Estimation of Nucleation and Growth Kinetics of Benzoic Acid by

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Estimation of Nucleation and Growth Kinetics of Benzoic Acid by Population Balance Modelling of a Continuous Cooling Mixed Suspension, Mixed Product Removal (MSMPR) Crystallizer Gary Morris, Graham Power, Steven Ferguson, Mark Barrett, Guangyang Hou, and Brian Glennon Org. Process Res. Dev., Just Accepted Manuscript • DOI: 10.1021/acs.oprd.5b00139 • Publication Date (Web): 09 Nov 2015 Downloaded from http://pubs.acs.org on November 13, 2015

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Organic Process Research & Development

Estimation of Nucleation and Growth Kinetics of Benzoic Acid by Population Balance Modelling of a Continuous Cooling Mixed Suspension, Mixed Product Removal (MSMPR) Crystallizer Gary Morris,†, § Graham Power,†, § Steven Ferguson,†, ∥ Mark Barrett,†, ‡ Guangyang Hou,†, § and Brian Glennon*, †, ‡ †

Synthesis and Solid State Pharmaceutical Centre, School of Chemical & Bioprocess

Engineering, University College Dublin, Belfield, Dublin 4, Ireland ‡

APC Ltd, NovaUCD, Belfield Innovation Park, Dublin 4, Ireland

ACS Paragon Plus Environment

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Table of Contents Graphic and Synopsis

Crystallization kinetics of benzoic acid were ascertained by population balance modelling of a continuous cooling mixed suspension, mixed product removal (MSMPR) crystallizer. Six kinetic parameters for nucleation and crystal growth were successfully estimated from a nonlinear optimization routine. The parameter estimates and model enable good prediction of critical process outputs of concentration and crystal size distribution (CSD).


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ABSTRACT: Nucleation and growth kinetics of benzoic acid crystallized by cooling from 1.5:1 (w/w) water:ethanol solutions were determined using a 500 mL continuous mixed suspension, mixed product removal (MSMPR) crystallizer and a mathematical model of the MSMPR process. The developed model solves the population, moment and mass balance relationships with the kinetic expressions for the system and is coupled with a nonlinear optimization routine for kinetic parameter estimation. Comprehensive experimental data for model fitting and parameter estimation was obtained by varying crystallizer operating conditions to induce changes in the rate affecting variables for growth and nucleation (temperature, supersaturation and suspension density). The size distribution was monitored in real-time using focused beam reflectance measurement (FBRM) to identify steady state, and hence enable on demand adjustment in operating conditions to transition multiple steady states for rapid acquisition of kinetic data. A Malvern Mastersizer was used to quantify the crystal size distribution (CSD) at steady state, whilst concentration was determined gravimetrically. Six kinetic parameters for crystal growth and nucleation were estimated from the model fitting procedure, which enabled accurate prediction of CSD and concentration results at steady state.

KEYWORDS: Crystallization kinetics; Population balance modelling; MSMPR crystallizer; Benzoic acid; Continuous processing


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1. INTRODUCTION Recent interest from the pharmaceutical industry on transitioning from batch to continuous drug substance manufacture has focused attention on the application of continuous reactor technologies to crystallization.1–5 The continuous mixed suspension, mixed product removal (MSMPR) crystallizer arguably represents the front running reactor device for enabling such transition within the industry, chiefly due to: (i) the broader application suitability of MSMPRs to accept a wider array of crystallization systems with fast or slow kinetic profiles and (ii) a tankbased design facilitating easier integration with existing batch plant equipment.6,7 Whilst understanding on the operation and modelling of these systems from a pharmaceutical context has grown significantly over the past decade,8–15 only few studies have attempted to leverage this knowledge to develop design strategies for the optimization of the MSMPR process,16,17 and contributions proposing firm methodologies for the optimal design of MSMPR systems are distinctly lacking, particularly with respect to the level of treatment the optimization of the batch crystallization process has received.18–25 Such an imbalance currently restricts the industry from progressing the introduction of MSMPR systems into pharmaceutical manufacture, as knowledge and familiarity with the design of batch crystallization processes remains superior. The development of strategies to optimize MSMPR processes for separate control over objective variables (e.g. particle size, supersaturation, product loss rate and productivity), all of which change depending on design variable (e.g. feed concentration, temperature and residence time) selection, will require understanding on the interaction of the operating conditions with the kinetics of the system over the process relevant supersaturation landscape. The interdependency of the variables at play here means that modelling provides an integral support tool in strengthening ones understanding of the system in order to prescribe operating conditions that


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target specific key process output variables (KPOVs) in terms of the crystal size distribution (CSD), supersaturation, product loss rate and productivity for example. The success of any crystallization model seeking to inform process design through simulation will ultimately hinge on the supply of accurate kinetic information to the model. Accordingly, the acquisition of reliable kinetic correlations that describe how rates of nucleation and growth vary over the design space for the crystallization process becomes of fundamental importance for enabling robust prediction of crystallizer performance. A commonly adopted approach for determining crystallization kinetics is to utilize the MSMPR crystallizer as a platform to induce changes in steady state nucleation and growth rates and hence correlate resolved rates with kinetic rate driving forces (e.g. temperature, supersaturation and suspension density). This method traditionally proceeds by first fitting the steady state population balance equation to acquired experimental population density data to deduce growth and nucleation rates, followed then by correlation of these determined rates with the kinetic rate driving forces. The ability to achieve steady state conditions, coupled with the typical assumptions made in the analysis of MSMPR crystallizers,26 allows the solution to the population balance to be simple and the direct recovery of nucleation and growth rates from experimental CSD data to be thus straightforward. This, in essence, is the power of the methodology, and the relative simplicity of the analysis has encouraged extensive application of the MSMPR crystallizer in determining and studying crystallization kinetics for a wide variety of systems.8–11,13,15,17,27–47 However, the manner by which the kinetic estimation procedure is compartmentalized into firstly kinetic rate retrieval from CSD data, followed by isolated fitting of correlations for nucleation and growth to driving force data, has the potential to overlook some of the variation in


