (cross) nucleation, secondary nucleation and growth of form I, as well as for the secondary nucleation and growth of form II. The primary nucleation of form II is neglected because the MSMPR is seeded with crystals of form II. In all the investigated conditions, the solute concentration (i.e., the paracetamol concentration in the liquid phase) remains above the solubility of form II, hence dissolution does not occur. The selected expressions of the nucleation (B) and growth (G) rates are given in Table 1, where cliq , c∗I , and c∗II denote the solute concentration, the solubility of form I, and the solubility of form II, respectively. The superscripts c and s stand for cross and secondary nucleation, respectively. Note that the primary nucleation of form I is considered here as proportional to the total surface of crystals of form II (through µII 2 , mathematically defined in Equation 7). Such “cross-nucleation” phenomenon has been observed for several systems 9



including ROY, 42 L-glutamic acid, 11,14 D-mannitol, 43 and polypivalolactone. 44 . However, it is worth mentioning that the nucleation parameters estimated from the comparison between experiments and model simulations should be regarded as lumped values, which average all primary nucleation phenomena together, including cross-nucleation, but also heterogeneous nucleation on the surfaces present in the crystallizer (e.g., walls of the vessel, impeller, probes). Table 1: Expressions of the nucleation and growth rates Form I Cross-nucleation Secondary nucleation Growth

liq nc c ln = µII 2 c∗I liq ns c BsI = ksI ln µI2 c∗I liq ng c I I G = kg ln c∗I BcI

Form II Secondary nucleation

BsII

Growth

GII

kcI

liq ns c = ln ∗ µII 2 cII liq ng c = kgII ln ∗ cII ksII

Table 2: Arrhenius dependence of kinetic parameters. 38 The parameter N denotes the agitation speed in rpm.

kc ks kg

k0 Ea k0 Ea k0 Ea

Form I [#/m /min] 1.16 × 1044 [kJ/mol] 191 [#/m2 /min] 8.6 × 1025 × N 3 [kJ/mol] 127 [m/min] 0.21 [kJ/mol] 22.2 3

Form II n.a. n.a. 2.53 × 1030 121 62.7 31.9

The power exponents are given by nc = 2, ns = 3, and ng = 2. 38 The reaction rate constants are instead given by an Arrhenius law of the type k = k0 exp(−Ea /(RT )), where R is the ideal gas constant, Ea the activation energy and k0 the prefactor. The prefactor of the secondary nucleation rate of form I was found to increase with the agitation speed N 10


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according to a cubic law. The numerical values of the prefactors and activation energies are summarized in Table 2. More details about the fitting procedure and the data supporting the increase of ksI with N 3 are given in the ESI. The population balance equations (PBE) in the MSMPR crystallizer are written as follows: ∂nI ∂ GI nI Q + + nI = (BcI + BsI ) × δ(L − L0 ) ∂t ∂L V ∂nII ∂ GII nII Q + + nII = BsII × δ(L − L0 ) ∂t ∂L V

(4) (5)

where t denotes the time and L the characteristic crystal length. The concentration of crystals comprised between L and L + dL is denoted n = n(t, L) and is expressed in #/m4 . The growth rates of both forms are assumed to be size independent. The symbol δ represents the Dirac delta function and indicates that nucleation produces crystals of size L0 only. Finally, Q and V denote the volumetric flow rate and reactor volume, respectively. The residence time is defined as τ = V /Q. In this study, the PBE were solved by using the methods of moments. The moments of order j of the crystal size distributions of both forms are defined as follows:

µIj

Z

∞

=

Lj nI (t, L)dL

(6)

Lj nII (t, L)dL

(7)

0

µII j

Z =

∞

0

By applying the above definitions, the PBE given in Equations 4 and 5 are written as: liq nc liq ns c dµI0 c 1 I I I I = ks ln µ2 + kc ln µII 2 − µ0 ∗ ∗ dt cI cI τ

liq ng dµIj c 1 I = j kg ln µIj−1 − µIj ∗ dt cI τ

11


j≥1

(8)

(9)