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the kinetic rate behaviour that may be explained though interactions between the CSD, crystallization kinetics, mass balance and operating conditions. Tadayon et al.48 suggest that the most accurate approach for the estimation of kinetic parameters is accomplished by coupling a full mathematical description of the crystallization process with reliable experimental data and employing a nonlinear optimization algorithm for parameter identification. The advantages of such an approach are twofold. Firstly, the use of a full mathematical model that requires concurrent solution of population, moment and mass balance relationships with the crystallization kinetics for the system and operating conditions, naturally enables any interactions between these features of the crystallization process to be appreciated in the parameter fitting. Secondly, as the intention is almost always to redeploy the resolved crystallization kinetics in modelling the process, then combining the parameter estimation procedure with the process model provides assurance that the determined crystallization kinetics are capable of predicting the KPOVs of interest. Recently, several MSMPR studies have emerged where the researchers have employed model-based optimization procedures to estimate crystallization kinetic parameters.10,11,13 In all examples the model predicted size distribution using the kinetic parameters estimated from this approach agreed reasonably well with the experimental equivalent, with growth and nucleation modelled as the principal kinetic mechanisms involved in shaping the CSD. Other kinetic factors of agglomeration and breakage can also be crucial in shaping the CSD and several studies have also considered inclusion of one or both phenomena in modelling crystallization systems.31,38,47,49–51 The aim of this paper is to investigate the variables affecting the nucleation and growth kinetics of benzoic acid crystallized from 1.5:1 (w/w) water:ethanol and hence develop and describe a model-based nonlinear parameter optimization procedure utilized for the determination of


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crystallization kinetics for benzoic acid. These crystallization kinetics are to be used in the formulation of a model-supported design strategy for the continuous cooling MSMPR process in subsequent work. The current work demonstrates the application of an MSMPR crystallizer for streamlined collection of a comprehensive kinetic dataset, by utilizing in-line focused beam reflectance measurement (FBRM) to identify steady state and enable on the fly adjustment in crystallizer operating conditions to transition multiple steady states. Crystal growth and nucleation rates are graphically resolved from analysis of steady state CSD data to examine dependency on rate affecting variables. The analysis highlights potential pitfalls that may be encountered when attempting to correlate kinetic rate expressions directly to acquired rate and driving force data, without a full mathematical model of the process being utilized to support in mapping some of the variation observed. This article thus describes the development and application of a full mathematical model providing simultaneous solution of population, moment and mass balance equations with kinetic rate expressions for nucleation and growth for retrieval of kinetic parameters for the benzoic acid crystallization process. The model and parameter fitting framework are constructed within Microsoft Excel, utilizing Solver and an inbuilt macro routine to determine optimum value estimates for the kinetic parameter set. 2. EXPERIMENTAL SECTION 2.1. Materials. Benzoic acid (99 % purity, Sigma-Aldrich) was crystallized from a 1.5 to 1 (w/w) solvent mixture of water (deionized) and ethanol (99.7 % purity, Fisher). The compound is highly soluble in ethanol and exhibits low solubility in water.52 For CSD analysis by Malvern, saturated solutions of benzoic acid dissolved in cyclohexane (> 99 % purity, Fisher) were prepared at room temperature for use as a dispersing agent for product crystals.


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2.2. Solubility. The solubility of benzoic acid in 1.5:1 (w/w) water:ethanol was determined using FBRM and the polythermal method.53,54 Figure 1 shows the solubility ascertained for the system.

Figure 1. Solubility of benzoic acid in a 1.5:1 (w/w) water:ethanol solvent mixture as a function of temperature. 2.3. Experimental Setup. The MSMPR platform developed to study the continuous cooling crystallization of benzoic acid from 1.5:1 (w/w) water:ethanol,14 was utilized to examine the variables affecting the nucleation and growth rates and hence estimate the kinetic parameters that describe the system. Figure 2 shows a schematic diagram of the experimental platform. Comprehensive detail on the experimental setup can be found in Supporting Information.


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Figure 2. Schematic diagram of the MSMPR experimental setup showing the process configured to operate in product recycle mode. 2.4. Measurement Techniques. FBRM (Mettler Toledo, G400) was employed in-line to monitor delivery of steady state in the MSMPR. Samples of the crystallizer suspension were taken at steady state to determine MSMPR solute concentration and isolate solid samples for both offline microscopy and CSD analysis. Details on the suspension sample extraction technique used are provided in Supporting Information. Isolated crystal samples were washed (water, ~ 5 °C) and oven dried overnight under vacuum (− 50 kPa, 35 °C). Filtered mother liquor samples were weighed and also placed in a vacuum oven (− 50 kPa, 25 °C) for several days to evaporate the solvent for gravimetric determination of solute concentration. After drying for 72 h, the isolated solids were measured several times until the mass was constant. An analytical balance (Mettler Toledo, AT201) with an accuracy of ± 0.01 mg was used to weigh mother liquor and isolated solid samples in the determination of crystallizer concentration. Samples of the dried crystalline material were analyzed under a microscope (Olympus, BH-2) to investigate the crystal size and shape at different operating conditions. Images were captured via a microscope camera (PAXcam, 2+) and associated software (PAX-it, version 7.1).


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A Malvern Mastersizer 2000 was used to generate CSD information for the isolated material. Samples of the dried crystal product were analyzed on a wet method basis using the Malvern instrument equipped with a Hydro 2000 SM dispersion unit. For analysis by wet method, saturated solutions of benzoic acid and cyclohexane were prepared for use as a sample dispersant. As a sampling routine, the Hydro 2000 SM and Malvern cell were flushed through with ethanol and followed by 2 ‒ 3 circulation washes with pure cyclohexane. Circulation was allowed to continue on the final cyclohexane wash for 5 ‒ 10 minutes and a mercury thermometer was used to measure the temperature of the circulating cyclohexane at intervals of 5 minutes. Once a consistent temperature was recorded, it was noted, and a solution of cyclohexane and benzoic acid prepared that was saturated at this temperature. Solid product samples were measured by Malvern in triplicate and an average measurement taken as the representative steady state CSD for a particular set of experimental operating conditions investigated. The unit rinsing procedure was repeated prior to each sample analysis. 2.5. Experimental Procedure. Multiple experiments were performed over the 0 ‒ 40 °C region of the phase diagram to induce changes in supersaturation, suspension density and crystallizer temperature for kinetic evaluation. The product recycle configuration (Figure 2) allowed several kinetic data points to be investigated using a reduced amount of starting material. Table 1 details the MSMPR process conditions investigated to retrieve steady state data for analysis and kinetic parameter estimation. Operating the MSMPR in product recycle mode necessitated performing a batch crystallization of the feed to the required MSMPR operating temperature in order to


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Table 1. Process conditions experimentally investigated to supply data for kinetic parameter estimation

Feed concentration [g/kg solvent] Saturated feed temp. [oC] MSMPR operating temp. [oC] Mean residence time [min] Volumetric flow rate [mL/min]

A1

A2

A3

Experimental operating conditions A4 A5 A6 A7 A8

A9

A10

A11

228

228

228

228

228

228

228

228

228

134

59

40

40

40

40

40

40

40

40

40

30

15

30

30

30

20

20

10

10

0

0

0

0

15

30

45

15

20

30

45

20

30

30

30

31.7

15.8

10.6

31.7

23.8

15.8

10.6

23.8

15.8

15.8

15.8

maintain the overall balance of solute mass between feed and MSMPR. The feed vessel was filled to a volume of 1000 mL with the crystallizer operating between 450 ‒ 500 mL. In advance of the experimental investigation, the feed pump (Watson-Marlow, 323S/D) was calibrated for the full range of required flow rates shown in Table 1, thus allowing flow rates to be adjusted on demand. Experiments A1 ‒ A9 were carried out such that a range of steady state points were transitioned in a single experimental run per temperature investigated. This was accomplished by sequentially adjusting the flow rate and product withdrawal sequence following attainment of steady state for one operating condition to access a new residence time. By doing this, a further steady state point could be reached in a significantly telescoped time frame versus restarting from batch, where a lengthy washout and transient period would be required.14,15 Figure 3 shows an example of the progressive steady state chord length distributions (CLDs) accessed for process conditions A1 ‒ A3 by successive adjustment in crystallizer throughput. In general, transition from one steady state to the next was observed to occur after approximately 1 – 2 mean residence times had elapsed at the new flow condition. Once consistent and stable


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Figure 3. Steady state unweighted CLDs for process conditions A1 ‒ A3, where feed flow rates and withdrawal frequencies were adjusted on the fly to transition between multiple steady states. trends were confirmed via in-line FBRM at each new operating condition, suspension samples were collected to determine the CSD, concentration and suspension density at the prevailing MSMPR conditions before transitioning to the next steady state point. At the end of a full experimental run the solubility of the remaining feed solution was checked with FBRM using the polythermal method to examine if any significant solvent loss from the feed vessel had occurred. By FBRM, the solubility of the end feed solution fell between 40.12 ‒ 40.88 °C, indicating that any potential solvent loss from the experimental setup had minimal effect on the starting feed concentration. Inspection of the cold trap at the conclusion of experimental investigations agreed with this finding, with unmeasurable amounts of escaping solvent captured in the trap. 3. THEORY 3.1. Population Balance Model. To estimate the kinetic parameters from experimental data a mathematical model for the continuous cooling MSMPR crystallization process was developed.