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liq ns dµII c 1 II 0 II = ks ln ∗ µII 2 − µ0 dt cII τ liq ng dµII c 1 II j II = j kg ln ∗ µII j−1 − µj dt cII τ

(10)

j≥1

(11)

These equations are coupled with a mass balance on the solute: dcliq cliq − cliq = 0 dt τ liq ng liq ng c c I II II I I µ2 − 3kv ρkg ln ∗ µII −3kv ρkg ln 2 ∗ cI cII

(12)

where ρ = 1332 kg/m3 is the density of the crystals (which can reasonably be assumed identical for both forms 35 ), whereas kvI = 0.866 and kvII ≈ 2 are the shape factors of form I and form II, respectively. 38,45 To analyze steady state stability, it is useful to define the Jacobian matrix of the system given by Equations 8 to 12 (for j = 1, 2), which is shown in Figure 1. The derivatives are organized as follows along the lines and columns of the Jacobian matrix: µI0 , µI1 , µI2 , II II liq µII 0 , µ1 , µ2 , c . A steady state is stable (i.e., it is not affected by "small" perturbations)

if all the eigenvalues of the Jacobian matrix at steady state have negative real parts. 46,47 This approach will be used later on to identify the theoretical design space of an MSMPR crystallizer allowing the production of a target polymorph.

Results and discussion Preliminary screening of operating conditions As a first step in this study, we selected several operating conditions varying in temperature, initial concentration, and agitation speed, which were found to lead to a satisfactory yield and polymorphic purity of form II in batch crystallizers. 38 Several residence times were also

12


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Figure 1: Jacobian matrix of the system given by Equations = 1, 2). 8 to 12 (for j liq The liq ∗ ∗ supersaturations of both forms are defined as SI = ln c /cI and SII = ln c /cII 

−1/τ

0

ksI SIns

0

0

kcI SInc

   I ng kg SI −1/τ 0 0 0 0    n  0 2kgI SI g −1/τ 0 0 0    ns  0 0 0 −1/τ 0 ksII SII J=   n  0 0 0 kgII SIIg −1/τ 0    n  0 −1/τ 0 0 0 2kgII SIIg    n n  0 0 −3kvI ρkgI SI g 0 0 −3kvII ρkgII SIIg 

ns ksI SIns −1

µII µI2 2 + nc kcI SInc −1 liq liq c c 



n −1

ng kgI SI g

µI0 cliq

ng kgI SI g

µI1 cliq

ns −1 ns ksII SII

µII 2 cliq

n −1

II n −1 µ0 cliq

ng kgII SIIg

II n −1 µ1 cliq

ng kgII SIIg

µI2 cliq II µ n −1 2 −3kvII ρkgII ng SIIg cliq n −1

−1/τ − 3kvI ρkgI ng SI g

selected and a summary of the investigated conditions is presented in Table 3. Table 3: Overview of the experimental runs (in the absence of metacetamol) Run a b c d e f g h

EtOH T [◦ C] cliq ] 0 [mg/g 0 280 0 280 0 280 -10 200 -10 250 -10 250 -10 250 -10 250

N [rpm] 400 400 400 400 400 400 400 200

τ [min] 25 40 50 100 125 100 50 50

As an example, Figure 2 shows the results of experimental run h (as indicated in Table 3). The measured solute concentration (i.e., the paracetamol concentration in the liquid phase), the percentage of form I and the percentage of form II are shown with black, blue, and red symbols, respectively, in Figure 2a. The blue and red dashed lines instead indicate the solubility of form I and form II, respectively. It is observed that the content of form II drops below 100% after approximately 5 h (approximately 6 residence times). Similarly, 13