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The model solves the population balance equation (PBE) with the mass balance, moment and kinetic rate relationships for the system simultaneously. As described by Randolph and Larsson,26 for a continuous MSMPR crystallizer operating at steady state with perfect back mixing, no particle agglomeration or breakage, representative product withdrawal, a crystal-free feed stream and crystals of uniform shape, the well-known analytical solution to the PBE is  L  n = n 0 exp −   Gτ 

(1)

where n is the population density [#/m3 µm], n 0 =

B0 represents the nuclei population density G

[#/m3 µm], B0 [#/m3 s] is the nucleation rate at steady state, G [µm /s] is the crystal growth rate at steady state, L [µm] is the characteristic size of the crystal and τ [min] is the mean residence time. By acknowledging equality between the time-averaged feed and product withdrawal flow rates, the relationship for the system mass balance is given by the following c0 − c = M T

(2)

where c0 [g/mL solution] represents the concentration of solute in the feed, c [g/mL solution] is the concentration of solute in the MSMPR mother liquor and M T [g/mL suspension] is the MSMPR suspension density. A balance on the third moment of the distribution then relates suspension density to the CSD and kinetics as follows M T = 6k v ρ c n 0 (Gτ ) 4

(3)

where crystal properties kv [-] and ρc [g/cm3] represent the volumetric shape factor and density of the crystals respectively. In this work, values of kv = 0.1 and ρc = 1.32 g/cm3 have been used,


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with a volumetric shape factor for that of a plate-like morphology selected which is representative of the continuous benzoic acid crystal habit experimentally obtained.55 Growth and nucleation kinetics were assumed to be described by commonly adopted powerlaw expressions of the form G = k g ∆C g

(4)

B 0 = k b M Tj ∆C b

(5)

with ∆C = C − C* and where C [g/g solvent] represents the concentration of solute in the MSMPR mother liquor, C* [g/g solvent] is the equilibrium MSMPR concentration and MT [g/mL suspension] is as before. The crystal growth rate coefficient kg [µm/s/(g/g solvent)g] is modelled as having an Arrhenius-type temperature dependency, expressed as

 Eg   k g = k g0 exp − RT  

(6)

where kg0 [µm/s/(g/g solvent)g] is the pre-exponential factor for the growth rate, Eg [kJ/mol] is the activation energy for growth, R [kJ/mol K] is the universal gas constant and T [K] is temperature. It is well known that secondary nucleation by collision is significant in MSMPR systems and can be the predominant source of nuclei generation. The inclusion of the suspension density term in Eq. 5 reflects this consideration. The birth of new crystals from solution is assumed to be a function principally of crystallizer supersaturation. Hence, Eq. 5 effectively models a total or overall nucleation rate. The nucleation rate coefficient kb [#/m3 s/{(g/mL suspension)j(g/g solvent)g}] is assumed to be mainly a function of the hydrodynamic conditions in the MSMPR suspension26 and temperature dependency on this parameter is not investigated in


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this work. Agitation speed was maintained at consistent levels throughout all experiments and is not included in Eq. 5. Solubility data experimentally determined for benzoic acid in 1.5:1 (w/w) water:ethanol (Figure 1) was fitted to a fourth degree polynomial equation in temperature, allowing computation of equilibrium concentration at given MSMPR temperatures over the 0 − 40 °C region of the design space C* = 2.03 × 10 −5 T 4 + 2.97 × 10 −4 T 3 + 4.70 × 10 −2 T 2 + 1.43T + 24.71

(7)

where T [oC] is temperature and C* [g/kg solvent] is equilibrium concentration. The equation was fitted using Microsoft Excel’s Solver function, returning a coefficient of determination of 0.998 for the goodness of fit between Eq. 7 and the original data. Description on the solution algorithm used for solving the mathematical process model can be found in Supporting Information. Model predictions of average steady state particle size are made using the mass-weighted mean size, which is calculated through the following ∞

∫ n L dL 4

L4, 3 =

0 ∞

= 4Gτ

(8)

∫ n L dL 3

0

where L4,3 [µm] represents the mass-weighted mean size of the CSD and is calculated from the ratio of the fourth and third moments of the size distribution.

3.2. Optimization Procedure for Parameter Estimation. The parameter estimation technique utilizes the mathematical model for the continuous cooling MSMPR process and a nonlinear


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optimization method to minimize the difference between experimental population density data and those predicted by the model. The set of kinetic parameters, θ = [kg0, Eg, g, kb, j, b], required for Eqs. 4 and 5 are estimated using a least squares optimization procedure, which consists of finding the values of θ that minimize the objective function Φ(θ) k Lmax

Φ(θ ) = ∑ ∑ (nexp − nˆ ) 2

(9)

1 Lmin

where Φ(θ) represents the total sum of the square error existing between model predicted population density n̂ and the experimental population density nexp values at sizes from Lmin ‒ Lmax for k number of experiments. For initial value estimates of the parameters in θ, the mathematical model is executed in sequence to predict population density data for every experiment using the particular experimental design variables, ψ = [C0, T, τ], applying in each case. Each model prediction of the population density is then compared to the corresponding experimental equivalent and the procedure repeated for subsequent iterations of parameter values in θ, until the objective function is minimized across k number of experimental datasets and an optimum set of kinetic parameter values thus returned. Eq. 10 is used to translate the experimental CSD information into population density data that is compared against the model predictions

nexp =

ν i (c0 − cexp ) k v ρ c ∆Li L

3 i, av

=

ν i M T, exp

(10)

k v ρ c ∆Li L3i, av

where νi is the volume fraction of crystals between sizes Li and Li+1, M T,

exp

[g/mL] is the

experimental MSMPR suspension density computed from measured solute concentration data cexp [g/mL solution] at steady state, ∆Li is the width of the bin between sizes Li and Li+1, Li, av is the average crystal size of bin i (Li, av = (Li + Li+1)/2), and c0, kv and ρc are as defined previously.


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The complete parameter estimation routine was implemented in Microsoft Excel by building all model equations into the spreadsheet workspace and utilizing Visual Basic for Applications (VBA) to construct a macro that for every iteration of θ, simultaneously solves the mathematical process model across all experimental datasets, whilst using Solver to globally control the parameter optimization process. The Solver routine targets the minimization of the objective function Φ(θ) by changing the values in θ, subject to the constraint that, within each mathematical model, the kinetic parameters allow for the following condition to hold true C0 > C > C*…for all k

(11)

The quadratic extrapolation approach, which is suited to solving nonlinear problems, was used within Solver to search for parameter estimates. The Solver routine is interrupted following an iteration of θ, allowing the equations in each mathematical model of the process to be solved simultaneously for the new estimates of the kinetic parameter values, before continuing with a subsequent iteration of θ. 4. RESULTS AND DISCUSSION

4.1. Impact of Crystallization Driving Forces on Kinetic Rates. Growth and nucleation rates were graphically estimated from investigated operating conditions A1 − A11 and the influence of crystallization driving forces (T, ∆C, MT) on these rates evaluated. Figure 4 illustrates a typical example of the population density distribution obtained following translation of steady state CSD data to population density values using Eq. 10 and plotting of Ln(nexp) versus L. In all population


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Figure 4. Population density distribution for MSMPR operating conditions A10. Linear regression for effective kinetic rates is based on the distribution from L > 65 µm. density plots, a minor deviation from the linear trend expected by Eq. 1 was observed and located exclusively towards the finer end of the distribution. Such deviations can result multiple phenomena, including: size-dependent growth, growth rate dispersion, product classification, agglomeration and particle attrition.26,54,55 Based on typical deviation profiles provided in Garside et al.,55 the particular deviation observed in this work may be indicative of secondary nucleation due to crystal collision, however, it is difficult to confidently distinguish between potential source causes via CSD analysis alone. To graphically resolve kinetic rates from obtained experimental population density distribution plots, the approach of estimating effective kinetic rates was employed.49,55–56 Using this method, linear fitting of Eq. 1 is focused on the section of the distribution of size larger than where the deviation is located. For the majority of experimental population density distributions this meant fitting Eq. 1 to data of size L > 65 µm. Extrapolation is then performed to zero size based on this linear fit to determine the nuclei density and the deviation is effectively ignored in the regression analysis.


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4.1.1. Growth Rate. The effect of temperature on the crystal growth rate was found to be quite pronounced, with higher MSMPR temperatures evidently providing enhanced conditions for crystal growth to dominate. A plot of the experimentally deduced crystal growth rates versus supersaturation is shown in Figure 5 for MSMPR operating conditions A1 ‒ A9. At the highest

Figure 5. Growth rate as function of supersaturation for data obtained in kinetic experiments investigating operating conditions A1 ‒ A9: (♦) A1 ‒ A3, (■) A4 ‒ A5, (●) A6 ‒ A7, (▲) A8 ‒ A9. investigated temperature of 30 °C growth rates are larger at much lower supersaturations compared to the other isothermal trends. The magnitude of the change in the growth rate is also much more significant and sensitive to small changes in supersaturation relative to data at lower temperatures. Moreover, a trend is clearly apparent that the slope of the growth rate as a function of supersaturation decreases as temperature is reduced. This suggests that at colder crystallizer temperatures the growth rate may almost become stagnant with respect to changes in supersaturation, potentially allowing nucleation to become the increasingly more dominant kinetic process under these conditions. Although for the three of the temperatures investigated only two data points are available for slope regression, and thus confirmation of linear


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dependence cannot be absolute, the analysis does provide an insight into the influence of temperature on the rate of crystal growth. Several microscope images of the isolated steady state crystal product for each temperature are shown in Figure 6, which correlate well with previous observations on the influence of temperature on the crystal growth rate. Crystal product formed at higher temperatures exhibited


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Figure 6. Microscope images of the crystal product obtained from kinetic experiments investigating operating conditions A1 ‒ A9: (A) 30 °C, (B) 20 °C, (C) 10 °C and (D) 0 °C. All images are 10x magnification. significant size increases along both length and width axes versus material generated at the lower temperature range. A higher presence of irregular shaped particles was observed in product examples obtained at 0 and 10 °C, compared to the more defined plate-like geometry evident at


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higher temperatures. This is potentially indicative of collision-based secondary nucleation effects featuring more prominently at the reduced temperatures, due to the increased levels of suspension density in the MSMPR and hence a greater propensity for crystal–crystal and crystal– vessel interactions. Figure 7 shows the considerable variation in CSD observed as a function of changes in MSMPR temperature for conditions of similar residence time. The substantial

Figure 7. Comparison of steady state CSDs obtained at different MSMPR temperatures and similar mean residence times over investigated operating conditions A1 ‒ A9: (♦) A2, (■) A5, (●) A6, (▲) A9. movement in the CSD towards the coarse end of the size range as temperature is increased further supports the observed trends in crystal growth rate and physical morphology. By the nature of the experimental approach adopted across A1 ‒ A9 it is difficult separate the mechanisms that are potentially driving particle size to vary to such an extent. It is likely that the significant reduction in particle size occurring at lower temperatures is more a summation effect of both a thermal limiting of the growth rate and an increase in secondary nucleation via higher suspension density levels. Similarly, the reverse is arguably valid for the higher temperature