                       


form I appeared in all the other experiments and none of the conditions reported in Table 3 allowed the production of form II in a continuous manner. Run h was in fact the experiment that produced form II for the longest time. It is worth mentioning that the operating EtOH conditions for run h (T = −10 ◦ C, cliq , N = 200 rpm) allowed a good 0 = 250 mg/g

polymorph control in batch since form II could be produced with both a high polymorphic purity and a high yield, namely a polymorphic purity in the order of 97% and a yield of 52% (which corresponds to 97% of the theoretical yield). 38 However, these conditions did not lead to a stable steady state of form II in MSMPR. Note that the observed change in polymorph content in MSMPR is not due to solvent-mediated transformation since the solute concentration remains above the solubility of form II at all times, which prevents the dissolution of form II. The change in polymorph content is instead due to the faster crystallization of form I with respect to form II, which is progressively washed out of the reactor. For illustrative purposes, Figure 2b shows inline pictures recorded after 30 min (when the crystallizer contains only form II) and after 7 h (when the crystallizer starts converting into form I). At 30 min, the crystals have a plate-like morphology with clear edges, characteristic of form II. On the other hand at 7 h, which corresponds to 8.4 residence times, it seems that the crystals have a more rounded morphology (even though the crystallizer still mainly contains form II). This illustrates that the crystal size and shape distributions approach steady state only after several residence times (although in that particular case the steady state of form II cannot be maintained due to the progressive conversion of the crystallizer into form I).

Theoretical design space The preliminary experiments conducted in the previous section show that selecting operating conditions allowing a satisfactory polymorph control in batch mode is not sufficient to control polymorphism in continuous mode. In this section, we try to rationalize this observation by 14


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(a) Solute concentration [mg/gEtOH]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


300

Form I

100

250 80

Solute

200

60 150 40 100 20 50 0

Polymorphic content [%]

Page 15 of 42

Form II 0 0

(b)

5

Time [h]

10

15

Figure 2: Example of an MSMPR run. The conditions are reported in line h of Table 3. (a) Solute concentration and polymorph content as a function of time. Symbols correspond to experimental data and lines to model predictions using the kinetic parameters determined in our previous study performed in batch. 38 The blue and red dashed lines indicate the solubility of form I and form II, respectively. (b) Inline images at 30 min and 7 h. identifying a theoretical design space where it is possible to obtain form II at steady state.

Neglecting cross-nucleation We start our analysis by neglecting cross-nucleation, which simplifies the mathematical treatment. Although the crystallizer is seeded with crystals of form II, it is very difficult to guarantee that there are no traces of form I in the seeds. Such traces may be sufficient to promote the crystallization of form I via secondary nucleation. This may potentially result in the presence of form I at steady state, either pure or as a mixture with form II. Besides, if the selected residence time is too short, the seeds may be washed out, resulting in the absence of crystals at steady state. There are thus four different steady states that can

15



Page 16 of 42

theoretically be reached: no crystals, pure form I, pure form II, or a mixture of forms I and II. Let us start with the situation where there are no crystals at steady state, which is referred to as the trivial steady state. In this case, all the moments are equal to 0 and the solute concentration is equal to the feed concentration cliq 0 . One approach to analyze the stability of this steady state is to compute the eigenvalues λ of the Jacobian matrix J at steady state. If all the eigenvalues are negative, the steady state is stable, i.e., it won’t be affected by "small" disturbances such as traces of form I in the seeds. The eigenvalues of the Jacobian matrix are the roots of the following equation:

det(J − λI) = 0

(13)

where I is the identity matrix of size 7 × 7. The Jacobian matrix J is given in Figure 1 and liq is evaluated at steady state (i.e., here µIj = µII = cliq 0 ). j = 0 for j = 0, 1, 2 and c

The seven roots of Equation 13 are given by:

λ1 = − 

1 τ

"

λ2 = λ3 = λ4 = 2ksI kgI

2

cliq 0 c∗I

ln



"

λ5 = λ6 = λ7 = 2ksII kgII

2

(14)

ln

!#2ng +ns 1/3

cliq 0 c∗II



−

1 τ

!#2ng +ns 1/3 

−

1 τ

(15)

(16)

While λ1 is always negative, the eigenvalues λ2 to λ7 are negative only if the two following constraints are satisfied: " I 2

2τ 3 ksI kg

ln

cliq 0 c∗I

" II 2

2τ 3 ksII kg

!#2ng +ns

cliq 0 c∗II

ln

16

Polymorph control in MSMPR crystallizers. A case study with

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