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conditions investigated, where the growth rate is enhanced and secondary nucleation is reduced as less material is crystallized, and the combination of these effects creates the environment for the significant particle size increase that is observed. To examine the interaction between crystal growth rate and secondary nucleation more closely, operating conditions A10 and A11 were experimentally investigated. Here, MSMPR temperature and mean residence time were maintained at 0 °C and 30 min respectively, with only feed concentration lowered in order to progressively reduce the suspension density level in the crystallizer. Results from A9 ‒ A11 can thus be compared together for significant changes occurring in suspension density. Figure 8 shows the growth rate as a function of supersaturation for MSMPR operating conditions A9 ‒ A11. Notably, the slope of the growth rate as a function of supersaturation at 0 °C is significantly

Figure 8. Growth rate as function of supersaturation for data obtained in kinetic experiments over investigated operating conditions A9 ‒ A11: (▲) A9, (+) A10, (X) A11. steeper in comparison to low temperature trends for 0 and 10 °C seen previously in Figure 5. This suggests that the growth rates observed at 0 °C in Figure 5 may be competitively limited by suspension density promoting increased levels of secondary nucleation. Figure 9 shows the CSD


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progressively shifting towards the larger particle size range as suspension density is reduced, with temperature and mean residence time consistent. Microscope images in Figure 10 support

Figure 9. Comparison of steady state CSDs obtained at different levels of MSMPR suspension density over investigated operating conditions A9 ‒ A11: (▲) A9, (+) A10, (X) A11. the observed trend in the size distribution, where image sets E and F can be compared with D in Figure 6 previously for significant changes occurring in feed concentration.


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Figure 10. Microscope images of the crystal product obtained from kinetic experiments investigating operating conditions A10 and A11: (E) C0 = 134 g/kg solvent (A10) and (F) C0 = 59 g/kg solvent (A11). All images are 10x magnification.

Across experimental conditions A9 ‒ A11 there is a sevenfold decrease in crystallizer suspension density (0.145 to 0.023 g/mL), which is sufficient to drive a change in the resulting size distribution at steady sate. It is expected that for a system in which secondary nucleation exhibits a linear dependence with suspension density, the growth rate shall not be affected by adjustment in suspension density and hence no change to the CSD should result.26 In the case of benzoic acid, however, the substantial variation in suspension density impacted the observed growth rate and steady sate size distribution, over a relatively narrow supersaturation range. This suggests the dependency of the rate of nucleation on suspension density to be nonlinear.26 Moreover, the observation highlights the potential for crystal growth rates to be affected in an indirect way by changes to suspension density that impact the rate of nucleation in the MSMPR.


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This raises an issue in relation to how successful attempting to separately correlate expressions for the growth and nucleation kinetics (Eqs. 4 and 5) with driving force data and graphically determined kinetic rates may be, without a full mathematical model involved in the fitting procedure to enable interaction between all underlying processes as shown in Section 3.1.

4.1.2. Nucleation Rate. The rate of nucleation was found to be strongly dependent on both supersaturation and suspension density as expected. Figures 11 and 12 respectively present the variation in nucleation rate as a function of these two variables using data derived from A1 ‒ A11. The magnitude of the nucleation rate obtained over the MSMPR operating conditions

Figure 11. Nucleation rate as function of supersaturation for data obtained over investigated operating conditions A1 ‒ A11: (♦) A1 ‒ A3, (■) A4 ‒ A5, (●) A6 ‒ A7, (▲) A8 ‒ A9, (+) A10, (X) A11. Nucleation rate values are plotted on a logarithmic scale. investigated varied considerably (1.1 × 107 ‒ 287.6 × 107 #/m3 s). Conditions where both supersaturation and suspension density were high (A8 ‒ A9) drove the nucleation rate to highest observed levels. Equally, lowest nucleation rates were obtained at conditions where the MSMPR operated with a reduced suspension density and supersaturation (A1 ‒ A3).


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Figure 12. Nucleation rate as function of suspension density for data obtained over investigated operating conditions A1 ‒ A11: (♦) A1 ‒ A3, (■) A4 ‒ A5, (●) A6 ‒ A7, (▲) A8 ‒ A9, (+) A10, (X) A11. Nucleation rate values are plotted on a logarithmic scale. In determining nucleation rate data by Eq. 1 a conflicting trend emerged in the nucleation rate as a function of suspension density. For conditions where feed and equilibrium concentration were fixed, but mean residence time was varied, a local trend of nucleation rate decreasing with increasing suspension density was apparent. Examples of this can be seen in Figure 12 by focussing exclusively on the matching data points at a time. Examining these individual datasets in isolation would suggest the nucleation rate to decrease with increasing suspension density. However, globally the nucleation rate is observed to increase with suspension density when the entire dataset is considered. This appears to be somewhat of a limitation of CSD analysis approach to determine nucleation rates and resolve an accurate understanding between the interacting driving forces that are inducing changes in the rate values obtained. As observed experimentally, an increase in mean residence time will decrease the dissolved concentration in the MSMPR, simultaneously inducing a reduction in supersaturation but also a minor increase in suspension density. Provided particle size increases with residence time or


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even remains unchanged, the nucleation rate graphically derived from CSD analysis using Eq. 1 will be seen to decrease. The difficulty apparent here is that the nucleation rate obtained is in effect an overall or total nucleation rate, which in its value estimation fails to discriminate between the relative contribution made from solution and particle collision sources of nuclei generation. In reality, the reduction in concentration has likely reduced the potential for nucleation from solution, but has also increased the potential for secondary nucleation from collision by augmenting the mass of crystals in suspension. Depending on how sensitive the overall nucleation rate is towards relative changes in supersaturation and suspension density, the reduction in supersaturation may have considerably more impact in reducing the overall nucleation rate than the corresponding increase in suspension density has in raising it, potentially resulting in a net reduction in the overall nucleation rate. This consideration may offer an explanation towards the experimentally observed trends in Figures 11 and 12, where on a local level for corresponding groups of data in both figures, the nucleation rate increases with superstation but decreases with suspension density, whilst over the complete dataset, where suspension density was varied significantly, the nucleation rate has a positive correlation with suspension density.

4.2. Kinetic Parameter Estimation. The preceding analysis highlights some of the difficulties and pitfalls that may be encountered in attempting to directly correlate graphically determined crystallization rate data with driving force variables, where process and kinetic interactions that may account for any nonlinearity observed in the kinetic rate behaviour have the potential to be omitted. To circumvent such difficulties and provide a means for the estimation of kinetic parameters for nucleation and growth with full appreciation of the interacting features of the crystallization process, the mathematical model and parameter estimation technique described in


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Section 3 was used in conjunction with experimental data from A1 ‒ A11. Similar to the approach of estimating effective kinetic rates as adopted for graphical determination of kinetic rates in Section 4.1, the model equations were fitted to the linear portion of the experimental population density distribution in each case. Table 2 provides the estimated kinetic parameters for Eqs. 4 and 5 as returned following application the procedure.

Table 2. Estimated kinetic parameter values from application of the nonlinear parameter optimization procedure and mathematical process model Kinetic Parameter kg0

Estimated Value 1.06 × 107

Units [µm/s/(g/g solvent)g]

Eg

40.05

[kJ/mol]

g

0.44

kb

9.16 × 10

[-] 12

3

[#/m s/{(g/mL suspension)j(g/g solvent)g}]

j

1.78

[-]

b

1.2

[-]

The estimated growth order with respect to supersaturation is at the lower range of values published in the literature for a variety of compounds (− 0.3 ≤ g ≤ 2.29),39,56 although sub-unity values are less common. In terms of the nucleation rate, the estimated order dependency on supersaturation also falls within range of values reported for other systems (− 0.34 ≤ b ≤ 10.85).37,56 Quantifying the relative kinetic order (i = b/g), a determined value of 2.73 that is greater than 1 indicates that crystal size should increase with residence time and agrees with experimental findings in this study and related work.14 Studies that have considered the influence of suspension density on the nucleation rate generally report values of j close to unity,27,55–57 although quite wide ranging values have also been observed (− 3.87 ≤ j ≤ 3.58).27,47 The estimated value of j = 1.78 suggests a relatively strong dependence of the nucleation rate on suspension


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density for the benzoic acid system. A close to second-order dependency on suspension destiny is potentially indicative of crystal–crystal interactions, as opposed to crystal–vessel interactions, being a dominant mechanism of secondary nuclei breeding in the benzoic acid system.50,58

4.3. Assessment of Parameter Confidence. To test parameter confidence, model predicted quantities of interest (growth/nucleation rates, CSD, particle size and concentration) were compared against experimental equivalents. In Figures 13 and 14 respectively, the model predicted growth and nucleation rates are compared to those directly measured from experimental CSD data. The fitted parameters and model display a reasonable level of accuracy

Figure 13. Correlation between the experimental and model predicted crystal growth rates using the optimized kinetic parameter estimates over the investigated operating conditions A1 ‒ A11.

in reproducing the experimentally measured crystal growth rate. A slope value of 0.755 suggests the model to misestimate the experimental crystal growth rate by almost 25 % on average. In terms of the nucleation rate, agreement between model predicted rate values and those derived directly from the experimental CSD is comparably lower. A slope value of 0.594 suggests


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Figure 14. Correlation between the experimental and model predicted nucleation rates using the optimized kinetic parameter estimates over the investigated operating conditions A1 ‒ A11. limited capability for the model to accurately predict nucleation rates derived directly from the experimental CSD, with such a value suggest almost 40 % error on average in predicting nucleation rates. Despite a lack of accuracy in predicting experimental growth and nucleation rates, notably, the fitted kinetic parameters and mathematical process model enable quite an accurate reproduction of the experimental CSD. This is highlighted through Figures 15 and 16, which present a comparison between the experimental and model predicted CSDs for operating conditions that yielded largest and smallest average particle sizes respectively. The process conditions


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Figure 15. Comparison between experimental and model simulated CSDs using optimized kinetic parameter estimates for investigated operating conditions that delivered the largest steady state particle size (operating conditions = A3).

Figure 16. Comparison between experimental and model simulated CSDs using optimized kinetic parameter estimates for investigated operating conditions that delivered the smallest steady state particle size (operating conditions = A8). investigated demonstrated a wide span of steady sate mean particle sizes (160 µm ≤ L4,3 ≤ 582 µm), which the mathematical model and optimized kinetic parameter estimates were able to


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predict to a satisfactory degree (Figure 17). A slope value of 0.890 observed in Figure 17 suggests that the model provides a reasonably accurate estimate of the true experimental particle size across the entire range of operating conditions investigated.

Figure 17. Correlation between the experimental and model predicted mass-weighted mean particle size values using the optimized kinetic parameter estimates for investigated operating conditions A1 ‒ A11. Model predicted estimates of the steady state concentration in the MSMPR were similarly compared to experimentally measured values to assess confidence in simulating this KPOV. Figure 18 displays the correlation between the model predicted concentration values and those measured from experiment. The resulting plot illustrates very good agreement between experimental and model predicted steady state concentration values, with a slope value of 0.960 confirming close mapping of the experimental results. By extension, accurate prediction of steady state concentration enables further pertinent quantities of interest (supersaturation, yield, product loss rate and productivity) to be reliably simulated by the mathematical process model.


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Figure 18. Correlation between the experimental and model predicted MSMPR solute concentration values using the optimized kinetic parameter estimates for operating conditions A1 ‒ A11. 5. CONCLUSIONS Crystallization kinetics of benzoic acid produced from aqueous ethanol solutions were studied using a continuous MSMPR crystallizer. The continuous operation enabled comprehensive interrogation of the design space for situations where both crystal growth and nucleation are occurring simultaneously. A strong interdependence between the crystallization kinetics and operating conditions was revealed, such that separate and direct correlation of experimental growth and nucleation rates with respective driving forces (temperature, supersaturation and suspension density) may be inherently difficult without a mathematical model to fully map process interactions. To therefore estimate kinetic parameters which capture such interactions that must be mapped for effective simulation of the process operation, a mathematical model was developed for fitting to the experimental data. Kinetic parameters for growth and nucleation were estimated from a nonlinear parameter optimization routine and simulation model solving


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population, moment and mass balance relationships with the kinetics and process operating conditions. Six kinetic parameters for crystal growth and nucleation were successfully estimated that enable good model prediction of CSD and concentration results from eleven varied conditions of MSMPR operation experimentally investigated. The overall nucleation rate was found to have a relatively high order dependency on suspension density. Whilst the MSMPR crystallizer provides a useful and unique means of capturing interaction between crystal growth and nucleation rates and driving force variables for model development, the concurrent nature of these processes at steady state makes segregation of individual kinetic contributions to growth and nucleation difficult. Potentially coupling MSMPR studies with well-known batch approaches to separate mechanisms of growth and nucleation may provide the most complete means of isolating the individual contribution of kinetic rate driving forces. ASSOCIATED CONTENT

Supporting Information Further details on the experimental setup, sampling technique and the solution algorithm used for the mathematical process model are provided in Supporting Information. AUTHOR INFORMATION

Corresponding Author *E-mail: [email protected]; Tel: +353-1-716-1954; Fax: +353-1-716-1177

Present Addresses §

Small Molecule Process Development, APC Ltd, Science Centre South, University College

Dublin, Belfield, Dublin 4, Ireland


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∥Chemical

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Process Research and Development, Biogen, 14 Cambridge Center, Cambridge,

Massachusetts 02142, United States ACKNOWLEDGMENT The authors express their sincere appreciation to Science Foundation Ireland, Eli Lilly and Company, GlaxoSmithKline, Janssen, Merck Sharp & Dohme, Pfizer and Roche for their funding of this research. The lead author would also like to personally acknowledge the financial support received from the Irish Research Council through the 'EMBARK Initiative' Postgraduate Scholarship Scheme (RS/2010/2358).


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NOMENCLATURE

Roman Symbols b

nucleation order on supersaturation ∆C [-]

B0

nucleation rate [#/m3 s]

c

crystallizer concentration [g/mL]

cexp

experimentally measured crystallizer concentration [g/mL]

c0

feed concentration [g/mL]

c*

equilibrium concentration [g/mL]

C

crystallizer concentration [g/g solvent]

Cexp

experimentally measured crystallizer concentration [g/g solvent]

C0

feed concentration [g/g solvent or g/kg solvent]

C*

equilibrium concentration [g/g solvent or g/kg solvent]

∆C

absolute supersaturation [g/g solvent]

Eb

activation energy for nucleation [kJ/mol]

Eg

activation energy for crystal growth [kJ/mol]

g

growth order on supersaturation ∆C [-]

G

crystal growth rate [µm/s]

j

nucleation order on suspension density MT [-]

k

number of experimental datasets included in the objective function Φ(θ)

kb

nucleation rate coefficient [#/m3 s/{(g/mL suspension)j(g/g solvent)g}]

kg

growth rate coefficient [µm/s/(g/g solvent)g]

kg0

pre-exponential factor for the growth rate [µm/s/(g/g solvent)g]

kv

volumetric shape factor [-]

L

characteristic crystal size [µm]

Li, Li+1 bounding sizes of bin i [µm] Li,av

average crystal size of bin i (Li, av = (Li + Li+1)/2) [µm]

Lmax

maximum crystal size [µm]


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Roman Symbols Lmin

minimum crystal size [µm]

L4,3

mass-weighed mean particle size [µm]

∆L

size interval [µm]

m3

third moment of the size distribution [m-4]

MT

suspension density [g/mL suspension]

MT, exp

experimental suspension density [g/mL suspension] population density [#/m3 µm]

exp

experimentally measured population density [#/m3 µm]

n̂

model predicted population density [#/m3 µm]

n0

nuclei density [#/m3 µm]

Q

volumetric flow rate [mL/min]

R

universal gas constant [kJ/mol K] temperature [°C or K]

Greek Symbols θ

kinetic parameter set θ = [kg0, Eg, g, kb, j, b]

νi

volume fraction of crystals between sizes Li and Li+1 [-]

ρc

crystal density [g/cm3]

τ

mean residence time [min]

Φ(θ)

objective function

ψ

process design variable set ψ = [C0, T, τ]


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Estimation of Nucleation and Growth Kinetics of Benzoic Acid by

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